formalverification / agda-native-air

An agent, Agda, and the tools between them

What a frontier model does with the agda-mcp server’s tools: eight real sessions from the committed archive, replayed with every call and every answer in full.

What this is

agda-native-air gives a coding agent a way to ask Agda questions. Its server, agda-mcp, offers tools over MCP that 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.

The five sessions below come from the project’s agent measurement (#154), as re-run on 2026-09-29 with fixture headers that no longer carry hints (#219): each of 55 proof obligations handed to one fresh, non-interactive Claude Code session with the server’s fourteen tools, Read and Edit on a single staged file and on the libraries’ sources, and nothing else (no shell, no settings, no project instructions, no memory), under caps of 30 turns, 900 seconds, and USD 3.00.

They are evidence of what a frontier model does with the tools: which ones it calls, what it asks, what the answers say, and what it writes from them. They are not evidence that the tools help. Every session in these arms had the server, so nothing in them compares the tools with their absence. That comparison is the control (#162): the same model, prompts, caps, and judge, with a shell and the same agda in place of the server, run the same day. The numbers below set it beside these sessions’ arms, and the guide to the results reads it in § 4.3. In short: the server did not come out ahead on the count, and the count does not settle the question, because every arm could read the library’s own proofs, and a proof in view can be written out rather than cited; and on a tier built to need several lemmas, with no proof on disk to copy, every configuration of both models wrote a checking proof of every row.

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 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.

mon→hom′ Opus 5 solved

A search finds the library's own lemma, and this session writes the proof out instead of citing it: a first probe with `_,_`, which the file does not import, comes back a type error, and the constructor named in full is accepted.

obligation
algebras-homs-mon-to-hom
tier
compositional
stratum
agda-algebras/wholesale
the library original
Setoid.Homomorphisms.Basic.mon→hom
model
claude-opus-5
turns
9
tool calls
8
wall
75 s
cost, list price
USD 0.24
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} →  mon 𝑨 𝑩 → hom 𝑨 𝑩
the session, 9 turns and 8 tool calls
  1. the model
    I'll start by reading the file.
  2. the work directory
    Read()
    answered
    1 -- Homs-mon-to-hom.agda
    the whole answer, as the agent saw it (721 characters of text)
    1	-- Homs-mon-to-hom.agda
2	--
3	-- File: data/benchmarks/agda-algebras-v0/obligations/Homs-mon-to-hom.agda
4	--
5	-- Benchmark obligation: algebras-homs-mon-to-hom
6	-- Difficulty: compositional
7	-- Import stratum: wholesale
8	--
9	module Homs-mon-to-hom where
10	
11	open import AgdaDojang.Debug
12	
13	open import Agda.Primitive   using ( Level )
14	open import Data.Product     using ( proj₂ )
15	
16	open import Overture             using ( Signature )
17	open import Setoid.Algebras      using ( Algebra )
18	open import Setoid.Homomorphisms
19	
20	mon→hom′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
21	           {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
22	  →  mon 𝑨 𝑩 → hom 𝑨 𝑩
23	mon→hom′ m = {!!}
24
  3. thought
    The model thought here, about 135 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.
  4. agda-mcp
    type_of(expr: mon)
    one of 2 calls the model issued in this turn
    answered
    type: {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)} (𝑨 : Algebra α ρᵃ) → Algebra β ρᵇ → Set (𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ)elapsedMs: 5401
    the whole answer, as the agent saw it (406 characters of JSON)
    {"checkedFromSource":true,"elapsedMs":5401,"expr":"mon","lane":{"load":"first","loadElapsedMs":5376},"project":{"root":"<work>","rootSource":"server-config"},"scope":"toplevel","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level}\n{𝑆 : Data.Product.Σ (Set 𝓞) (λ F → F → Set 𝓥)} (𝑨 : Algebra α ρᵃ) →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"}
  5. agda-mcp
    search_by_name(pattern: mon→hom)
    one of 2 calls the model issued in this turn
    answered
    results: 5
    the whole answer, as the agent saw it (2,155 characters of JSON)
