agda-native-air: an agent, Agda, and thirteen tools

What this is

agda-native-air gives a coding agent a way to ask Agda questions. Thirteen tools over MCP load a file, report a hole’s goal and context, infer the type of an expression, resolve a name, list a module’s exports, search a corpus, probe a candidate term into a hole, and type-check the result. To find out whether that helps, 55 proof obligations were each handed to one fresh, non-interactive Claude Code session with those thirteen tools, Read and Edit on a single staged file, and nothing else: no shell, no settings, no project instructions, no memory. Each session ran under a 30-turn cap, a 900-second cap, and a USD 3.00 cap.

Every fact the judge uses about the Agda in the file a session left behind is Agda’s own answer, asked through the same server and the same agda invocation the benchmark’s gold solutions are verified with: the elaborated type of the definition, the references in its body, the safe-flag refusals, the hole list, the exit code. One gate is textual and stays textual, because it asks about the file rather than about the Agda in it: whether the module line and every original import line are still there, read as a line diff with comments stripped on both sides. The marked listing at the foot of each session below is the page’s own diff of the obligation against the final file, for the reader; the judge never saw it.

The five sessions on this page are replayed out of the transcripts committed in this repository, under reports/agent-bench/. The replay types what the model typed and shows what the server answered; it invents nothing, and the page is complete and readable with JavaScript switched off.

The sessions

Five of the 55, chosen because each is one half of a contrast the archive already holds. A call types the way the model typed it; the answer under it appears whole, because the server answers in one response. Every answer is here in full, behind the control that says how long it is, so nothing is abbreviated in a way that could hide a failure.

`kercon′` Opus 5 solved

Four `type_of` queries, one `fill_hole` probe, one edit, one check. The congruence is assembled from the pieces the fixture's own `using` lists name, under a `where` block.

obligation: algebras-kernels-ker-con
tier: non-obvious
stratum: agda-algebras/using
the library original: Setoid.Homomorphisms.Kernels.kercon
model: claude-opus-5
turns: 10
tool calls: 9
wall: 67 s
cost, list price: USD 0.35

the hole’s goal, as get_goal reported it

{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩) →  Con 𝑨 ρᵇ

the session, 10 turns and 9 tool calls

the model
I'll start by reading the file.

the work directory

Read()

answered

1 -- Kernels-ker-con.agda

the whole answer, as the agent saw it (1,052 characters of text)

1	-- Kernels-ker-con.agda
2	--
3	-- File: data/benchmarks/agda-algebras-v0/obligations/Kernels-ker-con.agda
4	--
5	-- Benchmark obligation: algebras-kernels-ker-con
6	-- Difficulty: non-obvious
7	-- Source: Setoid.Homomorphisms.Kernels (agda-algebras)
8	-- Import stratum: using
9	-- Strategy: record-assembly
10	--
11	module Kernels-ker-con where
12	
13	open import AgdaDojang.Debug
14	
15	open import Agda.Primitive   using ( Level )
16	open import Data.Product     using ( _,_ ; proj₁ )
17	open import Relation.Binary  using ( Setoid )
18	open import Function.Bundles using ( Func )
19	
20	open import Overture              using ( Signature ; kerRel ; kerRelOfEquiv )
21	open import Setoid.Algebras       using ( Algebra ; 𝔻[_] )
22	open import Setoid.Congruences    using ( Con ; mkcon )
23	open import Setoid.Homomorphisms  using ( hom ; HomKerComp )
24	
25	kercon′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
26	          {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
27	          (h : hom 𝑨 𝑩)
28	  →  Con 𝑨 ρᵇ
29	kercon′ {𝑩 = 𝑩} h = {!!}
30

thought
The model thought here, about 250 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

get_goal(line: 29, column: 21)

one of 2 calls the model issued in this turn

answered

goal: Data.Product.Σ (Setoid.Carrier (Algebra.Domain 𝑨) → Setoid.Carrier (Algebra.Domain 𝑨) → Set ρᵇ) (Setoid.Congruences.IsCongruence 𝑨)elapsedMs: 5126context: 10

the whole answer, as the agent saw it (2,054 characters of JSON)

{"checkedFromSource":true,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"context":[{"name":"𝓞","type":"Level"},{"name":"𝓥","type":"Level"},{"name":"α","type":"Level"},{"name":"ρᵃ","type":"Level"},{"name":"β","type":"Level"},{"name":"ρᵇ","type":"Level"},{"name":"𝑆","type":"Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)"},{"name":"𝑨","type":"Algebra α ρᵃ"},{"name":"𝑩","type":"Algebra β ρᵇ"},{"name":"h","type":"Data.Product.Σ (Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑩))\n(Setoid.Homomorphisms.IsHom 𝑨 𝑩)"}],"elapsedMs":5126,"goal":"Data.Product.Σ\n(Setoid.Carrier (Algebra.Domain 𝑨) →\n Setoid.Carrier (Algebra.Domain 𝑨) → Set ρᵇ)\n(Setoid.Congruences.IsCongruence 𝑨)","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>\" None Direct (Cmd_show_version)","IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_load \"<work>/Kernels-ker-con.agda\" [\"-i\",\"agda-dojang/agda\",\"--library-file=agda/libraries\",\"-l\",\"agda-dojang\",\"-l\",\"standard-library\",\"-l\",\"agda-algebras\",\"-i\",\"<work>\"])","IOTCM \"<work>\" None Direct (Cmd_show_version)","IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_goal_type_context Normalised 0 noRange \"\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"first","loadElapsedMs":5087,"pid":2335111,"root":"<work>","spawned":true},"module":"Kernels-ker-con","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"]},"source":"interaction-lane"}

agda-mcp

type_of(expr: mkcon)

one of 2 calls the model issued in this turn

answered

type: (reflexive : {a₀ a₁ : Setoid.Carrier (Algebra.Domain _𝑨_31)} → (Algebra.Domain _𝑨_31 Setoid.≈ a₀) a₁ → _θ_33 a₀ a₁) (is-equivalence : Relation.Binary.IsEquivalence _θ_33) (is-compatible : (𝑓 : _𝑆_30 .proj₁) {x y …elapsedMs: 6

the whole answer, as the agent saw it (1,584 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":6,"expr":"mkcon","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"mkcon\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2335111,"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":"(reflexive\n : {a₀ a₁ : Setoid.Carrier (Algebra.Domain _𝑨_31)} →\n   (Algebra.Domain _𝑨_31 Setoid.≈ a₀) a₁ → _θ_33 a₀ a₁)\n(is-equivalence : Relation.Binary.IsEquivalence _θ_33)\n(is-compatible\n : (𝑓 : _𝑆_30 .proj₁)\n   {x y\n    : Overture.proj₂ _𝑆_30 𝑓 → Setoid.Carrier (Algebra.Domain _𝑨_31)} →\n   ((i : Overture.proj₂ _𝑆_30 𝑓) → _θ_33 (x i) (y i)) →\n   _θ_33 (Func.to (Algebra.Interp _𝑨_31) (𝑓 , x))\n   (Func.to (Algebra.Interp _𝑨_31) (𝑓 , y))) →\nSetoid.Congruences.IsCongruence _𝑨_31 _θ_33"}

thought
The model thought here, about 155 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

type_of(expr: kerRel)

answered

type: {a b : Level} {A : Set a} {B : Set b} {ρ : Level} → (B → B → Set ρ) → (A → B) → A → A → Set ρelapsedMs: 5

the whole answer, as the agent saw it (1,181 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":5,"expr":"kerRel","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"kerRel\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2335111,"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":"{a b : Level} {A : Set a} {B : Set b} {ρ : Level} →\n(B → B → Set ρ) → (A → B) → A → A → Set ρ"}

agda-mcp

type_of(expr: kerRelOfEquiv)

answered

type: {a b : Level} {A : Set a} {B : Set b} {ρ : Level} {R : B → B → Set ρ} → Relation.Binary.IsEquivalence R → (h : A → B) → Relation.Binary.IsEquivalence (λ x y → R (h x) (h y))elapsedMs: 3

the whole answer, as the agent saw it (1,277 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":3,"expr":"kerRelOfEquiv","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"kerRelOfEquiv\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2335111,"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":"{a b : Level} {A : Set a} {B : Set b} {ρ : Level}\n{R : B → B → Set ρ} →\nRelation.Binary.IsEquivalence R →\n(h : A → B) → Relation.Binary.IsEquivalence (λ x y → R (h x) (h y))"}

agda-mcp

type_of(expr: HomKerComp)

answered

type: {𝓞 = a : Level} {𝓥 = b : Level} {α = A : Level} {ρᵃ = B : Level} {β = ρ : Level} {ρᵇ = R : Level} {𝑆 = x : Data.Product.Σ (Set a) (λ h → h → Set b)} {𝑨 = y : Algebra A B} {𝑩 : Algebra ρ R} (.patternInTele0 : Data.Produc…elapsedMs: 5

the whole answer, as the agent saw it (1,884 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":5,"expr":"HomKerComp","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"HomKerComp\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2335111,"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 = a : Level} {𝓥 = b : Level} {α = A : Level} {ρᵃ = B : Level}\n{β = ρ : Level} {ρᵇ = R : Level}\n{𝑆 = x : Data.Product.Σ (Set a) (λ h → h → Set b)}\n{𝑨 = y : Algebra A B} {𝑩 : Algebra ρ R}\n(.patternInTele0\n : Data.Product.Σ (Func (Algebra.Domain y) (Algebra.Domain 𝑩))\n   (Setoid.Homomorphisms.IsHom y 𝑩))\n(𝑓 : x .proj₁)\n{x = x₁ : Overture.proj₂ x 𝑓 → Setoid.Carrier (Algebra.Domain y)}\n{y = y₁ : Overture.proj₂ x 𝑓 → Setoid.Carrier (Algebra.Domain y)} →\n((i : Overture.proj₂ x 𝑓) →\n (Algebra.Domain 𝑩 Setoid.≈ Func.to (.patternInTele0 .proj₁) (x₁ i))\n (Func.to (.patternInTele0 .proj₁) (y₁ i))) →\n(Algebra.Domain 𝑩 Setoid.≈\n Func.to (.patternInTele0 .proj₁)\n (Func.to (Algebra.Interp y) (𝑓 , x₁)))\n(Func.to (.patternInTele0 .proj₁)\n (Func.to (Algebra.Interp y) (𝑓 , y₁)))"}

thought
The model thought here, about 1,100 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

fill_hole(line: 29, column: 21, candidate: kerRel (Setoid._≈_ 𝔻[ 𝑩 ]) (Func.to (proj₁ h)) , mkcon (Func.cong (proj₁ h)) (kerRelOfEquiv (Setoid.isEquivalence 𝔻[ 𝑩 ]) (Func.to (proj₁ h))) (HomKerComp h))

answered

status: okremainingHoles: 0verdict.exitCode: 0elapsedMs: 4954holes: 0

the whole answer, as the agent saw it (2,153 characters of JSON)

