Demo¶
A proof assistant is a program that checks proofs. It cannot be talked into accepting one: it type-checks the text it is given and answers with an exit code. Here the proof assistant is Agda, the writer is a language model, and Agda's answer is the only judge of what the model wrote.
agda-mcp is a server that lets the model ask Agda questions while it works, the way a person does in an editor: what is the type of this expression, what does this module export, does this file check. This page shows the server doing that, with nothing invented. The clocks are measured from archived sessions, the sessions are replayed from their transcripts, and every figure comes from a file beside this page that names the run it was read from.
Three clocks¶
What the server changes is the price of a question, not the price of a verdict. A question about a file the server already holds is answered from the loaded file, in milliseconds. A verdict, whether the file type-checks, is a fresh run of Agda every time, on the server and off it, and costs seconds. In a real development the same verdict from a shell takes longer still: the third bar is one module's check in the field, and the whole-library check a development finally gates on runs from tens of seconds to minutes. The bars start together, and each stops when its measured time is up.
- A question about a file the server already holds3 ms
The median of 165 questions the interaction lane answered about a file it already held (p90 22 ms, longest 93 ms); the same questions when the file had to be loaded first took 4.8 s at the median. Measured on
arm-surface-mcp-1andarm-surface-mcp-2andarm-surface-both-1andarm-verdict-mcp-1andarm-verdict-mcp-2, Sonnet 5 on the server of 2026-09-26. - A verdict:
check_file, a fresh batch run of Agda2.0 sThe median of 283 calls of
check_fileon the same runs (p90 5.5 s, longest 7.9 s), each a run ofagdaover the file, answered from its exit code. - The same verdict from a shell, in a real development10 s
Not from the runs above: the archive holds no shell timings. This is the median of the single-module
agdachecks a shell ran in an agda-algebras session, as agda-native-air issue 83 recorded them and ADR 0002 § 2 quotes them; the whole-library check those sessions gate on runs 20 s to 10 min (ADR 0002 § 12).
One scale: a bar’s length is its time, and the bars start together. Every figure is in
numbers.json beside this page, with the run ids and the field it was read from.Six sessions, one lemma¶
The lemma is about algebras: if one algebra sits inside a second as a
subalgebra, and the second is isomorphic to a third, then the first sits
inside the third. The library it comes from already proves it. The
statement was put to the model with that proof removed and a hole in its
place, and the model was free to read the library. The same model met it
six times: with the server, with a shell that had Agda on it, and with both,
and each of those twice. The first time, the file's header carried two
comment lines the benchmark's fixtures used to carry, one naming the module
that holds the library's proof and one naming the proof's shape in a word,
pairing; every run archived before 2026-09-29 had such lines in view
(agda-native-air issue 219). The second time, eight days later, the header
had been stripped of them. The server had been trimmed and the shell's
prompt now named the library's directories in between, so the two rows are
not a controlled experiment on the header alone; the page shows what the
six sessions did and says no more than that. Each tab replays what the
model typed and what it was told back, in the order it happened. A call
types the way the model typed it; the answer under it appears whole,
because the tool answered in one response, and it is here in full behind a
control that says how long it is. The verdict at the end of each session
is the archive's judge's, not this page's.
| instrument | fixture header | the judge | turns | questions to Agda | corpus searches | library reads | edits | verdicts | each verdict, in order |
|---|---|---|---|---|---|---|---|---|---|
| server only | hints in view | solved | 12 | 5 | 2 | 1 | 1 | 1 | accepted |
| shell only | hints in view | solved | 15 | 0 | 0 | 9 | 2 | 2 | refused, accepted |
| server and shell | hints in view | solved | 14 | 0 | 0 | 8 | 2 | 2 | refused, accepted |
| server only | header-free | restated | 5 | 1 | 0 | 0 | 1 | 1 | accepted |
| shell only | header-free | restated | 9 | 0 | 0 | 5 | 1 | 1 | accepted |
| server and shell | header-free | restated | 13 | 0 | 0 | 8 | 1 | 1 | accepted |
Counted from each session’s steps: a question is a lane-answered tool (type_of, exports_of, definition_of, get_goal); a search is a corpus tool; a library read is a shell command the archive’s audit classed as one, or a read of a file outside the work directory; a verdict is check_file or agda run in the shell, accepted or refused as the tool answered. The judge’s column is the archive’s: a file that type-checks by calling the library’s own lemma for the statement is restated, never solved.
Six sessions on one lemma settle nothing about a sweep, and the score below
says how the sweeps went. What they show is the shape of the difference,
twice over. Across the instruments: with the server alone, the model asked
Agda what things were before it wrote, and its one verdict was accepted;
with a shell, alone or beside the server, it read the library through grep
and wrote from what it found, and with the hint in view it had its first
verdict refused for a name out of scope, which is the kind of question the
first session had asked. Across the header: with the proof's shape named,
the model wrote the proof out in all three sessions, and the judge counted
three solves; with the line gone, all three found the library's own lemma,
one by asking the server what a module exports and two by grepping the
sources, and called it, which the judge counts as a restatement and never
as a solve. The row did not get harder. agda-native-air's reading of the
rows that moved this way across its sweeps is that the line decided whether
a found answer was written out or cited, and that is what the two rows of
tabs show on this one. Opus 5 met the same row header-free, was shown by
one probe that the citation would do, and wrote the proof out anyway; that
session is
suite219-opus5-mcp-1
in the archive and is linked rather than replayed, because the tabs are one
model. These six are here because the row is in every archived arm of the
mined tier, and because the second row of tabs is the same model on the
same row with the same three instruments and the header gone.
≤-trans-≅′, server only, hints in view
Server only, with the hint in view. The corpus is searched, definition_of pays for loading the file, and the questions after it are answered on the loaded file in milliseconds. Then one edit that writes the proof out, and one verdict, which is a fresh run of Agda.
- model
- Sonnet 5 (claude-sonnet-5)
- instrument
- server only
- run
- arm162-mcp-1 2026-09-21
- fixture header
- hints in view:
Source: Setoid.Subalgebras.Properties (agda-algebras);Strategy: pairing - server
- the server of 2026-09-21 (fourteen tools, the libraries' sources readable on every arm)
- client
- 2.1.261 (Claude Code)
- turns
- 12
- tool calls
- 11
Read×2,search_by_name×1,search_by_type×1,definition_of×1,exports_of×2,type_of×2,Edit×1,check_file×1 - wall clock
- 48 s (indicative: subjects ran three at a time)
- cost, list price
- USD 0.39
the statement, as the obligation states it, with the proof removed
{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪the session, 12 turns and 11 tool calls
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- filesRead(file_path: <work>/Subalgebras-sub-trans-iso.agda)answered1 -- Subalgebras-sub-trans-iso.agda
the whole answer, as the model saw it (963 characters of text)
1 -- Subalgebras-sub-trans-iso.agda 2 -- 3 -- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda 4 -- 5 -- Benchmark obligation: algebras-subalgebras-sub-trans-iso 6 -- Difficulty: non-obvious 7 -- Source: Setoid.Subalgebras.Properties (agda-algebras) 8 -- Import stratum: wholesale 9 -- Strategy: pairing 10 -- 11 module Subalgebras-sub-trans-iso where 12 13 open import AgdaDojang.Debug 14 15 open import Agda.Primitive using ( Level ) 16 open import Data.Product using ( _,_ ; proj₁ ; proj₂ ) 17 18 open import Overture using ( Signature ) 19 open import Setoid.Algebras using ( Algebra ) 20 open import Setoid.Functions 21 open import Setoid.Homomorphisms 22 open import Setoid.Subalgebras 23 24 ≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} 25 {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} 26 → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 27 ≤-trans-≅′ p B≅C = {!!} 28 - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- agda-mcpsearch_by_name(pattern: ≤-trans-≅)answeredresults: 1
the whole answer, as the model saw it (507 characters of JSON)
[{"defKind":"function","hasBody":true,"module":"Setoid.Subalgebras.Properties","prettyQname":"Setoid.Subalgebras.Properties.≤-trans-≅","type":"{𝑆.𝓞 𝑆.𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ β ρᵇ γ ρᶜ : Agda.Primitive.Level}\n{𝑨 : Setoid.Algebras.Basic.Algebra α ρᵃ}\n{𝑩 : Setoid.Algebras.Basic.Algebra β ρᵇ}\n{𝑪 : Setoid.Algebras.Basic.Algebra γ ρᶜ} →\n𝑨 Setoid.Subalgebras.Basic._.≤ 𝑩 →\n𝑩 Setoid.Homomorphisms.Isomorphisms._.≅ 𝑪 →\n𝑨 Setoid.Subalgebras.Basic._.≤ 𝑪"}] - agda-mcpsearch_by_type(pattern: ≤, limit: 30)answeredresults: 30
the whole answer, as the model saw it (18,762 characters of JSON)
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The archive keeps the block’s signature and not its text, so nothing is quoted.
- agda-mcpdefinition_of(name: Setoid.Subalgebras.Properties.≤-trans-≅)answered5.8 s (loading the file first)elapsedMs: 5849definitions: 11
the whole answer, as the model saw it (5,760 characters of JSON)
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- filesRead(file_path: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md, offset: 1, limit: 95)answered1 ---
the whole answer, as the model saw it (4,347 characters of text)
1 --- 2 layout: default 3 title : "Setoid.Subalgebras.Properties module (The Agda Universal Algebra Library)" 4 date : "2021-07-18" 5 author: "agda-algebras development team" 6 --- 7 8 #### Properties of the subalgebra relation for setoid algebras 9 10 This is the [Setoid.Subalgebras.Properties][] module of the [Agda Universal Algebra Library][]. 11 12 This module establishes the order-theoretic facts about the subalgebra relation 13 `_≤_`{.AgdaFunction} that the rest of the library relies on: that it is a 14 preorder, that isomorphism refines it in either direction, that it survives 15 universe lifting, and that it is preserved by products. 16 17 The preorder is stated with respect to isomorphism rather than equality 18 (`≤-preorder`{.AgdaFunction}), which is the only sensible choice, since mutual 19 embeddings do not make two algebras equal. The universe-lifting group is the 20 part used most often, for the reason recorded in [Setoid.Algebras.Basic][]: 21 Agda's universes are not cumulative, so a `≤` established at one level has to be 22 transported explicitly before it can be used at another. 23 24 <!-- 25 ```agda 26 {-# OPTIONS --cubical-compatible --exact-split --safe #-} 27 28 module Setoid.Subalgebras.Properties where 29 30 open import Agda.Primitive using () renaming ( Set to Type ) 31 32 -- Imports from the Agda Standard Library ------------------------------------------- 33 open import Data.Product using ( _,_ ) 34 open import Function using ( _∘_ ) renaming ( Func to _⟶_ ) 35 open import Level using ( Level ; _⊔_ ) 36 open import Relation.Binary using ( Setoid ) 37 open import Relation.Unary using ( Pred ; _⊆_ ) 38 import Relation.Binary.Structures as RelStructs 39 40 import Relation.Binary.Reasoning.Setoid as SetoidReasoning 41 42 -- Imports from the Agda Universal Algebra Library ---------------------------------- 43 open import Overture using ( proj₁ ; proj₂ ; 𝓞 ; 𝓥 ; Signature ; 𝑆 ) 44 open import Setoid.Algebras using ( Algebra ; Lift-Algˡ ; Lift-Algʳ 45 ; Lift-Alg ; ov ; ⨅ ; 𝔻[_] ) 46 open import Setoid.Functions using ( id-is-injective ; IsInjective ; ⊙-injective ) 47 open import Setoid.Homomorphisms using ( hom ; IsHom ; 𝒾𝒹 ; ⊙-hom ; _≅_ 48 ; ≅toInjective ; ≅fromInjective ; mkiso 49 ; ≅-sym ; ≅-refl ; ≅-trans 50 ; Lift-≅ˡ ; Lift-≅ ; Lift-≅ʳ) 51 open import Setoid.Subalgebras.Basic using ( _≤_ ; _≥_ ; _≤c_ ) 52 53 private variable α ρᵃ β ρᵇ γ ρᶜ ι : Level 54 ``` 55 --> 56 57 The subalgebra relation is a *preorder*, i.e., a reflexive, transitive binary relation. 58 59 ```agda 60 open _≅_ 61 62 ≅→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 63 ≅→≤ φ = (to φ) , ≅toInjective φ 64 65 ≅→≥ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 66 ≅→≥ φ = (from φ) , ≅fromInjective φ 67 68 ≤-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 69 ≤-refl = ≅→≤ 70 71 ≥-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 72 ≥-refl = ≅→≤ ∘ ≅-sym 73 74 ≤-reflexive : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≤ 𝑨 75 ≤-reflexive {𝑨 = 𝑨} = 𝒾𝒹 , id-is-injective {𝑨 = 𝔻[ 𝑨 ]} 76 77 module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 78 ≤-trans : 𝑨 ≤ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 79 ≤-trans ( f , finj ) ( g , ginj ) = (⊙-hom f g) , ⊙-injective (proj₁ f) (proj₁ g) finj ginj 80 81 ≤-trans-≅ : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 82 ≤-trans-≅ (h , hinj) B≅C = 83 ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C) 84 85 ≅-trans-≤ : 𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 86 ≅-trans-≤ A≅B (h , hinj) = 87 ⊙-hom (to A≅B) h , ⊙-injective (proj₁ (to A≅B)) (proj₁ h) (≅toInjective A≅B) hinj 88 89 module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 90 ≥-trans : 𝑨 ≥ 𝑩 → 𝑩 ≥ 𝑪 → 𝑨 ≥ 𝑪 91 ≥-trans A≥B B≥C = ≤-trans B≥C A≥B 92 93 ≤→≤c→≤c : {𝑨 : Algebra {𝑆 = 𝑆} α α}{𝑩 : Algebra {𝑆 = 𝑆} α α}{𝒦 : Pred(Algebra {𝑆 = 𝑆} α α) (ov {𝑆 = 𝑆} α)} 94 → 𝑨 ≤ 𝑩 → 𝑩 ≤c 𝒦 → 𝑨 ≤c 𝒦 95 ≤→≤c→≤c A≤B sB = (proj₁ sB) , (proj₁ (proj₂ sB) , ≤-trans A≤B (proj₂ (proj₂ sB))) - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- agda-mcpexports_of(module: Setoid.Homomorphisms)answered56 ms (the file already loaded)elapsedMs: 56exports: 91modules: 4
the whole answer, as the model saw it (25,119 characters of JSON)
{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":56,"exports":[{"name":"_IsHomImageOf_","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {β ρᵇ α ρᵃ : Level}\n(𝑩 : Algebra β ρᵇ) (𝑨 : Algebra α ρᵃ) →\nSet\n(𝑆.𝓞 Agda.Primitive.⊔ 𝑆.𝓥 Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ\n Agda.Primitive.⊔ α\n Agda.Primitive.⊔ ρᵃ)"},{"name":"_≅_","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"Bijective→≅","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩) →\nIsInjective (proj₁ h) → IsSurjective (proj₁ h) → 𝑨 ≅ 𝑩"},{"name":"Cg⊆ker","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ β ρᵇ : Level} {ℓ = ℓ₁ : Level} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩)\n{R : Relation.Binary.Core.Rel Setoid.Algebras.𝕌[ 𝑨 ] ℓ₁} →\nR Relation.Binary.Core.⇒ proj₁ (kercon h) →\nSetoid.Congruences.Generation.Gen R Relation.Binary.Core.⇒\nproj₁ (kercon h)"},{"name":"FirstHomTheorem","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (hh : hom 𝑨 𝑩) →\nSetoid.Algebras.Σ-syntax (hom (kerquo hh) 𝑩)\n(λ .patternInTele0 →\n ((a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Setoid.Homomorphisms.Noether.h hh a)\n (Function.Bundles.Func.to (.patternInTele0 .proj₁)\n (Function.Bundles.Func.to (πker hh .proj₁) a)))\n Data.Product.× IsInjective (.patternInTele0 .proj₁))"},{"name":"FirstHomUnique","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (hh : hom 𝑨 𝑩)\n{.patternInTele0 .patternInTele1 : hom (kerquo hh) 𝑩} →\n((a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Setoid.Homomorphisms.Noether.h hh a)\n (Function.Bundles.Func.to (.patternInTele0 .proj₁)\n (Function.Bundles.Func.to (πker hh .proj₁) a))) →\n((a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Setoid.Homomorphisms.Noether.h hh a)\n (Function.Bundles.Func.to (.patternInTele1 .proj₁)\n (Function.Bundles.Func.to (πker hh .proj₁) a))) →\n([a]\n : Relation.Binary.Bundles.Setoid.Carrier\n (Algebra.Domain (kerquo hh))) →\n(Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to (.patternInTele0 .proj₁) [a])\n(Function.Bundles.Func.to (.patternInTele1 .proj₁) [a])"},{"name":"FromLift","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} {r : Level} →\nhom (Setoid.Algebras.Lift-Alg 𝑨 ℓ₁ r) 𝑨"},{"name":"FromLiftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} →\nhom (Setoid.Algebras.Lift-Algʳ 𝑨 ℓ₁) 𝑨"},{"name":"FromLiftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} →\nhom (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ₁) 𝑨"},{"name":"FromToLiftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level}\n(a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n(Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) a"},{"name":"FromToLiftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level}\n(a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n(Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) a"},{"name":"HomFactor","type":"{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} (.patternInTele0 : hom 𝑨 𝑩)\n(.patternInTele1 : hom 𝑨 𝑪) →\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑪 ])\n(Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1)\nRelation.Unary.⊆\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑩 ])\n(Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1) →\nIsSurjective (.patternInTele1 .proj₁) →\nSetoid.Algebras.Σ-syntax (hom 𝑪 𝑩)\n(λ .patternInTele2 →\n (a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1 a)\n (Function.Bundles.Func.to (.patternInTele2 .proj₁)\n (Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1 a)))"},{"name":"HomFactorEpi","type":"{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} (.patternInTele0 : hom 𝑨 𝑩)\n(.patternInTele1 : hom 𝑨 𝑪) →\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑪 ])\n(Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1)\nRelation.Unary.⊆\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑩 ])\n(Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1) →\nIsSurjective (.patternInTele1 .proj₁) →\nIsSurjective (.patternInTele0 .proj₁) →\nSetoid.Algebras.Σ-syntax (epi 𝑪 𝑩)\n(λ .patternInTele2 →\n (a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1 a)\n (Function.Bundles.Func.to (.patternInTele2 .proj₁)\n (Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1 a)))"},{"name":"HomImage-≅","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 𝑨' : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\n𝑨 IsHomImageOf 𝑨' → 𝑨 ≅ 𝑩 → 𝑩 IsHomImageOf 𝑨'"},{"name":"HomImage-≅'","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 𝑨' : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\n𝑨 IsHomImageOf 𝑨' → 𝑨' ≅ 𝑩 → 𝑨 IsHomImageOf 𝑩"},{"name":"HomImageOfClass","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level} →\nRelation.Unary.Pred (Algebra α ρᵃ) (Agda.Primitive.lsuc α) →\nSet\n(𝑆.𝓞 Agda.Primitive.⊔ 𝑆.𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ)"},{"name":"HomImageOf[_]","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 → Algebra (α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ) ρᵇ"},{"name":"HomImages","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level} →\nAlgebra α ρᵃ →\nSet\n(𝑆.𝓞 Agda.Primitive.⊔ 𝑆.𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ Agda.Primitive.lsuc β\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵇ)"},{"name":"HomKerComp","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (.patternInTele0 : hom 𝑨 𝑩) →\n𝑨 Setoid.Congruences.Basic.∣≈\nOverture.kerRel\n(Relation.Binary.Bundles.Setoid._≈_ (Algebra.Domain 𝑩))\n(Setoid.Homomorphisms.Kernels.h .patternInTele0)"},{"name":"IdHomImage","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} →\n𝑨 IsHomImageOf 𝑨"},{"name":"IsEpi","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ)\n(h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n Setoid.Algebras.𝔻[ 𝑩 ]) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"IsHom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ)\n(h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n Setoid.Algebras.𝔻[ 𝑩 ]) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ ρᵇ)"},{"name":"IsHomImageOfClass","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝒦 : Relation.Unary.Pred (Algebra α ρᵃ) (Agda.Primitive.lsuc α)} →\nAlgebra α ρᵃ →\nSet\n(𝑆.𝓞 Agda.Primitive.⊔ 𝑆.𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ)"},{"name":"IsMon","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ)\n(h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n Setoid.Algebras.𝔻[ 𝑩 ]) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"Lift-Alg-hom-imageˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (ℓᵃ ℓᵇ : Level) →\n𝑩 IsHomImageOf 𝑨 →\nSetoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ IsHomImageOf\nSetoid.Algebras.Lift-Algˡ 𝑨 ℓᵃ"},{"name":"Lift-Alg-iso","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ℓᵃ rᵃ ℓᵇ rᵇ : Level} →\n𝑨 ≅ 𝑩 →\nSetoid.Algebras.Lift-Alg 𝑨 ℓᵃ rᵃ ≅ Setoid.Algebras.Lift-Alg 𝑩 ℓᵇ rᵇ"},{"name":"Lift-Alg-isoʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ℓᵃ ℓᵇ : Level} →\n𝑨 ≅ 𝑩 →\nSetoid.Algebras.Lift-Algʳ 𝑨 ℓᵃ ≅ Setoid.Algebras.Lift-Algʳ 𝑩 ℓᵇ"},{"name":"Lift-Alg-isoˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ℓᵃ ℓᵇ : Level} →\n𝑨 ≅ 𝑩 →\nSetoid.Algebras.Lift-Algˡ 𝑨 ℓᵃ ≅ Setoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ"},{"name":"Lift-Alg-⨅≅ˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ γ β ρᵇ : Level} {𝓘 : Level} {I : Set 𝓘}\n{𝒜 : I → Algebra α ρᵃ} {ℬ : Level.Lift γ I → Algebra β ρᵇ} →\n((i : I) → 𝒜 i ≅ ℬ (Level.lift i)) →\nSetoid.Algebras.Lift-Algˡ (Setoid.Algebras.⨅ 𝒜) γ ≅\nSetoid.Algebras.⨅ ℬ"},{"name":"Lift-HomImage-lemma","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {γ : Level} →\nSetoid.Algebras.Lift-Alg 𝑨 γ γ IsHomImageOf 𝑩 → 𝑨 IsHomImageOf 𝑩"},{"name":"Lift-assoc","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} {ρ : Level} →\nSetoid.Algebras.Lift-Alg 𝑨 ℓ₁ ρ ≅\nSetoid.Algebras.Lift-Algʳ (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ₁) ρ"},{"name":"Lift-assoc'","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α : Level}\n{𝑨 : Algebra α α} {β γ : Level} →\nSetoid.Algebras.Lift-Alg 𝑨 (β Agda.Primitive.⊔ γ)\n(β Agda.Primitive.⊔ γ)\n≅ Setoid.Algebras.Lift-Alg (Setoid.Algebras.Lift-Alg 𝑨 β β) γ γ"},{"name":"Lift-assocʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ₁ ℓ₂ : Level} →\nSetoid.Algebras.Lift-Algʳ 𝑨 (ℓ₁ Agda.Primitive.⊔ ℓ₂) ≅\nSetoid.Algebras.Lift-Algʳ (Setoid.Algebras.Lift-Algʳ 𝑨 ℓ₁) ℓ₂"},{"name":"Lift-assocˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ₁ ℓ₂ : Level} →\nSetoid.Algebras.Lift-Algˡ 𝑨 (ℓ₁ Agda.Primitive.⊔ ℓ₂) ≅\nSetoid.Algebras.Lift-Algˡ (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ₁) ℓ₂"},{"name":"Lift-epi-is-epiˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩)\n(ℓᵃ ℓᵇ : Level) →\nIsSurjective (proj₁ h) → IsSurjective (proj₁ (Lift-homˡ h ℓᵃ ℓᵇ))"},{"name":"Lift-hom","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 →\n(ℓᵃ rᵃ ℓᵇ rᵇ : Level) →\nhom (Setoid.Algebras.Lift-Alg 𝑨 ℓᵃ rᵃ)\n(Setoid.Algebras.Lift-Alg 𝑩 ℓᵇ rᵇ)"},{"name":"Lift-hom-fst","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 → (ℓ₁ r : Level) → hom (Setoid.Algebras.Lift-Alg 𝑨 ℓ₁ r) 𝑩"},{"name":"Lift-hom-snd","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 → (ℓ₁ r : Level) → hom 𝑨 (Setoid.Algebras.Lift-Alg 𝑩 ℓ₁ r)"},{"name":"Lift-homʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 →\n(rᵃ rᵇ : Level) →\nhom (Setoid.Algebras.Lift-Algʳ 𝑨 rᵃ)\n(Setoid.Algebras.Lift-Algʳ 𝑩 rᵇ)"},{"name":"Lift-homˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 →\n(ℓᵃ ℓᵇ : Level) →\nhom (Setoid.Algebras.Lift-Algˡ 𝑨 ℓᵃ)\n(Setoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ)"},{"name":"Lift-≅","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} {ρ : Level} →\n𝑨 ≅ Setoid.Algebras.Lift-Alg 𝑨 ℓ₁ ρ"},{"name":"Lift-≅ʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} →\n𝑨 ≅ Setoid.Algebras.Lift-Algʳ 𝑨 ℓ₁"},{"name":"Lift-≅ˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} →\n𝑨 ≅ Setoid.Algebras.Lift-Algˡ 𝑨 ℓ₁"},{"name":"ToFromLift","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} {r : Level}\n{b\n : Relation.Binary.Bundles.Setoid.Carrier\n (Algebra.Domain (Setoid.Algebras.Lift-Alg 𝑨 ℓ₁ r))} →\n(Setoid.Algebras.