    [{"defKind":"function","hasBody":true,"module":"Legacy.Base.Homomorphisms.Basic","prettyQname":"Legacy.Base.Homomorphisms.Basic.mon→hom","type":"{𝓞 𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n{α β : Agda.Primitive.Level}\n(𝑨 : Legacy.Base.Algebras.Basic.Algebra α)\n{𝑩 : Legacy.Base.Algebras.Basic.Algebra β} →\nLegacy.Base.Homomorphisms.Basic.mon 𝑨 𝑩 →\nLegacy.Base.Homomorphisms.Basic.hom 𝑨 𝑩"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Homomorphisms","prettyQname":"Legacy.Base.Homomorphisms.mon→hom","type":"{𝓞 𝓥 : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n{α β : Agda.Primitive.Level}\n(𝑨 : Legacy.Base.Algebras.Basic.Algebra α)\n{𝑩 : Legacy.Base.Algebras.Basic.Algebra β} →\nLegacy.Base.Homomorphisms.Basic.mon 𝑨 𝑩 →\nLegacy.Base.Homomorphisms.Basic.hom 𝑨 𝑩"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Structures.Homs","prettyQname":"Legacy.Base.Structures.Homs.mon→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 : Agda.Primitive.Level}\n{𝑨 𝑩 : Legacy.Base.Structures.Basic.structure 𝐹 𝑅} →\nLegacy.Base.Structures.Homs.mon →\nLegacy.Base.Structures.Homs.hom 𝑨 𝑩"},{"defKind":"function","hasBody":true,"module":"Legacy.Base.Structures.Sigma.Homs","prettyQname":"Legacy.Base.Structures.Sigma.Homs.mon→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) →\nLegacy.Base.Structures.Sigma.Homs.mon 𝑨 𝑩 →\nLegacy.Base.Structures.Sigma.Homs.hom 𝑨 𝑩"},{"defKind":"function","hasBody":true,"module":"Setoid.Homomorphisms.Basic","prettyQname":"Setoid.Homomorphisms.Basic.mon→hom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Agda.Primitive.Level}\n{𝑆 : Overture.Signatures.Signature 𝓞 𝓥}\n(𝑨 : Setoid.Algebras.Basic.Algebra α ρᵃ)\n(𝑩 : Setoid.Algebras.Basic.Algebra β ρᵇ) →\nSetoid.Homomorphisms.Basic.mon 𝑨 𝑩 →\nSetoid.Homomorphisms.Basic.hom 𝑨 𝑩"}]
  6. thought
    The model thought here, about 269 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.
  7. agda-mcp
    normalize(expr: λ {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ) (𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ) → mon 𝑨 𝑩)
    answered
    normalForm: λ {𝓞} {𝓥} {α} {ρᵃ} {β} {ρᵇ} {𝑆} 𝑨 𝑩 → Data.Product.Σ (Function.Bundles.Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑩)) (IsMon 𝑨 𝑩)elapsedMs: 7
    the whole answer, as the agent saw it (412 characters of JSON)
    {"checkedFromSource":false,"elapsedMs":7,"expr":"λ {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ) (𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ) → mon 𝑨 𝑩","lane":{"load":"reused"},"normalForm":"λ {𝓞} {𝓥} {α} {ρᵃ} {β} {ρᵇ} {𝑆} 𝑨 𝑩 →\n  Data.Product.Σ\n  (Function.Bundles.Func (Algebra.Domain 𝑨) (Algebra.Domain 𝑩))\n  (IsMon 𝑨 𝑩)","project":{"root":"<work>","rootSource":"server-config"},"scope":"toplevel"}
  8. thought
    The model thought here, about 450 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.
  9. agda-mcp
    fill_hole(candidate: _ , IsMon.isHom (proj₂ m), line: 23, col: 15)
    answered
    status: type_errorremainingHoles: 0verdict.exitCode: 42elapsedMs: 5248holes: 0
    the whole answer, as the agent saw it (516 characters of JSON)
    {"candidate":"_ , IsMon.isHom (proj₂ m)","checkedFromSource":true,"elapsedMs":5248,"holes":[],"message":"Checking Homs-mon-to-hom (<work>/Homs-mon-to-hom.agda).\n<work>/Homs-mon-to-hom.agda:23.16-17: error: [NotInScope]\nNot in scope:\n  ,\n  at <work>/Homs-mon-to-hom.agda:23.16-17\n    (did you mean\n       'Data.Product._,_' or\n       'Setoid.Algebras._,_'?)\nwhen scope checking ,\n\n","project":{"root":"<work>","rootSource":"server-config"},"remainingHoles":0,"status":"type_error","verdict":{"exitCode":42}}
  10. thought
    The model thought here, about 186 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.
  11. agda-mcp
    fill_hole(candidate: Data.Product._,_ _ (IsMon.isHom (proj₂ m)), line: 23, col: 15)
    answered
    status: okremainingHoles: 0verdict.exitCode: 0elapsedMs: 4701holes: 0
    the whole answer, as the agent saw it (226 characters of JSON)
    {"candidate":"Data.Product._,_ _ (IsMon.isHom (proj₂ m))","checkedFromSource":true,"elapsedMs":4701,"holes":[],"project":{"root":"<work>","rootSource":"server-config"},"remainingHoles":0,"status":"ok","verdict":{"exitCode":0}}
  12. thought
    The model thought here, about 184 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.
  13. the work directory
    Edit(replace_all: false, old_string: mon→hom′ m = {!!}, new_string: -- A monomorphism is a homomorphism together with an injectivity proof, -- so the underlying hom is obtained by forgetting that extra structure. mon→hom′ m = Da…)
    answered
    The file <work>/Homs-mon-to-hom.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>/Homs-mon-to-hom.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  14. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5191diagnostics: 0holes: 0
    the whole answer, as the agent saw it (221 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5191,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  15. the model
    check_file reports success (exit code 0, no diagnostics, no holes) on the final file, where the hole is filled with `Data.Product._,_ _ (IsMon.isHom (proj₂ m))`.