{"candidate":"kerRel (Setoid._≈_ 𝔻[ 𝑩 ]) (Func.to (proj₁ h)) , mkcon (Func.cong (proj₁ h)) (kerRelOfEquiv (Setoid.isEquivalence 𝔻[ 𝑩 ]) (Func.to (proj₁ h))) (HomKerComp h)","checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","-i","<work>","<work>/Kernels-ker-con.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"elapsedMs":4954,"holes":[],"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"]},"remainingHoles":0,"status":"ok","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 -i <work> <work>/Kernels-ker-con.agda, run against that file with the hole at line 29, column 21 (index 0) replaced by the candidate (the file on disk is restored afterwards, byte for byte)","exitCode":0,"meaning":"status is \"ok\" if and only if that command exited 0, or failed with nothing but [UnsolvedInteractionMetas] (holes still open in the file), including any new sub-hole the candidate itself introduced, which is a successful refinement. Every other failure is \"type_error\", including [UnsolvedMetaVariables] and [UnsolvedConstraints]: a candidate that leaves a meta unsolved does not pass the build and is not ok here either. A run killed by --timeout is \"timeout\" (the candidate was never judged) and an agda binary that could not be started at all is \"crash\"."}}

the model
The candidate typechecks. Now write it into the file.
the work directory
Edit(replace_all: false, old_string: kercon′ {𝑩 = 𝑩} h = {!!}, new_string: kercon′ {𝑩 = 𝑩} h = kerRel _≈_ ∣h∣ , mkcon (Func.cong (proj₁ h)) (kerRelOfEquiv isEquivalence ∣h∣) (HomKerComp h) where open Setoid 𝔻[ 𝑩 ] using ( _≈_ ; isEqu…)
answered
The file <work>/Kernels-ker-con.agda has been updated successfully. (file state is current in your context — no need to…
the whole answer, as the agent saw it (133 characters of text)
The file <work>/Kernels-ker-con.agda has been updated successfully. (file state is current in your context — no need to Read it back)

agda-mcp

check_file()

answered

success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5096diagnostics: 0holes: 0

the whole answer, as the agent saw it (1,782 characters of JSON)

{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","-i","<work>","<work>/Kernels-ker-con.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5096,"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 -i <work> <work>/Kernels-ker-con.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 model
check_file reports success (exit code 0, no diagnostics, no holes) on the final file, where the hole is filled with `kerRel _≈_ ∣h∣ , mkcon (Func.cong (proj₁ h)) (kerRelOfEquiv isEquivalence ∣h∣) (HomKerComp h)` plus a `where` block opening `Setoid 𝔻[ 𝑩 ]`.

The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the server’s own, in full. The client billed 1,768 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 0

statement preserved yes

turns 10

the file the judge read, marked against the obligation it started as

 -- Kernels-ker-con.agda
 --
 -- File: data/benchmarks/agda-algebras-v0/obligations/Kernels-ker-con.agda
 --
 -- Benchmark obligation: algebras-kernels-ker-con
 -- Difficulty: non-obvious
 -- Source: Setoid.Homomorphisms.Kernels (agda-algebras)
 -- Import stratum: using
 -- Strategy: record-assembly
 --
 module Kernels-ker-con where
 
 open import AgdaDojang.Debug
 
 open import Agda.Primitive   using ( Level )
 open import Data.Product     using ( _,_ ; proj₁ )
 open import Relation.Binary  using ( Setoid )
 open import Function.Bundles using ( Func )
 
 open import Overture              using ( Signature ; kerRel ; kerRelOfEquiv )
 open import Setoid.Algebras       using ( Algebra ; 𝔻[_] )
 open import Setoid.Congruences    using ( Con ; mkcon )
 open import Setoid.Homomorphisms  using ( hom ; HomKerComp )
 
 kercon′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
           {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
           (h : hom 𝑨 𝑩)
   →  Con 𝑨 ρᵇ
-kercon′ {𝑩 = 𝑩} h = {!!}
+kercon′ {𝑩 = 𝑩} h = kerRel _≈_ ∣h∣ , mkcon (Func.cong (proj₁ h)) (kerRelOfEquiv isEquivalence ∣h∣) (HomKerComp h)
+ where
+ open Setoid 𝔻[ 𝑩 ] using ( _≈_ ; isEquivalence )
+ ∣h∣ = Func.to (proj₁ h)

reports/agent-bench/agent-opus5-1/subjects/algebras-kernels-ker-con/final/Kernels-ker-con.agda

`kercon′` Sonnet 5 restated

The same obligation and the same tools. This session finds the library's own lemma for the statement, imports it, and calls it. The file type-checks; the judge counts it restated, not solved.

obligation: algebras-kernels-ker-con
tier: non-obvious
stratum: agda-algebras/using
the library original: Setoid.Homomorphisms.Kernels.kercon
model: claude-sonnet-5
turns: 16
tool calls: 15
wall: 57 s
cost, list price: USD 0.28

the hole’s goal, as get_goal reported it

{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩) →  Con 𝑨 ρᵇ

the session, 16 turns and 15 tool calls

thought
The model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

the work directory

Read()

answered

1 -- Kernels-ker-con.agda

the whole answer, as the agent saw it (1,052 characters of text)

1	-- Kernels-ker-con.agda
2	--
3	-- File: data/benchmarks/agda-algebras-v0/obligations/Kernels-ker-con.agda
4	--
5	-- Benchmark obligation: algebras-kernels-ker-con
6	-- Difficulty: non-obvious
7	-- Source: Setoid.Homomorphisms.Kernels (agda-algebras)
8	-- Import stratum: using
9	-- Strategy: record-assembly
10	--
11	module Kernels-ker-con where
12	
13	open import AgdaDojang.Debug
14	
15	open import Agda.Primitive   using ( Level )
16	open import Data.Product     using ( _,_ ; proj₁ )
17	open import Relation.Binary  using ( Setoid )
18	open import Function.Bundles using ( Func )
19	
20	open import Overture              using ( Signature ; kerRel ; kerRelOfEquiv )
21	open import Setoid.Algebras       using ( Algebra ; 𝔻[_] )
22	open import Setoid.Congruences    using ( Con ; mkcon )
23	open import Setoid.Homomorphisms  using ( hom ; HomKerComp )
24	
25	kercon′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
26	          {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
27	          (h : hom 𝑨 𝑩)
28	  →  Con 𝑨 ρᵇ
29	kercon′ {𝑩 = 𝑩} h = {!!}
30

thought
The model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

type_of(expr: Con)

one of 5 calls the model issued in this turn

answered

type: {𝓞 𝓥 α ρ : Level} {𝑆 : Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)} → Algebra α ρ → (ℓ : Level) → Set (𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρ Agda.Primitive.⊔ Agda.Primitive.lsuc ℓ)elapsedMs: 5396

the whole answer, as the agent saw it (1,659 characters of JSON)

{"checkedFromSource":true,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":5396,"expr":"Con","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>\" None Direct (Cmd_show_version)","IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_load \"<work>/Kernels-ker-con.agda\" [\"-i\",\"agda-dojang/agda\",\"--library-file=agda/libraries\",\"-l\",\"agda-dojang\",\"-l\",\"standard-library\",\"-l\",\"agda-algebras\",\"-i\",\"<work>\"])","IOTCM \"<work>\" None Direct (Cmd_show_version)","IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"Con\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"first","loadElapsedMs":5357,"pid":2311594,"root":"<work>","spawned":true},"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 ρ : Level} {𝑆 : Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)} →\nAlgebra α ρ →\n(ℓ : Level) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρ\n Agda.Primitive.⊔ Agda.Primitive.lsuc ℓ)"}

agda-mcp

type_of(expr: mkcon)

one of 5 calls the model issued in this turn

answered

the whole answer, as the agent saw it (1,584 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":7,"expr":"mkcon","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"mkcon\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"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":"(reflexive\n : {a₀ a₁ : Setoid.Carrier (Algebra.Domain _𝑨_31)} →\n   (Algebra.Domain _𝑨_31 Setoid.≈ a₀) a₁ → _θ_33 a₀ a₁)\n(is-equivalence : Relation.Binary.IsEquivalence _θ_33)\n(is-compatible\n : (𝑓 : _𝑆_30 .proj₁)\n   {x y\n    : Overture.proj₂ _𝑆_30 𝑓 → Setoid.Carrier (Algebra.Domain _𝑨_31)} →\n   ((i : Overture.proj₂ _𝑆_30 𝑓) → _θ_33 (x i) (y i)) →\n   _θ_33 (Func.to (Algebra.Interp _𝑨_31) (𝑓 , x))\n   (Func.to (Algebra.Interp _𝑨_31) (𝑓 , y))) →\nSetoid.Congruences.IsCongruence _𝑨_31 _θ_33"}

agda-mcp

type_of(expr: kerRel)

one of 5 calls the model issued in this turn

answered

type: {a = 𝓞 : Level} {b = 𝓥 : Level} {A = α : Set 𝓞} {B = ρ : Set 𝓥} {ρ = F : Level} → (ρ → ρ → Set F) → (α → ρ) → α → α → Set FelapsedMs: 3

the whole answer, as the agent saw it (1,212 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":3,"expr":"kerRel","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"kerRel\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"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":"{a = 𝓞 : Level} {b = 𝓥 : Level} {A = α : Set 𝓞} {B = ρ : Set 𝓥}\n{ρ = F : Level} →\n(ρ → ρ → Set F) → (α → ρ) → α → α → Set F"}

agda-mcp

type_of(expr: kerRelOfEquiv)

one of 5 calls the model issued in this turn

answered

type: {a = 𝓞 : Level} {b = 𝓥 : Level} {A = α : Set 𝓞} {B = ρ : Set 𝓥} {ρ = F : Level} {R = 𝑆 : ρ → ρ → Set F} → Relation.Binary.IsEquivalence 𝑆 → (ℓ : α → ρ) → Relation.Binary.IsEquivalence (λ x y → 𝑆 (ℓ x) (ℓ y))elapsedMs: 3

the whole answer, as the agent saw it (1,311 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":3,"expr":"kerRelOfEquiv","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"kerRelOfEquiv\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"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":"{a = 𝓞 : Level} {b = 𝓥 : Level} {A = α : Set 𝓞} {B = ρ : Set 𝓥}\n{ρ = F : Level} {R = 𝑆 : ρ → ρ → Set F} →\nRelation.Binary.IsEquivalence 𝑆 →\n(ℓ : α → ρ) → Relation.Binary.IsEquivalence (λ x y → 𝑆 (ℓ x) (ℓ y))"}

agda-mcp

type_of(expr: hom)

one of 5 calls the model issued in this turn

answered

type: {𝓞 𝓥 α : Level} {ρᵃ = ρ : Level} {β = F : Level} {ρᵇ = 𝑆 : Level} {𝑆 = x : Data.Product.Σ (Set 𝓞) (λ ℓ → ℓ → Set 𝓥)} (y : Algebra α ρ) → Algebra F 𝑆 → Set (𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρ Agda…elapsedMs: 3

the whole answer, as the agent saw it (1,343 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":3,"expr":"hom","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"hom\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"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 : Level} {ρᵃ = ρ : Level} {β = F : Level} {ρᵇ = 𝑆 : Level}\n{𝑆 = x : Data.Product.Σ (Set 𝓞) (λ ℓ → ℓ → Set 𝓥)}\n(y : Algebra α ρ) →\nAlgebra F 𝑆 →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρ\n Agda.Primitive.⊔ F\n Agda.Primitive.⊔ 𝑆)"}