𝔻[ Setoid.Algebras.Lift-Alg 𝑨 ℓ₁ r ]\n Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to (ToLift .proj₁)\n (Function.Bundles.Func.to (FromLift .proj₁) b))\nb"},{"name":"ToFromLiftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level}\n(b\n : Relation.Binary.Bundles.Setoid.Carrier\n (Algebra.Domain (Setoid.Algebras.Lift-Algʳ 𝑨 ℓ₁))) →\n(Setoid.Algebras.𝔻[ Setoid.Algebras.Lift-Algʳ 𝑨 ℓ₁ ]\n Relation.Binary.Bundles.Setoid.≈ b)\nb"},{"name":"ToFromLiftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level}\n(b\n : Relation.Binary.Bundles.Setoid.Carrier\n (Algebra.Domain (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ₁))) →\n(Setoid.Algebras.𝔻[ Setoid.Algebras.Lift-Algˡ 𝑨 ℓ₁ ]\n Relation.Binary.Bundles.Setoid.≈ Level.lift (Level.Lift.lower b))\nb"},{"name":"ToLift","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} {r : Level} →\nhom 𝑨 (Setoid.Algebras.Lift-Alg 𝑨 ℓ₁ r)"},{"name":"ToLift-epi","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} {r : Level} →\nepi 𝑨 (Setoid.Algebras.Lift-Alg 𝑨 ℓ₁ r)"},{"name":"ToLiftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} →\nhom 𝑨 (Setoid.Algebras.Lift-Algʳ 𝑨 ℓ₁)"},{"name":"ToLiftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ = ℓ₁ : Level} →\nhom 𝑨 (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ₁)"},{"name":"compatible-map","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nFunction.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\nSetoid.Algebras.𝔻[ 𝑩 ] →\nSet (𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵇ)"},{"name":"compatible-map-op","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nFunction.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\nSetoid.Algebras.𝔻[ 𝑩 ] →\nOverture.OperationSymbolsOf 𝑆 →\nSet (𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵇ)"},{"name":"epi","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ) →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"epi→hom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nepi 𝑨 𝑩 → hom 𝑨 𝑩"},{"name":"epi→ontohom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nepi 𝑨 𝑩 →\nSetoid.Algebras.Σ-syntax (hom 𝑨 𝑩) (λ h → IsSurjective (proj₁ h))"},{"name":"hom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ) →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"ker-in-con","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩) {ℓ = ℓ₁ : Level}\n{θ : Setoid.Congruences.Basic.Con 𝑨 ℓ₁}\n{x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)} →\nkercon (πhom h θ) .proj₁ x y → θ .proj₁ x y"},{"name":"ker[_⇒_]_","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nhom 𝑨 𝑩 → Algebra α ρᵇ"},{"name":"kercon","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (.patternInTele0 : hom 𝑨 𝑩) →\nSetoid.Congruences.Basic.Con 𝑨 ρᵇ"},{"name":"kerquo","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (.patternInTele0 : hom 𝑨 𝑩) →\nAlgebra α ρᵇ"},{"name":"lift-hom-lemma","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩)\n(a : Setoid.Algebras.𝕌[ 𝑨 ]) (ℓᵃ ℓᵇ : Level) →\n(Setoid.Algebras.𝔻[ Setoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ ]\n Relation.Binary.Bundles.Setoid.≈\n Level.lift (Function.Bundles.Func.to (h .proj₁) a))\n(Function.Bundles.Func.to (Lift-homˡ h ℓᵃ ℓᵇ .proj₁)\n (Level.lift a))"},{"name":"mkIsHom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ}\n{h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n Setoid.Algebras.𝔻[ 𝑩 ]}\n(compatible : compatible-map 𝑨 𝑩 h) →\nIsHom 𝑨 𝑩 h"},{"name":"mkhom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ)\n(h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n Setoid.Algebras.𝔻[ 𝑩 ]) →\ncompatible-map 𝑨 𝑩 h → hom 𝑨 𝑩"},{"name":"mkiso","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (to : hom 𝑨 𝑩) (from : hom 𝑩 𝑨)\n(to∼from\n : (b : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑩)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to (to .proj₁)\n (Function.Bundles.Func.to (from .proj₁) b))\n b)\n(from∼to\n : (a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to (from .proj₁)\n (Function.Bundles.Func.to (to .proj₁) a))\n a) →\n𝑨 ≅ 𝑩"},{"name":"mon","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ) →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"mon→hom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nmon 𝑨 𝑩 → hom 𝑨 𝑩"},{"name":"mon→intohom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nmon 𝑨 𝑩 →\nSetoid.Algebras.Σ-syntax (hom 𝑨 𝑩) (λ h → IsInjective (proj₁ h))"},{"name":"πepi","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 →\n{ℓ = ℓ₁ : Level} (θ : Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nepi 𝑨 (𝑨 Setoid.Congruences.Basic._.╱ θ)"},{"name":"πhom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 →\n{ℓ = ℓ₁ : Level} (θ : Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nhom 𝑨 (𝑨 Setoid.Congruences.Basic._.╱ θ)"},{"name":"πker","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩) →\nepi 𝑨 (ker[ 𝑨 ⇒ 𝑩 ] h)"},{"name":"ℓ⨅≅⨅ℓ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ : Level} {ℓᵃ : Level} {I : Set ℓᵃ} {𝒜 : I → Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level} →\nSetoid.Algebras.Lift-Alg (Setoid.Algebras.⨅ 𝒜) ℓ₁ ℓ₁ ≅\nSetoid.Algebras.⨅\n(λ i → Setoid.Algebras.Lift-Alg (𝒜 (Level.Lift.lower i)) ℓ₁ ℓ₁)"},{"name":"≅-mkAlgebra","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ}\n(f\n : (o : Overture.OperationSymbolsOf 𝑆) →\n Overture.Op (Overture.ArityOf 𝑆 o) Setoid.Algebras.𝕌[ 𝑨 ])\n(cong-f\n : (o : Overture.OperationSymbolsOf 𝑆)\n {u v : Overture.ArityOf 𝑆 o → Setoid.Algebras.𝕌[ 𝑨 ]} →\n ((i : Overture.ArityOf 𝑆 o) →\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ u i)\n (v i)) →\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ f o u)\n (f o v)) →\n((o : 𝑆 .proj₁)\n (a\n : Overture.ArityOf 𝑆 o →\n Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈\n (o Setoid.Algebras.^ 𝑨) a)\n (f o a)) →\n𝑨 ≅ Setoid.Algebras.mkAlgebra Setoid.Algebras.𝔻[ 𝑨 ] f cong-f"},{"name":"≅-refl","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} →\n𝑨 ≅ 𝑨"},{"name":"≅-sym","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\n𝑨 ≅ 𝑩 → 𝑩 ≅ 𝑨"},{"name":"≅-trans","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≅ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≅ 𝑪"},{"name":"≅fromInjective","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (φ : 𝑨 ≅ 𝑩) →\nIsInjective (_≅_.from φ .proj₁)"},{"name":"≅toInjective","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (φ : 𝑨 ≅ 𝑩) →\nIsInjective (proj₁ (_≅_.to φ))"},{"name":"≅⨅-refl","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} →\n𝑨 ≅ Setoid.Algebras.⨅ (λ i → 𝑨)"},{"name":"≅⨅⁺-refl","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ : Level} {ι : Level} {𝑨 : Algebra α ρᵃ} →\n𝑨 ≅ Setoid.Algebras.⨅ (λ i → 𝑨)"},{"name":"⊙-epi","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\nepi 𝑨 𝑩 → epi 𝑩 𝑪 → epi 𝑨 𝑪"},{"name":"⊙-hom","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\nhom 𝑨 𝑩 → hom 𝑩 𝑪 → hom 𝑨 𝑪"},{"name":"⊙-is-epi","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}\n{g\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n Setoid.Algebras.𝔻[ 𝑩 ]}\n{h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑩 ]\n Setoid.Algebras.𝔻[ 𝑪 ]} →\nIsEpi 𝑨 𝑩 g → IsEpi 𝑩 𝑪 h → IsEpi 𝑨 𝑪 (h ⊙ g)"},{"name":"⊙-is-hom","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}\n{g\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n Setoid.Algebras.𝔻[ 𝑩 ]}\n{h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑩 ]\n Setoid.Algebras.𝔻[ 𝑪 ]} →\nIsHom 𝑨 𝑩 g → IsHom 𝑩 𝑪 h → IsHom 𝑨 𝑪 (h ⊙ g)"},{"name":"⨅-hom","type":"{𝓞 𝓥 𝓘 α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set 𝓘}\n(𝒜 : I → Algebra α ρ) {β ρᵇ : Level} (ℬ : I → Algebra β ρᵇ) →\n((i : I) → hom (𝒜 i) (ℬ i)) →\nhom (Setoid.Algebras.⨅ 𝒜) (Setoid.Algebras.⨅ ℬ)"},{"name":"⨅-hom-co","type":"{𝓞 𝓥 α ρ 𝓘 β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n{I : Set 𝓘} (ℬ : I → Algebra β ρᵇ) →\n((i : I) → hom 𝑨 (ℬ i)) → hom 𝑨 (Setoid.Algebras.⨅ ℬ)"},{"name":"⨅-proj","type":"{𝓞 𝓥 𝓘 α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set 𝓘}\n(𝒜 : I → Algebra α ρ) (i : I) →\nhom (Setoid.Algebras.⨅ 𝒜) (𝒜 i)"},{"name":"⨅≅","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ β ρᵇ : Level} {𝓘 : Level} {I : Set 𝓘} {𝒜 : I → Algebra α ρᵃ}\n{ℬ : I → Algebra β ρᵇ} →\n((i : I) → 𝒜 i ≅ ℬ i) → Setoid.Algebras.⨅ 𝒜 ≅ Setoid.Algebras.⨅ ℬ"},{"name":"⨅≅⨅lowerℓρ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ : Level} {ℓᵃ : Level} {I : Set ℓᵃ} {𝒜 : I → Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level} {ρ : Level} →\nSetoid.Algebras.⨅ 𝒜 ≅\nSetoid.Algebras.⨅\n(λ i → Setoid.Algebras.Lift-Alg (𝒜 (Level.Lift.lower i)) ℓ₁ ρ)"},{"name":"⨅≅⨅ℓ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ : Level} {𝓘 : Level} {I : Set 𝓘} {𝒜 : I → Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level} →\nSetoid.Algebras.⨅ 𝒜 ≅\nSetoid.Algebras.⨅\n(λ i → Setoid.Algebras.Lift-Alg (𝒜 (Level.Lift.lower i)) ℓ₁ ℓ₁)"},{"name":"⨅≅⨅ℓρ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ : Level} {ι : Level} {I : Set ι} {𝒜 : I → Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level} {ρ : Level} →\nSetoid.Algebras.⨅ 𝒜 ≅\nSetoid.Algebras.⨅ (λ i → Setoid.Algebras.Lift-Alg (𝒜 i) ℓ₁ ρ)"},{"name":"𝒾𝒹","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ} → hom 𝑨 𝑨"}],"lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_show_module_contents_toplevel Simplified \"Setoid.Homomorphisms\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":1368034,"root":"<work>","spawned":false},"module":"Setoid.Homomorphisms","modules":["_≅_","IsEpi","IsHom","IsMon"],"project":{"includePaths":["agda-dojang/agda","<work>"],"librariesFile":"<repo>/agda/libraries","registeredLibraries":[{"includes":["agda"],"libFile":"<repo>/agda-dojang/agda-dojang.agda-lib","name":"agda-dojang","root":"<repo>/agda-dojang"},{"includes":["src"],"libFile":"<nix>/standard-library-2.3/standard-library.agda-lib","name":"standard-library-2.3","root":"<nix>/standard-library-2.3"},{"includes":["src"],"libFile":"<nix>/agda-algebras-unstable-2026-09-07/agda-algebras.agda-lib","name":"agda-algebras","root":"<nix>/agda-algebras-unstable-2026-09-07"}],"root":"<work>","rootSource":"server-config","selectedLibraries":["agda-dojang","standard-library","agda-algebras"]}} - agda-mcpexports_of(module: Setoid.Functions)answered25 ms (the file already loaded)elapsedMs: 25exports: 51modules: 2
the whole answer, as the model saw it (14,833 characters of JSON)
{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":25,"exports":[{"name":"_⊙_","type":"{α ρᵃ β ρᵇ γ ρᶜ : Level} {A : Relation.Binary.Bundles.Setoid α ρᵃ}\n{B : Relation.Binary.Bundles.Setoid β ρᵇ}\n{C : Relation.Binary.Bundles.Setoid γ ρᶜ} →\nFunction.Bundles.Func B C →\nFunction.Bundles.Func A B → Function.Bundles.Func A C"},{"name":"BijInv","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(f : Function.Bundles.Func 𝑨 𝑩) →\nIsBijective f → Function.Bundles.Func 𝑩 𝑨"},{"name":"Dom","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 → Relation.Binary.Bundles.Setoid α ρᵇ"},{"name":"Image_∋_","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 →\nRelation.Binary.Bundles.Setoid.Carrier 𝑩 →\nSet (α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ)"},{"name":"Imagef∋f","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩}\n{a : Relation.Binary.Bundles.Setoid.Carrier 𝑨} →\nImage F ∋ Function.Bundles.Func.to F a"},{"name":"Image⊆Range","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩}\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} →\nImage F ∋ b → b Relation.Unary.∈ IsInRange F"},{"name":"Img_∋_","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\n(Relation.Binary.Bundles.Setoid.Carrier 𝑨 →\n Relation.Binary.Bundles.Setoid.Carrier 𝑩) →\nRelation.Binary.Bundles.Setoid.Carrier 𝑩 →\nSet (α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ)"},{"name":"Inv","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩)\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} →\nImage F ∋ b → Relation.Binary.Bundles.Setoid.Carrier 𝑨"},{"name":"Inv'","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩)\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} →\nb Relation.Unary.∈ IsInRange F →\nRelation.Binary.Bundles.Setoid.Carrier 𝑨"},{"name":"InvIsInverseʳ","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩}\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} (q : Image F ∋ b) →\n(𝑩 Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to F (Inv F q))\nb"},{"name":"InvIsInverseˡ","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩}\n{a : Relation.Binary.Bundles.Setoid.Carrier 𝑨} →\n(𝑨 Relation.Binary.Bundles.Setoid.≈ a) a"},{"name":"IsBijective","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 →\nSet (α Agda.Primitive.⊔ ρᵃ Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ)"},{"name":"IsInRange","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 →\nRelation.Unary.Pred (Relation.Binary.Bundles.Setoid.Carrier 𝑩)\n(α Agda.Primitive.⊔ ρᵇ)"},{"name":"IsInRange→IsInImage","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩}\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} →\nb Relation.Unary.∈ IsInRange F → Image F ∋ b"},{"name":"IsInjective","type":"{a α b β : Level} {𝑨 : Relation.Binary.Bundles.Setoid a α}\n{𝑩 : Relation.Binary.Bundles.Setoid b β} →\nFunction.Bundles.Func 𝑨 𝑩 →\nSet (a Agda.Primitive.⊔ α Agda.Primitive.⊔ β)"},{"name":"IsSurjective","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 →\nSet (α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ)"},{"name":"LeftInvPreserves≈","type":"{a α b β : Level} {𝑨 : Relation.Binary.Bundles.Setoid a α}\n{𝑩 : Relation.Binary.Bundles.Setoid b β}\n(F : Function.Bundles.Injection 𝑨 𝑩)\n{b₀ b₁ : Relation.Binary.Bundles.Setoid.Carrier 𝑩}\n(u : Image Function.Bundles.Injection.function F ∋ b₀)\n(v : Image Function.Bundles.Injection.function F ∋ b₁) →\n(𝑩 Relation.Binary.Bundles.Setoid.≈ b₀) b₁ →\n(𝑨 Relation.Binary.Bundles.Setoid.≈\n Inv (Function.Bundles.Injection.function F) u)\n(Inv (Function.Bundles.Injection.function F) v)"},{"name":"RRan","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 →\nRelation.Binary.Bundles.Setoid\n(α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ) (ρᵃ Agda.Primitive.⊔ ρᵇ)"},{"name":"Ran","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 →\nRelation.Binary.Bundles.Setoid\n(α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ) ρᵇ"},{"name":"SurjInv","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(f : Function.Bundles.Func 𝑨 𝑩) →\nIsSurjective f →\nRelation.Binary.Bundles.Setoid.Carrier 𝑩 →\nRelation.Binary.Bundles.Setoid.Carrier 𝑨"},{"name":"SurjInvIsInverseʳ","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(f : Function.Bundles.Func 𝑨 𝑩) (fE : IsSurjective f)\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} →\n(𝑩 Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to f (SurjInv f fE b))\nb"},{"name":"SurjectionIsSurjection","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Surjection 𝑨 𝑩 →\nSetoid.Algebras.Σ-syntax (Function.Bundles.Func 𝑨 𝑩)\n(λ g →\n Function.Structures.IsSurjection\n (Relation.Binary.Bundles.Setoid._≈_ 𝑨)\n (Relation.Binary.Bundles.Setoid._≈_ 𝑩)\n (Function.Bundles.Func.to g))"},{"name":"SurjectionIsSurjective","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Surjection 𝑨 𝑩 →\nSetoid.Algebras.Σ-syntax (Function.Bundles.Func 𝑨 𝑩) IsSurjective"},{"name":"[_]⁻¹","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩) →\nrange F → Relation.Binary.Bundles.Setoid.Carrier 𝑨"},{"name":"epic-factor","type":"{α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{𝑪 : Relation.Binary.Bundles.Setoid γ ρᶜ}\n(f : Function.Bundles.Func 𝑨 𝑩) (g : Function.Bundles.Func 𝑨 𝑪)\n(h : Function.Bundles.Func 𝑪 𝑩) →\nIsSurjective f →\n((i : Relation.Binary.Bundles.Setoid.Carrier 𝑨) →\n (𝑩 Relation.Binary.Bundles.Setoid.≈ Function.Bundles.Func.to f i)\n (Function.Bundles.Func.to h (Function.Bundles.Func.to g i))) →\nIsSurjective h"},{"name":"eq","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{f\n : Relation.Binary.Bundles.Setoid.Carrier 𝑨 →\n Relation.Binary.Bundles.Setoid.Carrier 𝑩}\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩}\n(a : Relation.Binary.Bundles.Setoid.Carrier 𝑨) →\n(𝑩 Relation.Binary.Bundles.Setoid.≈ b) (f a) → Img f ∋ b"},{"name":"eq","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩}\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩}\n(a : Relation.Binary.Bundles.Setoid.Carrier 𝑨) →\n(𝑩 Relation.Binary.Bundles.Setoid.≈ b)\n(Function.Bundles.Func.to F a) →\nImage F ∋ b"},{"name":"f∈range","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩} →\nRelation.Binary.Bundles.Setoid.Carrier 𝑨 → range F"},{"name":"id-is-injective","type":"{a α : Level} {𝑨 : Relation.Binary.Bundles.Setoid a α} →\nIsInjective 𝑖𝑑"},{"name":"image","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩) →\nrange F → Relation.Binary.Bundles.Setoid.Carrier 𝑩"},{"name":"inv","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(f\n : Relation.Binary.Bundles.Setoid.Carrier 𝑨 →\n Relation.Binary.Bundles.Setoid.Carrier 𝑩)\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} →\nImg f ∋ b → Relation.Binary.Bundles.Setoid.Carrier 𝑨"},{"name":"invIsInvʳ","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{f\n : Relation.Binary.Bundles.Setoid.Carrier 𝑨 →\n Relation.Binary.Bundles.Setoid.Carrier 𝑩}\n{b : Relation.Binary.Bundles.Setoid.Carrier 𝑩} (q : Img f ∋ b) →\n(𝑩 Relation.Binary.Bundles.Setoid.≈ f (inv f q)) b"},{"name":"isSurj","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\n(Relation.Binary.Bundles.Setoid.Carrier 𝑨 →\n Relation.Binary.Bundles.Setoid.Carrier 𝑩) →\nSet (α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ)"},{"name":"isSurj→IsSurjective","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩) →\nisSurj (Function.Bundles.Func.to F) → IsSurjective F"},{"name":"liftFunc","type":"{α ρᵃ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{ℓ = ℓ₁ : Level} →\nFunction.Bundles.Func 𝑨 (𝑙𝑖𝑓𝑡 ℓ₁)"},{"name":"lift∼lower","type":"{α ρᵃ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{β : Level}\n(a : Level.Lift β (Relation.Binary.Bundles.Setoid.Carrier 𝑨)) →\n(𝑨 Relation.Binary.Bundles.Setoid.≈ Level.Lift.lower a)\n(Level.Lift.lower a)"},{"name":"lower∼lift","type":"{α ρᵃ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{β : Level} (a : Relation.Binary.Bundles.Setoid.Carrier 𝑨) →\n(𝑨 Relation.Binary.Bundles.Setoid.≈ a) a"},{"name":"preimage","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩) →\nrange F → Relation.Binary.Bundles.Setoid.Carrier 𝑨"},{"name":"preimage≈image","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩) (r : range F) →\n(𝑩 Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to F (preimage F r))\n(image F r)"},{"name":"range","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ} →\nFunction.Bundles.Func 𝑨 𝑩 →\nSet (α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ)"},{"name":"⁻¹IsInverseʳ","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩} {bap : range F} →\n(𝑩 Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to F ([ F ]⁻¹ bap))\n(bap .proj₁)"},{"name":"⁻¹IsInverseˡ","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{F : Function.Bundles.Func 𝑨 𝑩}\n{a : Relation.Binary.Bundles.Setoid.Carrier 𝑨} →\n(𝑨 Relation.Binary.Bundles.Setoid.≈ [ F ]⁻¹ (f∈range a)) a"},{"name":"∘-epic","type":"{α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{𝑪 : Relation.Binary.Bundles.Setoid γ ρᶜ} →\nFunction.Bundles.Surjection 𝑨 𝑪 →\nFunction.Bundles.Surjection 𝑪 𝑩 → Function.Bundles.Surjection 𝑨 𝑩"},{"name":"∘-injective-bare","type":"{a α b β c γ : Level} {A : Set a}\n(_≈₁_ : Relation.Binary.Core.Rel A α) {B : Set b}\n(_≈₂_ : Relation.Binary.Core.Rel B β) {C : Set c}\n(_≈₃_ : Relation.Binary.Core.Rel C γ) {f : A → B} {g : B → C} →\nFunction.Definitions.Injective _≈₁_ _≈₂_ f →\nFunction.Definitions.Injective _≈₂_ _≈₃_ g →\nFunction.Definitions.Injective _≈₁_ _≈₃_ (g Function.Base.∘ f)"},{"name":"⊙-IsSurjective","type":"{α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n{𝑪 : Relation.Binary.Bundles.Setoid γ ρᶜ}\n{G : Function.Bundles.Func 𝑨 𝑪} {H : Function.Bundles.Func 𝑪 𝑩} →\nIsSurjective G → IsSurjective H → IsSurjective (H ⊙ G)"},{"name":"⊙-injection","type":"{a α b β c γ : Level} {𝑨 : Relation.Binary.Bundles.Setoid a α}\n{𝑩 : Relation.Binary.Bundles.Setoid b β}\n{𝑪 : Relation.Binary.Bundles.Setoid c γ} →\nFunction.Bundles.Injection 𝑨 𝑩 →\nFunction.Bundles.Injection 𝑩 𝑪 → Function.Bundles.Injection 𝑨 𝑪"},{"name":"⊙-injective","type":"{a α b β c γ : Level} {𝑨 : Relation.Binary.Bundles.Setoid a α}\n{𝑩 : Relation.Binary.Bundles.Setoid b β}\n{𝑪 : Relation.Binary.Bundles.Setoid c γ}\n(f : Function.Bundles.Func 𝑨 𝑩) (g : Function.Bundles.Func 𝑩 𝑪) →\nIsInjective f → IsInjective g → IsInjective (g ⊙ f)"},{"name":"⌜_⌝","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩) →\nRelation.Binary.Bundles.Setoid.Carrier 𝑨 → range F"},{"name":"⟦_⟧⁻¹","type":"{α ρᵃ β ρᵇ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n{𝑩 : Relation.Binary.Bundles.Setoid β ρᵇ}\n(F : Function.Bundles.Func 𝑨 𝑩) →\nFunction.Bundles.Func (Ran F) (Dom F)"},{"name":"𝑖𝑑","type":"{α ρᵃ : Level} {A : Relation.Binary.Bundles.Setoid α ρᵃ} →\nFunction.Bundles.Func A A"},{"name":"𝑙𝑖𝑓𝑡","type":"{α ρᵃ : Level} {𝑨 : Relation.Binary.Bundles.Setoid α ρᵃ}\n(ℓ₁ : Level) →\nRelation.Binary.Bundles.Setoid (α Agda.Primitive.⊔ ℓ₁) ρᵃ"}],"lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_show_module_contents_toplevel Simplified \"Setoid.Functions\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":1368034,"root":"<work>","spawned":false},"module":"Setoid.Functions","modules":["Image_∋_","Img_∋_"],"project":{"includePaths":["agda-dojang/agda","<work>"],"librariesFile":"<repo>/agda/libraries","registeredLibraries":[{"includes":["agda"],"libFile":"<repo>/agda-dojang/agda-dojang.agda-lib","name":"agda-dojang","root":"<repo>/agda-dojang"},{"includes":["src"],"libFile":"<nix>/standard-library-2.3/standard-library.agda-lib","name":"standard-library-2.3","root":"<nix>/standard-library-2.3"},{"includes":["src"],"libFile":"<nix>/agda-algebras-unstable-2026-09-07/agda-algebras.agda-lib","name":"agda-algebras","root":"<nix>/agda-algebras-unstable-2026-09-07"}],"root":"<work>","rootSource":"server-config","selectedLibraries":["agda-dojang","standard-library","agda-algebras"]}} - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- agda-mcptype_of(expr: _≅_.to)answered3 ms (the file already loaded)type: {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)} {𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} → 𝑨 ≅ 𝑩 → Data.Product.Σ (Function.Bundles.Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑩)) (IsHom 𝑨 𝑩)elapsedMs: 3
the whole answer, as the model saw it (1,315 characters of JSON)
{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":3,"expr":"_≅_.to","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_infer_toplevel Normalised \"_\\8773_.to\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":1368034,"root":"<work>","spawned":false},"project":{"includePaths":["agda-dojang/agda","<work>"],"librariesFile":"<repo>/agda/libraries","registeredLibraries":[{"includes":["agda"],"libFile":"<repo>/agda-dojang/agda-dojang.agda-lib","name":"agda-dojang","root":"<repo>/agda-dojang"},{"includes":["src"],"libFile":"<nix>/standard-library-2.3/standard-library.agda-lib","name":"standard-library-2.3","root":"<nix>/standard-library-2.3"},{"includes":["src"],"libFile":"<nix>/agda-algebras-unstable-2026-09-07/agda-algebras.agda-lib","name":"agda-algebras","root":"<nix>/agda-algebras-unstable-2026-09-07"}],"root":"<work>","rootSource":"server-config","selectedLibraries":["agda-dojang","standard-library","agda-algebras"]},"scope":"toplevel","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level}\n{𝑆 : Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} →\n𝑨 ≅ 𝑩 →\nData.Product.Σ\n(Function.Bundles.Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑩))\n(IsHom 𝑨 𝑩)"} - the modelGood, `_≅_.to` resolves fine. Now let's check the type of `_≤_` to confirm `p` unfolds to `(f , finj)` shape.