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 1,335 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 9
the file the judge read, marked against the obligation it started as
-- Homs-mon-to-hom.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Homs-mon-to-hom.agda
--
-- Benchmark obligation: algebras-homs-mon-to-hom
-- Difficulty: compositional
-- Import stratum: wholesale
--
module Homs-mon-to-hom where

open import AgdaDojang.Debug

open import Agda.Primitive   using ( Level )
open import Data.Product     using ( proj₂ )

open import Overture             using ( Signature )
open import Setoid.Algebras      using ( Algebra )
open import Setoid.Homomorphisms

mon→hom′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
           {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
  →  mon 𝑨 𝑩 → hom 𝑨 𝑩
-- A monomorphism is a homomorphism together with an injectivity proof,
-- so the underlying hom is obtained by forgetting that extra structure.
mon→hom′ m = {!!}
mon→hom′ m = Data.Product._,_ _ (IsMon.isHom (proj₂ m))

reports/agent-bench/suite219-opus5-mcp-1/subjects/algebras-homs-mon-to-hom/final/Homs-mon-to-hom.agda

mon→hom′ Sonnet 5 restated

The same obligation and the same tools. Two `exports_of` queries find `mon→hom`, and the session calls it. The file type-checks; the judge counts it restated, not solved.

obligation
algebras-homs-mon-to-hom
tier
compositional
stratum
agda-algebras/wholesale
the library original
Setoid.Homomorphisms.Basic.mon→hom
model
claude-sonnet-5
turns
6
tool calls
5
wall
32 s
cost, list price
USD 0.08
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} →  mon 𝑨 𝑩 → hom 𝑨 𝑩
the session, 6 turns and 5 tool calls
  1. 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.
  2. the work directory
    Read()
    answered
    1 -- Homs-mon-to-hom.agda
    the whole answer, as the agent saw it (721 characters of text)
    1	-- Homs-mon-to-hom.agda
2	--
3	-- File: data/benchmarks/agda-algebras-v0/obligations/Homs-mon-to-hom.agda
4	--
5	-- Benchmark obligation: algebras-homs-mon-to-hom
6	-- Difficulty: compositional
7	-- Import stratum: wholesale
8	--
9	module Homs-mon-to-hom where
10	
11	open import AgdaDojang.Debug
12	
13	open import Agda.Primitive   using ( Level )
14	open import Data.Product     using ( proj₂ )
15	
16	open import Overture             using ( Signature )
17	open import Setoid.Algebras      using ( Algebra )
18	open import Setoid.Homomorphisms
19	
20	mon→hom′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
21	           {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
22	  →  mon 𝑨 𝑩 → hom 𝑨 𝑩
23	mon→hom′ m = {!!}
24
  3. 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.
  4. agda-mcp
    exports_of(module: Setoid.Homomorphisms, pattern: mon)
    one of 2 calls the model issued in this turn
    answered
    elapsedMs: 4775exports: 4modules: 1
    the whole answer, as the agent saw it (1,119 characters of JSON)
    {"checkedFromSource":true,"elapsedMs":4775,"exports":[{"name":"IsMon","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ)\n(h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n   Setoid.Algebras.𝔻[ 𝑩 ]) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"mon","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ) →\nAlgebra β ρᵇ →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ β\n Agda.Primitive.⊔ ρᵇ)"},{"name":"mon→hom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nmon 𝑨 𝑩 → hom 𝑨 𝑩"},{"name":"mon→intohom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ) →\nmon 𝑨 𝑩 →\nSetoid.Algebras.Σ-syntax (hom 𝑨 𝑩)\n(λ h → Setoid.Functions.Injective.IsInjective (Overture.proj₁ h))"}],"lane":{"load":"first","loadElapsedMs":4694},"module":"Setoid.Homomorphisms","modules":["IsMon"],"project":{"root":"<work>","rootSource":"server-config"},"total":4,"truncated":false}
  5. agda-mcp
    exports_of(module: Setoid.Homomorphisms, pattern: hom)
    one of 2 calls the model issued in this turn
    answered
    elapsedMs: 50exports: 20modules: 1
    the whole answer, as the agent saw it (7,818 characters of JSON)
    {"checkedFromSource":false,"elapsedMs":50,"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":"FirstHomTheorem","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (hh : hom 𝑨 𝑩) →\nSetoid.Algebras.Σ-syntax (hom (kerquo hh) 𝑩)\n(λ .patternInTele0 →\n   ((a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n    (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n     Setoid.Homomorphisms.Noether.h hh a)\n    (Function.Bundles.Func.to (.patternInTele0 .Overture.proj₁)\n     (Function.Bundles.Func.to (πker hh .Overture.proj₁) a)))\n   Data.Product.×\n   Setoid.Functions.Injective.IsInjective\n   (.patternInTele0 .Overture.proj₁))"},{"name":"FirstHomUnique","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (hh : hom 𝑨 𝑩)\n{.patternInTele0 .patternInTele1 : hom (kerquo hh) 𝑩} →\n((a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n  Setoid.Homomorphisms.Noether.h hh a)\n (Function.Bundles.Func.to (.patternInTele0 .Overture.proj₁)\n  (Function.Bundles.Func.to (πker hh .Overture.proj₁) a))) →\n((a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n  Setoid.Homomorphisms.Noether.h hh a)\n (Function.Bundles.Func.to (.patternInTele1 .Overture.proj₁)\n  (Function.Bundles.Func.to (πker hh .Overture.proj₁) a))) →\n([a]\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Algebra.Domain (kerquo hh))) →\n(Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to (.patternInTele0 .Overture.proj₁) [a])\n(Function.Bundles.Func.to (.patternInTele1 .Overture.proj₁) [a])"},{"name":"HomFactor","type":"{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} (.patternInTele0 : hom 𝑨 𝑩)\n(.patternInTele1 : hom 𝑨 𝑪) →\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑪 ])\n(Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1)\nRelation.Unary.