agda-mcp

type_of(expr: HomKerComp)

answered

type: {𝓞 𝓥 α : Level} {ρᵃ = ρ : Level} {β = F : Level} {ρᵇ = 𝑆 : Level} {𝑆 = x : Data.Product.Σ (Set 𝓞) (λ ℓ → ℓ → Set 𝓥)} {𝑨 = y : Algebra α ρ} {𝑩 : Algebra F 𝑆} (.patternInTele0 : Data.Product.Σ (Func (Algebra.Domain y) (Al…elapsedMs: 5

the whole answer, as the agent saw it (1,851 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":5,"expr":"HomKerComp","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_infer_toplevel Normalised \"HomKerComp\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"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 : Level} {ρᵃ = ρ : Level} {β = F : Level} {ρᵇ = 𝑆 : Level}\n{𝑆 = x : Data.Product.Σ (Set 𝓞) (λ ℓ → ℓ → Set 𝓥)}\n{𝑨 = y : Algebra α ρ} {𝑩 : Algebra F 𝑆}\n(.patternInTele0\n : Data.Product.Σ (Func (Algebra.Domain y) (Algebra.Domain 𝑩))\n   (Setoid.Homomorphisms.IsHom y 𝑩))\n(𝑓 : x .proj₁)\n{x = x₁ : Overture.proj₂ x 𝑓 → Setoid.Carrier (Algebra.Domain y)}\n{y = y₁ : Overture.proj₂ x 𝑓 → Setoid.Carrier (Algebra.Domain y)} →\n((i : Overture.proj₂ x 𝑓) →\n (Algebra.Domain 𝑩 Setoid.≈ Func.to (.patternInTele0 .proj₁) (x₁ i))\n (Func.to (.patternInTele0 .proj₁) (y₁ i))) →\n(Algebra.Domain 𝑩 Setoid.≈\n Func.to (.patternInTele0 .proj₁)\n (Func.to (Algebra.Interp y) (𝑓 , x₁)))\n(Func.to (.patternInTele0 .proj₁)\n (Func.to (Algebra.Interp y) (𝑓 , y₁)))"}

thought
The model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

exports_of(module: Setoid.Congruences)

answered

Error: result (83,785 characters across 1 line) exceeds maximum allowed tokens. Output has been saved to ~/.claude/proje…

the whole answer, as the agent saw it (1,428 characters of text)

Error: result (83,785 characters across 1 line) exceeds maximum allowed tokens. Output has been saved to ~/.claude/projects/<work>-slug/2eaa267d-7543-441f-9c9c-b34136d7917c/tool-results/mcp-agda-exports_of-1789490690981.txt.
Format: Plain text
Use offset and limit parameters to read specific portions of the file, search within it for specific content, and jq to make structured queries.
REQUIREMENTS FOR SUMMARIZATION/ANALYSIS/REVIEW:
- You MUST read the content from the file at ~/.claude/projects/<work>-slug/2eaa267d-7543-441f-9c9c-b34136d7917c/tool-results/mcp-agda-exports_of-1789490690981.txt in sequential chunks until 100% of the content has been read.
- Note: this file's lines are too long for Read's offset/limit chunking. If a shell tool is available, slice by character range (e.g. python read()[A:B], dd, or cut -c) instead.
- If you receive truncation warnings when reading the file, reduce the chunk size until you have read 100% of the content without truncation.
- Before producing ANY summary or analysis, you MUST explicitly describe what portion of the content you have read. ***If you did not read the entire content, you MUST explicitly state this.***
- If after a few attempts you cannot read the file (file not found, lines too long for Read's offset/limit, no shell access), STOP retrying. Summarize what you were able to read, explicitly state which portion you could not read and why, and proceed.

agda-mcp

exports_of(module: Setoid.Homomorphisms)