- agda-mcptype_of(expr: _≤_)answered2 ms (the file already loaded)type: {𝓞 𝓥 : Level} {𝑆 = ρᵃ : Data.Product.Σ (Set 𝓞) (λ α → α → Set 𝓥)} {α = β : Level} {ρᵃ = ρᵇ : Level} {β = F : Level} {ρᵇ = 𝑆 : Level} → Algebra β ρᵇ → Algebra F 𝑆 → Set (𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ β Agda.Primit…elapsedMs: 2
the whole answer, as the model saw it (1,373 characters of JSON)
{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":2,"expr":"_≤_","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_infer_toplevel Normalised \"_\\8804_\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":1368034,"root":"<work>","spawned":false},"project":{"includePaths":["agda-dojang/agda","<work>"],"librariesFile":"<repo>/agda/libraries","registeredLibraries":[{"includes":["agda"],"libFile":"<repo>/agda-dojang/agda-dojang.agda-lib","name":"agda-dojang","root":"<repo>/agda-dojang"},{"includes":["src"],"libFile":"<nix>/standard-library-2.3/standard-library.agda-lib","name":"standard-library-2.3","root":"<nix>/standard-library-2.3"},{"includes":["src"],"libFile":"<nix>/agda-algebras-unstable-2026-09-07/agda-algebras.agda-lib","name":"agda-algebras","root":"<nix>/agda-algebras-unstable-2026-09-07"}],"root":"<work>","rootSource":"server-config","selectedLibraries":["agda-dojang","standard-library","agda-algebras"]},"scope":"toplevel","type":"{𝓞 𝓥 : Level} {𝑆 = ρᵃ : Data.Product.Σ (Set 𝓞) (λ α → α → Set 𝓥)}\n{α = β : Level} {ρᵃ = ρᵇ : Level} {β = F : Level}\n{ρᵇ = 𝑆 : Level} →\nAlgebra β ρᵇ →\nAlgebra F 𝑆 →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ\n Agda.Primitive.⊔ F\n Agda.Primitive.⊔ 𝑆)"} - the modelNow let's fill the hole using pattern matching on `p` directly, mirroring the library proof.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: ≤-trans-≅′ p B≅C = {!!}, new_string: ≤-trans-≅′ (h , hinj) B≅C = ⊙-hom h (_≅_.to B≅C) , ⊙-injective (proj₁ h) (proj₁ (_≅_.to B≅C)) hinj (≅toInjective B≅C))answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- agda-mcpcheck_file()answered5.5 ssuccess: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5485diagnostics: 0holes: 0
the whole answer, as the model saw it (1,818 characters of JSON)
{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","--safe","-i","<work>","<work>/Subalgebras-sub-trans-iso.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5485,"holes":[],"holesCount":0,"project":{"includePaths":["agda-dojang/agda","<work>"],"librariesFile":"<repo>/agda/libraries","registeredLibraries":[{"includes":["agda"],"libFile":"<repo>/agda-dojang/agda-dojang.agda-lib","name":"agda-dojang","root":"<repo>/agda-dojang"},{"includes":["src"],"libFile":"<nix>/standard-library-2.3/standard-library.agda-lib","name":"standard-library-2.3","root":"<nix>/standard-library-2.3"},{"includes":["src"],"libFile":"<nix>/agda-algebras-unstable-2026-09-07/agda-algebras.agda-lib","name":"agda-algebras","root":"<nix>/agda-algebras-unstable-2026-09-07"}],"root":"<work>","rootSource":"server-config","selectedLibraries":["agda-dojang","standard-library","agda-algebras"]},"success":true,"timedOut":false,"verdict":{"equivalentTo":"equivalent-to: <nix>/agdaWithPackages-2.8.0/bin/agda -i agda-dojang/agda --library-file=agda/libraries -l agda-dojang -l standard-library -l agda-algebras --safe -i <work> <work>/Subalgebras-sub-trans-iso.agda","exitCode":0,"meaning":"success is true if and only if that command exited 0, so it means exactly what green means in a batch build. Unsolved metavariables, unsolved constraints, and open holes all make agda exit non-zero and so make success false; there is no interaction mode anywhere in this server. The verdict is read from the exit code alone — never from the diagnostics text — so a change in Agda's message format can empty the diagnostics list but cannot turn a failing build green."}} - the modelcheck_file reports success (exit code 0, no diagnostics, no holes) on the final file.
The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the tool’s own, in full. The client billed 1,030 thinking tokens over the session; their text is not in the archive.
solvedthe judge, in outcome.json
Every gate passed, and the definition does not refer to the library’s own lemma for this statement, so the row counts as a solve.
Agda’s exit code on the file 0
statement preserved yes
last verdict taken from
check_filethe library’s own proof in view before the edit yes, the model read the file that holds it
the file the judge read, marked against the obligation it started as
-- Subalgebras-sub-trans-iso.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda
--
-- Benchmark obligation: algebras-subalgebras-sub-trans-iso
-- Difficulty: non-obvious
-- Source: Setoid.Subalgebras.Properties (agda-algebras)
-- Import stratum: wholesale
-- Strategy: pairing
--
module Subalgebras-sub-trans-iso where
open import AgdaDojang.Debug
open import Agda.Primitive using ( Level )
open import Data.Product using ( _,_ ; proj₁ ; proj₂ )
open import Overture using ( Signature )
open import Setoid.Algebras using ( Algebra )
open import Setoid.Functions
open import Setoid.Homomorphisms
open import Setoid.Subalgebras
≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥}
{𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}
→ 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪
≤-trans-≅′ p B≅C = {!!}
≤-trans-≅′ (h , hinj) B≅C =
⊙-hom h (_≅_.to B≅C) , ⊙-injective (proj₁ h) (proj₁ (_≅_.to B≅C)) hinj (≅toInjective B≅C)≤-trans-≅′, shell only, hints in view
Shell only, with the hint in view. The same model finds the same lemmas with grep and find over the library's sources, writes the proof out, and takes its verdicts from agda in the shell, which the archive does not time: the first is refused, the second accepted.
- model
- Sonnet 5 (claude-sonnet-5)
- instrument
- shell only
- run
- arm162-shell-1 2026-09-21
- fixture header
- hints in view:
Source: Setoid.Subalgebras.Properties (agda-algebras);Strategy: pairing - server
- the server of 2026-09-21 (fourteen tools, the libraries' sources readable on every arm)
- client
- 2.1.261 (Claude Code)
- turns
- 15
- tool calls
- 14
Read×1,Bash×11,Edit×2 - wall clock
- 1.2 min (indicative: subjects ran three at a time)
- cost, list price
- USD 0.17
the statement, as the obligation states it, with the proof removed
{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪the session, 15 turns and 14 tool calls
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- filesRead(file_path: <work>/Subalgebras-sub-trans-iso.agda)answered1 -- Subalgebras-sub-trans-iso.agda
the whole answer, as the model saw it (963 characters of text)
1 -- Subalgebras-sub-trans-iso.agda 2 -- 3 -- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda 4 -- 5 -- Benchmark obligation: algebras-subalgebras-sub-trans-iso 6 -- Difficulty: non-obvious 7 -- Source: Setoid.Subalgebras.Properties (agda-algebras) 8 -- Import stratum: wholesale 9 -- Strategy: pairing 10 -- 11 module Subalgebras-sub-trans-iso where 12 13 open import AgdaDojang.Debug 14 15 open import Agda.Primitive using ( Level ) 16 open import Data.Product using ( _,_ ; proj₁ ; proj₂ ) 17 18 open import Overture using ( Signature ) 19 open import Setoid.Algebras using ( Algebra ) 20 open import Setoid.Functions 21 open import Setoid.Homomorphisms 22 open import Setoid.Subalgebras 23 24 ≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} 25 {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} 26 → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 27 ≤-trans-≅′ p B≅C = {!!} 28 - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "≤-trans-≅\|_≤_\|_≅_" -r <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/*.agda | head -80)answeredugrep: warning: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/*.agda: No such file or directory
the whole answer, as the model saw it (112 characters of text)
ugrep: warning: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/*.agda: No such file or directory
- shellBash(command: find <nix>/agda-algebras-unstable-2026-09-07 -iname "*Subalgebras*")answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md
the whole answer, as the model saw it (818 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/_build/2.8.0/agda/src/Setoid/Subalgebras.agdai <nix>/agda-algebras-unstable-2026-09-07/_build/2.8.0/agda/src/Setoid/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/_build/2.8.0/agda/src/Legacy/Base/Subalgebras.agdai <nix>/agda-algebras-unstable-2026-09-07/_build/2.8.0/agda/src/Legacy/Base/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/_build/2.8.0/agda/src/Legacy/Base/Subalgebras/Subalgebras.agdai
- shellBash(command: find <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras -iname "*")answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras
the whole answer, as the model saw it (915 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subuniverses.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/CompleteLattice.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Basic.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Finite.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/BirkhoffSI.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Irreducible.lagda.md
- shellBash(command: grep -n "≤-trans-≅\|≅\|≤" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md | head -100)answered13:`_≤_`{.AgdaFunction} that the rest of the library relies on: that it is a
the whole answer, as the model saw it (4,561 characters of text)
13:`_≤_`{.AgdaFunction} that the rest of the library relies on: that it is a 18:(`≤-preorder`{.AgdaFunction}), which is the only sensible choice, since mutual 21:Agda's universes are not cumulative, so a `≤` established at one level has to be 47:open import Setoid.Homomorphisms using ( hom ; IsHom ; 𝒾𝒹 ; ⊙-hom ; _≅_ 48: ; ≅toInjective ; ≅fromInjective ; mkiso 49: ; ≅-sym ; ≅-refl ; ≅-trans 50: ; Lift-≅ˡ ; Lift-≅ ; Lift-≅ʳ) 51:open import Setoid.Subalgebras.Basic using ( _≤_ ; _≥_ ; _≤c_ ) 60:open _≅_ 62:≅→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 63:≅→≤ φ = (to φ) , ≅toInjective φ 65:≅→≥ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 66:≅→≥ φ = (from φ) , ≅fromInjective φ 68:≤-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 69:≤-refl = ≅→≤ 71:≥-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 72:≥-refl = ≅→≤ ∘ ≅-sym 74:≤-reflexive : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≤ 𝑨 75:≤-reflexive {𝑨 = 𝑨} = 𝒾𝒹 , id-is-injective {𝑨 = 𝔻[ 𝑨 ]} 78: ≤-trans : 𝑨 ≤ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 79: ≤-trans ( f , finj ) ( g , ginj ) = (⊙-hom f g) , ⊙-injective (proj₁ f) (proj₁ g) finj ginj 81: ≤-trans-≅ : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 82: ≤-trans-≅ (h , hinj) B≅C = 83: ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C) 85: ≅-trans-≤ : 𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 86: ≅-trans-≤ A≅B (h , hinj) = 87: ⊙-hom (to A≅B) h , ⊙-injective (proj₁ (to A≅B)) (proj₁ h) (≅toInjective A≅B) hinj 91: ≥-trans A≥B B≥C = ≤-trans B≥C A≥B 93:≤→≤c→≤c : {𝑨 : Algebra {𝑆 = 𝑆} α α}{𝑩 : Algebra {𝑆 = 𝑆} α α}{𝒦 : Pred(Algebra {𝑆 = 𝑆} α α) (ov {𝑆 = 𝑆} α)} 94: → 𝑨 ≤ 𝑩 → 𝑩 ≤c 𝒦 → 𝑨 ≤c 𝒦 95:≤→≤c→≤c A≤B sB = (proj₁ sB) , (proj₁ (proj₂ sB) , ≤-trans A≤B (proj₂ (proj₂ sB))) 99: open RelStructs {a = ov {𝑆 = 𝑆} (α ⊔ ρᵃ)} {ℓ = 𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ} (_≅_ {α = α}{ρᵃ = ρᵃ}{𝑆 = 𝑆}) 102: ≤-preorder : IsPreorder _≤_ 103: isEquivalence ≤-preorder = record { refl = ≅-refl ; sym = ≅-sym ; trans = ≅-trans } 104: reflexive ≤-preorder = ≤-refl 105: trans ≤-preorder A≤B B≤C = ≤-trans A≤B B≤C 108: A≥B×B≅C→A≥C : 𝑨 ≥ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≥ 𝑪 109: A≥B×B≅C→A≥C A≥B B≅C = ≥-trans A≥B (≅→≥ B≅C) 111: A≤B×B≅C→A≤C : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 112: A≤B×B≅C→A≤C A≤B B≅C = ≤-trans A≤B (≅→≤ B≅C) 114: A≅B×B≥C→A≥C : 𝑨 ≅ 𝑩 → 𝑩 ≥ 𝑪 → 𝑨 ≥ 𝑪 115: A≅B×B≥C→A≥C A≅B B≥C = ≥-trans (≅→≥ A≅B) B≥C 117: A≅B×B≤C→A≤C : 𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 118: A≅B×B≤C→A≤C A≅B B≤C = ≤-trans (≅→≤ A≅B) B≤C 123: (φ : 𝑨 ≅ 𝑩) → IsInjective (proj₁ (to φ)) 132:≤-mono : {𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝒦 𝒦' : Pred (Algebra {𝑆 = 𝑆} α ρᵃ) γ} 133: → 𝒦 ⊆ 𝒦' → 𝑩 ≤c 𝒦 → 𝑩 ≤c 𝒦' 134:≤-mono KK' (𝑨 , (KA , B≤A)) = 𝑨 , ((KK' KA) , B≤A) 139:Ten lemmas of one shape: transport a `≤` or a `≥` across the isomorphism between 141:of a mixed transitivity lemma from above to one of the `Lift-≅`{.AgdaFunction} 143:`Lift-≤-Lift`{.AgdaFunction} variants then compose two of those. Either way no 152:+ `≤-Liftˡ`{.AgdaFunction}, `≤-Liftʳ`{.AgdaFunction} and 153: `≤-Lift`{.AgdaFunction} lift the algebra on the right of `≤`, the larger one; 156:+ `Lift-≤-Liftˡ`{.AgdaFunction}, `Lift-≤-Liftʳ`{.AgdaFunction} and 157: `Lift-≤-Lift`{.AgdaFunction} lift both algebras at once, and are literally the 166: → 𝑩 ≤c 𝒦 → (Lift-Algˡ 𝑩 ℓ) ≤c 𝒦 167:Lift-is-sub (𝑨 , (KA , B≤A)) = 𝑨 , (KA , A≥B×B≅C→A≥C B≤A Lift-≅ˡ) 170: ≤-Liftˡ : {ℓ : Level} → 𝑨 ≤ 𝑩 → 𝑨 ≤ Lift-Algˡ 𝑩 ℓ 171: ≤-Liftˡ A≤B = A≤B×B≅C→A≤C A≤B Lift-≅ˡ 173: ≤-Liftʳ : {ρ : Level} → 𝑨 ≤ 𝑩 → 𝑨 ≤ Lift-Algʳ 𝑩 ρ 174: ≤-Liftʳ A≤B = A≤B×B≅C→A≤C A≤B Lift-≅ʳ 176: ≤-Lift : {ℓ ρ : Level} → 𝑨 ≤ 𝑩 → 𝑨 ≤ Lift-Alg 𝑩 ℓ ρ 177: ≤-Lift A≤B = A≤B×B≅C→A≤C A≤B Lift-≅ 180: ≥-Liftˡ A≥B = A≥B×B≅C→A≥C A≥B Lift-≅ˡ 183: ≥-Liftʳ A≥B = A≥B×B≅C→A≥C A≥B Lift-≅ʳ 186: ≥-Lift A≥B = A≥B×B≅C→A≥C A≥B Lift-≅ 189: Lift-≤-Liftˡ : {ℓᵃ ℓᵇ : Level} → 𝑨 ≤ 𝑩 → Lift-Algˡ 𝑨 ℓᵃ ≤ Lift-Algˡ 𝑩 ℓᵇ 190: Lift-≤-Liftˡ A≤B = ≥-Liftˡ (≤-Liftˡ A≤B) 192: Lift-≤-Liftʳ : {rᵃ rᵇ : Level} → 𝑨 ≤ 𝑩 → Lift-Algʳ 𝑨 rᵃ ≤ Lift-Algʳ 𝑩 rᵇ 193: Lift-≤-Liftʳ A≤B = ≥-Liftʳ (≤-Liftʳ A≤B) 195: Lift-≤-Lift : {a rᵃ b rᵇ : Level} 196: → 𝑨 ≤ 𝑩 → Lift-Alg 𝑨 a rᵃ ≤ Lift-Alg 𝑩 b rᵇ 197: Lift-≤-Lift A≤B = ≥-Lift (≤-Lift A≤B) 202:`⨅-≤`{.AgdaFunction} is the compatibility of the subalgebra relation with 203:products: if `ℬ i ≤ 𝒜 i` for every index `i`, then `⨅ ℬ ≤ ⨅ 𝒜`. The embedding is 221: ⨅-≤ : (∀ i → ℬ i ≤ 𝒜 i) → ⨅ ℬ ≤ ⨅ 𝒜 222: ⨅-≤ B≤A = h , hM 228: homAt = λ i → proj₁ (B≤A i) 241: hM = λ xy i → (proj₂ (B≤A i)) (xy i) - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "⊙-hom\|⊙-injective\b" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/*.lagda.md | head -30 echo --- grep -n "^_≤_\|_≤_ " <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md | head -20)answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:34:+ **composition**, `⊙-hom`{.AgdaFunc…
the whole answer, as the model saw it (2,433 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:34:+ **composition**, `⊙-hom`{.AgdaFunction}, and the homomorphisms that witness <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:39: ; lift-hom-lemma ; ⊙-hom ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:182: Lift-HomImage-lemma {γ} φ = ⊙-hom (φ .proj₁) (from Lift-≅) , <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:188: HomImage-≅ φ A≅B = ⊙-hom (φ .proj₁) (to A≅B) , ⊙-IsSurjective (φ .proj₂) (toIsSurjective A≅B) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:191: HomImage-≅' φ A'≅B = (⊙-hom (from A'≅B) (proj₁ φ)) , ⊙-IsSurjective (fromIsSurjective A'≅B) (φ .proj₂) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:66:open import Setoid.Homomorphisms.Properties using ( ⊙-hom ; ToLiftˡ ; FromLiftˡ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:145: f = ⊙-hom (to ab) (to bc) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:148: g = ⊙-hom (from bc) (from ab) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:73:+ `⊙-hom`{.AgdaFunction} and `⊙-epi`{.AgdaFunction} are the bundled forms. <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:78:`⊙-hom f g` is read as "apply `f`, then `g`". <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:102: ⊙-hom : hom 𝑨 𝑩 → hom 𝑩 𝑪 → hom 𝑨 𝑪 <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:103: ⊙-hom (h , hhom) (g , ghom) = g ⊙ h , ⊙-is-hom hhom ghom <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:182: ToLift = ⊙-hom ToLiftˡ ToLiftʳ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:185: FromLift = ⊙-hom FromLiftʳ FromLiftˡ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:255: Lift-hom-fst φ _ _ = ⊙-hom FromLift φ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Properties.lagda.md:258: Lift-hom-snd φ _ _ = ⊙-hom φ ToLift --- 102: _≤_ -- alias for subalgebra relation - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "⊙-injective" -r <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/ 2>/dev/null | head -20 echo --- sed -n '1,40p' <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md)answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Functions/Injective.lagda.md:91: ⊙-injective : (f : 𝑨 ⟶ 𝑩)(g : 𝑩 ⟶…
the whole answer, as the model saw it (3,246 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Functions/Injective.lagda.md:91: ⊙-injective : (f : 𝑨 ⟶ 𝑩)(g : 𝑩 ⟶ 𝑪) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Functions/Injective.lagda.md:95: ⊙-injective _ _ finj ginj = finj ∘ ginj <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Functions/Injective.lagda.md:101: ; injective = ⊙-injective (function fi) (function gi) (injective fi) (injective gi) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md:46:open import Setoid.Functions using ( id-is-injective ; IsInjective ; ⊙-injective ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md:79: ≤-trans ( f , finj ) ( g , ginj ) = (⊙-hom f g) , ⊙-injective (proj₁ f) (proj₁ g) finj ginj <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md:83: ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md:87: ⊙-hom (to A≅B) h , ⊙-injective (proj₁ (to A≅B)) (proj₁ h) (≅toInjective A≅B) hinj --- --- layout: default title : "Setoid.Subalgebras.Basic module (The Agda Universal Algebra Library)" date : "2021-07-17" author: "agda-algebras development team" --- #### Subalgebras of setoid algebras This is the [Setoid.Subalgebras.Basic][] module of the [Agda Universal Algebra Library][]. `𝑨` is a **subalgebra** of `𝑩`, written `𝑨 ≤ 𝑩`, just in case `𝑨` can be *homomorphically embedded* in `𝑩`: there is a homomorphism from `𝑨` to `𝑩` whose underlying map is injective. Note that the notion of subalgebra that we define here is different from the standard textbook definition, which defines a *subuniverse* to be a subset of the carrier of an algebra that is closed under the operations (as we do in the [Setoid.Subalgebras.Subuniverses][] module); a subalgebra is then a subuniverse together with the operations of the larger algebra restricted to it. Nothing in our definition requires the carrier of a subalgebra of `𝑩` to be a subset of the carrier of `𝑩`.[^1] Our subalgebra relation `_≤_`{.AgdaFunction} is clearly reflexive and transitive, so it is a *preorder*; that is exactly what [Setoid.Subalgebras.Properties][] proves, as `≤-preorder`{.AgdaFunction}, and it is the only order-theoretic fact about `_≤_`{.AgdaFunction} that this subtree establishes. It is not a partial order. Antisymmetry would require that `𝑨 ≤ 𝑩` and `𝑩 ≤ 𝑨` imply `𝑨 ≈ 𝑩` for some suitable `≈`, and read as "on the nose" equality that plainly fails. The tempting repair is to read `≈` as isomorphism, but that fails too: `𝑨 ≤ 𝑩` and `𝑩 ≤ 𝑨` supply injective homomorphisms in both directions, and a pair of injections is not an isomorphism. Neither map need invert the other, and in general no isomorphism need exist at all — the free groups on two and on three generators embed in each other yet are not isomorphic, as their abelianizations `ℤ²` and `ℤ³` already show. Extracting a bijection from two injections is Cantor–Bernstein, which is unavailable constructively, and the bijection it yields classically need not be a homomorphism. For **finite** algebras the repair does work, and that is the setting much of this - the modelNow let me write the fix, adding needed imports and the definition.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: ≤-trans-≅′ p B≅C = {!!}, new_string: ≤-trans-≅′ (h , hinj) B≅C = ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C))answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "⊙-hom\|⊙-injective\|≅toInjective\|^open\|to \b" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms.lagda.md echo ---FUNCTIONS--- grep -n "^open\|⊙-injective" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Functions.lagda.md)answered25:+ [Setoid.Homomorphisms.Products][]: the homomorphism into a product induced by a
the whole answer, as the model saw it (852 characters of text)
25:+ [Setoid.Homomorphisms.Products][]: the homomorphism into a product induced by a 40:open import Setoid.Homomorphisms.Basic public 41:open import Setoid.Homomorphisms.Kernels public 42:open import Setoid.Homomorphisms.Products public 43:open import Setoid.Homomorphisms.Noether public 44:open import Setoid.Homomorphisms.Factor public 45:open import Setoid.Homomorphisms.Isomorphisms public 46:open import Setoid.Homomorphisms.HomomorphicImages public 47:open import Setoid.Homomorphisms.Properties public ---FUNCTIONS--- 45:open import Setoid.Functions.Basic public 46:open import Setoid.Functions.Bijective public 47:open import Setoid.Functions.Injective public 48:open import Setoid.Functions.Inverses public 49:open import Setoid.Functions.Surjective public
- the modelGood, those are all re-exported publicly. Let's check `≅toInjective` and `to` field, and `_≅_` module in Isomorphisms.lagda.md, then typecheck.