⊆\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑩 ])\n(Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1) →\nSetoid.Functions.Surjective.IsSurjective\n(.patternInTele1 .Overture.proj₁) →\nSetoid.Algebras.Σ-syntax (hom 𝑪 𝑩)\n(λ .patternInTele2 →\n   (a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n   (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n    Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1 a)\n   (Function.Bundles.Func.to (.patternInTele2 .Overture.proj₁)\n    (Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1 a)))"},{"name":"HomFactorEpi","type":"{𝓞 𝓥 α ρᵃ β ρᵇ γ ρᶜ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} {𝑪 : Algebra γ ρᶜ} (.patternInTele0 : hom 𝑨 𝑩)\n(.patternInTele1 : hom 𝑨 𝑪) →\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑪 ])\n(Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1)\nRelation.Unary.⊆\nOverture.kernelRel\n(Relation.Binary.Bundles.Setoid._≈_ Setoid.Algebras.𝔻[ 𝑩 ])\n(Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1) →\nSetoid.Functions.Surjective.IsSurjective\n(.patternInTele1 .Overture.proj₁) →\nSetoid.Functions.Surjective.IsSurjective\n(.patternInTele0 .Overture.proj₁) →\nSetoid.Algebras.Σ-syntax (epi 𝑪 𝑩)\n(λ .patternInTele2 →\n   (a : Relation.Binary.Bundles.Setoid.Carrier (Algebra.Domain 𝑨)) →\n   (Setoid.Algebras.𝔻[ 𝑩 ] Relation.Binary.Bundles.Setoid.≈\n    Setoid.Homomorphisms.Factor.g .patternInTele0 .patternInTele1 a)\n   (Function.Bundles.Func.to (.patternInTele2 .Overture.proj₁)\n    (Setoid.Homomorphisms.Factor.h .patternInTele0 .patternInTele1 a)))"},{"name":"HomImage-≅","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 𝑨' : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\n𝑨 IsHomImageOf 𝑨' → 𝑨 ≅ 𝑩 → 𝑩 IsHomImageOf 𝑨'"},{"name":"HomImage-≅'","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 𝑨' : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\n𝑨 IsHomImageOf 𝑨' → 𝑨' ≅ 𝑩 → 𝑨 IsHomImageOf 𝑩"},{"name":"HomImageOfClass","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level} →\nRelation.Unary.Pred (Algebra α ρᵃ) (Agda.Primitive.lsuc α) →\nSet\n(𝑆.𝓞 Agda.Primitive.⊔ 𝑆.𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ)"},{"name":"HomImageOf[_]","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} →\nhom 𝑨 𝑩 → Algebra (α Agda.Primitive.⊔ β Agda.Primitive.⊔ ρᵇ) ρᵇ"},{"name":"HomImages","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level} →\nAlgebra α ρᵃ →\nSet\n(𝑆.𝓞 Agda.Primitive.⊔ 𝑆.𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ Agda.Primitive.lsuc β\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵇ)"},{"name":"HomKerComp","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra α ρᵃ}\n{𝑩 : Algebra β ρᵇ} (.patternInTele0 : hom 𝑨 𝑩) →\n𝑨 Setoid.Congruences.Basic.∣≈\nOverture.kerRel\n(Relation.Binary.Bundles.Setoid._≈_ (Algebra.Domain 𝑩))\n(Setoid.Homomorphisms.Kernels.h .patternInTele0)"},{"name":"IdHomImage","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝑨 : Algebra α ρᵃ} →\n𝑨 IsHomImageOf 𝑨"},{"name":"IsHom","type":"{𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥} (𝑨 : Algebra α ρᵃ)\n(𝑩 : Algebra β ρᵇ)\n(h\n : Function.Bundles.Func Setoid.Algebras.𝔻[ 𝑨 ]\n   Setoid.Algebras.𝔻[ 𝑩 ]) →\nSet\n(𝓞 Agda.Primitive.⊔ 𝓥 Agda.Primitive.⊔ α Agda.Primitive.⊔ ρᵃ\n Agda.Primitive.⊔ ρᵇ)"},{"name":"IsHomImageOfClass","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ : Level}\n{𝒦 : Relation.Unary.Pred (Algebra α ρᵃ) (Agda.Primitive.lsuc α)} →\nAlgebra α ρᵃ →\nSet\n(𝑆.𝓞 Agda.Primitive.⊔ 𝑆.𝓥 Agda.Primitive.⊔ Agda.Primitive.lsuc α\n Agda.Primitive.⊔ Agda.Primitive.lsuc ρᵃ)"},{"name":"Lift-Alg-hom-imageˡ","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} (ℓᵃ ℓᵇ : Level) →\n𝑩 IsHomImageOf 𝑨 →\nSetoid.Algebras.Lift-Algˡ 𝑩 ℓᵇ IsHomImageOf\nSetoid.Algebras.Lift-Algˡ 𝑨 ℓᵃ"},{"name":"Lift-HomImage-lemma","type":"{𝑆.𝓞 𝑆.𝓥 : Level} {𝑆 : Signature 𝑆.𝓞 𝑆.𝓥} {α ρᵃ β ρᵇ : Level}\n{𝑨 : Algebra α ρᵃ} {𝑩 : Algebra β ρᵇ} {γ : Level} →\nSetoid.Algebras.Lift-Alg 𝑨 γ γ IsHomImageOf 𝑩 → 𝑨 IsHomImageOf 𝑩"},{"name":"Lift-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)"}],"lane":{"load":"reused"},"module":"Setoid.Homomorphisms","modules":["IsHom"],"nextOffset":20,"project":{"root":"<work>","rootSource":"server-config"},"remaining":["Lift-homˡ","epi→hom","epi→ontohom","hom","lift-hom-lemma","mkIsHom","mkhom","mon→hom","mon→intohom","πhom","⊙-hom","⊙-is-hom","⨅-hom","⨅-hom-co"],"total":34,"truncated":true}
  6. the model
    There's already a `mon→hom` function with exactly the type we need (applied to 𝑨 𝑩 implicit args). Let's just use it.
  7. the work directory
    Edit(replace_all: false, old_string: mon→hom′ m = {!!}, new_string: mon→hom′ m = mon→hom _ _ m)
    answered
    The file <work>/Homs-mon-to-hom.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>/Homs-mon-to-hom.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  8. 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.
  9. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5473diagnostics: 0holes: 0
    the whole answer, as the agent saw it (221 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5473,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  10. 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 90 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.Basic.mon→hom`: 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 6
restatement evidence ref Setoid.Homomorphisms.Basic.mon→hom
the file the judge read, marked against the obligation it started as
-- Homs-mon-to-hom.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Homs-mon-to-hom.agda
--
-- Benchmark obligation: algebras-homs-mon-to-hom
-- Difficulty: compositional
-- Import stratum: wholesale
--
module Homs-mon-to-hom where