answered

elapsedMs: 64exports: 91modules: 4

the whole answer, as the agent saw it (24,615 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":64,"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) →\nSetoid.Functions.Injective.IsInjective (proj₁ h) →\nSetoid.Functions.Surjective.IsSurjective (proj₁ h) →\n𝑨 Setoid.Homomorphisms.≅ 𝑩"},{"name":"Cg⊆ker","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ ℓ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (h : hom 𝑨 𝑩)\n{R : Relation.Binary.Rel Setoid.Algebras.𝕌[ 𝑨 ] ℓ} →\nR Relation.Binary.⇒ proj₁ (Setoid.Homomorphisms.kercon h) →\nSetoid.Congruences.Gen R Relation.Binary.⇒\nproj₁ (Setoid.Homomorphisms.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 (Setoid.Homomorphisms.kerquo hh) 𝑩)\n(λ .patternInTele0 →\n   ((a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n    (𝔻[ 𝑩 ] Setoid.≈ Setoid.Homomorphisms.Noether.h hh a)\n    (Func.to (.patternInTele0 .proj₁)\n     (Func.to (Setoid.Homomorphisms.πker hh .proj₁) a)))\n   Data.Product.×\n   Setoid.Functions.Injective.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\n : hom (Setoid.Homomorphisms.kerquo hh) 𝑩} →\n((a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n (𝔻[ 𝑩 ] Setoid.≈ Setoid.Homomorphisms.Noether.h hh a)\n (Func.to (.patternInTele0 .proj₁)\n  (Func.to (Setoid.Homomorphisms.πker hh .proj₁) a))) →\n((a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n (𝔻[ 𝑩 ] Setoid.≈ Setoid.Homomorphisms.Noether.h hh a)\n (Func.to (.patternInTele1 .proj₁)\n  (Func.to (Setoid.Homomorphisms.πker hh .proj₁) a))) →\n([a]\n : Setoid.Carrier\n   (Algebra.Domain (Setoid.Homomorphisms.kerquo hh))) →\n(𝔻[ 𝑩 ] Setoid.≈ Func.to (.patternInTele0 .proj₁) [a])\n(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 r : Level} →\nhom (Setoid.Algebras.Lift-Alg 𝑨 ℓ 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 : Level} →\nhom (Setoid.Algebras.Lift-Algʳ 𝑨 ℓ) 𝑨"},{"name":"FromLiftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ : Level} →\nhom (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ) 𝑨"},{"name":"FromToLiftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ : Level}\n(a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n(𝔻[ 𝑨 ] 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 : Level}\n(a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n(𝔻[ 𝑨 ] 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 (Setoid._≈_ 𝔻[ 𝑪 ])\n(Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1)\nRelation.Unary.⊆\nOverture.kernelRel (Setoid._≈_ 𝔻[ 𝑩 ])\n(Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1) →\nSetoid.Functions.Surjective.IsSurjective (.patternInTele1 .proj₁) →\nSetoid.Algebras.Σ-syntax (hom 𝑪 𝑩)\n(λ .patternInTele2 →\n   (a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n   (𝔻[ 𝑩 ] Setoid.≈\n    Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1 a)\n   (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 (Setoid._≈_ 𝔻[ 𝑪 ])\n(Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1)\nRelation.Unary.⊆\nOverture.kernelRel (Setoid._≈_ 𝔻[ 𝑩 ])\n(Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1) →\nSetoid.Functions.Surjective.IsSurjective (.patternInTele1 .proj₁) →\nSetoid.Functions.Surjective.IsSurjective (.patternInTele0 .proj₁) →\nSetoid.Algebras.Σ-syntax (Setoid.Homomorphisms.epi 𝑪 𝑩)\n(λ .patternInTele2 →\n   (a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n   (𝔻[ 𝑩 ] Setoid.≈\n    Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1 a)\n   (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 Setoid.Homomorphisms.IsHomImageOf 𝑨' →\n𝑨 Setoid.Homomorphisms.≅ 𝑩 → 𝑩 Setoid.Homomorphisms.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 Setoid.Homomorphisms.IsHomImageOf 𝑨' →\n𝑨' Setoid.Homomorphisms.≅ 𝑩 → 𝑨 Setoid.Homomorphisms.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.∣≈\nkerRel (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 Setoid.Homomorphisms.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) (h : Func 𝔻[ 𝑨 ] 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) (h : Func 𝔻[ 𝑨 ] 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) (h : Func 𝔻[ 𝑨 ] 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 Setoid.Homomorphisms.IsHomImageOf 𝑨 →\nSetoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ Setoid.Homomorphisms.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 Setoid.Homomorphisms.≅ 𝑩 →\nSetoid.Algebras.Lift-Alg 𝑨 ℓᵃ rᵃ Setoid.Homomorphisms.≅\nSetoid.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 Setoid.Homomorphisms.≅ 𝑩 →\nSetoid.Algebras.Lift-Algʳ 𝑨 ℓᵃ Setoid.Homomorphisms.≅\nSetoid.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 Setoid.Homomorphisms.≅ 𝑩 →\nSetoid.Algebras.Lift-Algˡ 𝑨 ℓᵃ Setoid.Homomorphisms.≅\nSetoid.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 Setoid.Homomorphisms.≅ ℬ (Level.lift i)) →\nSetoid.Algebras.Lift-Algˡ (Setoid.Algebras.⨅ 𝒜) γ\nSetoid.Homomorphisms.≅ Setoid.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 Setoid.Homomorphisms.IsHomImageOf\n𝑩 →\n𝑨 Setoid.Homomorphisms.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 ρ : Level} →\nSetoid.Algebras.Lift-Alg 𝑨 ℓ ρ Setoid.Homomorphisms.≅\nSetoid.Algebras.Lift-Algʳ (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ) 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)\nSetoid.Homomorphisms.≅\nSetoid.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.Homomorphisms.≅\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.Homomorphisms.≅\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) →\nSetoid.Functions.Surjective.IsSurjective (proj₁ h) →\nSetoid.Functions.Surjective.IsSurjective\n(proj₁ (Setoid.Homomorphisms.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 → (ℓ r : Level) → hom (Setoid.Algebras.Lift-Alg 𝑨 ℓ 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 → (ℓ r : Level) → hom 𝑨 (Setoid.Algebras.Lift-Alg 𝑩 ℓ 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 ρ : Level} →\n𝑨 Setoid.Homomorphisms.≅ Setoid.Algebras.Lift-Alg 𝑨 ℓ 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 : Level} →\n𝑨 Setoid.Homomorphisms.≅ Setoid.Algebras.Lift-Algʳ 𝑨 ℓ"},{"name":"Lift-≅ˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ : Level} →\n𝑨 Setoid.Homomorphisms.≅ Setoid.Algebras.Lift-Algˡ 𝑨 ℓ"},{"name":"ToFromLift","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ r : Level}\n{b\n : Setoid.Carrier\n   (Algebra.Domain (Setoid.Algebras.Lift-Alg 𝑨 ℓ r))} →\n(𝔻[ Setoid.Algebras.Lift-Alg 𝑨 ℓ r ] Setoid.≈\n Func.to (Setoid.Homomorphisms.ToLift .proj₁)\n (Func.to (Setoid.Homomorphisms.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 : Level}\n(b\n : Setoid.Carrier\n   (Algebra.Domain (Setoid.Algebras.Lift-Algʳ 𝑨 ℓ))) →\n(𝔻[ Setoid.Algebras.Lift-Algʳ 𝑨 ℓ ] Setoid.≈ b) b"},{"name":"ToFromLiftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ : Level}\n(b\n : Setoid.Carrier\n   (Algebra.Domain (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ))) →\n(𝔻[ Setoid.Algebras.Lift-Algˡ 𝑨 ℓ ] Setoid.≈\n 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 r : Level} →\nhom 𝑨 (Setoid.Algebras.Lift-Alg 𝑨 ℓ 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 r : Level} →\nSetoid.Homomorphisms.epi 𝑨 (Setoid.Algebras.Lift-Alg 𝑨 ℓ r)"},{"name":"ToLiftʳ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ : Level} →\nhom 𝑨 (Setoid.Algebras.Lift-Algʳ 𝑨 ℓ)"},{"name":"ToLiftˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} {ℓ : Level} →\nhom 𝑨 (Setoid.Algebras.Lift-Algˡ 𝑨 ℓ)"},{"name":"compatible-map","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nFunc 𝔻[ 𝑨 ] 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) →\nFunc 𝔻[ 𝑨 ] 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) →\nSetoid.Homomorphisms.epi 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) →\nSetoid.Homomorphisms.epi 𝑨 𝑩 →\nSetoid.Algebras.Σ-syntax (hom 𝑨 𝑩)\n(λ h → Setoid.Functions.Surjective.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 : Level} {θ : Con 𝑨 ℓ}\n{x y : Setoid.Carrier (Algebra.Domain 𝑨)} →\nSetoid.Homomorphisms.kercon (Setoid.Homomorphisms.πhom h θ) .proj₁\nx y →\nθ .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) →\nCon 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(𝔻[ Setoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ ] Setoid.≈\n Level.lift (Func.to (h .proj₁) a))\n(Func.to (Setoid.Homomorphisms.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} {h : Func 𝔻[ 𝑨 ] 𝔻[ 𝑩 ]}\n(compatible : Setoid.Homomorphisms.compatible-map 𝑨 𝑩 h) →\nSetoid.Homomorphisms.IsHom 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) (h : Func 𝔻[ 𝑨 ] 𝔻[ 𝑩 ]) →\nSetoid.Homomorphisms.compatible-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 : Setoid.Carrier (Algebra.Domain 𝑩)) →\n   (𝔻[ 𝑩 ] Setoid.≈ Func.to (to .proj₁) (Func.to (from .proj₁) b)) b)\n(from∼to\n : (a : Setoid.Carrier (Algebra.Domain 𝑨)) →\n   (𝔻[ 𝑨 ] Setoid.≈ Func.to (from .proj₁) (Func.to (to .proj₁) a))\n   a) →\n𝑨 Setoid.Homomorphisms.≅ 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) →\nSetoid.Homomorphisms.mon 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) →\nSetoid.Homomorphisms.mon 𝑨 𝑩 →\nSetoid.Algebras.Σ-syntax (hom 𝑨 𝑩)\n(λ h → Setoid.Functions.Injective.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 : Level} (θ : Con 𝑨 ℓ) →\nSetoid.Homomorphisms.epi 𝑨 (𝑨 Setoid.Congruences.╱ 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 : Level} (θ : Con 𝑨 ℓ) → hom 𝑨 (𝑨 Setoid.Congruences.╱ 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) →\nSetoid.Homomorphisms.epi 𝑨 (Setoid.Homomorphisms.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 : Level} →\nSetoid.Algebras.Lift-Alg (Setoid.Algebras.⨅ 𝒜) ℓ ℓ\nSetoid.Homomorphisms.≅\nSetoid.Algebras.⨅\n(λ i → Setoid.Algebras.Lift-Alg (𝒜 (Level.Lift.lower i)) ℓ ℓ)"},{"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) → (𝔻[ 𝑨 ] Setoid.≈ u i) (v i)) →\n   (𝔻[ 𝑨 ] Setoid.≈ f o u) (f o v)) →\n((o : 𝑆 .proj₁)\n (a : Overture.ArityOf 𝑆 o → Setoid.Carrier (Algebra.Domain 𝑨)) →\n (𝔻[ 𝑨 ] Setoid.≈ (o Setoid.Algebras.^ 𝑨) a) (f o a)) →\n𝑨 Setoid.Homomorphisms.≅ Setoid.Algebras.mkAlgebra 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 Setoid.Homomorphisms.≅ 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 Setoid.Homomorphisms.≅ 𝑩 → 𝑩 Setoid.Homomorphisms.≅ 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 Setoid.Homomorphisms.≅ 𝑩 →\n𝑩 Setoid.Homomorphisms.≅ 𝑪 → 𝑨 Setoid.Homomorphisms.≅ 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}\n(φ : 𝑨 Setoid.Homomorphisms.≅ 𝑩) →\nSetoid.Functions.Injective.IsInjective\n(Setoid.Homomorphisms._≅_.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}\n(φ : 𝑨 Setoid.Homomorphisms.≅ 𝑩) →\nSetoid.Functions.Injective.IsInjective\n(proj₁ (Setoid.Homomorphisms._≅_.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.Homomorphisms.≅ 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.Homomorphisms.≅ 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} →\nSetoid.Homomorphisms.epi 𝑨 𝑩 →\nSetoid.Homomorphisms.epi 𝑩 𝑪 → Setoid.Homomorphisms.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 : Func 𝔻[ 𝑨 ] 𝔻[ 𝑩 ]} {h : Func 𝔻[ 𝑩 ] 𝔻[ 𝑪 ]} →\nSetoid.Homomorphisms.IsEpi 𝑨 𝑩 g →\nSetoid.Homomorphisms.IsEpi 𝑩 𝑪 h →\nSetoid.Homomorphisms.IsEpi 𝑨 𝑪 (h Setoid.Functions.Basic.⊙ 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 : Func 𝔻[ 𝑨 ] 𝔻[ 𝑩 ]} {h : Func 𝔻[ 𝑩 ] 𝔻[ 𝑪 ]} →\nSetoid.Homomorphisms.IsHom 𝑨 𝑩 g →\nSetoid.Homomorphisms.IsHom 𝑩 𝑪 h →\nSetoid.Homomorphisms.IsHom 𝑨 𝑪 (h Setoid.Functions.Basic.⊙ 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 Setoid.Homomorphisms.≅ ℬ i) →\nSetoid.Algebras.⨅ 𝒜 Setoid.Homomorphisms.≅ 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 ρ : Level} →\nSetoid.Algebras.⨅ 𝒜 Setoid.Homomorphisms.≅\nSetoid.Algebras.⨅\n(λ i → Setoid.Algebras.Lift-Alg (𝒜 (Level.Lift.lower i)) ℓ 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 : Level} →\nSetoid.Algebras.⨅ 𝒜 Setoid.Homomorphisms.≅\nSetoid.Algebras.⨅\n(λ i → Setoid.Algebras.Lift-Alg (𝒜 (Level.Lift.lower i)) ℓ ℓ)"},{"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 ρ : Level} →\nSetoid.Algebras.⨅ 𝒜 Setoid.Homomorphisms.≅\nSetoid.Algebras.⨅ (λ i → Setoid.Algebras.Lift-Alg (𝒜 i) ℓ 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>/Kernels-ker-con.agda\" None Direct (Cmd_show_module_contents_toplevel Simplified \"Setoid.Homomorphisms\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"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-mcp

definition_of(name: Con)

answered

elapsedMs: 1definitions: 1

the whole answer, as the agent saw it (1,274 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"definitions":[{"col":3,"endCol":6,"endLine":96,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Congruences/Basic.lagda.md","line":96,"qualified":"Setoid.Congruences.Basic._.Con"}],"elapsedMs":1,"lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_why_in_scope_toplevel \"Con\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"root":"<work>","spawned":false},"name":"Con","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","unlocated":[]}

agda-mcp

definition_of(name: hom)

answered

elapsedMs: 1definitions: 1

the whole answer, as the agent saw it (1,278 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"definitions":[{"col":3,"endCol":6,"endLine":98,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md","line":98,"qualified":"Setoid.Homomorphisms.Basic._.hom"}],"elapsedMs":1,"lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Kernels-ker-con.agda\" None Direct (Cmd_why_in_scope_toplevel \"hom\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2311594,"root":"<work>","spawned":false},"name":"hom","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","unlocated":[]}

thought
The model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.
the work directory
Read(offset: 70, limit: 40)
refused
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Congruences/Basic.lagda.md is outside <work>; --restricted confines t…
the whole answer, as the agent saw it (159 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Congruences/Basic.lagda.md is outside <work>; --restricted confines the file tools to the working directory.
the work directory
Read(offset: 80, limit: 160)
refused
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md is outside <work>; --restricted confines…
the whole answer, as the agent saw it (161 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Homomorphisms/Basic.lagda.md is outside <work>; --restricted confines the file tools to the working directory.
thought
The model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.
the work directory
Edit(replace_all: false, old_string: open import Setoid.Homomorphisms using ( hom ; HomKerComp ) kercon′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Al…, new_string: open import Setoid.Homomorphisms using ( hom ; HomKerComp ) open import Setoid.Homomorphisms using ( kercon ) kercon′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signatur…)
answered
The file <work>/Kernels-ker-con.agda has been updated successfully. (file state is current in your context — no need to…
the whole answer, as the agent saw it (133 characters of text)
The file <work>/Kernels-ker-con.agda has been updated successfully. (file state is current in your context — no need to Read it back)

agda-mcp

check_file()

answered

success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5185diagnostics: 0holes: 0

the whole answer, as the agent saw it (1,782 characters of JSON)

{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","-i","<work>","<work>/Kernels-ker-con.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5185,"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 -i <work> <work>/Kernels-ker-con.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 model
check_file reports success (exitCode 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 server’s own, in full. The client billed 1,846 thinking tokens over the session; their text is not in the archive.

restatedthe judge, in outcome.json

Every gate passed, including the type-check. The judge's restatement evidence is `ref Setoid.Homomorphisms.Kernels.kercon`: the definition refers to the library's own lemma for this statement, so the row goes in the restated column and never into the solve count.