- shellBash(command: grep -n "≅toInjective\|record _≅_\|to :\|to \b" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md | head -20)answered13:both directions that are mutually inverse up to the setoid equalities of the two
the whole answer, as the model saw it (1,407 characters of text)
13:both directions that are mutually inverse up to the setoid equalities of the two 27:+ `≅toInjective`{.AgdaFunction}: the forward map of an isomorphism is injective; 31:+ `Lift-≅`{.AgdaFunction}: an algebra is isomorphic to each of its universe 45:open import Agda.Primitive using () renaming ( Set to Type ) 49:open import Function using () renaming ( Func to _⟶_ ) 71:open _⟶_ using ( cong ) renaming ( to to _⟨$⟩_ ) 87:However, with four components, an equivalent record type is easier to work with. 91: open Setoid 𝔻[ 𝑨 ] using ( sym ; trans ) renaming ( _≈_ to _≈₁_ ) 92: open Setoid 𝔻[ 𝑩 ] using () renaming ( _≈_ to _≈₂_ ; sym to sym₂ ; trans to trans₂) 94: record _≅_ : Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ β ⊔ ρᵃ ⊔ ρᵇ ) where 97: to : hom 𝑨 𝑩 100: from∼to : ∀ a → from .proj₁ ⟨$⟩ (to .proj₁ ⟨$⟩ a) ≈₁ a 111: Goal = trans (sym (from∼to x)) (trans ξ (from∼to y)) 114: fromIsSurjective {y} = eq (to .proj₁ ⟨$⟩ y) (sym (from∼to y)) 126:and forth between them which compose to the identity map. 145: f = ⊙-hom (to ab) (to bc) 150: open Setoid 𝔻[ 𝑪 ] using () renaming ( _≈_ to _≈₃_ ; trans to trans₃ ) 152: τ b = trans₃ (cong (to bc .proj₁) (to∼from ab (from bc .proj₁ ⟨$⟩ b))) (to∼from bc b) 154: open Setoid 𝔻[ 𝑨 ] using () renaming ( _≈_ to _≈₁_ ; trans to trans₁ ) 156: ν a = trans₁ (cong (from ab .proj₁) (from∼to bc (to ab .proj₁ ⟨$⟩ a))) (from∼to ab a) - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "open _≅_\|module _≅_\|open.*_≅_" -r <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms* <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms.lagda.md)answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:37:open import Setoid.Homomo…
the whole answer, as the model saw it (479 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:37:open import Setoid.Homomorphisms.Isomorphisms using ( _≅_ ; Lift-≅ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:180: open _≅_ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/HomomorphicImages.lagda.md:186: open _≅_ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:132:open _≅_
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: agda --no-default-libraries --library-file <repo>/agda/libraries --library standard-library --library agda-dojang --library agda-algebras --safe -i . Subalgebras-sub-trans-iso.agda)refusedno timing in the archiveExit code 42
the whole answer, as the model saw it (326 characters of text)
Exit code 42 Checking Subalgebras-sub-trans-iso (<work>/Subalgebras-sub-trans-iso.agda). <work>/Subalgebras-sub-trans-iso.agda:28.12-14: error: [NotInScope] Not in scope: to at <work>/Subalgebras-sub-trans-iso.agda:28.12-14 (did you mean '_≅_.to' or 'Setoid.Homomorphisms._≅_.to'?) when scope checking to - the modelNeed to open `_≅_`. Let me add that import/open line.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: open import Setoid.Subalgebras ≤-trans-≅′, new_string: open import Setoid.Subalgebras open _≅_ ≤-trans-≅′)answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- shellBash(command: agda --no-default-libraries --library-file <repo>/agda/libraries --library standard-library --library agda-dojang --library agda-algebras --safe -i . Subalgebras-sub-trans-iso.agda)answeredno timing in the archiveChecking Subalgebras-sub-trans-iso (<work>/Subalgebras-sub-trans-iso.agda).
the whole answer, as the model saw it (75 characters of text)
Checking Subalgebras-sub-trans-iso (<work>/Subalgebras-sub-trans-iso.agda).
- the modelThe file now type-checks successfully with the exact judging command, confirming success.
The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the tool’s own, in full. The client billed 704 thinking tokens over the session; their text is not in the archive.
solvedthe judge, in outcome.json
Every gate passed, and the definition does not refer to the library’s own lemma for this statement, so the row counts as a solve.
Agda’s exit code on the file 0
statement preserved yes
last verdict taken from
agda in the shellthe library’s own proof in view before the edit yes, the model read the file that holds it
the file the judge read, marked against the obligation it started as
-- Subalgebras-sub-trans-iso.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda
--
-- Benchmark obligation: algebras-subalgebras-sub-trans-iso
-- Difficulty: non-obvious
-- Source: Setoid.Subalgebras.Properties (agda-algebras)
-- Import stratum: wholesale
-- Strategy: pairing
--
module Subalgebras-sub-trans-iso where
open import AgdaDojang.Debug
open import Agda.Primitive using ( Level )
open import Data.Product using ( _,_ ; proj₁ ; proj₂ )
open import Overture using ( Signature )
open import Setoid.Algebras using ( Algebra )
open import Setoid.Functions
open import Setoid.Homomorphisms
open import Setoid.Subalgebras
open _≅_
≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥}
{𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}
→ 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪
≤-trans-≅′ p B≅C = {!!}
≤-trans-≅′ (h , hinj) B≅C =
⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C)≤-trans-≅′, server and shell, hints in view
Server and shell, with the hint in view. Given both, the model reads the library through the shell, writes the proof out, and takes its verdicts from check_file: the first says no and names what is out of scope, the second says yes.
- model
- Sonnet 5 (claude-sonnet-5)
- instrument
- server and shell
- run
- arm162-both-1 2026-09-21
- fixture header
- hints in view:
Source: Setoid.Subalgebras.Properties (agda-algebras);Strategy: pairing - server
- the server of 2026-09-21 (fourteen tools, the libraries' sources readable on every arm)
- client
- 2.1.261 (Claude Code)
- turns
- 14
- tool calls
- 13
Read×3,Bash×6,Edit×2,check_file×2 - wall clock
- 46 s (indicative: subjects ran three at a time)
- cost, list price
- USD 0.19
the statement, as the obligation states it, with the proof removed
{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪the session, 14 turns and 13 tool calls
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- filesRead(file_path: <work>/Subalgebras-sub-trans-iso.agda)answered1 -- Subalgebras-sub-trans-iso.agda
the whole answer, as the model saw it (963 characters of text)
1 -- Subalgebras-sub-trans-iso.agda 2 -- 3 -- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda 4 -- 5 -- Benchmark obligation: algebras-subalgebras-sub-trans-iso 6 -- Difficulty: non-obvious 7 -- Source: Setoid.Subalgebras.Properties (agda-algebras) 8 -- Import stratum: wholesale 9 -- Strategy: pairing 10 -- 11 module Subalgebras-sub-trans-iso where 12 13 open import AgdaDojang.Debug 14 15 open import Agda.Primitive using ( Level ) 16 open import Data.Product using ( _,_ ; proj₁ ; proj₂ ) 17 18 open import Overture using ( Signature ) 19 open import Setoid.Algebras using ( Algebra ) 20 open import Setoid.Functions 21 open import Setoid.Homomorphisms 22 open import Setoid.Subalgebras 23 24 ≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} 25 {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} 26 → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 27 ≤-trans-≅′ p B≅C = {!!} 28 - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "≤-trans-≅\|_≤_\|_≅_" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/*.agda 2>/dev/null | head -80)answered(Bash completed with no output)
the whole answer, as the model saw it (31 characters of text)
(Bash completed with no output)
- shellBash(command: find <nix>/agda-algebras-unstable-2026-09-07/src/Setoid -iname "*Subalgebra*")answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md
the whole answer, as the model saw it (134 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras
- shellBash(command: ls <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/)answeredBasic.lagda.md
the whole answer, as the model saw it (110 characters of text)
Basic.lagda.md CompleteLattice.lagda.md Properties.lagda.md Subdirect Subdirect.lagda.md Subuniverses.lagda.md
- shellBash(command: grep -n "≤-trans-≅\|^_≤_\|_≤_ \|≅-refl\|module\|open import" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md)answered3:title : "Setoid.Subalgebras.Properties module (The Agda Universal Algebra Library)"
the whole answer, as the model saw it (1,855 characters of text)
3:title : "Setoid.Subalgebras.Properties module (The Agda Universal Algebra Library)" 10:This is the [Setoid.Subalgebras.Properties][] module of the [Agda Universal Algebra Library][]. 12:This module establishes the order-theoretic facts about the subalgebra relation 28:module Setoid.Subalgebras.Properties where 30:open import Agda.Primitive using () renaming ( Set to Type ) 33:open import Data.Product using ( _,_ ) 34:open import Function using ( _∘_ ) renaming ( Func to _⟶_ ) 35:open import Level using ( Level ; _⊔_ ) 36:open import Relation.Binary using ( Setoid ) 37:open import Relation.Unary using ( Pred ; _⊆_ ) 43:open import Overture using ( proj₁ ; proj₂ ; 𝓞 ; 𝓥 ; Signature ; 𝑆 ) 44:open import Setoid.Algebras using ( Algebra ; Lift-Algˡ ; Lift-Algʳ 46:open import Setoid.Functions using ( id-is-injective ; IsInjective ; ⊙-injective ) 47:open import Setoid.Homomorphisms using ( hom ; IsHom ; 𝒾𝒹 ; ⊙-hom ; _≅_ 49: ; ≅-sym ; ≅-refl ; ≅-trans 51:open import Setoid.Subalgebras.Basic using ( _≤_ ; _≥_ ; _≤c_ ) 77:module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 81: ≤-trans-≅ : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 82: ≤-trans-≅ (h , hinj) B≅C = 89:module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 97:module _ {𝑆 : Signature 𝓞 𝓥} {α ρᵃ ρ : Level} where 103: isEquivalence ≤-preorder = record { refl = ≅-refl ; sym = ≅-sym ; trans = ≅-trans } 107:module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 169:module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} where 188:module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} where 214:module _ - the modelFound the existing `≤-trans-≅` proof. Let's read it.
- filesRead(file_path: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md, offset: 70, limit: 40)answered70
the whole answer, as the model saw it (1,777 characters of text)
70 71 ≥-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 72 ≥-refl = ≅→≤ ∘ ≅-sym 73 74 ≤-reflexive : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≤ 𝑨 75 ≤-reflexive {𝑨 = 𝑨} = 𝒾𝒹 , id-is-injective {𝑨 = 𝔻[ 𝑨 ]} 76 77 module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 78 ≤-trans : 𝑨 ≤ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 79 ≤-trans ( f , finj ) ( g , ginj ) = (⊙-hom f g) , ⊙-injective (proj₁ f) (proj₁ g) finj ginj 80 81 ≤-trans-≅ : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 82 ≤-trans-≅ (h , hinj) B≅C = 83 ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C) 84 85 ≅-trans-≤ : 𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 86 ≅-trans-≤ A≅B (h , hinj) = 87 ⊙-hom (to A≅B) h , ⊙-injective (proj₁ (to A≅B)) (proj₁ h) (≅toInjective A≅B) hinj 88 89 module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 90 ≥-trans : 𝑨 ≥ 𝑩 → 𝑩 ≥ 𝑪 → 𝑨 ≥ 𝑪 91 ≥-trans A≥B B≥C = ≤-trans B≥C A≥B 92 93 ≤→≤c→≤c : {𝑨 : Algebra {𝑆 = 𝑆} α α}{𝑩 : Algebra {𝑆 = 𝑆} α α}{𝒦 : Pred(Algebra {𝑆 = 𝑆} α α) (ov {𝑆 = 𝑆} α)} 94 → 𝑨 ≤ 𝑩 → 𝑩 ≤c 𝒦 → 𝑨 ≤c 𝒦 95 ≤→≤c→≤c A≤B sB = (proj₁ sB) , (proj₁ (proj₂ sB) , ≤-trans A≤B (proj₂ (proj₂ sB))) 96 97 module _ {𝑆 : Signature 𝓞 𝓥} {α ρᵃ ρ : Level} where 98 99 open RelStructs {a = ov {𝑆 = 𝑆} (α ⊔ ρᵃ)} {ℓ = 𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ} (_≅_ {α = α}{ρᵃ = ρᵃ}{𝑆 = 𝑆}) 100 open IsPreorder 101 102 ≤-preorder : IsPreorder _≤_ 103 isEquivalence ≤-preorder = record { refl = ≅-refl ; sym = ≅-sym ; trans = ≅-trans } 104 reflexive ≤-preorder = ≤-refl 105 trans ≤-preorder A≤B B≤C = ≤-trans A≤B B≤C 106 107 module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where 108 A≥B×B≅C→A≥C : 𝑨 ≥ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≥ 𝑪 109 A≥B×B≅C→A≥C A≥B B≅C = ≥-trans A≥B (≅→≥ B≅C) - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "_≤_\|_≥_" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md | head -20)answered24:Our subalgebra relation `_≤_`{.AgdaFunction} is clearly reflexive and transitive,
the whole answer, as the model saw it (743 characters of text)
24:Our subalgebra relation `_≤_`{.AgdaFunction} is clearly reflexive and transitive, 27:about `_≤_`{.AgdaFunction} that this subtree establishes. 45:finite algebras `_≤_`{.AgdaFunction} is antisymmetric up to isomorphism, hence a 49:This module defines the subalgebra relation `_≤_`{.AgdaFunction} and the several 81:+ `_IsSubalgebraOf_`{.AgdaFunction}, with the infix alias `_≤_`{.AgdaFunction}, 84:+ `_IsSupalgebraOf_`{.AgdaFunction}, aliased `_≥_`{.AgdaFunction}, is the 98: _≥_ -- alias for supalgebra (aka overalgebra) 102: _≤_ -- alias for subalgebra relation 181: same symbol, `_≤_`{.AgdaFunction}. The two do not collide, because the 186: as `_≤_`{.AgdaFunction} only where `Sublattice`{.AgdaModule} is opened, as in - the modelGood, matches. Now fill the hole using the same pattern.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: ≤-trans-≅′ p B≅C = {!!}, new_string: ≤-trans-≅′ (h , hinj) B≅C = ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C))answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- agda-mcpcheck_file()answered5.1 ssuccess: falseholesCount: 0diagnosticsTotal: 1verdict.exitCode: 42elapsedMs: 5060diagnostics: 1holes: 0
the whole answer, as the model saw it (2,245 characters of JSON)
{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","--safe","-i","<work>","<work>/Subalgebras-sub-trans-iso.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[{"code":"NotInScope","col":12,"file":"<work>/Subalgebras-sub-trans-iso.agda","involved":{"candidates":["_≅_.to","Setoid.Homomorphisms._≅_.to"]},"line":28,"message":"Not in scope:\n to\n at <work>/Subalgebras-sub-trans-iso.agda:28.12-14\n (did you mean\n '_≅_.to' or\n 'Setoid.Homomorphisms._≅_.to'?)\nwhen scope checking to","range":{"endCol":14,"endLine":28,"startCol":12,"startLine":28},"severity":"error"}],"diagnosticsTotal":1,"elapsedMs":5060,"holes":[],"holesCount":0,"project":{"includePaths":["agda-dojang/agda","<work>"],"librariesFile":"<repo>/agda/libraries","registeredLibraries":[{"includes":["agda"],"libFile":"<repo>/agda-dojang/agda-dojang.agda-lib","name":"agda-dojang","root":"<repo>/agda-dojang"},{"includes":["src"],"libFile":"<nix>/standard-library-2.3/standard-library.agda-lib","name":"standard-library-2.3","root":"<nix>/standard-library-2.3"},{"includes":["src"],"libFile":"<nix>/agda-algebras-unstable-2026-09-07/agda-algebras.agda-lib","name":"agda-algebras","root":"<nix>/agda-algebras-unstable-2026-09-07"}],"root":"<work>","rootSource":"server-config","selectedLibraries":["agda-dojang","standard-library","agda-algebras"]},"success":false,"timedOut":false,"verdict":{"equivalentTo":"equivalent-to: <nix>/agdaWithPackages-2.8.0/bin/agda -i agda-dojang/agda --library-file=agda/libraries -l agda-dojang -l standard-library -l agda-algebras --safe -i <work> <work>/Subalgebras-sub-trans-iso.agda","exitCode":42,"meaning":"success is true if and only if that command exited 0, so it means exactly what green means in a batch build. Unsolved metavariables, unsolved constraints, and open holes all make agda exit non-zero and so make success false; there is no interaction mode anywhere in this server. The verdict is read from the exit code alone — never from the diagnostics text — so a change in Agda's message format can empty the diagnostics list but cannot turn a failing build green."}} - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "record _≅_\|to \|from \|≅toInjective\|module.*≅" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/*.lagda.md 2>/dev/null | grep -i "to\|≅" | head -40)answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:14:`h`: there is a homomorphism `φ` fro…
the whole answer, as the model saw it (6,291 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:14:`h`: there is a homomorphism `φ` from the codomain of `h` to the codomain of `g` <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:31:-- Imports from Agda and the Agda Standard Library ------------------------------------------------- <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:33:open import Function renaming ( Func to _⟶_ ) using ( _∘_ ; _$_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:40:-- Imports from the Agda Universal Algebra Library ------------------------------------------------ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:72: open Setoid 𝔻[ 𝑩 ] using () renaming ( _≈_ to _≈₂_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:73: open Setoid 𝔻[ 𝑪 ] using ( trans ) renaming ( _≈_ to _≈₃_ ; sym to sym₃ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:74: open _⟶_ using ( cong ) renaming ( to to _⟨$⟩_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:119: open _⟶_ φ using () renaming (cong to φcong) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Factor.lagda.md:157: is obtained from `HomFactor`{.AgdaFunction}. <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:12:A **homomorphism** from `𝑨` to `𝑩` is a setoid function `h : 𝔻[ 𝑨 ] ⟶ 𝔻[ 𝑩 ]` <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:17:Two things distinguish this from an ordinary type-based definition, where <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:21: cannot fail to send equal arguments to equal results. <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:22:2. Compatibility is asserted up to the equality *of the codomain*, not up to <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:25: appeal to extensionality. <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:50:-- Imports from Agda and the Agda Standard Library ------------------------------ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:51:open import Agda.Primitive using () renaming ( Set to Type ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:53:open import Function.Bundles using () renaming ( Func to _⟶_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:58:-- Imports from the Agda Universal Algebra Library --------------------------- <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:75:setoid function `h` from the domain of `𝑨` to that of `𝑩`, together with a proof <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:79:`h` and a `compatible-map`{.AgdaFunction} proof never has to write the `Σ`-pair <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:84: open _⟶_ {a = α}{ρᵃ}{β}{ρᵇ}{From = 𝔻[ 𝑨 ]}{To = 𝔻[ 𝑩 ]} renaming (to to _⟨$⟩_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:88: where open Setoid 𝔻[ 𝑩 ] using() renaming ( _≈_ to _≈₂_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md:128:monomorphism into a subalgebra and an epimorphism into a homomorphic image. <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:13:both directions that are mutually inverse up to the setoid equalities of the two <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:18:`from∼to`{.AgdaField}, and derives from it that each direction is both injective <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:27:+ `≅toInjective`{.AgdaFunction}: the forward map of an isomorphism is injective; <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:31:+ `Lift-≅`{.AgdaFunction}: an algebra is isomorphic to each of its universe <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:44:-- Imports from Agda (builtin/primitive) and the Agda Standard Library --------------------- <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:45:open import Agda.Primitive using () renaming ( Set to Type ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:47:open import Data.Unit.Polymorphic.Base using () renaming ( ⊤ to 𝟙 ; tt to ∗ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:49:open import Function using () renaming ( Func to _⟶_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:55:-- Imports from the Agda Universal Algebra Library ----------------------------------------- <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:71:open _⟶_ using ( cong ) renaming ( to to _⟨$⟩_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:87:However, with four components, an equivalent record type is easier to work with. <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:91: open Setoid 𝔻[ 𝑨 ] using ( sym ; trans ) renaming ( _≈_ to _≈₁_ ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:92: open Setoid 𝔻[ 𝑩 ] using () renaming ( _≈_ to _≈₂_ ; sym to sym₂ ; trans to trans₂) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:94: record _≅_ : Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ β ⊔ ρᵃ ⊔ ρᵇ ) where <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:97: to : hom 𝑨 𝑩 <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:98: from : hom 𝑩 𝑨 <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:99: to∼from : ∀ b → to .proj₁ ⟨$⟩ (from .proj₁ ⟨$⟩ b) ≈₂ b - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- filesRead(file_path: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md, offset: 40, limit: 30)answered40 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
the whole answer, as the model saw it (1,390 characters of text)
40 import Relation.Binary.Reasoning.Setoid as SetoidReasoning 41 42 -- Imports from the Agda Universal Algebra Library ---------------------------------- 43 open import Overture using ( proj₁ ; proj₂ ; 𝓞 ; 𝓥 ; Signature ; 𝑆 ) 44 open import Setoid.Algebras using ( Algebra ; Lift-Algˡ ; Lift-Algʳ 45 ; Lift-Alg ; ov ; ⨅ ; 𝔻[_] ) 46 open import Setoid.Functions using ( id-is-injective ; IsInjective ; ⊙-injective ) 47 open import Setoid.Homomorphisms using ( hom ; IsHom ; 𝒾𝒹 ; ⊙-hom ; _≅_ 48 ; ≅toInjective ; ≅fromInjective ; mkiso 49 ; ≅-sym ; ≅-refl ; ≅-trans 50 ; Lift-≅ˡ ; Lift-≅ ; Lift-≅ʳ) 51 open import Setoid.Subalgebras.Basic using ( _≤_ ; _≥_ ; _≤c_ ) 52 53 private variable α ρᵃ β ρᵇ γ ρᶜ ι : Level 54 ``` 55 --> 56 57 The subalgebra relation is a *preorder*, i.e., a reflexive, transitive binary relation. 58 59 ```agda 60 open _≅_ 61 62 ≅→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 63 ≅→≤ φ = (to φ) , ≅toInjective φ 64 65 ≅→≥ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 66 ≅→≥ φ = (from φ) , ≅fromInjective φ 67 68 ≤-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 69 ≤-refl = ≅→≤ - the modelThey have `open _≅_` at top level. I'll use `_≅_.to` directly instead of adding an open, to avoid extra top-level opens interfering (though allowed). Simplest: use qualified `_≅_.to`.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: ≤-trans-≅′ (h , hinj) B≅C = ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C), new_string: ≤-trans-≅′ (h , hinj) B≅C = ⊙-hom h (_≅_.to B≅C) , ⊙-injective (proj₁ h) (proj₁ (_≅_.to B≅C)) hinj (≅toInjective B≅C))answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- agda-mcpcheck_file()answered6.0 ssuccess: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5982diagnostics: 0holes: 0
the whole answer, as the model saw it (1,818 characters of JSON)
{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","--safe","-i","<work>","<work>/Subalgebras-sub-trans-iso.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5982,"holes":[],"holesCount":0,"project":{"includePaths":["agda-dojang/agda","<work>"],"librariesFile":"<repo>/agda/libraries","registeredLibraries":[{"includes":["agda"],"libFile":"<repo>/agda-dojang/agda-dojang.agda-lib","name":"agda-dojang","root":"<repo>/agda-dojang"},{"includes":["src"],"libFile":"<nix>/standard-library-2.3/standard-library.agda-lib","name":"standard-library-2.3","root":"<nix>/standard-library-2.3"},{"includes":["src"],"libFile":"<nix>/agda-algebras-unstable-2026-09-07/agda-algebras.agda-lib","name":"agda-algebras","root":"<nix>/agda-algebras-unstable-2026-09-07"}],"root":"<work>","rootSource":"server-config","selectedLibraries":["agda-dojang","standard-library","agda-algebras"]},"success":true,"timedOut":false,"verdict":{"equivalentTo":"equivalent-to: <nix>/agdaWithPackages-2.8.0/bin/agda -i agda-dojang/agda --library-file=agda/libraries -l agda-dojang -l standard-library -l agda-algebras --safe -i <work> <work>/Subalgebras-sub-trans-iso.agda","exitCode":0,"meaning":"success is true if and only if that command exited 0, so it means exactly what green means in a batch build. Unsolved metavariables, unsolved constraints, and open holes all make agda exit non-zero and so make success false; there is no interaction mode anywhere in this server. The verdict is read from the exit code alone — never from the diagnostics text — so a change in Agda's message format can empty the diagnostics list but cannot turn a failing build green."}} - the modelThe verdict on the final file is success (exit code 0, no diagnostics).
The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the tool’s own, in full. The client billed 496 thinking tokens over the session; their text is not in the archive.
solvedthe judge, in outcome.json
Every gate passed, and the definition does not refer to the library’s own lemma for this statement, so the row counts as a solve.
Agda’s exit code on the file 0
statement preserved yes
last verdict taken from
check_filethe library’s own proof in view before the edit yes, the model read the file that holds it
the file the judge read, marked against the obligation it started as
-- Subalgebras-sub-trans-iso.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda
--
-- Benchmark obligation: algebras-subalgebras-sub-trans-iso
-- Difficulty: non-obvious
-- Source: Setoid.Subalgebras.Properties (agda-algebras)
-- Import stratum: wholesale
-- Strategy: pairing
--
module Subalgebras-sub-trans-iso where
open import AgdaDojang.Debug
open import Agda.Primitive using ( Level )
open import Data.Product using ( _,_ ; proj₁ ; proj₂ )
open import Overture using ( Signature )
open import Setoid.Algebras using ( Algebra )
open import Setoid.Functions
open import Setoid.Homomorphisms
open import Setoid.Subalgebras
≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥}
{𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}
→ 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪
≤-trans-≅′ p B≅C = {!!}
≤-trans-≅′ (h , hinj) B≅C =
⊙-hom h (_≅_.to B≅C) , ⊙-injective (proj₁ h) (proj₁ (_≅_.to B≅C)) hinj (≅toInjective B≅C)≤-trans-≅′, server only, header-free
Server only, header-free. One question, exports_of on a module the file imports, lists a lemma of exactly this type; the model calls it, and the one verdict is accepted. A proof that names the library's own lemma for the statement is counted restated by the judge, never solved.