open import AgdaDojang.Debug

open import Agda.Primitive   using ( Level )
open import Data.Product     using ( proj₂ )

open import Overture             using ( Signature )
open import Setoid.Algebras      using ( Algebra )
open import Setoid.Homomorphisms

mon→hom′ : {𝓞 𝓥 α ρᵃ β ρᵇ : Level} {𝑆 : Signature 𝓞 𝓥}
           {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} {𝑩 : Algebra β ρᵇ}
  →  mon 𝑨 𝑩 → hom 𝑨 𝑩
mon→hom′ m = {!!}
mon→hom′ m = mon→hom _ _ m

reports/agent-bench/suite219-sonnet5-mcp-1/subjects/algebras-homs-mon-to-hom/final/Homs-mon-to-hom.agda

≤-reflexive′ Opus 5 solved

A wholesale row, where the fixture names nothing useful. The session reads the definition of `_≤_` where `definition_of` points, and asks `search_in_scope` for the identity homomorphism: one row in scope, and it is the one the proof uses.

obligation
algebras-subalgebras-sub-reflexive
tier
non-obvious
stratum
agda-algebras/wholesale
the library original
Setoid.Subalgebras.Properties.≤-reflexive
model
claude-opus-5
turns
11
tool calls
10
wall
58 s
cost, list price
USD 0.34
the hole’s goal, as get_goal reported it
{𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥} {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ} →  𝑨 ≤ 𝑨
the session, 11 turns and 10 tool calls
  1. the model
    I'll start by reading the file.
  2. the work directory
    Read()
    answered
    1 -- Subalgebras-sub-reflexive.agda
    the whole answer, as the agent saw it (814 characters of text)
    1	-- Subalgebras-sub-reflexive.agda
2	--
3	-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-reflexive.agda
4	--
5	-- Benchmark obligation: algebras-subalgebras-sub-reflexive
6	-- Difficulty: non-obvious
7	-- Import stratum: wholesale
8	--
9	module Subalgebras-sub-reflexive where
10	
11	open import AgdaDojang.Debug
12	
13	open import Agda.Primitive   using ( Level )
14	open import Data.Product     using ( _,_ )
15	
16	open import Overture              using ( Signature )
17	open import Setoid.Algebras       using ( Algebra ; 𝔻[_] )
18	open import Setoid.Functions
19	open import Setoid.Homomorphisms
20	open import Setoid.Subalgebras
21	
22	≤-reflexive′ : {𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥}
23	               {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}
24	  →  𝑨 ≤ 𝑨
25	≤-reflexive′ {𝑨 = 𝑨} = {!!}
26
  3. 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.
  4. agda-mcp
    get_goal(line: 25, column: 23)
    refused
    No hole at line 25, column 23 in <work>/Subalgebras-sub-reflexive.agda (a position addresses the hole whose span contain…
    the whole answer, as the agent saw it (516 characters of text)
    No hole at line 25, column 23 in <work>/Subalgebras-sub-reflexive.agda (a position addresses the hole whose span contains it; starting at it counts).
  nearest holes (1 in the file):
    index 0 at line 25, column 24
  Address a hole by the (line, column) its own listing reports — get_diagnostics.holes, check_file.holes, or the holes list every fill_hole response carries.
  Take that listing from the latest response: a fill above a hole moves it when the candidate is a different length, and renumbers it always.
  5. agda-mcp
    definition_of(name: _≤_)
    answered
    elapsedMs: 5695definitions: 1
    the whole answer, as the agent saw it (386 characters of JSON)
    {"checkedFromSource":true,"definitions":[{"col":3,"endCol":6,"endLine":102,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Setoid/Subalgebras/Basic.lagda.md","line":102,"qualified":"Setoid.Subalgebras.Basic._._≤_"}],"elapsedMs":5695,"lane":{"load":"first","loadElapsedMs":5666},"name":"_≤_","project":{"root":"<work>","rootSource":"server-config"},"scope":"toplevel","unlocated":[]}
  6. the work directory
    Read(offset: 70, limit: 60)
    answered
    70 open import Overture using ( proj₁ ; proj₂ ; Signature ; 𝓞 ; 𝓥 )
    the whole answer, as the agent saw it (2,900 characters of text)
    70	open import Overture              using ( proj₁ ; proj₂ ; Signature ; 𝓞 ; 𝓥 )
71	open import Setoid.Algebras       using ( Algebra ; ov )
72	open import Setoid.Functions      using ( IsInjective )
73	open import Setoid.Homomorphisms  using ( hom ; mon ; mon→intohom ; kerquo ; FirstHomTheorem )
74	
75	private variable α ρᵃ β ρᵇ ℓ : Level
76	```
77	-->
78	
79	The relation comes in two directions and three packagings.
80	
81	+  `_IsSubalgebraOf_`{.AgdaFunction}, with the infix alias `_≤_`{.AgdaFunction},
82	   is the relation itself: `𝑨 ≤ 𝑩` is the type of pairs `(h , inj)` with
83	   `h : hom 𝑨 𝑩` and `inj` a proof that the underlying map of `h` is injective.
84	+  `_IsSupalgebraOf_`{.AgdaFunction}, aliased `_≥_`{.AgdaFunction}, is the
85	   converse, and is *definitionally* the same type read the other way round:
86	   `𝑨 ≤ 𝑩` and `𝑩 ≥ 𝑨` both unfold to an injective homomorphism from `𝑨` into
87	   `𝑩`.
88	+  `mon→≤`{.AgdaFunction} turns a monomorphism into a proof of the subalgebra
89	   relation.  It is immediate, because `mon→intohom`{.AgdaFunction} of
90	   [Setoid.Homomorphisms.Basic][] already produces exactly this type.
91	+  `SubalgebraOf`{.AgdaRecord} bundles both algebras and the proof into a single
92	   record.  `Subalgebra`{.AgdaFunction} instead fixes the larger algebra and
93	   collects the smaller one with its embedding, so an inhabitant of
94	   `Subalgebra 𝑨` is a pair `(𝑩 , p)` with `p : 𝑩 ≤ 𝑨`.
95	
96	```agda
97	module _ {𝑆 : Signature 𝓞 𝓥} where
98	  _≥_   -- alias for supalgebra (aka overalgebra)
99	    _IsSupalgebraOf_ : Algebra {𝑆 = 𝑆} α ρᵃ → Algebra {𝑆 = 𝑆} β ρᵇ → Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ ⊔ β ⊔ ρᵇ)
100	  𝑨 IsSupalgebraOf 𝑩 = Σ[ h ∈ hom 𝑩 𝑨 ] IsInjective (proj₁ h)
101	
102	  _≤_   -- alias for subalgebra relation