Agda’s exit code 0

statement preserved yes

turns 16

import lines added open import Setoid.Homomorphisms using ( kercon )

restatement evidence ref Setoid.Homomorphisms.Kernels.kercon

the file the judge read, marked against the obligation it started as

 -- Kernels-ker-con.agda
 --
 -- File: data/benchmarks/agda-algebras-v0/obligations/Kernels-ker-con.agda
 --
 -- Benchmark obligation: algebras-kernels-ker-con
 -- Difficulty: non-obvious
 -- Source: Setoid.Homomorphisms.Kernels (agda-algebras)
 -- Import stratum: using
 -- Strategy: record-assembly
 --
 module Kernels-ker-con where
 
 open import AgdaDojang.Debug
 
 open import Agda.Primitive   using ( Level )
 open import Data.Product     using ( _,_ ; proj₁ )
 open import Relation.Binary  using ( Setoid )
 open import Function.Bundles using ( Func )
 
 open import Overture              using ( Signature ; kerRel ; kerRelOfEquiv )
 open import Setoid.Algebras       using ( Algebra ; 𝔻[_] )
 open import Setoid.Congruences    using ( Con ; mkcon )
 open import Setoid.Homomorphisms  using ( hom ; HomKerComp )
+open import Setoid.Homomorphisms  using ( kercon )
 
 kercon′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
           {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
           (h : hom 𝑨 𝑩)
   →  Con 𝑨 ρᵇ
-kercon′ {𝑩 = 𝑩} h = {!!}
+kercon′ {𝑩 = 𝑩} h = kercon h

reports/agent-bench/agent-sonnet5-1/subjects/algebras-kernels-ker-con/final/Kernels-ker-con.agda

`≤-trans-≅′` Opus 5 solved

A wholesale row, where the fixture names nothing useful. Two queries answer `NotInScope`, and the second one's error carries the name that ends up in the proof.

obligation: algebras-subalgebras-sub-trans-iso
tier: non-obvious
stratum: agda-algebras/wholesale
the library original: Setoid.Subalgebras.Properties.≤-trans-≅
model: claude-opus-5
turns: 17
tool calls: 16
wall: 84 s
cost, list price: USD 0.70

the hole’s goal, as get_goal reported it

{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

the session, 17 turns and 16 tool calls

the model
I'll start by reading the file.

the work directory

Read()

answered

1 -- Subalgebras-sub-trans-iso.agda

the whole answer, as the agent 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

thought
The model thought here, about 200 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

get_goal(line: 27, column: 21)

one of 2 calls the model issued in this turn

answered

goal: Data.Product.Σ (Data.Product.Σ (Function.Bundles.Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑪)) (IsHom 𝑨 𝑪)) (λ h → {x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)} → (Algebra.Domain 𝑪 Relation.Bina…elapsedMs: 5637context: 14

the whole answer, as the agent saw it (2,776 characters of JSON)

{"checkedFromSource":true,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"context":[{"name":"𝓞","type":"Level"},{"name":"𝓥","type":"Level"},{"name":"α","type":"Level"},{"name":"ρᵃ","type":"Level"},{"name":"β","type":"Level"},{"name":"ρᵇ","type":"Level"},{"name":"γ","type":"Level"},{"name":"ρᶜ","type":"Level"},{"name":"𝑆","type":"Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)"},{"name":"𝑨","type":"Algebra α ρᵃ"},{"name":"𝑩","type":"Algebra β ρᵇ"},{"name":"𝑪","type":"Algebra γ ρᶜ"},{"name":"p","type":"Data.Product.Σ\n(Data.Product.Σ\n (Function.Bundles.Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑩))\n (IsHom 𝑨 𝑩))\n(λ h →\n   {x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)} →\n   (Algebra.Domain 𝑩 Relation.Binary.Bundles.Setoid.≈\n    Function.Bundles.Func.to (proj₁ h) x)\n   (Function.Bundles.Func.to (proj₁ h) y) →\n   (Algebra.Domain 𝑨 Relation.Binary.Bundles.Setoid.≈ x) y)"},{"name":"B≅C","type":"𝑩 ≅ 𝑪"}],"elapsedMs":5637,"goal":"Data.Product.Σ\n(Data.Product.Σ\n (Function.Bundles.Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑪))\n (IsHom 𝑨 𝑪))\n(λ h →\n   {x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)} →\n   (Algebra.Domain 𝑪 Relation.Binary.Bundles.Setoid.≈\n    Function.Bundles.Func.to (proj₁ h) x)\n   (Function.Bundles.Func.to (proj₁ h) y) →\n   (Algebra.Domain 𝑨 Relation.Binary.Bundles.Setoid.≈ x) y)","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>\" None Direct (Cmd_show_version)","IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_load \"<work>/Subalgebras-sub-trans-iso.agda\" [\"-i\",\"agda-dojang/agda\",\"--library-file=agda/libraries\",\"-l\",\"agda-dojang\",\"-l\",\"standard-library\",\"-l\",\"agda-algebras\",\"-i\",\"<work>\"])","IOTCM \"<work>\" None Direct (Cmd_show_version)","IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_goal_type_context Normalised 0 noRange \"\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"first","loadElapsedMs":5606,"pid":2334713,"root":"<work>","spawned":true},"module":"Subalgebras-sub-trans-iso","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"]},"source":"interaction-lane"}

agda-mcp

search_by_name(pattern: ≤-trans)

one of 2 calls the model issued in this turn

answered

results: 5

the whole answer, as the agent saw it (2,589 characters of JSON)

[{"defKind":"function","hasBody":true,"module":"Classical.Properties.Lattice.Lattice-Order","prettyQname":"Classical.Properties.Lattice.Lattice-Order.≤-trans","type":"{α ρ : Agda.Primitive.Level}\n(𝑳 : Classical.Structures.Lattice.Basic.Lattice α ρ)\n{x y z\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(𝑳 Classical.Properties.Lattice.Lattice-Order.≤ x) y →\n(𝑳 Classical.Properties.Lattice.Lattice-Order.≤ y) z →\n(𝑳 Classical.Properties.Lattice.Lattice-Order.≤ x) z"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Subalgebras.Properties","prettyQname":"Legacy.Base.Subalgebras.Properties.≤-trans","type":"{𝓞 𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n{α β γ : Agda.Primitive.Level}\n(𝑨 : Legacy.Base.Algebras.Basic.Algebra α)\n{𝑩 : Legacy.Base.Algebras.Basic.Algebra β}\n(𝑪 : Legacy.Base.Algebras.Basic.Algebra γ) →\n𝑨 Legacy.Base.Subalgebras.Subalgebras.≤ 𝑩 →\n𝑩 Legacy.Base.Subalgebras.Subalgebras.≤ 𝑪 →\n𝑨 Legacy.Base.Subalgebras.Subalgebras.≤ 𝑪"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Subalgebras","prettyQname":"Legacy.Base.Subalgebras.≤-trans","type":"{𝓞 𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n{α β γ : Agda.Primitive.Level}\n(𝑨 : Legacy.Base.Algebras.Basic.Algebra α)\n{𝑩 : Legacy.Base.Algebras.Basic.Algebra β}\n(𝑪 : Legacy.Base.Algebras.Basic.Algebra γ) →\n𝑨 Legacy.Base.Subalgebras.Subalgebras.≤ 𝑩 →\n𝑩 Legacy.Base.Subalgebras.Subalgebras.≤ 𝑪 →\n𝑨 Legacy.Base.Subalgebras.Subalgebras.≤ 𝑪"},{"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.Subalgebras.Basic._.≤ 𝑪 → 𝑨 Setoid.Subalgebras.Basic._.≤ 𝑪"},{"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"}]

thought
The model thought here, about 250 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

get_dependencies(name: Setoid.Subalgebras.Properties.≤-trans-≅)

answered

type: {𝑆.𝓞 𝑆.𝓥 : Agda.Primitive.Level} {𝑆 : Overture.Signatures.Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ γ ρᶜ : Agda.Primitive.Level} {𝑨 : Setoid.Algebras.Basic.Algebra α ρᵃ} {𝑩 : Setoid.Algebras.Basic.Algebra β ρᵇ} {𝑪 : Setoid.Algebras.…dependencies: 10neighbors: 0

the whole answer, as the agent saw it (636 characters of JSON)

{"dependencies":["Agda.Primitive.Level","Overture.Signatures.Signature","Setoid.Algebras.Basic.Algebra","Setoid.Homomorphisms.Isomorphisms._.","Setoid.Subalgebras.Basic._.","ρᵃ","ρᵇ","ρᶜ","𝑆.𝓞","𝑆.𝓥"],"name":"Setoid.Subalgebras.Properties.≤-trans-≅","neighbors":[],"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-mcp

resolve_name(name: ≤-trans-≅)

answered

inScope: trueelapsedMs: 1candidates: 1

the whole answer, as the agent saw it (1,852 characters of JSON)

{"candidates":[{"definition":{"col":3,"endCol":12,"endLine":81,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md","line":81,"qualified":"Setoid.Subalgebras.Properties._.≤-trans-≅"},"description":"a defined name Setoid.Subalgebras.Properties._.≤-trans-≅","provenance":[{"site":{"col":13,"endCol":31,"endLine":22,"file":"<work>/Subalgebras-sub-trans-iso.agda","line":22},"step":"the opening of Setoid.Subalgebras"},{"step":"the opening of Setoid.Subalgebras.Properties"},{"site":{"col":3,"endCol":12,"endLine":81,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md","line":81},"step":"its definition"}],"qualified":"Setoid.Subalgebras.Properties._.≤-trans-≅"}],"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":1,"inScope":true,"lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_why_in_scope_toplevel \"\\8804-trans-\\8773\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2334713,"root":"<work>","spawned":false},"name":"≤-trans-≅","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"}

thought
The model thought here, about 149 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.
the work directory
Read()
refused
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md is outside <work>; --restricted confi…
the whole answer, as the agent saw it (164 characters of text)
<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Properties.lagda.md is outside <work>; --restricted confines the file tools to the working directory.
thought
The model thought here, about 300 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

search_by_name(pattern: ∘-hom)

answered

results: 5

the whole answer, as the agent saw it (2,576 characters of JSON)