- model
- Sonnet 5 (claude-sonnet-5)
- instrument
- server only
- run
- suite219-sonnet5-mcp-1 2026-09-29
- fixture header
- header-free: no header line named a source, a needle or the proof’s shape
- server
- the server of 2026-09-29 (the same surface, frozen for the header-free arms (PR 223))
- client
- 2.1.282 (Claude Code)
- turns
- 5
- tool calls
- 4
Read×1,exports_of×1,Edit×1,check_file×1 - wall clock
- 32 s (indicative: subjects ran three at a time)
- cost, list price
- USD 0.18
the statement, as the obligation states it, with the proof removed
{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪the session, 5 turns and 4 tool calls
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- filesRead(file_path: <work>/Subalgebras-sub-trans-iso.agda)answered1 -- Subalgebras-sub-trans-iso.agda
the whole answer, as the model saw it (879 characters of text)
1 -- Subalgebras-sub-trans-iso.agda 2 -- 3 -- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda 4 -- 5 -- Benchmark obligation: algebras-subalgebras-sub-trans-iso 6 -- Difficulty: non-obvious 7 -- Import stratum: wholesale 8 -- 9 module Subalgebras-sub-trans-iso where 10 11 open import AgdaDojang.Debug 12 13 open import Agda.Primitive using ( Level ) 14 open import Data.Product using ( _,_ ; proj₁ ; proj₂ ) 15 16 open import Overture using ( Signature ) 17 open import Setoid.Algebras using ( Algebra ) 18 open import Setoid.Functions 19 open import Setoid.Homomorphisms 20 open import Setoid.Subalgebras 21 22 ≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} 23 {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} 24 → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 25 ≤-trans-≅′ p B≅C = {!!} 26 - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- agda-mcpexports_of(module: Setoid.Subalgebras, limit: 0)answered5.7 s (loading the file first)elapsedMs: 5679exports: 125modules: 8
the whole answer, as the model saw it (38,695 characters of JSON)
{"checkedFromSource":true,"elapsedMs":5679,"exports":[{"name":"_IsSubalgebraOf_","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ β ρᵇ : Level} →\nAlgebra α ρᵃ →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"_IsSubalgebraOfClass_","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥}\n{β ρᵇ α ρᵃ : Level} {ℓ = ℓ₁ : Level} →\nAlgebra β ρᵇ →\nRelation.Unary.Pred (Algebra α ρᵃ) ℓ₁ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ\n Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ\n Agda.Primitive.⊔ ℓ₁)"},{"name":"_IsSupalgebraOf_","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ β ρᵇ : Level} →\nAlgebra α ρᵃ →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"_∈?_","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(.patternInTele0\n : Data.Fin.Base.Fin (𝑭 .Setoid.Algebras.FiniteAlgebra.card)\n Data.Product.×\n Data.Fin.Base.Fin (𝑭 .Setoid.Algebras.FiniteAlgebra.card))\n(d : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)) →\nRelation.Nullary.Decidable.Core.Dec\n(Setoid.Congruences.Finite.Basic.ConRel d\n (𝑭 .Setoid.Algebras.FiniteAlgebra.enum (.patternInTele0 .proj₁))\n (𝑭 .Setoid.Algebras.FiniteAlgebra.enum (.patternInTele0 .proj₂)))"},{"name":"_≤_","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ β ρᵇ : Level} →\nAlgebra α ρᵃ →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"_≤c_","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥}\n{β ρᵇ α ρᵃ : Level} {ℓ = ℓ₁ : Level} →\nAlgebra β ρᵇ →\nRelation.Unary.Pred (Algebra α ρᵃ) ℓ₁ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ\n Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ\n Agda.Primitive.⊔ ℓ₁)"},{"name":"_≥_","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ β ρᵇ : Level} →\nAlgebra α ρᵃ →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"_⊆ᶜ_","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨) →\nSetoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪) →\nSetoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪) →\nSet (α Agda.Primitive.⊔ ρ Agda.Primitive.⊔ 𝓞 Agda.Primitive.⊔ 𝓥)"},{"name":"_⊆ᶜ?_","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(d e : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)) →\nRelation.Nullary.Decidable.Core.Dec ((𝑭 ⊆ᶜ 𝑪) d e)"},{"name":"A≅B×B≤C→A≤C","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪"},{"name":"A≅B×B≥C→A≥C","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≅ 𝑩 → 𝑩 ≥ 𝑪 → 𝑨 ≥ 𝑪"},{"name":"A≤B×B≅C→A≤C","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪"},{"name":"A≥B×B≅C→A≥C","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≥ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≥ 𝑪"},{"name":"B-onlyif-TermImageB","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{B : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρᵃ} →\nB Relation.Unary.⊆ TermImage B"},{"name":"Birkhoff-subdirect","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρ : Level} {ℓ = ℓ₁ : Level} {ι : Level} →\n((𝑨 : Algebra α ρ) → SubdirectSIRep 𝑨 ℓ₁ ι) →\n(𝑨 : Algebra α ρ) → SubdirectlyRepresentable 𝑨 ℓ₁ ι"},{"name":"FirstHomCorollary","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (hh : hom 𝑨 𝑩) →\nkerquo hh IsSubalgebraOf 𝑩"},{"name":"IsSubdirectEmbedding","type":"{𝓞 𝓥 ι β ρᵇ α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑩 : Algebra β ρᵇ} (𝒜 : I → Algebra α ρ)\n(h : hom 𝑩 (Setoid.Algebras.⨅ 𝒜)) →\nSet\n(ι Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ Agda.Primitive.⊔ α\n Agda.Primitive.⊔ ρ)"},{"name":"IsoToFactor","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι} →\nAlgebra α ρ →\n(I → Algebra αᵃ ρ) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ ι Agda.Primitive.⊔ α\n Agda.Primitive.⊔ ρ\n Agda.Primitive.⊔ αᵃ)"},{"name":"Lift-is-sub","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝒦 : Relation.Unary.Pred (Algebra α ρᵃ) (Setoid.Algebras.ov α)}\n{𝑩 : Algebra β ρᵇ} {ℓ = ℓ₁ : Level} →\n𝑩 ≤c 𝒦 → Setoid.Algebras.Lift-Algˡ 𝑩 ℓ₁ ≤c 𝒦"},{"name":"Lift-≤-Lift","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {a rᵃ b rᵇ : Level} →\n𝑨 ≤ 𝑩 →\nSetoid.Algebras.Lift-Alg 𝑨 a rᵃ ≤ Setoid.Algebras.Lift-Alg 𝑩 b rᵇ"},{"name":"Lift-≤-Liftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {rᵃ rᵇ : Level} →\n𝑨 ≤ 𝑩 →\nSetoid.Algebras.Lift-Algʳ 𝑨 rᵃ ≤ Setoid.Algebras.Lift-Algʳ 𝑩 rᵇ"},{"name":"Lift-≤-Liftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ℓᵃ ℓᵇ : Level} →\n𝑨 ≤ 𝑩 →\nSetoid.Algebras.Lift-Algˡ 𝑨 ℓᵃ ≤ Setoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ"},{"name":"Q","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nAlgebra α (ℓ 𝑭 𝑪)"},{"name":"Q→A","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nSetoid.Congruences.Basic.Con (Q 𝑭 𝑪 a b a≢b) (ℓ 𝑭 𝑪) →\nSetoid.Congruences.Basic.Con 𝑨 (ℓ 𝑭 𝑪)"},{"name":"Rₐᵦ","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nRelation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ] →\nRelation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ] →\nSet α"},{"name":"SI-Q","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nSetoid.Congruences.Monolith.IsSubdirectlyIrreducible\n(Q 𝑭 𝑪 a b a≢b)"},{"name":"SIRep→Representable","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρ : Level} {ℓ = ℓ₁ : Level} {ι : Level} {𝑨 : Algebra α ρ} →\nSubdirectSIRep 𝑨 ℓ₁ ι → SubdirectlyRepresentable 𝑨 ℓ₁ ι"},{"name":"Separates","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ} →\n(I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nSet (ι Agda.Primitive.⊔ α Agda.Primitive.⊔ ρ Agda.Primitive.⊔ ℓ₁)"},{"name":"Sg","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n{ℓ = ℓ₁ : Level} →\nRelation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ℓ₁ →\nSetoid.Subalgebras.Subuniverses.A 𝑨 →\nSet (𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ℓ₁)"},{"name":"SgB-onlyif-TermImageB","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n(B : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρᵃ) →\nSg 𝑨 B Relation.Unary.⊆ TermImage B"},{"name":"Subalgebra","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ : Level} →\nAlgebra α ρᵃ →\n{β ρᵇ : Level} →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ Agda.Primitive.lsuc β\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵇ)"},{"name":"SubalgebraOf","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α β ρᵃ ρᵇ : Level} →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc β\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵇ)"},{"name":"SubalgebraOfClass","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥}\n{α β ρᵃ ρᵇ : Level} {ℓ = ℓ₁ : Level} →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc β\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵇ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ℓ₁)"},{"name":"SubalgebraOfClass'","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥}\n{α β ρᵃ ρᵇ : Level} {ℓ = ℓ₁ : Level} →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc β\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵇ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ℓ₁)"},{"name":"SubalgebrasOfClass","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ : Level} {ℓ = ℓ₁ : Level} →\nRelation.Unary.Pred (Algebra α ρᵃ) ℓ₁ →\n{β ρᵇ : Level} →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ\n Agda.Primitive.⊔ ℓ₁\n Agda.Primitive.⊔ Agda.Primitive.lsuc β\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵇ)"},{"name":"SubdirectEmbedding","type":"{𝓞 𝓥 ι β ρᵇ α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑩 : Algebra β ρᵇ} →\n(I → Algebra α ρ) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ ι Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ\n Agda.Primitive.⊔ α\n Agda.Primitive.⊔ ρ)"},{"name":"SubdirectSIRep","type":"{𝓞 𝓥 α ρ : Level} {𝑆 : Signature 𝓞 𝓥} →\nAlgebra α ρ →\n(ℓ₁ ι : Level) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ℓ₁\n Agda.Primitive.⊔ Agda.Primitive.lsuc ι)"},{"name":"SubdirectlyRepresentable","type":"{𝓞 𝓥 α ρ : Level} {𝑆 : Signature 𝓞 𝓥} →\nAlgebra α ρ →\n(ℓ₁ ι : Level) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ ρ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ℓ₁\n Agda.Primitive.⊔ Agda.Primitive.lsuc ι)"},{"name":"Subuniverse","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n{ℓ = ℓ₁ : Level} →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ℓ₁)"},{"name":"Subuniverses","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n{ℓ = ℓ₁ : Level} →\nRelation.Unary.Pred\n(Relation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ℓ₁)\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ℓ₁)"},{"name":"TermImage","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ} →\nRelation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρᵃ →\nSetoid.Subalgebras.Subuniverses.A →\nSet (𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ)"},{"name":"TermImageB-onlyif-SgB","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n(B : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρᵃ) →\nTermImage B Relation.Unary.⊆ Sg 𝑨 B"},{"name":"TermImageIsSub","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{B : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρᵃ} →\nTermImage B Relation.Unary.∈ Subuniverses 𝑨"},{"name":"app","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level}\n{G : Relation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ℓ₁}\n(f : 𝑆 .proj₁)\n(a : Overture.ArityOf 𝑆 f → Setoid.Subalgebras.Subuniverses.A 𝑨) →\nOverture.Im a ⊆ Sg 𝑨 G →\n(f Setoid.Algebras.^ 𝑨) a Relation.Unary.∈ Sg 𝑨 G"},{"name":"app","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{B : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρᵃ}\n(f : Overture.OperationSymbolsOf 𝑆)\n(ts\n : Overture.ArityOf 𝑆 f →\n Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n((i : Overture.ArityOf 𝑆 f) → ts i Relation.Unary.∈ TermImage B) →\n(f Setoid.Algebras.^ 𝑨) ts Relation.Unary.∈ TermImage B"},{"name":"a≢bCons","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nAgda.Builtin.List.List\n(Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪))"},{"name":"below→coord-inj","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) {i : I} →\nSetoid.Congruences.Monolith.BelowDiagonal 𝑨 (kerfam 𝒜 h i) →\nIsInjective (proj₁ (coord 𝒜 h i))"},{"name":"carrier-lift","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(R S : Setoid.Congruences.Basic.Con 𝑨 (ℓ 𝑭 𝑪)) →\n((i j\n : Data.Fin.Base.Fin (𝑭 .Setoid.Algebras.FiniteAlgebra.card)) →\n proj₁ R (𝑭 .Setoid.Algebras.FiniteAlgebra.enum i)\n (𝑭 .Setoid.Algebras.FiniteAlgebra.enum j) →\n proj₁ S (𝑭 .Setoid.Algebras.FiniteAlgebra.enum i)\n (𝑭 .Setoid.Algebras.FiniteAlgebra.enum j)) →\nR Setoid.Congruences.Lattice._.⊆ S"},{"name":"coord","type":"{𝓞 𝓥 ι β ρᵇ α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑩 : Algebra β ρᵇ} (𝒜 : I → Algebra α ρ) →\nhom 𝑩 (Setoid.Algebras.⨅ 𝒜) → (i : I) → hom 𝑩 (𝒜 i)"},{"name":"coord-inj→below","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) {i : I} →\nIsInjective (proj₁ (coord 𝒜 h i)) →\nSetoid.Congruences.Monolith.BelowDiagonal 𝑨 (kerfam 𝒜 h i)"},{"name":"coord-iso","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) {i : I} →\nIsInjective (proj₁ (coord 𝒜 h i)) →\nIsSurjective (proj₁ (coord 𝒜 h i)) → 𝑨 ≅ 𝒜 i"},{"name":"count","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨) →\nSetoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪) →\nAgda.Builtin.Nat.Nat"},{"name":"count-mono","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(d e : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)) →\nproj₁ d Setoid.Congruences.Lattice._.⊆ proj₁ e →\ncount 𝑭 𝑪 d Data.Nat.Base.≤ count 𝑭 𝑪 e"},{"name":"count-strict","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(d e : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪))\n(i j : Data.Fin.Base.Fin (𝑭 .Setoid.Algebras.FiniteAlgebra.card)) →\nproj₁ d Setoid.Congruences.Lattice._.⊆ proj₁ e →\nSetoid.Congruences.Finite.Basic.ConRel e\n(𝑭 .Setoid.Algebras.FiniteAlgebra.enum i)\n(𝑭 .Setoid.Algebras.FiniteAlgebra.enum j) →\nRelation.Nullary.Negation.Core.¬\nSetoid.Congruences.Finite.Basic.ConRel d\n(𝑭 .Setoid.Algebras.FiniteAlgebra.enum i)\n(𝑭 .Setoid.Algebras.FiniteAlgebra.enum j) →\ncount 𝑭 𝑪 d Data.Nat.Base.< count 𝑭 𝑪 e"},{"name":"embed-inj","type":"{𝓞 𝓥 ι β ρᵇ α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑩 : Algebra β ρᵇ} {𝒜 : I → Algebra α ρ}\n{h : hom 𝑩 (Setoid.Algebras.⨅ 𝒜)} →\nIsSubdirectEmbedding 𝒜 h → IsInjective (proj₁ h)"},{"name":"embed→separates","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) →\nIsInjective (proj₁ h) → Separates (kerfam 𝒜 h)"},{"name":"embed↔separates","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) →\nIsInjective (proj₁ h) Agda.Builtin.Equality.≡\nSeparates (kerfam 𝒜 h)"},{"name":"finite-Birkhoff","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨) →\nSubdirectlyRepresentable 𝑨 (ℓ 𝑭 𝑪) (α Agda.Primitive.⊔ ρ)"},{"name":"finiteSubdirectSIRep","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨) →\nSubdirectSIRep 𝑨 (ℓ 𝑭 𝑪) (α Agda.Primitive.⊔ ρ)"},{"name":"hom-unique","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{β ρᵇ : Level} {𝑩 : Algebra β ρᵇ} (gh hh : hom 𝑨 𝑩)\n{ℓ = ℓ₁ : Level}\n(G : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ℓ₁) →\n((x : Setoid.Subalgebras.Subuniverses.A) →\n x Relation.Unary.∈ G →\n (Algebra.Domain 𝑩 Relation.Binary.Bundles.Setoid.≈\n Setoid.Subalgebras.Subuniverses.g gh hh x)\n (Setoid.Subalgebras.Subuniverses.h gh hh x)) →\n(a : Setoid.Subalgebras.Subuniverses.A) →\na Relation.Unary.∈ Sg 𝑨 G →\n(Algebra.Domain 𝑩 Relation.Binary.Bundles.Setoid.≈\n Setoid.Subalgebras.Subuniverses.g gh hh a)\n(Setoid.Subalgebras.Subuniverses.h gh hh a)"},{"name":"injective↔0kernel","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) {i : I} →\nIsInjective (proj₁ (coord 𝒜 h i)) Agda.Builtin.Equality.≡\nSetoid.Congruences.Monolith.BelowDiagonal 𝑨 (kerfam 𝒜 h i)"},{"name":"injective⇔0kernel","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) →\n(λ {i} → IsInjective (proj₁ (coord 𝒜 h i))) Agda.Builtin.Equality.≡\n(λ {i} →\n Setoid.Congruences.Monolith.BelowDiagonal 𝑨 (kerfam 𝒜 h i))"},{"name":"iso-coord⟹¬all-proper","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} {𝒜 : I → Algebra αᵃ ρ}\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) →\nData.Product.∃-syntax (λ i → IsInjective (proj₁ (coord 𝒜 h i))) →\nRelation.Nullary.Negation.Core.¬\n((i : I) → Setoid.Congruences.Monolith.Nonzero 𝑨 (kerfam 𝒜 h i))"},{"name":"iso→injective","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n(𝑨 : Algebra α ρᵃ) {𝑩 : Algebra β ρᵇ} (φ : 𝑨 ≅ 𝑩) →\nIsInjective (proj₁ (_≅_.to φ))"},{"name":"kerfam","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) →\nI → Setoid.Congruences.Basic.Con 𝑨 ρ"},{"name":"mksub","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level}\n(sset\n : Relation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ℓ₁)\n(isSub : sset Relation.Unary.∈ Subuniverses 𝑨) →\nSubuniverse 𝑨"},{"name":"monolith⇒¬all-nonzero","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρ : Level}\n{𝑨 : Algebra α ρ} {ι : Level} →\nSetoid.Congruences.Monolith.HasMonolith 𝑨 →\n{I : Set ι} (θ : I → Setoid.Congruences.Basic.Con 𝑨 ρ) →\nSeparates θ →\nRelation.Nullary.Negation.Core.¬\n((i : I) → Setoid.Congruences.Monolith.Nonzero 𝑨 (θ i))"},{"name":"mon→≤","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nmon 𝑨 𝑩 → 𝑨 ≤ 𝑩"},{"name":"natmap","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nhom 𝑨 (Setoid.Algebras.⨅ (𝑨╱ θ))"},{"name":"natmap-injective","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nSeparates θ → IsInjective (proj₁ (natmap θ))"},{"name":"natmap-proj-onto","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) (i : I) →\nIsSurjective (proj₁ (coord (𝑨╱ θ) (natmap θ) i))"},{"name":"natmap-separates","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nIsInjective (proj₁ (natmap θ)) → Separates θ"},{"name":"notRel?","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ]) →\nRelation.Nullary.Negation.Core.¬\n(Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b →\n(d : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)) →\nRelation.Nullary.Decidable.Core.Dec\n(Relation.Nullary.Negation.Core.¬\n Setoid.Congruences.Finite.Basic.ConRel d a b)"},{"name":"pairs","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨) →\nAgda.Builtin.List.List\n(Data.Fin.Base.Fin (𝑭 .Setoid.Algebras.FiniteAlgebra.card)\n Data.Product.×\n Data.Fin.Base.Fin (𝑭 .Setoid.Algebras.FiniteAlgebra.card))"},{"name":"proj-onto","type":"{𝓞 𝓥 ι β ρᵇ α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑩 : Algebra β ρᵇ} {𝒜 : I → Algebra α ρ}\n{h : hom 𝑩 (Setoid.Algebras.⨅ 𝒜)} →\nIsSubdirectEmbedding 𝒜 h →\n(i : I) → IsSurjective (proj₁ (coord 𝒜 h i))"},{"name":"separates→embed","type":"{𝓞 𝓥 ι α ρ αᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑨 : Algebra α ρ} (𝒜 : I → Algebra αᵃ ρ)\n(h : hom 𝑨 (Setoid.Algebras.⨅ 𝒜)) →\nSeparates (kerfam 𝒜 h) → IsInjective (proj₁ h)"},{"name":"separates≡below-meet","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρ : Level}\n{𝑨 : Algebra α ρ} {I : Set ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ρ) →\nSeparates θ Agda.Builtin.Equality.≡\nSetoid.Congruences.Monolith.BelowDiagonal 𝑨\n(Setoid.Congruences.Monolith.⋂ 𝑨 θ)"},{"name":"separating→SubdirectEmbedding","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nSeparates θ → SubdirectEmbedding (𝑨╱ θ)"},{"name":"separating→subdirect","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nSeparates θ → IsSubdirectEmbedding (𝑨╱ θ) (natmap θ)"},{"name":"sgIsSmallest","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n{ℓ = ℓ₁ : Level} {ρᵇ : Level}\n{G : Relation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ℓ₁}\n(B\n : Relation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ρᵇ) →\nB Relation.Unary.∈ Subuniverses 𝑨 →\nG Relation.Unary.⊆ B → Sg 𝑨 G Relation.Unary.⊆ B"},{"name":"sgIsSub","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n{ℓ = ℓ₁ : Level}\n{G\n : Relation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ℓ₁} →\nSg 𝑨 G Relation.Unary.∈ Subuniverses 𝑨"},{"name":"si⇒iso-coord","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρ αᵃ : Level}\n{n : Agda.Builtin.Nat.Nat} {𝑨 : Algebra α ρ}\n{𝒜 : Data.Fin.Base.Fin n → Algebra αᵃ ρ}\n(.patternInTele0\n : Setoid.Congruences.Monolith.IsSubdirectlyIrreducible 𝑨)\n(.patternInTele1 : SubdirectEmbedding 𝒜)\n(dec\n : (i : Data.Fin.Base.Fin n) →\n Relation.Nullary.Decidable.Core.Dec\n (Setoid.Congruences.Monolith.BelowDiagonal 𝑨\n (kerfam 𝒜 (.patternInTele1 .proj₁) i))) →\nIsoToFactor 𝑨 𝒜"},{"name":"si⇒¬all-proper","type":"{ι 𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρ αᵃ : Level}\n{I : Set ι} {𝑨 : Algebra α ρ} {𝒜 : I → Algebra αᵃ ρ}\n(si : Setoid.Congruences.Monolith.IsSubdirectlyIrreducible 𝑨)\n(sub : SubdirectEmbedding 𝒜) →\nRelation.Nullary.Negation.Core.¬\n((i : I) →\n Setoid.Congruences.Monolith.Nonzero 𝑨\n (kerfam 𝒜 (Setoid.Subalgebras.Subdirect.Irreducible.h si sub) i))"},{"name":"si⇒¬no-iso-coord","type":"{ι 𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρ αᵃ : Level}\n{I : Set ι} {𝑨 : Algebra α ρ} {𝒜 : I → Algebra αᵃ ρ}\n(si : Setoid.Congruences.Monolith.IsSubdirectlyIrreducible 𝑨)\n(sub : SubdirectEmbedding 𝒜) →\nRelation.Nullary.Negation.Core.¬\n((i : I) → Relation.Nullary.Negation.Core.¬ (𝑨 ≅ 𝒜 i))"},{"name":"sub-separates","type":"{ι 𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρ αᵃ : Level}\n{I : Set ι} {𝑨 : Algebra α ρ} {𝒜 : I → Algebra αᵃ ρ}\n(si : Setoid.Congruences.Monolith.IsSubdirectlyIrreducible 𝑨)\n(sub : SubdirectEmbedding 𝒜) →\nSeparates\n(kerfam 𝒜 (Setoid.Subalgebras.Subdirect.Irreducible.h si sub))"},{"name":"sub-term-closed","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level} {X.χ : Level} {X : Set X.χ}\n(B : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ℓ₁) →\nB Relation.Unary.∈ Subuniverses 𝑨 →\n(t : Overture.Term X)\n(b\n : Relation.Binary.Bundles.Setoid.Carrier\n (Setoid.Terms.Basic.Environment.Env 𝑨 X)) →\n((x : X) → b x Relation.Unary.∈ B) →\nFunction.Bundles.Func.to (Setoid.Terms.Basic.Environment.⟦ 𝑨 ⟧ t) b\nRelation.Unary.∈ B"},{"name":"subdirect→≤","type":"{𝓞 𝓥 ι β ρᵇ α ρ : Level} {𝑆 : Signature 𝓞 𝓥} {I : Set ι}\n{𝑩 : Algebra β ρᵇ} (𝒜 : I → Algebra α ρ) →\nSubdirectEmbedding 𝒜 → 𝑩 ≤ Setoid.Algebras.⨅ 𝒜"},{"name":"var","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{ℓ = ℓ₁ : Level}\n{G : Relation.Unary.Pred (Setoid.Subalgebras.Subuniverses.A 𝑨) ℓ₁}\n{v : Setoid.Subalgebras.Subuniverses.A 𝑨} →\nv Relation.Unary.∈ G → v Relation.Unary.∈ Sg 𝑨 G"},{"name":"var","type":"{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{B : Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρᵃ}\n{b : Setoid.Subalgebras.Subuniverses.A} →\nb Relation.Unary.∈ B → b Relation.Unary.∈ TermImage B"},{"name":"¬Δab","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nRelation.Nullary.Negation.Core.¬\nSetoid.Congruences.Finite.Basic.ConRel\n(Setoid.Congruences.Finite.Basic.FiniteCongruences.witness 𝑪\n (Δ 𝑭 𝑪))\na b"},{"name":"¬Θab","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nRelation.Nullary.Negation.Core.¬ proj₁ (Θ 𝑭 𝑪 a b a≢b) a b"},{"name":"Δ","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨) →\nSetoid.Congruences.Basic.Con 𝑨 (ℓ 𝑭 𝑪)"},{"name":"Δ∈a≢bCons","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\n(Relation.Binary.PropositionalEquality.Properties.setoid\n (Data.Product.Σ\n (Setoid.Congruences.Basic.Con 𝑨\n (α Agda.Primitive.⊔ ρ Agda.Primitive.⊔ 𝓞 Agda.Primitive.⊔ 𝓥))\n (λ .patternInTele0 →\n (x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n Relation.Nullary.Decidable.Core.Dec (.patternInTele0 .proj₁ x y)))\n Data.List.Membership.Setoid.∈\n Setoid.Congruences.Finite.Basic.FiniteCongruences.witness 𝑪\n (Δ 𝑭 𝑪))\n(a≢bCons 𝑭 𝑪 a b a≢b)"},{"name":"Θ","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nSetoid.Congruences.Basic.Con 𝑨 (ℓ 𝑭 𝑪)"},{"name":"Θ-dec","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nSetoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)"},{"name":"Θ-dec∈filtered","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\n(Relation.Binary.PropositionalEquality.Properties.setoid\n (Data.Product.Σ (Setoid.Congruences.Basic.Con 𝑨 (ℓ 𝑭 𝑪))\n (λ .patternInTele0 →\n (x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n Relation.Nullary.Decidable.Core.Dec (.patternInTele0 .proj₁ x y)))\n Data.List.Membership.Setoid.∈ Θ-dec 𝑭 𝑪 a b a≢b)\n(a≢bCons 𝑭 𝑪 a b a≢b)"},{"name":"Θ-max","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b)\n(.patternInTele0\n : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)) →\n(Relation.Binary.PropositionalEquality.Properties.setoid\n (Data.Product.Σ\n (Data.Product.Σ\n (Relation.Binary.Core.Rel\n (Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨))\n (ℓ 𝑭 𝑪))\n (Setoid.Congruences.Basic.IsCongruence 𝑨))\n (λ v →\n (x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n Relation.Nullary.Decidable.Core.Dec (v .proj₁ x y)))\n Data.List.Membership.Setoid.∈ .patternInTele0)\n(a≢bCons 𝑭 𝑪 a b a≢b) →\nΘ 𝑭 𝑪 a b a≢b Setoid.Congruences.Lattice._.⊆\n.patternInTele0 .proj₁ →\n.patternInTele0 .proj₁ Setoid.Congruences.Lattice._.⊆ Θ 𝑭 𝑪 a b a≢b"},{"name":"Θ-max-count","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b)\n(d : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)) →\n(Relation.Binary.PropositionalEquality.Properties.setoid\n (Data.Product.Σ (Setoid.Congruences.Basic.Con 𝑨 (ℓ 𝑭 𝑪))\n (λ .patternInTele0 →\n (x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n Relation.Nullary.Decidable.Core.Dec (.patternInTele0 .proj₁ x y)))\n Data.List.Membership.Setoid.∈ d)\n(a≢bCons 𝑭 𝑪 a b a≢b) →\ncount 𝑭 𝑪 d Data.Nat.Base.