103	    _IsSubalgebraOf_ : Algebra {𝑆 = 𝑆} α ρᵃ → Algebra {𝑆 = 𝑆} β ρᵇ → Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ ρᵃ ⊔ β ⊔ ρᵇ)
104	  𝑨 IsSubalgebraOf 𝑩 = Σ[ h ∈ hom 𝑨 𝑩 ] IsInjective (proj₁ h)
105	
106	  -- Syntactic sugar for sup/sub-algebra relations.
107	  𝑨 ≥ 𝑩 = 𝑨 IsSupalgebraOf 𝑩
108	  𝑨 ≤ 𝑩 = 𝑨 IsSubalgebraOf 𝑩
109	
110	  mon→≤ : {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}{𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ} → mon 𝑨 𝑩 → 𝑨 ≤ 𝑩
111	  mon→≤ {𝑨 = 𝑨}{𝑩} x = mon→intohom 𝑨 𝑩 x
112	
113	  record SubalgebraOf : Type (ov {𝑆 = 𝑆} (α ⊔ β ⊔ ρᵃ ⊔ ρᵇ)) where
114	    field
115	      algebra : Algebra {𝑆 = 𝑆} α ρᵃ
116	      subalgebra : Algebra {𝑆 = 𝑆} β ρᵇ
117	      issubalgebra : subalgebra ≤ algebra
118	
119	  Subalgebra : Algebra {𝑆 = 𝑆} α ρᵃ → {β ρᵇ : Level} → Type _
120	  Subalgebra 𝑨 {β}{ρᵇ} = Σ[ 𝑩 ∈ (Algebra β ρᵇ) ] 𝑩 ≤ 𝑨
121	
122	{- usage note: for 𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ, an inhabitant of `Subalgebra 𝑨` is a pair
123	   `(𝑩 , p) : Subalgebra 𝑨`  providing
124	   - `𝑩 : Algebra {𝑆 = 𝑆} β ρᵇ` and
125	   - `p : 𝑩 ≤ 𝑨`, a proof that 𝑩 is a subalgebra of 𝐴. -}
126	```
127	
128	From now on we will use `𝑩 ≤ 𝑨` to express the assertion that `𝑩` is a subalgebra of `𝑨`.
129
  7. agda-mcp
    get_goal(line: 25, column: 24)
    answered
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    the whole answer, as the agent saw it (833 characters of JSON)
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  8. 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.
  9. agda-mcp
    search_in_scope(query: {"name": "𝒾𝒹"})
    answered
    elapsedMs: 14
    the whole answer, as the agent saw it (1,425 characters of JSON)
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  10. agda-mcp
    search_by_name(pattern: id, limit: 40)
    answered
    results: 40
    the whole answer, as the agent saw it (15,892 characters of JSON)
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Classical.Bundles.Monoid.⟨ .patternInTele0 ⟩ᵐᵒ ⟫ᵐᵒ\n  Classical.Structures.Monoid.Monoid-Op.∙ a)\n b)\n((.patternInTele0 Classical.Structures.Monoid.Monoid-Op.∙ a) b)"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Monoid","prettyQname":"Classical.Bundles.Monoid.setoid","type":"{α ρ : Agda.Primitive.Level} (M : Algebra.Bundles.Monoid α ρ) →\nRelation.Binary.Bundles.Setoid α ρ"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Monoid","prettyQname":"Classical.Bundles.Monoid.ε","type":"{α ρ : Agda.Primitive.Level}\n{.patternInTele0 : Classical.Structures.Monoid.Monoid α ρ} →\nSetoid.Algebras.Basic.𝕌[\nClassical.Structures.Monoid.Monoid-Op.𝑨\nClassical.Bundles.Monoid.⟪\nClassical.Bundles.Monoid.⟨ .patternInTele0 ⟩ᵐᵒ ⟫ᵐᵒ\n]"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Monoid","prettyQname":"Classical.Bundles.Monoid.⟨_⟩ᵐᵒ","type":"{α ρ : Agda.Primitive.Level} →\nClassical.Structures.Monoid.Monoid α ρ → Algebra.Bundles.Monoid α ρ"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Monoid","prettyQname":"Classical.Bundles.Monoid.⟪_⟫ᵐᵒ","type":"{α ρ : Agda.Primitive.Level} →\nAlgebra.Bundles.Monoid α ρ → Classical.Structures.Monoid.Monoid α ρ"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Monoid","prettyQname":"Classical.Bundles.Monoid.𝑨","type":"{α ρ : Agda.Primitive.Level} (M : Algebra.Bundles.Monoid α ρ) →\nSetoid.Algebras.Basic.Algebra α ρ"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Ring","prettyQname":"Classical.Bundles.Ring.setoid","type":"{α ρ : Agda.Primitive.Level} (R : Algebra.Bundles.Ring α ρ) →\nRelation.Binary.Bundles.Setoid α ρ"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Semigroup","prettyQname":"Classical.Bundles.Semigroup.setoid","type":"{α ρ : Agda.Primitive.Level} (S : Algebra.Bundles.Semigroup α ρ) →\nRelation.Binary.Bundles.Setoid α ρ"},{"defKind":"function","hasBody":true,"module":"Classical.Bundles.Semilattice","prettyQname":"Classical.Bundles.Semilattice.setoid","type":"{α ρ : Agda.Primitive.Level}\n(S : Algebra.Lattice.Bundles.Semilattice α ρ) →\nRelation.Binary.Bundles.Setoid α ρ"},{"defKind":"function","hasBody":true,"module":"Classical.Categories.AdjoinUnit.AdjoinUnit","prettyQname":"Classical.Categories.AdjoinUnit.AdjoinUnit.∙¹-idʳ","type":"{α ρ : Agda.Primitive.Level}\n(𝑺 : Classical.Structures.Semigroup.Semigroup α ρ) →\nAlgebra.Definitions.RightIdentity\n(Classical.Categories.AdjoinUnit.AdjoinUnit._≈¹_ 𝑺)\nAgda.Builtin.Maybe.Maybe.nothing\n(Classical.Categories.AdjoinUnit.AdjoinUnit._∙¹_ 𝑺)"}]
  11. 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.
  12. agda-mcp
    fill_hole(line: 25, col: 24, candidate: 𝒾𝒹 , (λ z → z))
    answered
    status: okremainingHoles: 0verdict.exitCode: 0elapsedMs: 5392holes: 0
    the whole answer, as the agent saw it (198 characters of JSON)
    {"candidate":"𝒾𝒹 , (λ z → z)","checkedFromSource":true,"elapsedMs":5392,"holes":[],"project":{"root":"<work>","rootSource":"server-config"},"remainingHoles":0,"status":"ok","verdict":{"exitCode":0}}
  13. the work directory
    Edit(replace_all: false, old_string: ≤-reflexive′ {𝑨 = 𝑨} = {!!}, new_string: ≤-reflexive′ {𝑨 = 𝑨} = 𝒾𝒹 , (λ z → z))
    answered
    The file <work>/Subalgebras-sub-reflexive.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-reflexive.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  14. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5374diagnostics: 0holes: 0
    the whole answer, as the agent saw it (222 characters of JSON)
    {"checkedFromSource":false,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5374,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  15. the model
    check_file reports success (exit code 0, no diagnostics, no holes) on the final file, with the hole filled by `𝒾𝒹 , (λ z → z)`.