[{"defKind":"function","hasBody":true,"module":"Examples.Demos.HSP","prettyQname":"Examples.Demos.HSP.∘-hom","type":"{𝓞 𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n{α ρᵃ β ρᵇ γ ρᶜ : Agda.Primitive.Level}\n{𝑨 : Examples.Demos.HSP.Algebra α ρᵃ}\n{𝑩 : Examples.Demos.HSP.Algebra β ρᵇ}\n{𝑪 : Examples.Demos.HSP.Algebra γ ρᶜ} →\nExamples.Demos.HSP.hom 𝑨 𝑩 →\nExamples.Demos.HSP.hom 𝑩 𝑪 → Examples.Demos.HSP.hom 𝑨 𝑪"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Homomorphisms.Properties","prettyQname":"Legacy.Base.Homomorphisms.Properties.∘-hom","type":"{𝓞 𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n{α β γ : Agda.Primitive.Level}\n(𝑨 : Legacy.Base.Algebras.Basic.Algebra α)\n{𝑩 : Legacy.Base.Algebras.Basic.Algebra β}\n(𝑪 : Legacy.Base.Algebras.Basic.Algebra γ) →\nLegacy.Base.Homomorphisms.Basic.hom 𝑨 𝑩 →\nLegacy.Base.Homomorphisms.Basic.hom 𝑩 𝑪 →\nLegacy.Base.Homomorphisms.Basic.hom 𝑨 𝑪"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Homomorphisms","prettyQname":"Legacy.Base.Homomorphisms.∘-hom","type":"{𝓞 𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n{α β γ : Agda.Primitive.Level}\n(𝑨 : Legacy.Base.Algebras.Basic.Algebra α)\n{𝑩 : Legacy.Base.Algebras.Basic.Algebra β}\n(𝑪 : Legacy.Base.Algebras.Basic.Algebra γ) →\nLegacy.Base.Homomorphisms.Basic.hom 𝑨 𝑩 →\nLegacy.Base.Homomorphisms.Basic.hom 𝑩 𝑪 →\nLegacy.Base.Homomorphisms.Basic.hom 𝑨 𝑪"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Structures.Homs","prettyQname":"Legacy.Base.Structures.Homs.∘-hom","type":"{𝐹.𝓞₀ 𝐹.𝓥₀ : Agda.Primitive.Level}\n{𝐹 : Legacy.Base.Structures.Basic.signature 𝐹.𝓞₀ 𝐹.𝓥₀}\n{𝑅.𝓞₁ 𝑅.𝓥₁ : Agda.Primitive.Level}\n{𝑅 : Legacy.Base.Structures.Basic.signature 𝑅.𝓞₁ 𝑅.𝓥₁}\n{α ρᵃ β ρᵇ γ ρᶜ : Agda.Primitive.Level}\n{𝑨 𝑩 𝑪 : Legacy.Base.Structures.Basic.structure 𝐹 𝑅} →\nLegacy.Base.Structures.Homs.hom 𝑨 𝑩 →\nLegacy.Base.Structures.Homs.hom 𝑩 𝑪 →\nLegacy.Base.Structures.Homs.hom 𝑨 𝑪"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Structures.Sigma.Homs","prettyQname":"Legacy.Base.Structures.Sigma.Homs.∘-hom","type":"{𝑅 𝐹 : Legacy.Base.Structures.Sigma.Basic.Signature}\n{α ρᵃ : Agda.Primitive.Level}\n(𝑨 : Legacy.Base.Structures.Sigma.Basic.Structure 𝑅 𝐹)\n{β ρᵇ : Agda.Primitive.Level}\n{𝑩 : Legacy.Base.Structures.Sigma.Basic.Structure 𝑅 𝐹}\n{γ ρᶜ : Agda.Primitive.Level}\n(𝑪 : Legacy.Base.Structures.Sigma.Basic.Structure 𝑅 𝐹) →\nLegacy.Base.Structures.Sigma.Homs.hom 𝑨 𝑩 →\nLegacy.Base.Structures.Sigma.Homs.hom 𝑩 𝑪 →\nLegacy.Base.Structures.Sigma.Homs.hom 𝑨 𝑪"}]

agda-mcp

search_by_name(pattern: Injective, limit: 40)

answered

results: 40

the whole answer, as the agent saw it (22,905 characters of JSON)

[{"defKind":"function","hasBody":true,"module":"Classical.Structures.Group.IndexAction.ActionKernel","prettyQname":"Classical.Structures.Group.IndexAction.ActionKernel.act-injective","type":"{ι β σ ℓ : Agda.Primitive.Level} {I : Agda.Primitive.Set ι}\n(𝒢 : Classical.Structures.Group.Basic.Group β σ)\n(H\n : Relation.Unary.Pred\n   Setoid.Algebras.Basic.𝕌[ 𝒢 .Agda.Builtin.Sigma.Σ.fst ] ℓ)\n(H-sg : Classical.Structures.Group.Subgroups.IsSubgroup 𝒢 H)\n(A : Classical.Structures.Group.IndexAction.RightAction I 𝒢)\n(spec : Classical.Structures.Group.IndexAction.IsCosetAction A H)\n(x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (𝒢 .Agda.Builtin.Sigma.Σ.fst)))\n{i j : I} →\nClassical.Structures.Group.IndexAction.RightAction.act A x i\nAgda.Builtin.Equality.≡\nClassical.Structures.Group.IndexAction.RightAction.act A x j →\ni Agda.Builtin.Equality.≡ j"},{"defKind":"function","hasBody":true,"module":"Classical.Structures.Group.IndexAction.IsCosetAction","prettyQname":"Classical.Structures.Group.IndexAction.IsCosetAction.act-injective","type":"{ι β σ ℓ : Agda.Primitive.Level} {I : Agda.Primitive.Set ι}\n{𝒢 : Classical.Structures.Group.Basic.Group β σ}\n{A : Classical.Structures.Group.IndexAction.RightAction I 𝒢}\n{H\n : Relation.Unary.Pred\n   Setoid.Algebras.Basic.𝕌[ 𝒢 .Agda.Builtin.Sigma.Σ.fst ] ℓ}\n(r : Classical.Structures.Group.IndexAction.IsCosetAction A H)\n(x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (𝒢 .Agda.Builtin.Sigma.Σ.fst)))\n{i j : I} →\nClassical.Structures.Group.IndexAction.RightAction.act A x i\nAgda.Builtin.Equality.≡\nClassical.Structures.Group.IndexAction.RightAction.act A x j →\ni Agda.Builtin.Equality.≡ j"},{"defKind":"function","hasBody":true,"module":"Classical.Structures.Group.IndexAction.RightAction","prettyQname":"Classical.Structures.Group.IndexAction.RightAction.act-injective","type":"{ι β σ : Agda.Primitive.Level} {I : Agda.Primitive.Set ι}\n{𝒢 : Classical.Structures.Group.Basic.Group β σ}\n(r : Classical.Structures.Group.IndexAction.RightAction I 𝒢)\n(x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (𝒢 .Agda.Builtin.Sigma.Σ.fst)))\n{i j : I} →\nClassical.Structures.Group.IndexAction.RightAction.act r x i\nAgda.Builtin.Equality.≡\nClassical.Structures.Group.IndexAction.RightAction.act r x j →\ni Agda.Builtin.Equality.≡ j"},{"defKind":"function","hasBody":true,"module":"Classical.Structures.Group.IndexAction","prettyQname":"Classical.Structures.Group.IndexAction.act-injective","type":"{ι β σ ℓ : Agda.Primitive.Level} {I : Agda.Primitive.Set ι}\n{𝒢 : Classical.Structures.Group.Basic.Group β σ}\n(A : Classical.Structures.Group.IndexAction.RightAction I 𝒢)\n(H\n : Relation.Unary.Pred\n   Setoid.Algebras.Basic.𝕌[ 𝒢 .Agda.Builtin.Sigma.Σ.fst ] ℓ)\n(x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (𝒢 .Agda.Builtin.Sigma.Σ.fst)))\n{i j : I} →\nClassical.Structures.Group.IndexAction.RightAction.act A x i\nAgda.Builtin.Equality.≡\nClassical.Structures.Group.IndexAction.RightAction.act A x j →\ni Agda.Builtin.Equality.≡ j"},{"defKind":"function","hasBody":true,"module":"Classical.Structures.Group.PartitionSubgroup.PartitionSubgroups","prettyQname":"Classical.Structures.Group.PartitionSubgroup.PartitionSubgroups.K-injective","type":"{α ρ : Agda.Primitive.Level} (n : Agda.Builtin.Nat.Nat)\n(𝒢 : Classical.Structures.Group.Basic.Group α ρ)\n(s : Setoid.Algebras.Basic.𝕌[ 𝒢 .Agda.Builtin.Sigma.Σ.fst ]) →\nRelation.Nullary.Negation.Core.¬\n(Setoid.Algebras.Basic.𝔻[ 𝒢 .Agda.Builtin.Sigma.Σ.fst ]\n Relation.Binary.Bundles.Setoid.≈ 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Setoid.Homomorphisms.Isomorphisms._.≅ 𝑩) →\nSetoid.Functions.Injective.IsInjective\n(Agda.Builtin.Sigma.Σ.fst\n (Setoid.Homomorphisms.Isomorphisms._≅_.to φ))"},{"defKind":"function","hasBody":true,"module":"Setoid.Subalgebras.Properties","prettyQname":"Setoid.Subalgebras.Properties.iso→injective","type":"{𝑆.𝓞 𝑆.𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝑆.𝓞 𝑆.𝓥}\n{α ρᵃ β ρᵇ : Agda.Primitive.Level}\n(𝑨 : Setoid.Algebras.Basic.Algebra α ρᵃ)\n{𝑩 : Setoid.Algebras.Basic.Algebra β ρᵇ}\n(φ : 𝑨 Setoid.Homomorphisms.Isomorphisms._.≅ 𝑩) →\nSetoid.Functions.Injective.IsInjective\n(Agda.Builtin.Sigma.Σ.fst\n (Setoid.Homomorphisms.Isomorphisms._≅_.to φ))"},{"defKind":"function","hasBody":true,"module":"Setoid.Subalgebras.Subdirect.Basic","prettyQname":"Setoid.Subalgebras.Subdirect.Basic.natmap-injective","type":"{𝓞 𝓥 ι α ρ ℓ : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥} {I : Agda.Primitive.Set ι}\n{𝑨 : Setoid.Algebras.Basic.Algebra α ρ}\n(θ : I → Setoid.Congruences.Basic.Con 𝑨 ℓ) →\nSetoid.Subalgebras.Subdirect.Basic.Separates θ →\nSetoid.Functions.Injective.IsInjective\n(Agda.Builtin.Sigma.Σ.fst\n (Setoid.Subalgebras.Subdirect.Basic.natmap θ))"},{"defKind":"function","hasBody":true,"module":"Setoid.Subalgebras.Subdirect.Irreducible","prettyQname":"Setoid.Subalgebras.Subdirect.Irreducible.injective↔0kernel","type":"{𝓞 𝓥 ι α ρ αᵃ : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥} {I : Agda.Primitive.Set ι}\n{𝑨 : Setoid.Algebras.Basic.Algebra α ρ}\n(𝒜 : I → Setoid.Algebras.Basic.Algebra αᵃ ρ)\n(h\n : Setoid.Homomorphisms.Basic.hom 𝑨 (Setoid.Algebras.Products.⨅ 𝒜))\n{i : I} →\nSetoid.Functions.Injective.IsInjective\n(Agda.Builtin.Sigma.Σ.fst\n (Setoid.Subalgebras.Subdirect.Basic.coord 𝒜 h i))\nAgda.Builtin.Equality.≡\nSetoid.Congruences.Monolith.BelowDiagonal 𝑨\n(Setoid.Subalgebras.Subdirect.Irreducible.kerfam 𝒜 h i)"}]

thought
The model thought here, about 163 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.
agda-mcp
search_by_type(pattern: Setoid.Homomorphisms.Basic.hom 𝑩 𝑪 -> Setoid.Homomorphisms.Basic.hom 𝑨 𝑪)
answered
results: 0
the whole answer, as the agent saw it (2 characters of JSON)
[]

agda-mcp

exports_of(module: Setoid.Homomorphisms.Properties)

answered

error.code: NotInScopeelapsedMs: 18

the whole answer, as the agent saw it (1,336 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":18,"error":{"code":"NotInScope","message":"1.1-32: error: [NotInScope]\nNot in scope:\n  Setoid.Homomorphisms.Properties at 1.1-32\nwhen scope checking Setoid.Homomorphisms.Properties","stage":"module"},"lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_show_module_contents_toplevel Simplified \"Setoid.Homomorphisms.Properties\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2334713,"root":"<work>","spawned":false},"module":"Setoid.Homomorphisms.Properties","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"]}}

thought
The model thought here, about 156 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

type_of(expr: ∘-hom)