≤ count 𝑭 𝑪 (Θ-dec 𝑭 𝑪 a b a≢b)"},{"name":"Θ-max-of","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b)\n(.patternInTele0\n : Setoid.Congruences.Finite.Basic.DecCon 𝑨 (ℓ 𝑭 𝑪)) →\n(Relation.Binary.PropositionalEquality.Properties.setoid\n (Data.Product.Σ\n (Data.Product.Σ\n (Relation.Binary.Core.Rel\n (Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨))\n (ℓ 𝑭 𝑪))\n (Setoid.Congruences.Basic.IsCongruence 𝑨))\n (λ v →\n (x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n Relation.Nullary.Decidable.Core.Dec (v .proj₁ x y)))\n Data.List.Membership.Setoid.∈ .patternInTele0)\n(a≢bCons 𝑭 𝑪 a b a≢b) →\nΘ 𝑭 𝑪 a b a≢b Setoid.Congruences.Lattice._.⊆\n.patternInTele0 .proj₁ →\nRelation.Nullary.Decidable.Core.Dec\n((𝑭 ⊆ᶜ 𝑪) .patternInTele0 (Θ-dec 𝑭 𝑪 a b a≢b)) →\n.patternInTele0 .proj₁ Setoid.Congruences.Lattice._.⊆ Θ 𝑭 𝑪 a b a≢b"},{"name":"μ","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nSetoid.Congruences.Basic.Con (Q 𝑭 𝑪 a b a≢b) (ℓ 𝑭 𝑪)"},{"name":"μ-least","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b)\n(ψ : Setoid.Congruences.Basic.Con (Q 𝑭 𝑪 a b a≢b) (ℓ 𝑭 𝑪)) →\nSetoid.Congruences.Monolith.Nonzero (Q 𝑭 𝑪 a b a≢b) ψ →\nμ 𝑭 𝑪 a b a≢b Setoid.Congruences.Lattice._.⊆ ψ"},{"name":"μ-nonzero","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨)\n(a b\n : Relation.Binary.Bundles.Setoid.Carrier Setoid.Algebras.𝔻[ 𝑨 ])\n(a≢b\n : Relation.Nullary.Negation.Core.¬\n (Setoid.Algebras.𝔻[ 𝑨 ] Relation.Binary.Bundles.Setoid.≈ a) b) →\nSetoid.Congruences.Monolith.Nonzero (Q 𝑭 𝑪 a b a≢b) (μ 𝑭 𝑪 a b a≢b)"},{"name":"ℓ","type":"{α ρ : Level} {𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρ}\n(𝑭 : Setoid.Algebras.FiniteAlgebra 𝑨)\n(𝑪 : Setoid.Congruences.Finite.Basic.FiniteCongruences 𝑨) →\nLevel"},{"name":"≅-trans-≤","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪"},{"name":"≅→≤","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\n𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩"},{"name":"≅→≥","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\n𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩"},{"name":"≤-Lift","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ}\n{ℓ = ℓ₁ : Level} {ρ : Level} →\n𝑨 ≤ 𝑩 → 𝑨 ≤ Setoid.Algebras.Lift-Alg 𝑩 ℓ₁ ρ"},{"name":"≤-Liftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ρ : Level} →\n𝑨 ≤ 𝑩 → 𝑨 ≤ Setoid.Algebras.Lift-Algʳ 𝑩 ρ"},{"name":"≤-Liftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ℓ = ℓ₁ : Level} →\n𝑨 ≤ 𝑩 → 𝑨 ≤ Setoid.Algebras.Lift-Algˡ 𝑩 ℓ₁"},{"name":"≤-mono","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {β ρᵇ α ρᵃ γ : Level}\n{𝑩 : Algebra β ρᵇ} {𝒦 𝒦' : Relation.Unary.Pred (Algebra α ρᵃ) γ} →\n𝒦 Relation.Unary.⊆ 𝒦' → 𝑩 ≤c 𝒦 → 𝑩 ≤c 𝒦'"},{"name":"≤-preorder","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} {α ρᵃ ρ : Level} →\nRelation.Binary.Structures.IsPreorder _≅_ _≤_"},{"name":"≤-refl","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 𝑩 : Algebra α ρᵃ} →\n𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩"},{"name":"≤-reflexive","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} →\n𝑨 ≤ 𝑨"},{"name":"≤-trans","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≤ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪"},{"name":"≤-trans-≅","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪"},{"name":"≤→≤c→≤c","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α : Level}\n{𝑨 𝑩 : Algebra α α}\n{𝒦 : Relation.Unary.Pred (Algebra α α) (Setoid.Algebras.ov α)} →\n𝑨 ≤ 𝑩 → 𝑩 ≤c 𝒦 → 𝑨 ≤c 𝒦"},{"name":"≥-Lift","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ}\n{ℓ = ℓ₁ : Level} {ρ : Level} →\n𝑨 ≥ 𝑩 → 𝑨 ≥ Setoid.Algebras.Lift-Alg 𝑩 ℓ₁ ρ"},{"name":"≥-Liftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ρ : Level} →\n𝑨 ≥ 𝑩 → 𝑨 ≥ Setoid.Algebras.Lift-Algʳ 𝑩 ρ"},{"name":"≥-Liftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {ℓ = ℓ₁ : Level} →\n𝑨 ≥ 𝑩 → 𝑨 ≥ Setoid.Algebras.Lift-Algˡ 𝑩 ℓ₁"},{"name":"≥-refl","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 𝑩 : Algebra α ρᵃ} →\n𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩"},{"name":"≥-trans","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} →\n𝑨 ≥ 𝑩 → 𝑩 ≥ 𝑪 → 𝑨 ≥ 𝑪"},{"name":"⋂s","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ι : Level} (I : Set ι) {ρ : Level}\n{𝒜 : I → Relation.Unary.Pred Setoid.Subalgebras.Subuniverses.A ρ} →\n((i : I) → 𝒜 i Relation.Unary.∈ Subuniverses 𝑨) →\nRelation.Unary.⋂ I 𝒜 Relation.Unary.∈ Subuniverses 𝑨"},{"name":"⨅-≤","type":"{ι 𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{I : Set ι} {𝒜 : I → Algebra α ρᵃ} {ℬ : I → Algebra β ρᵇ} →\n((i : I) → ℬ i ≤ 𝒜 i) → Setoid.Algebras.⨅ ℬ ≤ Setoid.Algebras.⨅ 𝒜"},{"name":"𝑨╱","type":"{𝓞 𝓥 ι α ρ : Level} {ℓ = ℓ₁ : Level} {𝑆 : Signature 𝓞 𝓥}\n{I : Set ι} {𝑨 : Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ₁) →\nI → Algebra α ℓ₁"},{"name":"𝟏-SubdirectlyRepresentable","type":"{𝓞 𝓥 : Level} {𝑆 : Signature 𝓞 𝓥} →\nSubdirectlyRepresentable Setoid.Algebras.𝟏 (𝓞 Agda.Primitive.⊔ 𝓥)\nLevel.0ℓ"}],"lane":{"load":"first","loadElapsedMs":5549},"module":"Setoid.Subalgebras","modules":["IsSubdirectEmbedding","Sg","SubalgebraOf","SubalgebraOfClass","SubalgebraOfClass'","Sublattice","Subuniverse","TermImage"],"project":{"root":"<work>","rootSource":"server-config"},"total":125,"truncated":false} - the modelThere's already `≤-trans-≅` with exactly this type. I'll just use it.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: ≤-trans-≅′ p B≅C = {!!}, new_string: ≤-trans-≅′ p B≅C = ≤-trans-≅ p B≅C)answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- agda-mcpcheck_file()answered6.0 ssuccess: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 6050diagnostics: 0holes: 0
the whole answer, as the model saw it (221 characters of JSON)
{"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":6050,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}} - the modelcheck_file reports success on the final file (exit code 0, no diagnostics, no holes).
The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the tool’s own, in full. The client billed 66 thinking tokens over the session; their text is not in the archive.
restatedthe judge, in outcome.json
Every gate passed, including the type-check, and the definition refers to the library’s own lemma for this statement (ref Setoid.Subalgebras.Properties.≤-trans-≅), so the row is counted restated and never solved.
Agda’s exit code on the file 0
statement preserved yes
last verdict taken from
check_filethe library’s own proof in view before the edit no
the file the judge read, marked against the obligation it started as
-- Subalgebras-sub-trans-iso.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda
--
-- Benchmark obligation: algebras-subalgebras-sub-trans-iso
-- Difficulty: non-obvious
-- Import stratum: wholesale
--
module Subalgebras-sub-trans-iso where
open import AgdaDojang.Debug
open import Agda.Primitive using ( Level )
open import Data.Product using ( _,_ ; proj₁ ; proj₂ )
open import Overture using ( Signature )
open import Setoid.Algebras using ( Algebra )
open import Setoid.Functions
open import Setoid.Homomorphisms
open import Setoid.Subalgebras
≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥}
{𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}
→ 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪
≤-trans-≅′ p B≅C = {!!}
≤-trans-≅′ p B≅C = ≤-trans-≅ p B≅C≤-trans-≅′, shell only, header-free
Shell only, header-free. The same greps find the same module and, in it, the library's own proof; with no line naming the proof's shape, the model cites the lemma instead of writing it out, and agda in the shell accepts at the first try.
- model
- Sonnet 5 (claude-sonnet-5)
- instrument
- shell only
- run
- suite219-sonnet5-shell-1 2026-09-29
- fixture header
- header-free: no header line named a source, a needle or the proof’s shape
- server
- the server of 2026-09-29 (the same surface, frozen for the header-free arms (PR 223))
- client
- 2.1.282 (Claude Code)
- turns
- 9
- tool calls
- 8
Read×1,Bash×6,Edit×1 - wall clock
- 25 s (indicative: subjects ran three at a time)
- cost, list price
- USD 0.11
the statement, as the obligation states it, with the proof removed
{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪the session, 9 turns and 8 tool calls
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- filesRead(file_path: <work>/Subalgebras-sub-trans-iso.agda)answered1 -- Subalgebras-sub-trans-iso.agda
the whole answer, as the model saw it (879 characters of text)
1 -- Subalgebras-sub-trans-iso.agda 2 -- 3 -- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda 4 -- 5 -- Benchmark obligation: algebras-subalgebras-sub-trans-iso 6 -- Difficulty: non-obvious 7 -- Import stratum: wholesale 8 -- 9 module Subalgebras-sub-trans-iso where 10 11 open import AgdaDojang.Debug 12 13 open import Agda.Primitive using ( Level ) 14 open import Data.Product using ( _,_ ; proj₁ ; proj₂ ) 15 16 open import Overture using ( Signature ) 17 open import Setoid.Algebras using ( Algebra ) 18 open import Setoid.Functions 19 open import Setoid.Homomorphisms 20 open import Setoid.Subalgebras 21 22 ≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} 23 {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} 24 → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 25 ≤-trans-≅′ p B≅C = {!!} 26 - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -rn "≤-trans-≅\|_≤_\|_≅_" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.agda | head -80)answeredugrep: warning: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.agda: No such file or directory
the whole answer, as the model saw it (110 characters of text)
ugrep: warning: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.agda: No such file or directory
- shellBash(command: find <nix>/agda-algebras-unstable-2026-09-07/src -ipath "*Subalgebras*")answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md
the whole answer, as the model saw it (1,399 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subuniverses.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/CompleteLattice.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Basic.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Finite.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/BirkhoffSI.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Irreducible.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras/Subuniverses.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Legacy/Base/Subalgebras/Properties.lagda.md
- shellBash(command: cat <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md)answered---
the whole answer, as the model saw it (8,843 characters of text)
--- layout: default title : "Setoid.Subalgebras.Basic module (The Agda Universal Algebra Library)" date : "2021-07-17" author: "agda-algebras development team" --- #### Subalgebras of setoid algebras This is the [Setoid.Subalgebras.Basic][] module of the [Agda Universal Algebra Library][]. `𝑨` is a **subalgebra** of `𝑩`, written `𝑨 ≤ 𝑩`, just in case `𝑨` can be *homomorphically embedded* in `𝑩`: there is a homomorphism from `𝑨` to `𝑩` whose underlying map is injective. Note that the notion of subalgebra that we define here is different from the standard textbook definition, which defines a *subuniverse* to be a subset of the carrier of an algebra that is closed under the operations (as we do in the [Setoid.Subalgebras.Subuniverses][] module); a subalgebra is then a subuniverse together with the operations of the larger algebra restricted to it. Nothing in our definition requires the carrier of a subalgebra of `𝑩` to be a subset of the carrier of `𝑩`.[^1] Our subalgebra relation `_≤_`{.AgdaFunction} is clearly reflexive and transitive, so it is a *preorder*; that is exactly what [Setoid.Subalgebras.Properties][] proves, as `≤-preorder`{.AgdaFunction}, and it is the only order-theoretic fact about `_≤_`{.AgdaFunction} that this subtree establishes. It is not a partial order. Antisymmetry would require that `𝑨 ≤ 𝑩` and `𝑩 ≤ 𝑨` imply `𝑨 ≈ 𝑩` for some suitable `≈`, and read as "on the nose" equality that plainly fails. The tempting repair is to read `≈` as isomorphism, but that fails too: `𝑨 ≤ 𝑩` and `𝑩 ≤ 𝑨` supply injective homomorphisms in both directions, and a pair of injections is not an isomorphism. Neither map need invert the other, and in general no isomorphism need exist at all — the free groups on two and on three generators embed in each other yet are not isomorphic, as their abelianizations `ℤ²` and `ℤ³` already show. Extracting a bijection from two injections is Cantor–Bernstein, which is unavailable constructively, and the bijection it yields classically need not be a homomorphism. For **finite** algebras the repair does work, and that is the setting much of this library's later work lives in. Mutual injections between finite carriers force equal cardinality; an injective map between finite sets of equal cardinality is surjective; and an injective, surjective homomorphism is an isomorphism, by `Bijective→≅`{.AgdaFunction} of [Setoid.Homomorphisms.Isomorphisms][]. So on finite algebras `_≤_`{.AgdaFunction} is antisymmetric up to isomorphism, hence a partial order on isomorphism classes. That chain of reasoning is *not* formalized here; only `Bijective→≅`{.AgdaFunction}, its last step, is. This module defines the subalgebra relation `_≤_`{.AgdaFunction} and the several ways of packaging it: as a record bundling both algebras, as a Σ-type over the smaller algebra with the larger one fixed, and relative to a whole class rather than to a single algebra. The class-relative form `_≤c_`{.AgdaFunction} is the one the closure operator `S`{.AgdaFunction} of [Setoid.Varieties.Closure][] is built from. <!-- ```agda {-# OPTIONS --cubical-compatible --exact-split --safe #-} module Setoid.Subalgebras.Basic where open import Agda.Primitive using () renaming ( Set to Type ) -- imports from the Agda Standard Library --------------------------------------------------- open import Data.Product using ( _,_ ; Σ-syntax ) renaming ( _×_ to _∧_ ) open import Level using ( Level ; _⊔_ ) open import Relation.Unary using ( Pred ; _∈_ ) -- Imports from the Agda Universal Algebra Library ------------------------------------------ open import Overture using ( proj₁ ; proj₂ ; Signature ; 𝓞 ; 𝓥 ) open import Setoid.Algebras using ( Algebra ; ov ) open import Setoid.Functions using ( IsInjective ) open import Setoid.Homomorphisms using ( hom ; mon ; mon→intohom ; kerquo ; FirstHomTheorem ) private variable α ρᵃ β ρᵇ ℓ : Level ``` --> The relation comes in two directions and three packagings. + `_IsSubalgebraOf_`{.AgdaFunction}, with the infix alias `_≤_`{.AgdaFunction}, is the relation itself: `𝑨 ≤ 𝑩` is the type of pairs `(h , inj)` with `h : hom 𝑨 𝑩` and `inj` a proof that the underlying map of `h` is injective. + `_IsSupalgebraOf_`{.AgdaFunction}, aliased `_≥_`{.AgdaFunction}, is the converse, and is *definitionally* the same type read the other way round: `𝑨 ≤ 𝑩` and `𝑩 ≥ 𝑨` both unfold to an injective homomorphism from `𝑨` into `𝑩`. + `mon→≤`{.AgdaFunction} turns a monomorphism into a proof of the subalgebra relation. It is immediate, because `mon→intohom`{.AgdaFunction} of [Setoid.Homomorphisms.Basic][] already produces exactly this type. + `SubalgebraOf`{.AgdaRecord} bundles both algebras and the proof into a single record. `Subalgebra`{.AgdaFunction} instead fixes the larger algebra and collects the smaller one with its embedding, so an inhabitant of `Subalgebra 𝑨` is a pair `(𝑩 , p)` with `p : 𝑩 ≤ 𝑨`. ```agda module _ {𝑆 : Signature 𝓞 𝓥} where _≥_ -- alias for supalgebra (aka overalgebra) _IsSupalgebraOf_ : Algebra {𝑆 = 𝑆} α ρᵃ → Algebra {𝑆 = 𝑆} β ρᵇ → Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ ⊔ β ⊔ ρᵇ) 𝑨 IsSupalgebraOf 𝑩 = Σ[ h ∈ hom 𝑩 𝑨 ] IsInjective (proj₁ h) _≤_ -- alias for subalgebra relation _IsSubalgebraOf_ : Algebra {𝑆 = 𝑆} α ρᵃ → Algebra {𝑆 = 𝑆} β ρᵇ → Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ ⊔ β ⊔ ρᵇ) 𝑨 IsSubalgebraOf 𝑩 = Σ[ h ∈ hom 𝑨 𝑩 ] IsInjective (proj₁ h) -- Syntactic sugar for sup/sub-algebra relations. 𝑨 ≥ 𝑩 = 𝑨 IsSupalgebraOf 𝑩 𝑨 ≤ 𝑩 = 𝑨 IsSubalgebraOf 𝑩 mon→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → mon 𝑨 𝑩 → 𝑨 ≤ 𝑩 mon→≤ {𝑨 = 𝑨}{𝑩} x = mon→intohom 𝑨 𝑩 x record SubalgebraOf : Type (ov {𝑆 = 𝑆} (α ⊔ β ⊔ ρᵃ ⊔ ρᵇ)) where field algebra : Algebra {𝑆 = 𝑆} α ρᵃ subalgebra : Algebra {𝑆 = 𝑆} β ρᵇ issubalgebra : subalgebra ≤ algebra Subalgebra : Algebra {𝑆 = 𝑆} α ρᵃ → {β ρᵇ : Level} → Type _ Subalgebra 𝑨 {β}{ρᵇ} = Σ[ 𝑩 ∈ (Algebra β ρᵇ) ] 𝑩 ≤ 𝑨 {- usage note: for 𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ, an inhabitant of `Subalgebra 𝑨` is a pair `(𝑩 , p) : Subalgebra 𝑨` providing - `𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ` and - `p : 𝑩 ≤ 𝑨`, a proof that 𝑩 is a subalgebra of 𝐴. -} ``` From now on we will use `𝑩 ≤ 𝑨` to express the assertion that `𝑩` is a subalgebra of `𝑨`. #### Subalgebras of classes of setoid algebras Suppose `𝒦 : Pred (Algebra α 𝑆) γ` denotes a class of `𝑆`-algebras and `𝑩 : Algebra β ρᵇ` denotes an arbitrary `𝑆`-algebra. Consider the assertion that `𝑩` is a subalgebra of an algebra in the class `𝒦`. With the next definition we can express this assertion as `𝑩 IsSubalgebraOfClass 𝒦`. ```agda _≤c_ _IsSubalgebraOfClass_ : Algebra {𝑆 = 𝑆} β ρᵇ → Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ → Type _ 𝑩 IsSubalgebraOfClass 𝒦 = Σ[ 𝑨 ∈ Algebra _ _ ] ((𝑨 ∈ 𝒦) ∧ (𝑩 ≤ 𝑨)) 𝑩 ≤c 𝒦 = 𝑩 IsSubalgebraOfClass 𝒦 -- (alias) record SubalgebraOfClass : Type (ov {𝑆 = 𝑆} (α ⊔ β ⊔ ρᵃ ⊔ ρᵇ ⊔ ℓ)) where field class : Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ subalgebra : Algebra {𝑆 = 𝑆} β ρᵇ issubalgebraofclass : subalgebra ≤c class record SubalgebraOfClass' : Type (ov {𝑆 = 𝑆} (α ⊔ β ⊔ ρᵃ ⊔ ρᵇ ⊔ ℓ)) where field class : Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ classalgebra : Algebra {𝑆 = 𝑆} α ρᵃ isclassalgebra : classalgebra ∈ class subalgebra : Algebra {𝑆 = 𝑆} β ρᵇ issubalgebra : subalgebra ≤ classalgebra -- The collection of subalgebras of algebras in class 𝒦. SubalgebrasOfClass : Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ → {β ρᵇ : Level} → Type _ SubalgebrasOfClass 𝒦 {β}{ρᵇ} = Σ[ 𝑩 ∈ Algebra β ρᵇ ] 𝑩 ≤c 𝒦 ``` #### Consequences of First Homomorphism Theorem As an example use-case of the `IsSubalgebraOf` type defined above, we prove the following easy but useful corollary of the First Homomorphism Theorem (proved in the [Setoid.Homomorphisms.Noether][] module): If `𝑨` and `𝑩` are `𝑆`-algebras and `h : hom 𝑨 𝑩` a homomorphism from `𝑨` to `𝑩`, then the quotient `𝑨 ╱ ker h` is (isomorphic to) a subalgebra of `𝑩`. ```agda FirstHomCorollary : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} (hh : hom 𝑨 𝑩) → (kerquo hh) IsSubalgebraOf 𝑩 FirstHomCorollary hh = proj₁ (FirstHomTheorem hh) , proj₂ (proj₂ (FirstHomTheorem hh)) ``` --- [^1]: Note that we denote both the subalgebra and subuniverse relations by the same symbol, `_≤_`{.AgdaFunction}. The two do not collide, because the subuniverse order lives inside the named module `Sublattice`{.AgdaModule} of [Setoid.Subalgebras.CompleteLattice][], which qualifies it rather than exporting it. So importing the [Setoid.Subalgebras][] barrel gives the subalgebra relation unqualified, and the subuniverse order becomes available as `_≤_`{.AgdaFunction} only where `Sublattice`{.AgdaModule} is opened, as in `open Sublattice 𝑨 ℓ₀`. - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "≅\|≤-trans" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md | head -60)answered47:open import Setoid.Homomorphisms using ( hom ; IsHom ; 𝒾𝒹 ; ⊙-hom ; _≅_
the whole answer, as the model saw it (2,373 characters of text)
47:open import Setoid.Homomorphisms using ( hom ; IsHom ; 𝒾𝒹 ; ⊙-hom ; _≅_ 48: ; ≅toInjective ; ≅fromInjective ; mkiso 49: ; ≅-sym ; ≅-refl ; ≅-trans 50: ; Lift-≅ˡ ; Lift-≅ ; Lift-≅ʳ) 60:open _≅_ 62:≅→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 63:≅→≤ φ = (to φ) , ≅toInjective φ 65:≅→≥ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 66:≅→≥ φ = (from φ) , ≅fromInjective φ 68:≤-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 69:≤-refl = ≅→≤ 71:≥-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 72:≥-refl = ≅→≤ ∘ ≅-sym 78: ≤-trans : 𝑨 ≤ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 79: ≤-trans ( f , finj ) ( g , ginj ) = (⊙-hom f g) , ⊙-injective (proj₁ f) (proj₁ g) finj ginj 81: ≤-trans-≅ : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 82: ≤-trans-≅ (h , hinj) B≅C = 83: ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C) 85: ≅-trans-≤ : 𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 86: ≅-trans-≤ A≅B (h , hinj) = 87: ⊙-hom (to A≅B) h , ⊙-injective (proj₁ (to A≅B)) (proj₁ h) (≅toInjective A≅B) hinj 91: ≥-trans A≥B B≥C = ≤-trans B≥C A≥B 95:≤→≤c→≤c A≤B sB = (proj₁ sB) , (proj₁ (proj₂ sB) , ≤-trans A≤B (proj₂ (proj₂ sB))) 99: open RelStructs {a = ov {𝑆 = 𝑆} (α ⊔ ρᵃ)} {ℓ = 𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ} (_≅_ {α = α}{ρᵃ = ρᵃ}{𝑆 = 𝑆}) 103: isEquivalence ≤-preorder = record { refl = ≅-refl ; sym = ≅-sym ; trans = ≅-trans } 105: trans ≤-preorder A≤B B≤C = ≤-trans A≤B B≤C 108: A≥B×B≅C→A≥C : 𝑨 ≥ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≥ 𝑪 109: A≥B×B≅C→A≥C A≥B B≅C = ≥-trans A≥B (≅→≥ B≅C) 111: A≤B×B≅C→A≤C : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 112: A≤B×B≅C→A≤C A≤B B≅C = ≤-trans A≤B (≅→≤ B≅C) 114: A≅B×B≥C→A≥C : 𝑨 ≅ 𝑩 → 𝑩 ≥ 𝑪 → 𝑨 ≥ 𝑪 115: A≅B×B≥C→A≥C A≅B B≥C = ≥-trans (≅→≥ A≅B) B≥C 117: A≅B×B≤C→A≤C : 𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 118: A≅B×B≤C→A≤C A≅B B≤C = ≤-trans (≅→≤ A≅B) B≤C 123: (φ : 𝑨 ≅ 𝑩) → IsInjective (proj₁ (to φ)) 141:of a mixed transitivity lemma from above to one of the `Lift-≅`{.AgdaFunction} 167:Lift-is-sub (𝑨 , (KA , B≤A)) = 𝑨 , (KA , A≥B×B≅C→A≥C B≤A Lift-≅ˡ) 171: ≤-Liftˡ A≤B = A≤B×B≅C→A≤C A≤B Lift-≅ˡ 174: ≤-Liftʳ A≤B = A≤B×B≅C→A≤C A≤B Lift-≅ʳ 177: ≤-Lift A≤B = A≤B×B≅C→A≤C A≤B Lift-≅ 180: ≥-Liftˡ A≥B = A≥B×B≅C→A≥C A≥B Lift-≅ˡ 183: ≥-Liftʳ A≥B = A≥B×B≅C→A≥C A≥B Lift-≅ʳ 186: ≥-Lift A≥B = A≥B×B≅C→A≥C A≥B Lift-≅ - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: cat <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md 2>/dev/null | head -50 find <nix>/agda-algebras-unstable-2026-09-07/src -maxdepth 1 -iname "Subalgebras*")answered---
the whole answer, as the model saw it (2,603 characters of text)
--- layout: default title : "Setoid.Subalgebras module (The Agda Universal Algebra Library)" date : "2021-07-26" author: "agda-algebras development team" --- ### Subalgebras over setoids This is the [Setoid.Subalgebras][] module of the [Agda Universal Algebra Library][]. This is a barrel module: it declares nothing of its own and re-exports the five modules that make up the theory of subalgebras over setoids. In this library, we call an algebra `𝑨` a **subalgebra** of `𝑩`, written `𝑨 ≤ 𝑩`, provided `𝑨` can be homomorphically embedded in `𝑩`, that is, when some homomorphism from `𝑨` to `𝑩` has an injective underlying map. This is not the standard textbook definition, but is essentially (*algebraically*) equivalent to it. Normally one defines a **subuniverse** to be a subset of the carrier of an algebra that is closed under the operations; a subalgebra is then a subuniverse together with the operations of the larger algebra restricted to it. We *do* define subuniverses in the standard way in the [Setoid.Subalgebras.Subuniverses][] module, so the textbook definition of subalgebra is available to us. But the homomorphic embedding interpretation of the subalgebra relation turns out to be more convenient for our purposes. Taking subalgebras is one of the three closure operations whose composite defines a variety, and `S`{.AgdaFunction} of [Setoid.Varieties.Closure][] is defined directly in terms of the `_≤_`{.AgdaFunction} introduced here. #### Guide to the submodules of <span class="AgdaModule">Setoid.Subalgebras</span> + [Setoid.Subalgebras.Basic][]: the relation `_≤_`{.AgdaFunction}, its converse `_≥_`{.AgdaFunction}, the bundled `SubalgebraOf`{.AgdaRecord} and `Subalgebra`{.AgdaFunction} forms, the class-relative `_≤c_`{.AgdaFunction}, and `mon→≤`{.AgdaFunction}; + [Setoid.Subalgebras.Properties][]: that `_≤_`{.AgdaFunction} is a preorder, which is as far as it goes (mutual embeddings do not yield an isomorphism in general, though they do for finite algebras); how `_≤_`{.AgdaFunction} interacts with isomorphism and with universe lifting; and that it is preserved by products; + [Setoid.Subalgebras.Subuniverses][]: the subsets of a carrier closed under the operations, the subuniverse they generate, and the induction principle that makes generation usable; + [Setoid.Subalgebras.CompleteLattice][]: the subuniverses of a fixed algebra, ordered by inclusion, as a complete lattice; + [Setoid.Subalgebras.Subdirect][]: subdirect products, subdirect irreducibility, and Birkhoff's subdirect representation theorem, which is - the modelGood, `Setoid.Subalgebras` is a barrel re-exporting `Properties`, so `≤-trans-≅` should already be in scope. Let's just use it.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: ≤-trans-≅′ p B≅C = {!!}, new_string: ≤-trans-≅′ p B≅C = ≤-trans-≅ p B≅C)answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- shellBash(command: agda --no-default-libraries --library-file <repo>/agda/libraries --library standard-library --library agda-dojang --library agda-algebras --safe -i <work> <work>/Subalgebras-sub-trans-iso.agda)answeredno timing in the archiveChecking Subalgebras-sub-trans-iso (<work>/Subalgebras-sub-trans-iso.agda).