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 291 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 11
the file the judge read, marked against the obligation it started as
-- Subalgebras-sub-reflexive.agda
--
-- File: data/benchmarks/agda-algebras-v0/obligations/Subalgebras-sub-reflexive.agda
--
-- Benchmark obligation: algebras-subalgebras-sub-reflexive
-- Difficulty: non-obvious
-- Import stratum: wholesale
--
module Subalgebras-sub-reflexive where

open import AgdaDojang.Debug

open import Agda.Primitive   using ( Level )
open import Data.Product     using ( _,_ )

open import Overture              using ( Signature )
open import Setoid.Algebras       using ( Algebra ; 𝔻[_] )
open import Setoid.Functions
open import Setoid.Homomorphisms
open import Setoid.Subalgebras

≤-reflexive′ : {𝓞 𝓥 α ρᵃ : Level} {𝑆 : Signature 𝓞 𝓥}
               {𝑨 : Algebra {𝑆 = 𝑆} α ρᵃ}
  →  𝑨 ≤ 𝑨
≤-reflexive′ {𝑨 = 𝑨} = {!!}
≤-reflexive′ {𝑨 = 𝑨} = 𝒾𝒹 , (λ z → z)

reports/agent-bench/suite219-opus5-mcp-1/subjects/algebras-subalgebras-sub-reflexive/final/Subalgebras-sub-reflexive.agda

*-comm Sonnet 5 refused

A correct induction, and 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
9
tool calls
8
wall
61 s
cost, list price
USD 0.12
the hole’s goal, as get_goal reported it
∀ (m n : ℕ) → m * n ≡ n * m
the session, 9 turns and 8 tool calls
  1. 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.
  2. the work directory
    Read()
    answered
    1 -- Nat-mul-comm.agda
    the whole answer, as the agent saw it (602 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	--
8	module Nat-mul-comm where
9	
10	open import AgdaDojang.Debug
11	
12	open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ )
13	open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym )
14	
15	-- Prerequisites (provided, not to be proved here)
16	open import Data.Nat.Properties using ( *-zeroʳ ; *-suc )
17	
18	*-comm : ∀ (m n : ℕ) → m * n ≡ n * m
19	*-comm m n = {!!}
20
  3. 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.
  4. the work directory
    Edit(replace_all: false, old_string: open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ ) open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ) -- Prerequis…, new_string: open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ ) open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ; trans ) -- P…)
    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)
  5. 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.
  6. agda-mcp
    check_file()
    answered
    success: falseholesCount: 0diagnosticsTotal: 1verdict.exitCode: 42elapsedMs: 2110diagnostics: 1holes: 0
    the whole answer, as the agent saw it (537 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[{"code":"UnequalTerms","col":69,"file":"<work>/Nat-mul-comm.agda","involved":{"actual":"n","expected":"n * m"},"line":21,"message":"n != n * m of type ℕ\nwhen checking that the expression *-suc n m has type\nn * suc m ≡ n * m + n","range":{"endCol":78,"endLine":21,"startCol":69,"startLine":21},"severity":"error"}],"diagnosticsTotal":1,"elapsedMs":2110,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":false,"timedOut":false,"verdict":{"exitCode":42}}
  7. 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.
  8. agda-mcp
    type_of(expr: *-suc)
    answered
    error.code: UnequalTermselapsedMs: 2103
    the whole answer, as the agent saw it (394 characters of JSON)
    {"checkedFromSource":true,"elapsedMs":2103,"error":{"code":"UnequalTerms","message":"<work>/Nat-mul-comm.agda:21.69-78: error: [UnequalTerms]\nn != n * m of type ℕ\nwhen checking that the expression *-suc n m has type\nn * suc m ≡ n * m + n","stage":"load"},"expr":"*-suc","lane":{"load":"first","loadElapsedMs":2059},"project":{"root":"<work>","rootSource":"server-config"},"scope":"toplevel"}
  9. 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.
  10. agda-mcp
    search_by_type(pattern: suc n ≡)
    answered
    results: 0
    the whole answer, as the agent saw it (2 characters of JSON)
    []
  11. agda-mcp
    search_by_name(pattern: *-suc)
    answered
    results: 3
    the whole answer, as the agent saw it (866 characters of JSON)
    [{"defKind":"function","hasBody":true,"module":"Data.Integer.Properties","prettyQname":"Data.Integer.Properties.*-suc","type":"(i j : Agda.Builtin.Int.Int) →\n(i Data.Integer.Base.* Data.Integer.Base.suc j)\nAgda.Builtin.Equality.≡\n(i Data.Integer.Base.+ (i Data.Integer.Base.* j))"},{"defKind":"function","hasBody":true,"module":"Data.Nat.Binary.Properties","prettyQname":"Data.Nat.Binary.Properties.*-suc","type":"(x y : Data.Nat.Binary.Base.ℕᵇ) →\n(x Data.Nat.Binary.Base.* Data.Nat.Binary.Base.suc y)\nAgda.Builtin.Equality.≡\n(x Data.Nat.Binary.Base.+ (x Data.Nat.Binary.Base.* y))"},{"defKind":"function","hasBody":true,"module":"Data.Nat.Properties","prettyQname":"Data.Nat.Properties.*-suc","type":"(m n : Agda.Builtin.Nat.Nat) →\n(m Agda.Builtin.Nat.* Agda.Builtin.Nat.Nat.suc n)\nAgda.Builtin.Equality.≡\n(m Agda.Builtin.Nat.+ (m Agda.Builtin.Nat.* n))"}]
  12. 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.
  13. the work directory
    Edit(replace_all: false, old_string: open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ ) open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ; trans ) -- P…, new_string: open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ ) open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym ; trans ) -- P…)
    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)
  14. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 1837diagnostics: 0holes: 0
    the whole answer, as the agent saw it (221 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":1837,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  15. 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 3,016 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 9
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)
--
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/suite219-sonnet5-mcp-1/subjects/stdlib-nat-mul-comm/final/Nat-mul-comm.agda