answered

error.code: NotInScopeelapsedMs: 8

the whole answer, as the agent saw it (1,376 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":8,"error":{"code":"NotInScope","message":"1.1-6: error: [NotInScope]\nNot in scope:\n  ∘-hom at 1.1-6\n    (did you mean\n       'Setoid.Homomorphisms.⊙-hom' or\n       'Setoid.Homomorphisms.⨅-hom' or\n       '⊙-hom' or\n       '⨅-hom'?)\nwhen scope checking ∘-hom","stage":"expression"},"expr":"∘-hom","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_infer_toplevel Normalised \"\\8728-hom\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2334713,"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"}

agda-mcp

type_of(expr: ⊙-injective)

answered

type: {a α b β c γ : Level} {𝑨 : Relation.Binary.Bundles.Setoid a α} {𝑩 : Relation.Binary.Bundles.Setoid b β} {𝑪 : Relation.Binary.Bundles.Setoid c γ} (f : Function.Bundles.Func 𝑨 𝑩) (g : Function.Bundles.Func 𝑩 𝑪) → ({x y : R…elapsedMs: 5

the whole answer, as the agent saw it (1,984 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":5,"expr":"⊙-injective","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_infer_toplevel Normalised \"\\8857-injective\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2334713,"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":"{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) →\n({x y : Relation.Binary.Bundles.Setoid.Carrier 𝑨} →\n (𝑩 Relation.Binary.Bundles.Setoid.≈ Function.Bundles.Func.to f x)\n (Function.Bundles.Func.to f y) →\n (𝑨 Relation.Binary.Bundles.Setoid.≈ x) y) →\n({x y : Relation.Binary.Bundles.Setoid.Carrier 𝑩} →\n (𝑪 Relation.Binary.Bundles.Setoid.≈ Function.Bundles.Func.to g x)\n (Function.Bundles.Func.to g y) →\n (𝑩 Relation.Binary.Bundles.Setoid.≈ x) y) →\n{x y : Relation.Binary.Bundles.Setoid.Carrier 𝑨} →\n(𝑪 Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to g (Function.Bundles.Func.to f x))\n(Function.Bundles.Func.to g (Function.Bundles.Func.to f y)) →\n(𝑨 Relation.Binary.Bundles.Setoid.≈ x) y"}

agda-mcp

type_of(expr: ≅toInjective)

answered

type: {𝑆.𝓞 = a : Level} {𝑆.𝓥 = α : Level} {𝑆 = β : Data.Product.Σ (Set a) (λ b → b → Set α)} {α = c : Level} {ρᵃ = γ : Level} {β = 𝑨 : Level} {ρᵇ = 𝑩 : Level} {𝑨 = 𝑪 : Algebra c γ} {𝑩 = f : Algebra 𝑨 𝑩} (g : 𝑪 ≅ f) {x y : Rela…elapsedMs: 3

the whole answer, as the agent saw it (1,603 characters of JSON)

{"checkedFromSource":false,"command":{"args":["--interaction-json"],"binary":"agda","cwd":"<repo>"},"elapsedMs":3,"expr":"≅toInjective","lane":{"agdaVersion":"2.8.0","iotcm":["IOTCM \"<work>/Subalgebras-sub-trans-iso.agda\" None Direct (Cmd_infer_toplevel Normalised \"\\8773toInjective\")","IOTCM \"<work>\" None Direct (Cmd_show_version)"],"load":"reused","pid":2334713,"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":"{𝑆.𝓞 = a : Level} {𝑆.𝓥 = α : Level}\n{𝑆 = β : Data.Product.Σ (Set a) (λ b → b → Set α)}\n{α = c : Level} {ρᵃ = γ : Level} {β = 𝑨 : Level} {ρᵇ = 𝑩 : Level}\n{𝑨 = 𝑪 : Algebra c γ} {𝑩 = f : Algebra 𝑨 𝑩} (g : 𝑪 ≅ f)\n{x y : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑪)} →\n(Algebra.Domain f Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to (proj₁ (_≅_.to g)) x)\n(Function.Bundles.Func.to (proj₁ (_≅_.to g)) y) →\n(Algebra.Domain 𝑪 Relation.Binary.Bundles.Setoid.≈ x) y"}

thought
The model thought here, about 250 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

agda-mcp

fill_hole(line: 27, column: 21, candidate: ⊙-hom (proj₁ p) (_≅_.to B≅C) , ⊙-injective (proj₁ (proj₁ p)) (proj₁ (_≅_.to B≅C)) (proj₂ p) (≅toInjective B≅C))

answered

status: okremainingHoles: 0verdict.exitCode: 0elapsedMs: 5531holes: 0

the whole answer, as the agent saw it (2,126 characters of JSON)

{"candidate":"⊙-hom (proj₁ p) (_≅_.to B≅C) , ⊙-injective (proj₁ (proj₁ p)) (proj₁ (_≅_.to B≅C)) (proj₂ p) (≅toInjective B≅C)","checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","-i","<work>","<work>/Subalgebras-sub-trans-iso.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"elapsedMs":5531,"holes":[],"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"]},"remainingHoles":0,"status":"ok","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 -i <work> <work>/Subalgebras-sub-trans-iso.agda, run against that file with the hole at line 27, column 20 (index 0) replaced by the candidate (the file on disk is restored afterwards, byte for byte)","exitCode":0,"meaning":"status is \"ok\" if and only if that command exited 0, or failed with nothing but [UnsolvedInteractionMetas] (holes still open in the file), including any new sub-hole the candidate itself introduced, which is a successful refinement. Every other failure is \"type_error\", including [UnsolvedMetaVariables] and [UnsolvedConstraints]: a candidate that leaves a meta unsolved does not pass the build and is not ok here either. A run killed by --timeout is \"timeout\" (the candidate was never judged) and an agda binary that could not be started at all is \"crash\"."}}

the model
Works. Now writing it into the file.
the work directory
Edit(replace_all: false, old_string: ≤-trans-≅′ p B≅C = {!!}, new_string: ≤-trans-≅′ p B≅C = ⊙-hom (proj₁ p) (_≅_.to B≅C) , ⊙-injective (proj₁ (proj₁ p)) (proj₁ (_≅_.to B≅C)) (proj₂ p) (…)
answered
The file <work>/Subalgebras-sub-trans-iso.agda has been updated successfully. (file state is current in your context — n…
the whole answer, as the agent 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-mcp

check_file()

answered

success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5707diagnostics: 0holes: 0

the whole answer, as the agent saw it (1,802 characters of JSON)

{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","-i","<work>","<work>/Subalgebras-sub-trans-iso.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5707,"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 -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 model
check_file reports success (exit code 0, no diagnostics, no holes) on the final file, where the hole is filled by composing the embedding with the isomorphism's `to` homomorphism (`⊙-hom`) and combining injectivity via `⊙-injective` and `≅toInjective`.

The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the server’s own, in full. The client billed 1,685 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 0

statement preserved yes

turns 17

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-≅′ p B≅C = ⊙-hom (proj₁ p) (_≅_.to B≅C)
+                 , ⊙-injective (proj₁ (proj₁ p)) (proj₁ (_≅_.to B≅C))
+                               (proj₂ p) (≅toInjective B≅C)

reports/agent-bench/agent-opus5-1/subjects/algebras-subalgebras-sub-trans-iso/final/Subalgebras-sub-trans-iso.agda

`*-comm` Sonnet 5 refused

Four turns, no query, a correct induction. Agda exits 0 and the row is still not a solve: the session reached for `trans` by editing the fixture's own import line.

obligation: stdlib-nat-mul-comm
tier: non-obvious
stratum: agda-stdlib
model: claude-sonnet-5
turns: 4
tool calls: 3
wall: 28 s
cost, list price: USD 0.05

the hole’s goal, as get_goal reported it

∀ (m n : ℕ) → m * n ≡ n * m

the session, 4 turns and 3 tool calls

thought
The model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.

the work directory

Read()

answered

1 -- Nat-mul-comm.agda

the whole answer, as the agent saw it (933 characters of text)

1	-- Nat-mul-comm.agda
2	--
3	-- File: data/benchmarks/agda-stdlib-v0/obligations/Nat-mul-comm.agda
4	--
5	-- Benchmark obligation: stdlib-nat-mul-comm
6	-- Difficulty: non-obvious (Tier 3)
7	-- Source: Data.Nat.Properties
8	-- Strategy: induction on m; base sym (*-zeroʳ n), step an equational chain using *-suc
9	--
10	-- Note: *-zeroʳ and *-suc are provided as imports; the challenge is the
11	-- non-obvious composition (the recursive call sits under cong, and the final
12	-- step rewrites with *-suc backwards).
13	--
14	module Nat-mul-comm where
15	
16	open import AgdaDojang.Debug
17	
18	open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ )
19	open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym )
20	
21	-- Prerequisites (provided, not to be proved here)
22	open import Data.Nat.Properties using ( *-zeroʳ ; *-suc )
23	
24	*-comm : ∀ (m n : ℕ) → m * n ≡ n * m
25	*-comm m n = {!!}
26

thought
The model thought here, of unrecorded length. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.
the work directory
Edit(replace_all: false, old_string: open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ) -- Prerequisites (provided, not to be proved here) open import Data.Nat.Pro…, new_string: open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ; trans ) -- Prerequisites (provided, not to be proved here) open import Data…)
answered
The file <work>/Nat-mul-comm.agda has been updated successfully. (file state is current in your context — no need to Rea…
the whole answer, as the agent saw it (130 characters of text)
The file <work>/Nat-mul-comm.agda has been updated successfully. (file state is current in your context — no need to Read it back)

agda-mcp

check_file()

answered

success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 3083diagnostics: 0holes: 0

the whole answer, as the agent saw it (1,776 characters of JSON)

{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","-i","<work>","<work>/Nat-mul-comm.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":3083,"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 -i <work> <work>/Nat-mul-comm.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 model
check_file reports success (exitCode 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 server’s own, in full. The client billed 640 thinking tokens over the session; their text is not in the archive.

refusedthe judge, in outcome.json

Refused at the preservation gate: import line changed or missing: open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ). Agda itself exited 0 on this file.