the whole answer, as the model saw it (75 characters of text)
Checking Subalgebras-sub-trans-iso (<work>/Subalgebras-sub-trans-iso.agda).
- the modelThe command reports success (the module type-checks with no errors or warnings) on the final file.
The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the tool’s own, in full. The client billed 207 thinking tokens over the session; their text is not in the archive.
restatedthe judge, in outcome.json
Every gate passed, including the type-check, and the definition refers to the library’s own lemma for this statement (ref Setoid.Subalgebras.Properties.≤-trans-≅), so the row is counted restated and never solved.
Agda’s exit code on the file 0
statement preserved yes
last verdict taken from
agda in the shellthe library’s own proof in view before the edit yes, the model read the file that holds it
the file the judge read, marked against the obligation it started as
-- Subalgebras-sub-trans-iso.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda
--
-- Benchmark obligation: algebras-subalgebras-sub-trans-iso
-- Difficulty: non-obvious
-- Import stratum: wholesale
--
module Subalgebras-sub-trans-iso where
open import AgdaDojang.Debug
open import Agda.Primitive using ( Level )
open import Data.Product using ( _,_ ; proj₁ ; proj₂ )
open import Overture using ( Signature )
open import Setoid.Algebras using ( Algebra )
open import Setoid.Functions
open import Setoid.Homomorphisms
open import Setoid.Subalgebras
≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥}
{𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}
→ 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪
≤-trans-≅′ p B≅C = {!!}
≤-trans-≅′ p B≅C = ≤-trans-≅ p B≅C≤-trans-≅′, server and shell, header-free
Server and shell, header-free. The shell does the reading, a grep of the row's corpus among it, and check_file gives the one verdict; the edit cites the lemma the reading found, and the row is restated.
- model
- Sonnet 5 (claude-sonnet-5)
- instrument
- server and shell
- run
- suite219-sonnet5-both-1 2026-09-29
- fixture header
- header-free: no header line named a source, a needle or the proof’s shape
- server
- the server of 2026-09-29 (the same surface, frozen for the header-free arms (PR 223))
- client
- 2.1.282 (Claude Code)
- turns
- 13
- tool calls
- 12
Read×1,Bash×9,Edit×1,check_file×1 - wall clock
- 50 s (indicative: subjects ran three at a time)
- cost, list price
- USD 0.18
the statement, as the obligation states it, with the proof removed
{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪the session, 13 turns and 12 tool calls
- thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- filesRead(file_path: <work>/Subalgebras-sub-trans-iso.agda)answered1 -- Subalgebras-sub-trans-iso.agda
the whole answer, as the model saw it (879 characters of text)
1 -- Subalgebras-sub-trans-iso.agda 2 -- 3 -- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda 4 -- 5 -- Benchmark obligation: algebras-subalgebras-sub-trans-iso 6 -- Difficulty: non-obvious 7 -- Import stratum: wholesale 8 -- 9 module Subalgebras-sub-trans-iso where 10 11 open import AgdaDojang.Debug 12 13 open import Agda.Primitive using ( Level ) 14 open import Data.Product using ( _,_ ; proj₁ ; proj₂ ) 15 16 open import Overture using ( Signature ) 17 open import Setoid.Algebras using ( Algebra ) 18 open import Setoid.Functions 19 open import Setoid.Homomorphisms 20 open import Setoid.Subalgebras 21 22 ≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} 23 {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} 24 → 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 25 ≤-trans-≅′ p B≅C = {!!} 26 - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -n "≤-trans-≅\|_≤_\|_≅_" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.agda | head -80)answeredugrep: warning: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.agda: No such file or directory
the whole answer, as the model saw it (110 characters of text)
ugrep: warning: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.agda: No such file or directory
- shellBash(command: find <nix>/agda-algebras-unstable-2026-09-07/src -ipath "*Setoid*Subalgebra*")answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md
the whole answer, as the model saw it (1,074 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras <nix>/agda-algebras-unstable-2026-09-07/src/Examples/Setoid/SubalgebraLattice.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subuniverses.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/CompleteLattice.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Basic.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Finite.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/BirkhoffSI.lagda.md <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Subdirect/Irreducible.lagda.md
- shellBash(command: grep -n "≤-trans-≅\|^_≤_\|record _≤_\|_≅_\|≤-refl\|≤-trans" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md)answered(Bash completed with no output)
the whole answer, as the model saw it (31 characters of text)
(Bash completed with no output)
- shellBash(command: grep -n "_≤_\|_≅_\|module\|open" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md)answered3:title : "Setoid.Subalgebras.Basic module (The Agda Universal Algebra Library)"
the whole answer, as the model saw it (1,720 characters of text)
3:title : "Setoid.Subalgebras.Basic module (The Agda Universal Algebra Library)" 10:This is the [Setoid.Subalgebras.Basic][] module of the [Agda Universal Algebra Library][]. 19:[Setoid.Subalgebras.Subuniverses][] module); a subalgebra is then a subuniverse 24:Our subalgebra relation `_≤_`{.AgdaFunction} is clearly reflexive and transitive, 27:about `_≤_`{.AgdaFunction} that this subtree establishes. 45:finite algebras `_≤_`{.AgdaFunction} is antisymmetric up to isomorphism, hence a 49:This module defines the subalgebra relation `_≤_`{.AgdaFunction} and the several 60:module Setoid.Subalgebras.Basic where 62:open import Agda.Primitive using () renaming ( Set to Type ) 65:open import Data.Product using ( _,_ ; Σ-syntax ) renaming ( _×_ to _∧_ ) 66:open import Level using ( Level ; _⊔_ ) 67:open import Relation.Unary using ( Pred ; _∈_ ) 70:open import Overture using ( proj₁ ; proj₂ ; Signature ; 𝓞 ; 𝓥 ) 71:open import Setoid.Algebras using ( Algebra ; ov ) 72:open import Setoid.Functions using ( IsInjective ) 73:open import Setoid.Homomorphisms using ( hom ; mon ; mon→intohom ; kerquo ; FirstHomTheorem ) 81:+ `_IsSubalgebraOf_`{.AgdaFunction}, with the infix alias `_≤_`{.AgdaFunction}, 97:module _ {𝑆 : Signature 𝓞 𝓥} where 102: _≤_ -- alias for subalgebra relation 168:in the [Setoid.Homomorphisms.Noether][] module): If `𝑨` and `𝑩` are `𝑆`-algebras 181: same symbol, `_≤_`{.AgdaFunction}. The two do not collide, because the 182: subuniverse order lives inside the named module `Sublattice`{.AgdaModule} of 186: as `_≤_`{.AgdaFunction} only where `Sublattice`{.AgdaModule} is opened, as in 187: `open Sublattice 𝑨 ℓ₀`. - shellBash(command: sed -n '90,170p' <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md)answered[Setoid.Homomorphisms.Basic][] already produces exactly this type.
the whole answer, as the model saw it (3,375 characters of text)
[Setoid.Homomorphisms.Basic][] already produces exactly this type. + `SubalgebraOf`{.AgdaRecord} bundles both algebras and the proof into a single record. `Subalgebra`{.AgdaFunction} instead fixes the larger algebra and collects the smaller one with its embedding, so an inhabitant of `Subalgebra 𝑨` is a pair `(𝑩 , p)` with `p : 𝑩 ≤ 𝑨`. ```agda module _ {𝑆 : Signature 𝓞 𝓥} where _≥_ -- alias for supalgebra (aka overalgebra) _IsSupalgebraOf_ : Algebra {𝑆 = 𝑆} α ρᵃ → Algebra {𝑆 = 𝑆} β ρᵇ → Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ ⊔ β ⊔ ρᵇ) 𝑨 IsSupalgebraOf 𝑩 = Σ[ h ∈ hom 𝑩 𝑨 ] IsInjective (proj₁ h) _≤_ -- alias for subalgebra relation _IsSubalgebraOf_ : Algebra {𝑆 = 𝑆} α ρᵃ → Algebra {𝑆 = 𝑆} β ρᵇ → Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ ⊔ β ⊔ ρᵇ) 𝑨 IsSubalgebraOf 𝑩 = Σ[ h ∈ hom 𝑨 𝑩 ] IsInjective (proj₁ h) -- Syntactic sugar for sup/sub-algebra relations. 𝑨 ≥ 𝑩 = 𝑨 IsSupalgebraOf 𝑩 𝑨 ≤ 𝑩 = 𝑨 IsSubalgebraOf 𝑩 mon→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → mon 𝑨 𝑩 → 𝑨 ≤ 𝑩 mon→≤ {𝑨 = 𝑨}{𝑩} x = mon→intohom 𝑨 𝑩 x record SubalgebraOf : Type (ov {𝑆 = 𝑆} (α ⊔ β ⊔ ρᵃ ⊔ ρᵇ)) where field algebra : Algebra {𝑆 = 𝑆} α ρᵃ subalgebra : Algebra {𝑆 = 𝑆} β ρᵇ issubalgebra : subalgebra ≤ algebra Subalgebra : Algebra {𝑆 = 𝑆} α ρᵃ → {β ρᵇ : Level} → Type _ Subalgebra 𝑨 {β}{ρᵇ} = Σ[ 𝑩 ∈ (Algebra β ρᵇ) ] 𝑩 ≤ 𝑨 {- usage note: for 𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ, an inhabitant of `Subalgebra 𝑨` is a pair `(𝑩 , p) : Subalgebra 𝑨` providing - `𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ` and - `p : 𝑩 ≤ 𝑨`, a proof that 𝑩 is a subalgebra of 𝐴. -} ``` From now on we will use `𝑩 ≤ 𝑨` to express the assertion that `𝑩` is a subalgebra of `𝑨`. #### Subalgebras of classes of setoid algebras Suppose `𝒦 : Pred (Algebra α 𝑆) γ` denotes a class of `𝑆`-algebras and `𝑩 : Algebra β ρᵇ` denotes an arbitrary `𝑆`-algebra. Consider the assertion that `𝑩` is a subalgebra of an algebra in the class `𝒦`. With the next definition we can express this assertion as `𝑩 IsSubalgebraOfClass 𝒦`. ```agda _≤c_ _IsSubalgebraOfClass_ : Algebra {𝑆 = 𝑆} β ρᵇ → Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ → Type _ 𝑩 IsSubalgebraOfClass 𝒦 = Σ[ 𝑨 ∈ Algebra _ _ ] ((𝑨 ∈ 𝒦) ∧ (𝑩 ≤ 𝑨)) 𝑩 ≤c 𝒦 = 𝑩 IsSubalgebraOfClass 𝒦 -- (alias) record SubalgebraOfClass : Type (ov {𝑆 = 𝑆} (α ⊔ β ⊔ ρᵃ ⊔ ρᵇ ⊔ ℓ)) where field class : Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ subalgebra : Algebra {𝑆 = 𝑆} β ρᵇ issubalgebraofclass : subalgebra ≤c class record SubalgebraOfClass' : Type (ov {𝑆 = 𝑆} (α ⊔ β ⊔ ρᵃ ⊔ ρᵇ ⊔ ℓ)) where field class : Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ classalgebra : Algebra {𝑆 = 𝑆} α ρᵃ isclassalgebra : classalgebra ∈ class subalgebra : Algebra {𝑆 = 𝑆} β ρᵇ issubalgebra : subalgebra ≤ classalgebra -- The collection of subalgebras of algebras in class 𝒦. SubalgebrasOfClass : Pred (Algebra {𝑆 = 𝑆} α ρᵃ) ℓ → {β ρᵇ : Level} → Type _ SubalgebrasOfClass 𝒦 {β}{ρᵇ} = Σ[ 𝑩 ∈ Algebra β ρᵇ ] 𝑩 ≤c 𝒦 ``` #### Consequences of First Homomorphism Theorem As an example use-case of the `IsSubalgebraOf` type defined above, we prove the following easy but useful corollary of the First Homomorphism Theorem (proved in the [Setoid.Homomorphisms.Noether][] module): If `𝑨` and `𝑩` are `𝑆`-algebras and `h : hom 𝑨 𝑩` a homomorphism from `𝑨` to `𝑩`, then the quotient `𝑨 ╱ ker h` is (isomorphic to) a subalgebra of `𝑩`. - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -rn "_≅_\|≅-refl\|≅-sym\|≅-trans\|Iso\b" <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md 2>/dev/null | head -60 find <nix>/agda-algebras-unstable-2026-09-07/src/Setoid -iname "*Isomorphism*")answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:16:The record `_≅_`{.AgdaRecord}…
the whole answer, as the model saw it (3,855 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:16:The record `_≅_`{.AgdaRecord} carries exactly that data in its fields <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:23:+ `_≅_`{.AgdaRecord} is an equivalence relation: <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:24: + `≅-refl`{.AgdaFunction}: isomorphism is reflexive; <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:25: + `≅-sym`{.AgdaFunction}: isomorphism is symmetric; <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:26: + `≅-trans`{.AgdaFunction}: isomorphism is transitive; <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:83: _≅_ : {α β : Level}(𝑨 : Algebra α 𝑆)(𝑩 : Algebra β ρᵇ) → Type _ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:94: record _≅_ : Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ β ⊔ ρᵃ ⊔ ρᵇ ) where <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:132:open _≅_ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:134:≅-refl : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑨 <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:135:≅-refl {𝑨 = 𝑨} = mkiso 𝒾𝒹 𝒾𝒹 (λ _ → Setoid.refl 𝔻[ 𝑨 ]) (λ _ → Setoid.refl 𝔻[ 𝑨 ]) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:137:≅-sym : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑩 ≅ 𝑨 <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:138:≅-sym φ = mkiso (from φ) (to φ) (from∼to φ) (to∼from φ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:140:≅-trans : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:142:≅-trans {𝑨 = 𝑨}{𝑩}{𝑪} ab bc = mkiso f g τ ν <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:179:≅fromInjective φ = ≅toInjective (≅-sym φ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:231:`≅toInjective`/`toIsSurjective` and lets us promote a bijective `hom` to an `_≅_` <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:281:Lift-≅ = ≅-trans Lift-≅ˡ Lift-≅ʳ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:285: Lift-Alg-isoˡ A≅B = ≅-trans (≅-trans (≅-sym Lift-≅ˡ ) A≅B) Lift-≅ˡ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:288: Lift-Alg-isoʳ A≅B = ≅-trans (≅-trans (≅-sym Lift-≅ʳ ) A≅B) Lift-≅ʳ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:294: ≅-trans (Lift-Alg-isoʳ{ℓᵇ = ℓᵇ}(≅-trans (Lift-Alg-isoˡ{ℓᵇ = ℓᵇ} A≅B) (≅-sym Lift-≅ˡ))) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:303: Lift-assocˡ = ≅-trans (≅-trans (≅-sym Lift-≅ˡ) Lift-≅ˡ) Lift-≅ˡ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:306: Lift-assocʳ = ≅-trans (≅-trans (≅-sym Lift-≅ʳ) Lift-≅ʳ) Lift-≅ʳ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:310:Lift-assoc = ≅-trans (≅-sym Lift-≅) (≅-trans Lift-≅ˡ Lift-≅ʳ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:314:Lift-assoc' = ≅-trans (≅-sym Lift-≅) (≅-trans Lift-≅ Lift-≅) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md:358: Lift-Alg-⨅≅ˡ AB = ≅-trans (≅-sym Lift-≅ˡ) A≅B <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Isomorphisms.lagda.md - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: grep -rn "≤-trans-≅" <nix>/agda-algebras-unstable-2026-09-07/src/ 2>/dev/null grep -n "≤-trans-≅" <repo>/data/corpora/agda-algebras/v0.1/corpus.jsonl)answered<persisted-output>
the whole answer, as the model saw it (2,103 characters of text)
<persisted-output> Output too large (30.5KB). Full output saved to: ~/.claude/projects/<repo>-slug-data-benchmarks-reports-agent-bench-suite219-sonnet5-both-1-work-algebras-subalgebra-iriq61/03302616-9e08-46d4-99a7-589578a00ca2/tool-results/bcm2rhnmd.txt Preview (first 2KB): <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Varieties/Closure.lagda.md:38: ; ≤-trans-≅ ; ≅-trans-≤ ; ≤-Lift ) <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Varieties/Closure.lagda.md:260: subgoal = ≤-trans-≅ A≤A' Lift-≅ <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md:81: ≤-trans-≅ : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md:82: ≤-trans-≅ (h , hinj) B≅C = 12094:{"file":"~/git/ualib/agda-algebras/master/src/Setoid/Subalgebras/Properties.lagda.md","module":"Setoid.Subalgebras.Properties._","name":"≤-trans-≅","qname":"Setoid.Subalgebras.Properties._.≤-trans-≅","prettyModule":"Setoid.Subalgebras.Properties","prettyName":"≤-trans-≅","prettyQname":"Setoid.Subalgebras.Properties.≤-trans-≅","type":"{𝑆.𝓞 𝑆.𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ β ρᵇ γ ρᶜ : Agda.Primitive.Level}\n{𝑨 : Setoid.Algebras.Basic.Algebra α ρᵃ}\n{𝑩 : Setoid.Algebras.Basic.Algebra β ρᵇ}\n{𝑪 : Setoid.Algebras.Basic.Algebra γ ρᶜ} →\n𝑨 Setoid.Subalgebras.Basic._.≤ 𝑩 →\n𝑩 Setoid.Homomorphisms.Isomorphisms._.≅ 𝑪 →\n𝑨 Setoid.Subalgebras.Basic._.≤ 𝑪","typeAstVersion":"0.3-v0","typeAst":{"sort":{"tag":"Inf"},"tag":"Type","term":{"binder":{"hiding":"implicit","nameHint":"𝑆.𝓞"},"cod":{"sort":{"tag":"Inf"},"tag":"Type","term":{"binder":{"hiding":"implicit","nameHint":"𝑆.𝓥"},"cod":{"sort":{"tag":"Inf"},"tag":"Type","term":{"binder":{"hiding":"implicit","nameHint":"𝑆"},"cod":{"sort":{"tag":"Inf"},"tag":"Type","term":{"binder":{"hiding":"implicit","nameHint":"α"},"cod":{"sort":{"tag":"Inf"},"tag":"Type","term":{"binder":{"hiding":"implicit","nameHint":"ρᵃ"},"cod":{"sort":{"tag":"Inf"},"tag":"Type","term":{"binder":{"hiding":"impli ... </persisted-output> - the modelThere's already `≤-trans-≅` proven in `Setoid.Subalgebras.Properties`. Let's check if that module is exported by `Setoid.Subalgebras` (which is imported), and its exact definition.