Agda’s exit code 0

statement preserved yes

turns 4

the file the judge read, marked against the obligation it started as

 -- Nat-mul-comm.agda
 --
 -- File: data/benchmarks/agda-stdlib-v0/obligations/Nat-mul-comm.agda
 --
 -- Benchmark obligation: stdlib-nat-mul-comm
 -- Difficulty: non-obvious (Tier 3)
 -- Source: Data.Nat.Properties
 -- Strategy: induction on m; base sym (*-zeroʳ n), step an equational chain using *-suc
 --
 -- Note: *-zeroʳ and *-suc are provided as imports; the challenge is the
 -- non-obvious composition (the recursive call sits under cong, and the final
 -- step rewrites with *-suc backwards).
 --
 module Nat-mul-comm where
 
 open import AgdaDojang.Debug
 
 open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ )
-open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym )
+open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ; trans )
 
 -- Prerequisites (provided, not to be proved here)
 open import Data.Nat.Properties using ( *-zeroʳ ; *-suc )
 
 *-comm : ∀ (m n : ℕ) → m * n ≡ n * m
-*-comm m n = {!!}
+*-comm zero n = sym (*-zeroʳ n)
+*-comm (suc m) n = trans (cong (n +_) (*-comm m n)) (sym (*-suc n m))

reports/agent-bench/agent-sonnet5-1/subjects/stdlib-nat-mul-comm/final/Nat-mul-comm.agda

`*-comm` Opus 5 solved

The same obligation, the same four turns, the same need for `trans`. This session added an import line instead of editing one, which is what the preservation gate allows.

obligation: stdlib-nat-mul-comm
tier: non-obvious
stratum: agda-stdlib
model: claude-opus-5
turns: 4
tool calls: 3
wall: 24 s
cost, list price: USD 0.11

the hole’s goal, as get_goal reported it

∀ (m n : ℕ) → m * n ≡ n * m

the session, 4 turns and 3 tool calls

the work directory

Read()

answered

1 -- Nat-mul-comm.agda

the whole answer, as the agent saw it (933 characters of text)

1	-- Nat-mul-comm.agda
2	--
3	-- File: data/benchmarks/agda-stdlib-v0/obligations/Nat-mul-comm.agda
4	--
5	-- Benchmark obligation: stdlib-nat-mul-comm
6	-- Difficulty: non-obvious (Tier 3)
7	-- Source: Data.Nat.Properties
8	-- Strategy: induction on m; base sym (*-zeroʳ n), step an equational chain using *-suc
9	--
10	-- Note: *-zeroʳ and *-suc are provided as imports; the challenge is the
11	-- non-obvious composition (the recursive call sits under cong, and the final
12	-- step rewrites with *-suc backwards).
13	--
14	module Nat-mul-comm where
15	
16	open import AgdaDojang.Debug
17	
18	open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ )
19	open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym )
20	
21	-- Prerequisites (provided, not to be proved here)
22	open import Data.Nat.Properties using ( *-zeroʳ ; *-suc )
23	
24	*-comm : ∀ (m n : ℕ) → m * n ≡ n * m
25	*-comm m n = {!!}
26

thought
The model thought here, about 350 tokens by the client’s running estimate. The archive keeps the block’s signature and not its text, so there is nothing to quote and nothing is quoted.
the work directory
Edit(replace_all: false, old_string: *-comm : ∀ (m n : ℕ) → m * n ≡ n * m *-comm m n = {!!}, new_string: open import Relation.Binary.PropositionalEquality using ( trans ) *-comm : ∀ (m n : ℕ) → m * n ≡ n * m *-comm zero n = sym (*-zeroʳ n) *-comm (suc m) n = tr…)
answered
The file <work>/Nat-mul-comm.agda has been updated successfully. (file state is current in your context — no need to Rea…
the whole answer, as the agent saw it (130 characters of text)
The file <work>/Nat-mul-comm.agda has been updated successfully. (file state is current in your context — no need to Read it back)

agda-mcp

check_file()

answered

success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 2998diagnostics: 0holes: 0

the whole answer, as the agent saw it (1,776 characters of JSON)

{"checkedFromSource":true,"command":{"args":["-i","agda-dojang/agda","--library-file=agda/libraries","-l","agda-dojang","-l","standard-library","-l","agda-algebras","-i","<work>","<work>/Nat-mul-comm.agda"],"binary":"<nix>/agdaWithPackages-2.8.0/bin/agda","cwd":"<repo>"},"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":2998,"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 -i <work> <work>/Nat-mul-comm.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 model
check_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 server’s own, in full. The client billed 449 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 0

statement preserved yes

turns 4

import lines added open import Relation.Binary.PropositionalEquality using ( trans )

the file the judge read, marked against the obligation it started as

 -- Nat-mul-comm.agda
 --
 -- File: data/benchmarks/agda-stdlib-v0/obligations/Nat-mul-comm.agda
 --
 -- Benchmark obligation: stdlib-nat-mul-comm
 -- Difficulty: non-obvious (Tier 3)
 -- Source: Data.Nat.Properties
 -- Strategy: induction on m; base sym (*-zeroʳ n), step an equational chain using *-suc
 --
 -- Note: *-zeroʳ and *-suc are provided as imports; the challenge is the
 -- non-obvious composition (the recursive call sits under cong, and the final
 -- step rewrites with *-suc backwards).
 --
 module Nat-mul-comm where
 
 open import AgdaDojang.Debug
 
 open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ )
 open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym )
 
 -- Prerequisites (provided, not to be proved here)
 open import Data.Nat.Properties using ( *-zeroʳ ; *-suc )
 
+open import Relation.Binary.PropositionalEquality using ( trans )
+
 *-comm : ∀ (m n : ℕ) → m * n ≡ n * m
-*-comm m n = {!!}
+*-comm zero    n = sym (*-zeroʳ n)
+*-comm (suc m) n = trans (cong (n +_) (*-comm m n)) (sym (*-suc n m))

reports/agent-bench/agent-opus5-1/subjects/stdlib-nat-mul-comm/final/Nat-mul-comm.agda

What restated means

A file that type-checks, keeps the statement it was given, and proves it by calling the library’s own lemma for that statement is not counted as a solve. It goes in a column of its own.

The benchmark’s obligations were mined from two Agda libraries, so for most of them the library already contains a proof. An agent that finds that proof and calls it has done something useful and has not done the thing being measured, which is whether the tools help a model build a proof. So the judge separates the two mechanically: it reads the references in the finished definition off Agda’s internal terms, through the agda-strux extractor, and compares them with the original the obligation was mined from, which each row of the benchmark index names in a restates: tag. A match is a restatement, whatever else the file earned.

The first and second tabs above are one obligation, given to two models with the same tools. Both files type-check. One is a solve and one is a restatement, and nothing in the difference is a matter of opinion.

The numbers

Both arms ran on 2026-09-15, one subject per obligation, three at a time. The loop columns are a different instrument on the same suite and the same verifier: agda-native-air’s own proof-search loop, first over a fixed space of candidate terms and then with corpus retrieval composed around it. They come from ADR 0001 § 9, which is their record; every other column on this page is regenerated from the runs’ own report.json at build time and compared with that ADR cell by cell, so a number that drifts fails the build.

Solved and restated per stratum, beside the proof-search loop on the same suite and the same verifier.
stratum	n	loop, fixed space	loop, retrieval	Sonnet 5 solved	Sonnet 5 restated	Opus 5 solved	Opus 5 restated
agda-stdlib	22	6	6	21	0	22	0
agda-stdlib/haystack	12	0	6	12	0	12	0
agda-algebras/using	11	2	2	9	2	11	0
agda-algebras/wholesale	10	0	0	4	6	9	1
total	55	8	14	46	8	54	1

`claude-sonnet-5`

run agent-sonnet5-1, Claude Code 2.1.261 (Claude Code)

46 of 55 solved
8 restated; 1 refused at the preservation gate
327 turns, 272 tool calls
USD 4.62 at list price
0 anomalies

`claude-opus-5`

run agent-opus5-1, Claude Code 2.1.261 (Claude Code)

54 of 55 solved
1 restated
325 turns, 270 tool calls
USD 9.14 at list price
0 anomalies

By difficulty tier.
tier	n	Sonnet 5 solved	Opus 5 solved
routine	16	15	16
compositional	25	20	24
non-obvious	14	11	14

Turns and tool calls are the comparable columns. Subjects ran three at a time, so the wall clocks are indicative only, and the costs are the client’s own list-price accounting. Neither arm produced an anomaly, and no subject reached a cap: every session stopped because the model stopped.

Every tool call of both arms, by tool. `get_diagnostics` and `check_project` were presented and never called.
tool	Sonnet 5	Opus 5
`Read`	66	57
`Edit`	58	56
`check_file`	56	55
`fill_hole`	12	41
`type_of`	24	21
`get_goal`	10	17
`search_by_name`	15	8
`definition_of`	13	4
`exports_of`	12	4
`get_dependencies`	4	2
`normalize`	0	3
`search_by_type`	2	1
`resolve_name`	0	1

Two readings the table does not show on its own. Sonnet’s eight restatements are all agda-algebras rows, six of them from the stratum whose fixtures import whole modules and name nothing useful. And the one row that is neither solved nor restated is the fourth tab above, *-comm as Sonnet 5 left it: a file Agda accepts, refused because the session reached for trans by editing the fixture’s own import line instead of adding one. The gate is the protocol’s, the file is fine, and the row stays unsolved.

One tier that measures nothing about the model

Twelve of the 55 obligations are a tier built to defeat retrieval by construction: each gold applies one standard-library lemma that the fixture imports but does not name in its using list, so the needle is never handed over in the answer key. The search loop’s fixed space solves none of the twelve; with corpus retrieval it solves six, its first solves under target exclusion anywhere in the project.

Both model arms solve all twelve. Sonnet 5 solves eleven of them in four turns with no Agda query at all (read the file, edit it, check it), naming the needle qualified from memory; the twelfth takes one type_of on Data.Nat.Properties.+-∸-assoc. These are standard-library lemmas, and they are in every frontier model’s training data.

The tier measures a ranker. It does not measure a model, and a page that reported its twelve solves as a result about the agent would be reporting a result about the corpus the tier was designed to stress.

How this page is built

Two Makefile targets and no network. make demo-data reads the committed archive and writes one small JSON per replay and one for the table; make demo-site renders this page from those. Neither the data nor the page is committed: the archive is, and the page is rebuilt from it. The archive itself is 1,051 files and 14 MB, and none of it is shipped to your browser; what is here is 5 sessions and a table.

Three things the build refuses to do. It will not write a page whose table disagrees with ADR 0001 § 9. It will not write a page carrying an absolute path from the machine the sweep ran on: every path here is anchored to <repo>, <work> (the one directory a session could see), or <nix>, and a path left pointing into a home directory, a Nix store, or a per-user runtime directory fails the build. And it quotes no reasoning: the archived transcripts carry each thinking block’s signature and an empty string in place of its text, so the replays mark where the model thought and say that the text is not there.

The evidence for every claim on this page is a file you can open:

reports/agent-bench/README.md: the protocol, the judge’s gates, and what a run directory holds.
docs/adr/0001-proof-search-on-agda-mcp.md § 9: the decision record this table is checked against.
data/benchmarks/README.md: the 55 obligations, their gold solutions, and the index row behind each panel’s facts.
scripts/python/demo/: the generator, and scripts/python/tests/ its tests, which run in CI.

The seam left open

This page shows what Agda answered in 2026. It cannot let you ask it anything, because there is no Agda here: the page is static and makes no request off its own origin. Whether a real type-checker could join it, as WebAssembly in the browser, is a separate question with its own measurement. The seam is ready for the answer: each session panel above carries its obligation’s path and its final file’s path as data attributes, and both files’ text is already in the page, so a lane that re-checked them in the browser would need no new data and no server.

What this is

The sessions

kercon′ Opus 5 solved

kercon′ Sonnet 5 restated

≤-trans-≅′ Opus 5 solved

*-comm Sonnet 5 refused

*-comm Opus 5 solved

What restated means

The numbers

claude-sonnet-5

claude-opus-5

One tier that measures nothing about the model

How this page is built

The seam left open

`kercon′` Opus 5 solved

`kercon′` Sonnet 5 restated

`≤-trans-≅′` Opus 5 solved

`*-comm` Sonnet 5 refused

`*-comm` Opus 5 solved

`claude-sonnet-5`

`claude-opus-5`