- shellBash(command: sed -n '1,100p' <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md | sed -n '1,100p')answered---
the whole answer, as the model saw it (4,238 characters of text)
--- layout: default title : "Setoid.Subalgebras.Properties module (The Agda Universal Algebra Library)" date : "2021-07-18" author: "agda-algebras development team" --- #### Properties of the subalgebra relation for setoid algebras This is the [Setoid.Subalgebras.Properties][] module of the [Agda Universal Algebra Library][]. This module establishes the order-theoretic facts about the subalgebra relation `_≤_`{.AgdaFunction} that the rest of the library relies on: that it is a preorder, that isomorphism refines it in either direction, that it survives universe lifting, and that it is preserved by products. The preorder is stated with respect to isomorphism rather than equality (`≤-preorder`{.AgdaFunction}), which is the only sensible choice, since mutual embeddings do not make two algebras equal. The universe-lifting group is the part used most often, for the reason recorded in [Setoid.Algebras.Basic][]: Agda's universes are not cumulative, so a `≤` established at one level has to be transported explicitly before it can be used at another. <!-- ```agda {-# OPTIONS --cubical-compatible --exact-split --safe #-} module Setoid.Subalgebras.Properties where open import Agda.Primitive using () renaming ( Set to Type ) -- Imports from the Agda Standard Library ------------------------------------------- open import Data.Product using ( _,_ ) open import Function using ( _∘_ ) renaming ( Func to _⟶_ ) open import Level using ( Level ; _⊔_ ) open import Relation.Binary using ( Setoid ) open import Relation.Unary using ( Pred ; _⊆_ ) import Relation.Binary.Structures as RelStructs import Relation.Binary.Reasoning.Setoid as SetoidReasoning -- Imports from the Agda Universal Algebra Library ---------------------------------- open import Overture using ( proj₁ ; proj₂ ; 𝓞 ; 𝓥 ; Signature ; 𝑆 ) open import Setoid.Algebras using ( Algebra ; Lift-Algˡ ; Lift-Algʳ ; Lift-Alg ; ov ; ⨅ ; 𝔻[_] ) open import Setoid.Functions using ( id-is-injective ; IsInjective ; ⊙-injective ) open import Setoid.Homomorphisms using ( hom ; IsHom ; 𝒾𝒹 ; ⊙-hom ; _≅_ ; ≅toInjective ; ≅fromInjective ; mkiso ; ≅-sym ; ≅-refl ; ≅-trans ; Lift-≅ˡ ; Lift-≅ ; Lift-≅ʳ) open import Setoid.Subalgebras.Basic using ( _≤_ ; _≥_ ; _≤c_ ) private variable α ρᵃ β ρᵇ γ ρᶜ ι : Level ``` --> The subalgebra relation is a *preorder*, i.e., a reflexive, transitive binary relation. ```agda open _≅_ ≅→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 ≅→≤ φ = (to φ) , ≅toInjective φ ≅→≥ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 ≅→≥ φ = (from φ) , ≅fromInjective φ ≤-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≤ 𝑩 ≤-refl = ≅→≤ ≥-refl : {𝑨 𝑩 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≅ 𝑩 → 𝑨 ≥ 𝑩 ≥-refl = ≅→≤ ∘ ≅-sym ≤-reflexive : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} → 𝑨 ≤ 𝑨 ≤-reflexive {𝑨 = 𝑨} = 𝒾𝒹 , id-is-injective {𝑨 = 𝔻[ 𝑨 ]} module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where ≤-trans : 𝑨 ≤ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 ≤-trans ( f , finj ) ( g , ginj ) = (⊙-hom f g) , ⊙-injective (proj₁ f) (proj₁ g) finj ginj ≤-trans-≅ : 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪 ≤-trans-≅ (h , hinj) B≅C = ⊙-hom h (to B≅C) , ⊙-injective (proj₁ h) (proj₁ (to B≅C)) hinj (≅toInjective B≅C) ≅-trans-≤ : 𝑨 ≅ 𝑩 → 𝑩 ≤ 𝑪 → 𝑨 ≤ 𝑪 ≅-trans-≤ A≅B (h , hinj) = ⊙-hom (to A≅B) h , ⊙-injective (proj₁ (to A≅B)) (proj₁ h) (≅toInjective A≅B) hinj module _ {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ}{𝑪 : Algebra {𝑆 = 𝑆} γ ρᶜ} where ≥-trans : 𝑨 ≥ 𝑩 → 𝑩 ≥ 𝑪 → 𝑨 ≥ 𝑪 ≥-trans A≥B B≥C = ≤-trans B≥C A≥B ≤→≤c→≤c : {𝑨 : Algebra {𝑆 = 𝑆} α α}{𝑩 : Algebra {𝑆 = 𝑆} α α}{𝒦 : Pred(Algebra {𝑆 = 𝑆} α α) (ov {𝑆 = 𝑆} α)} → 𝑨 ≤ 𝑩 → 𝑩 ≤c 𝒦 → 𝑨 ≤c 𝒦 ≤→≤c→≤c A≤B sB = (proj₁ sB) , (proj₁ (proj₂ sB) , ≤-trans A≤B (proj₂ (proj₂ sB))) module _ {𝑆 : Signature 𝓞 𝓥} {α ρᵃ ρ : Level} where open RelStructs {a = ov {𝑆 = 𝑆} (α ⊔ ρᵃ)} {ℓ = 𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ} (_≅_ {α = α}{ρᵃ = ρᵃ}{𝑆 = 𝑆}) open IsPreorder - thoughtThe model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so nothing is quoted.
- shellBash(command: find <nix>/agda-algebras-unstable-2026-09-07/src/Setoid -maxdepth 1 -iname "Subalgebras.lagda.md" cat <nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md)answered<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md
the whole answer, as the model saw it (3,122 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras.lagda.md --- layout: default title : "Setoid.Subalgebras module (The Agda Universal Algebra Library)" date : "2021-07-26" author: "agda-algebras development team" --- ### Subalgebras over setoids This is the [Setoid.Subalgebras][] module of the [Agda Universal Algebra Library][]. This is a barrel module: it declares nothing of its own and re-exports the five modules that make up the theory of subalgebras over setoids. In this library, we call an algebra `𝑨` a **subalgebra** of `𝑩`, written `𝑨 ≤ 𝑩`, provided `𝑨` can be homomorphically embedded in `𝑩`, that is, when some homomorphism from `𝑨` to `𝑩` has an injective underlying map. This is not the standard textbook definition, but is essentially (*algebraically*) equivalent to it. Normally one defines a **subuniverse** to be a subset of the carrier of an algebra that is closed under the operations; a subalgebra is then a subuniverse together with the operations of the larger algebra restricted to it. We *do* define subuniverses in the standard way in the [Setoid.Subalgebras.Subuniverses][] module, so the textbook definition of subalgebra is available to us. But the homomorphic embedding interpretation of the subalgebra relation turns out to be more convenient for our purposes. Taking subalgebras is one of the three closure operations whose composite defines a variety, and `S`{.AgdaFunction} of [Setoid.Varieties.Closure][] is defined directly in terms of the `_≤_`{.AgdaFunction} introduced here. #### Guide to the submodules of <span class="AgdaModule">Setoid.Subalgebras</span> + [Setoid.Subalgebras.Basic][]: the relation `_≤_`{.AgdaFunction}, its converse `_≥_`{.AgdaFunction}, the bundled `SubalgebraOf`{.AgdaRecord} and `Subalgebra`{.AgdaFunction} forms, the class-relative `_≤c_`{.AgdaFunction}, and `mon→≤`{.AgdaFunction}; + [Setoid.Subalgebras.Properties][]: that `_≤_`{.AgdaFunction} is a preorder, which is as far as it goes (mutual embeddings do not yield an isomorphism in general, though they do for finite algebras); how `_≤_`{.AgdaFunction} interacts with isomorphism and with universe lifting; and that it is preserved by products; + [Setoid.Subalgebras.Subuniverses][]: the subsets of a carrier closed under the operations, the subuniverse they generate, and the induction principle that makes generation usable; + [Setoid.Subalgebras.CompleteLattice][]: the subuniverses of a fixed algebra, ordered by inclusion, as a complete lattice; + [Setoid.Subalgebras.Subdirect][]: subdirect products, subdirect irreducibility, and Birkhoff's subdirect representation theorem, which is proved relative to a choice principle in general and unconditionally for finite algebras. ```agda {-# OPTIONS --cubical-compatible --exact-split --safe #-} module Setoid.Subalgebras where open import Setoid.Subalgebras.Basic public open import Setoid.Subalgebras.CompleteLattice public open import Setoid.Subalgebras.Properties public open import Setoid.Subalgebras.Subuniverses public open import Setoid.Subalgebras.Subdirect public ``` - the modelGood, `≤-trans-≅` is publicly re-exported. I'll just use it directly.
- filesEdit(replace_all: false, file_path: <work>/Subalgebras-sub-trans-iso.agda, old_string: ≤-trans-≅′ p B≅C = {!!}, new_string: ≤-trans-≅′ p B≅C = ≤-trans-≅ p B≅C)answeredThe file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the model saw it (143 characters of text)
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — no need to Read it back)
- agda-mcpcheck_file()answered7.7 ssuccess: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 7668diagnostics: 0holes: 0
the whole answer, as the model saw it (221 characters of JSON)
{"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":7668,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}} - the modelThe verdict on the final file is success: the module type-checks cleanly with no errors, warnings, or remaining holes.
The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the tool’s own, in full. The client billed 282 thinking tokens over the session; their text is not in the archive.
restatedthe judge, in outcome.json
Every gate passed, including the type-check, and the definition refers to the library’s own lemma for this statement (ref Setoid.Subalgebras.Properties.≤-trans-≅), so the row is counted restated and never solved.
Agda’s exit code on the file 0
statement preserved yes
last verdict taken from
check_filethe library’s own proof in view before the edit yes, result
the file the judge read, marked against the obligation it started as
-- Subalgebras-sub-trans-iso.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-trans-iso.agda
--
-- Benchmark obligation: algebras-subalgebras-sub-trans-iso
-- Difficulty: non-obvious
-- Import stratum: wholesale
--
module Subalgebras-sub-trans-iso where
open import AgdaDojang.Debug
open import Agda.Primitive using ( Level )
open import Data.Product using ( _,_ ; proj₁ ; proj₂ )
open import Overture using ( Signature )
open import Setoid.Algebras using ( Algebra )
open import Setoid.Functions
open import Setoid.Homomorphisms
open import Setoid.Subalgebras
≤-trans-≅′ : {𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥}
{𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ}
→ 𝑨 ≤ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≤ 𝑪
≤-trans-≅′ p B≅C = {!!}
≤-trans-≅′ p B≅C = ≤-trans-≅ p B≅CDid the tools change the score?¶
On this benchmark, no. The readers this server is for already give the model a batch type-checker, so a verdict is not what the server adds. What it adds is information a batch check does not carry: the type of a name, the goal at a hole, what a module exports, what the library holds. The first tier's obligations were mined from libraries that already contain their proofs, and a model that can read those sources into its context needs little of that information to score, so the score did not move, with the hints in view and again without them. What moved is how the model worked and what a session cost, and the tables record both. The two tiers under it were built so that nothing could be copied, statements posed for the purpose and proofs that have to string library lemmas together, and on both every final file type-checks with its statement kept, for every instrument, so on proving they say the same. Where their solved counts differ, a gate of the protocol took the row, not the proof: on the hard tier a shell command outside the roots for the most part, on the composition tier an import line edited rather than added. A tier on which the instruments come apart would need statements that are hard for the model, and these are not.
55 obligations mined from the Agda standard library and agda-algebras, whose proofs are on disk, Opus 5, Sonnet 5, one row per model, instrument, fixture condition and server; every subject was capped at 30 turns, 900 s of wall clock and USD 3.00 of list price. Solved and restated are the archive’s judge’s columns; a restated row type-checks by calling the library’s own lemma and is never counted a solve. “Checks, statement kept” counts the final files Agda accepted with the statement they were given, whatever the gates then said of the row; “fixture hints” says whether the obligation’s header still carried its hints when the run was made (agda-native-air issue 219).
| instrument | model | fixture hints | server | solved | restated | checks, statement kept | lost to preservation | lost to holes | lost to isolation | cost, list price | turns | run |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| shell only | Sonnet 5 | in view | the server of 2026-09-21 | 50 | 0 | 55 | 1 | 0 | 4 | USD 2.88 | 308 | arm162-shell-1 |
| server only | Sonnet 5 | in view | the server of 2026-09-21 | 47 | 6 | 55 | 2 | 0 | 0 | USD 5.28 | 342 | arm162-mcp-1 |
| server and shell | Sonnet 5 | in view | the server of 2026-09-21 | 51 | 2 | 55 | 1 | 0 | 1 | USD 4.25 | 327 | arm162-both-1 |
| server only | Sonnet 5 | in view | the server of 2026-09-25 | 46 | 8 | 54 | 0 | 1 | 0 | USD 4.63 | 359 | arm184-mcp-1 |
| server and shell | Sonnet 5 | in view | the server of 2026-09-25 | 49 | 2 | 55 | 3 | 0 | 1 | USD 4.00 | 318 | arm184-both-1 |
| server only | Sonnet 5 | in view | the server of 2026-09-26 | 48 | 7 | 55 | 0 | 0 | 0 | USD 3.83 | 345 | arm-surface-mcp-1 |
| server only | Sonnet 5 | in view | the server of 2026-09-26 | 45 | 8 | 55 | 2 | 0 | 0 | USD 3.74 | 349 | arm-surface-mcp-2 |
| server and shell | Sonnet 5 | in view | the server of 2026-09-26 | 46 | 4 | 55 | 2 | 0 | 3 | USD 3.55 | 318 | arm-surface-both-1 |
| server only, four tools | Sonnet 5 | in view | the server of 2026-09-26 | 52 | 1 | 55 | 2 | 0 | 0 | USD 3.78 | 458 | arm-verdict-mcp-1 |
| server only, four tools | Sonnet 5 | in view | the server of 2026-09-26 | 53 | 2 | 55 | 0 | 0 | 0 | USD 3.84 | 461 | arm-verdict-mcp-2 |
| shell only | Sonnet 5 | none | the server of 2026-09-29 | 47 | 4 | 55 | 1 | 0 | 3 | USD 3.42 | 345 | suite219-sonnet5-shell-1 |
| server only | Sonnet 5 | none | the server of 2026-09-29 | 44 | 9 | 55 | 2 | 0 | 0 | USD 3.99 | 347 | suite219-sonnet5-mcp-1 |
| server and shell | Sonnet 5 | none | the server of 2026-09-29 | 48 | 5 | 55 | 2 | 0 | 0 | USD 4.13 | 353 | suite219-sonnet5-both-1 |
| server only | Opus 5 | none | the server of 2026-09-29 | 55 | 0 | 55 | 0 | 0 | 0 | USD 7.07 | 323 | suite219-opus5-mcp-1 |
- Sonnet 5, hints in view. The shell alone solved 50, inside the 45 to 53 of 55 the server arms solved: on these obligations the tools changed how the model worked, not what it solved.
- Sonnet 5, header-free. The shell alone solved 47, inside the 44 to 48 of 55 the server arms solved: on these obligations the tools changed how the model worked, not what it solved.
- Opus 5, header-free. 55 of 55 solved with the server; no shell arm was run under this condition, so the comparison is not made here.
- Final files that type-check with their statement kept: 55 of 55 in 13 of the 14 arms, 54 in
arm184-mcp-1. The gates took preservation 18 (an import line edited rather than added, or the statement changed); holes 1 (a hole left in the file); isolation 12 (a command or a read outside the roots, or one the audit cannot see through) across the arms. - Every arm is one seed. Where a configuration ran twice (
arm-surface-mcp-1witharm-surface-mcp-2andarm-verdict-mcp-1witharm-verdict-mcp-2), its seeds differ by 1 to 3 solved, and a difference inside that spread is not a finding. - Offered the server and a shell together, the model took all 55 verdicts from
check_filein each of the 4 arms that had both (arm162-both-1,arm184-both-1,arm-surface-both-1,suite219-sonnet5-both-1), and never ranagdain the shell: the verdict is the tool it would not give up. - For Sonnet 5 a sweep with the server cost more: USD 3.55 to USD 5.28 across the 11 arms that had it, against USD 2.88 to USD 3.42 with the shell alone in
arm162-shell-1andsuite219-sonnet5-shell-1. agda-native-air’s reading, from these arms, is that the tools’ descriptions, re-read on every turn, were part of the difference, and that cutting them closed that part and not the whole; the rest is output and turns. - Waiting on Agda was a small part of a session: 8 % to 11 % of its wall clock at the median, at most 52 %. The rest is the model’s own turns, which is where a faster server cannot help and a smaller one might.
12 composition obligations from agda-algebras, each proved by stringing two to four library lemmas together, with no proof of the statement on disk, Opus 5, Sonnet 5, one row per model, instrument, fixture condition and server; every subject was capped at 60 turns, 1,800 s of wall clock and USD 6.00 of list price. Solved is the archive’s judge’s column. No statement of this tier has a proof on disk to restate, so the judge’s restated column is empty here and is not shown. “Checks, statement kept” counts the final files Agda accepted with the statement they were given, whatever the gates then said of the row; “fixture hints” says whether the obligation’s header still carried its hints when the run was made (agda-native-air issue 219).
| instrument | model | fixture hints | server | solved | checks, statement kept | lost to preservation | lost to isolation | cost, list price | turns | run |
|---|---|---|---|---|---|---|---|---|---|---|
| shell only | Opus 5 | none | the server of 2026-09-29 | 10 | 12 | 0 | 2 | USD 4.13 | 130 | comp-opus5-shell-1 |
| server only | Opus 5 | none | the server of 2026-09-29 | 12 | 12 | 0 | 0 | USD 4.04 | 128 | comp-opus5-mcp-1 |
| server and shell | Opus 5 | none | the server of 2026-09-29 | 10 | 12 | 0 | 2 | USD 4.87 | 148 | comp-opus5-both-1 |
| shell only | Sonnet 5 | none | the server of 2026-09-29 | 6 | 12 | 5 | 1 | USD 2.60 | 157 | comp-sonnet5-shell-1 |
| server only | Sonnet 5 | none | the server of 2026-09-29 | 3 | 12 | 9 | 0 | USD 3.55 | 206 | comp-sonnet5-mcp-1 |
| server and shell | Sonnet 5 | none | the server of 2026-09-29 | 5 | 12 | 5 | 2 | USD 3.39 | 179 | comp-sonnet5-both-1 |
- Every final file type-checks with its statement kept, 12 of 12 in each of the 6 arms. A row missing from a solved count lost a gate (preservation 19 (an import line edited rather than added, or the statement changed); isolation 7 (a command or a read outside the roots, or one the audit cannot see through) across the arms), never the proof: no row of this tier was proved in one arm and not in another.
- Opus 5, header-free. Shell 10, server 12, both 10 of 12 solved, and 12 of 12 final files check in each arm: the counts differ by the gates alone.
- Sonnet 5, header-free. Shell 6, server 3, both 5 of 12 solved, and 12 of 12 final files check in each arm: the counts differ by the gates alone.
- Every arm is one seed, and on 12 rows a difference of a row or two is not a finding.
- Offered the server and a shell together, the model took 11 of 12 verdicts from
check_fileand 1 fromagdain the shell incomp-opus5-both-1, runningagdathere 1 time; 12 of 12 verdicts fromcheck_fileincomp-sonnet5-both-1, runningagdathere 1 time. - For Opus 5 the server did not decide what a sweep cost: USD 4.04 to USD 4.87 across the 2 arms that had it, against USD 4.13 with the shell alone in
comp-opus5-shell-1. - For Sonnet 5 a sweep with the server cost more: USD 3.39 to USD 3.55 across the 2 arms that had it, against USD 2.60 with the shell alone in
comp-sonnet5-shell-1. - Waiting on Agda was a small part of a session: 13 % to 25 % of its wall clock at the median, at most 36 %. The rest is the model’s own turns, which is where a faster server cannot help and a smaller one might.
14 obligations posed for the purpose, in group theory and universal algebra, with no proof on disk, Opus 5, one row per model, instrument, fixture condition and server; every subject was capped at 60 turns, 1,800 s of wall clock and USD 6.00 of list price. Solved is the archive’s judge’s column. No statement of this tier has a proof on disk to restate, so the judge’s restated column is empty here and is not shown. “Checks, statement kept” counts the final files Agda accepted with the statement they were given, whatever the gates then said of the row; “fixture hints” says whether the obligation’s header still carried its hints when the run was made (agda-native-air issue 219).
| instrument | model | fixture hints | server | solved | checks, statement kept | lost to preservation | lost to isolation | cost, list price | turns | run |
|---|---|---|---|---|---|---|---|---|---|---|
| shell only | Opus 5 | in view | the server of 2026-09-27 | 8 | 14 | 1 | 5 | USD 6.41 | 174 | hard-opus5-shell-1 |
| server only | Opus 5 | in view | the server of 2026-09-27 | 14 | 14 | 0 | 0 | USD 7.82 | 143 | hard-opus5-mcp-1 |
| server and shell | Opus 5 | in view | the server of 2026-09-27 | 8 | 14 | 0 | 6 | USD 6.62 | 162 | hard-opus5-both-1 |
| shell only | Opus 5 | none | the server of 2026-09-29 | 10 | 14 | 0 | 4 | USD 6.31 | 148 | hard219-opus5-shell-1 |
| server only | Opus 5 | none | the server of 2026-09-29 | 14 | 14 | 0 | 0 | USD 7.71 | 139 | hard219-opus5-mcp-1 |
| server and shell | Opus 5 | none | the server of 2026-09-29 | 10 | 14 | 0 | 4 | USD 6.56 | 174 | hard219-opus5-both-1 |
- Every final file type-checks with its statement kept, 14 of 14 in each of the 6 arms. A row missing from a solved count lost a gate (preservation 1 (an import line edited rather than added, or the statement changed); isolation 19 (a command or a read outside the roots, or one the audit cannot see through) across the arms), never the proof: no row of this tier was proved in one arm and not in another.
- Opus 5, hints in view. Shell 8, server 14, both 8 of 14 solved, and 14 of 14 final files check in each arm: the counts differ by the gates alone.
- Opus 5, header-free. Shell 10, server 14, both 10 of 14 solved, and 14 of 14 final files check in each arm: the counts differ by the gates alone.
- Every arm is one seed, and on 14 rows a difference of a row or two is not a finding.
- Offered the server and a shell together, the model took 10 of 14 verdicts from
check_fileand 4 fromagdain the shell inhard-opus5-both-1, runningagdathere 4 times; 9 of 14 verdicts fromcheck_fileand 5 fromagdain the shell inhard219-opus5-both-1, runningagdathere 5 times. - For Opus 5 a sweep with the server cost more: USD 6.56 to USD 7.82 across the 4 arms that had it, against USD 6.31 to USD 6.41 with the shell alone in
hard-opus5-shell-1andhard219-opus5-shell-1. - Waiting on Agda was a small part of a session: 3 % to 8 % of its wall clock at the median, at most 34 %. The rest is the model’s own turns, which is where a faster server cannot help and a smaller one might.
What this does not show¶
- That the server makes the model smarter. On every tier here the final files check about the same whichever instrument the model had, and the tables above are the whole of the evidence. The page is about what the server makes cheap, what it keeps expensive and why, and what a model does with the difference.
- That the model found the proof unaided. In the three sessions with
the hint in view, the header named the proof's shape, and the library's
own proof of this lemma came into view before the model's edit, which the
judge records beside each verdict. In the three without it, the model
found the library's lemma and cited it. The sessions show how each
instrument is used, not a proof found from nothing. Opus 5's header-free
session on this row, linked above, wrote the proof out after
fill_holehad accepted the citation, with the library's proof in view. - A comparison with no type-checker at all. That would measure the checker, which every reader here already grants, and not the server.
- What the fixture headers were worth beyond this row. Every run
archived before 2026-09-29 had its obligation's header in view, with a
Strategy:line sketching the proof on every row and, on the mined tiers, aSource:line naming the module that holds the original; three archived subjects quote such a line in their own words. The headers were stripped, and the four arms behind agda-native-air's own figures were run again; the corrected numbers are in the first table above, under "fixture hints", and the comparison, with what a difference of a few rows can mean when every arm is one seed, is agda-native-air's reading guide, § 4.6. - That a tier with no proof on disk separates the instruments. Two were built for the question, and both sit at the models' ceiling: every final file type-checks with its statement kept, for both models in every arm, and the solved counts differ only by the gates. The library lemmas the composition tier's proofs needed were found in tool answers and source reads, not recalled from memory, with three exceptions the guide names, and the two models took different routes to the same files, which the guide's § 4.7 records per row. What remains open is a tier that can tell the instruments apart, which needs relations a proof cannot unfold, longer chains, or weaker subjects; it will be reported here the way these were, with its run ids, whichever way it goes.
Go deeper¶
- agda-native-air's demo
replays five sessions from the same archive on other rows, both models
among them, and carries the full benchmark table; the archive itself is
reports/agent-bench/, with every transcript, verdict and final file this page was built from. - ADR 0002 is the server's design record, with the field measurements the third clock quotes; ADR 0001 § 9 is the benchmark's, and reading the results says for every number in it what it is a win or a loss for.
- The agent that never called the server is the story of why the verdict travels with the command it is equivalent to, and the typechecker as a reward signal is what the exit code is worth to a loop that trains or evaluates a model.
- The field reports are the sessions in real developments, written up in both directions, where the whole-library check is the gate and the server's verdict is the instrument.