*-comm Opus 5 solved

The same obligation and the same need for `trans`, in four turns. 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
27 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
  1. the model
    I'll start by reading the file.
  2. the work directory
    Read()
    answered
    1 -- Nat-mul-comm.agda
    the whole answer, as the agent saw it (602 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	--
8	module Nat-mul-comm where
9	
10	open import AgdaDojang.Debug
11	
12	open import Data.Nat.Base using ( ℕ ; zero ; suc ; _+_ ; _*_ )
13	open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; cong ; sym )
14	
15	-- Prerequisites (provided, not to be proved here)
16	open import Data.Nat.Properties using ( *-zeroʳ ; *-suc )
17	
18	*-comm : ∀ (m n : ℕ) → m * n ≡ n * m
19	*-comm m n = {!!}
20
  3. thought
    The model thought here, about 600 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.
  4. 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)
  5. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 2512diagnostics: 0holes: 0
    the whole answer, as the agent saw it (221 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":2512,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  6. 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 734 thinking tokens over the session; their text is not in the archive.

solvedthe judge, in outcome.json

Every gate passed, so the row counts as a solve. Its index entry names no library original, so the judge looks for no citation of one.

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)
--
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/suite219-opus5-mcp-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 model can build the 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 the judge told them apart from Agda’s own terms, not from a reading of the text.

What the rule cannot see

The rule reads references, so it catches a proof that cites the library’s lemma and not one that transcribes the lemma’s own proof under a new name: a transcription passes as a solve. So on every row with an original the judge also reports, beside its verdict and deciding nothing, whether a line of the original’s own proof came back in one of the session’s tool answers before its last edit of the file (#188). For the sessions on this page that reading is as follows:

The other two sessions are on one row whose index entry names no original, so the judge makes no such reading there. None of the three sessions on a row with an original had its proof in view: each of their files, solve or restatement, was written without the library’s proof in front of the model. Every arm on this page could read the libraries’ sources; the control says what that does to the counts.

The numbers

The sessions above come from two arms, one subject per obligation, run three at a time on 2026-09-29. Every figure below is read from the archive at build time, and the two tables ADR 0001 § 9 states, this one and the control’s, are also compared with the ADR cell by cell, so a number in either that drifts fails the build. The loop’s two columns are the exception: no report in the repository carries them, so they are read from the ADR, which is their record, and the build checks only that the ADR’s totals for them are the sums of its rows.

The loop is a different instrument: agda-native-air’s own proof-search loop, with no model in it, first over a fixed space of candidate terms and then with corpus retrieval composed around it, and judged by its own final check rather than by the agents’ judge. It is not a baseline for the agents (the guide’s § 1); its columns stand apart at the right of the table, for the tier below that was built to test it.

Solved and restated per stratum, for the two arms the sessions above come from, checked against ADR 0001 § 9’s agent table. The last two columns are the proof-search loop’s solves: a different instrument, with no model in it.
stratumnSonnet 5, with the serverOpus 5, with the serverthe search loop, no model
solvedrestatedsolvedrestatedfixed spaceretrieval
agda-stdlib2221022066
agda-stdlib/haystack1212012006
agda-algebras/using119111022
agda-algebras/wholesale102810000
total55449550814

These columns say what a model does with the server, and nothing about whether the server helps, since every subject had it. The control, at the end of this section, measures that.

claude-sonnet-5

run suite219-sonnet5-mcp-1, Claude Code 2.1.282

  • 44 of 55 solved
  • 9 restated; 2 refused at the preservation gate
  • 347 turns, 292 tool calls
  • USD 3.99 at list price
  • 0 anomalies

claude-opus-5

run suite219-opus5-mcp-1, Claude Code 2.1.282

  • 55 of 55 solved
  • 0 restated
  • 323 turns, 268 tool calls
  • USD 7.07 at list price
  • 0 anomalies
By difficulty tier.
tiernSonnet 5 solvedOpus 5 solved
routine161516
compositional252025
non-obvious14914

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. check_project and get_dependencies and get_diagnostics and resolve_name were presented and never called.
toolSonnet 5Opus 5
Read6664
Edit6956
check_file6155
fill_hole1142
type_of2317
search_by_name217
exports_of230
definition_of1110
get_goal514
normalize02
search_by_type20
search_in_scope01

Two readings the table does not show on its own. Sonnet’s nine restatements are all agda-algebras rows, eight of them from the stratum whose fixtures import whole modules and name nothing useful. And the two rows that are neither solved nor restated include 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.

The control

Every subject of those two arms had the server, so the tables above cannot say whether it helps. The control (#162) can: Sonnet 5 again, over the same 55 obligations with the same prompts, caps, and judge, in three arms run one at a time on 2026-09-29. The shell arm has Bash with the same pinned agda and no server, the mcp arm has the server, and the both arm has both. The mcp arm is the Sonnet arm above, so its column repeats that arm’s counts. Their counts are regenerated from their reports and checked against ADR 0001 § 9 the same way.

The three arms and the Opus arm above share one protocol: the server’s fourteen tools, the libraries’ own sources readable on every arm, the judge’s --safe on the subjects’ servers, and fixture headers that carry no hints (#219). One difference is deliberate: the prompts of the two arms with a shell name the directory holding each library’s sources, which a subject with only the server’s file tools could not search anyway.

The control, solved and restated per stratum, checked against ADR 0001 § 9’s attribution table. Sonnet 5 in every column; the mcp column is the Sonnet arm above.
stratumnshellmcpboth
solvedrestatedsolvedrestatedsolvedrestated
agda-stdlib22220210210
agda-stdlib/haystack12110120110
agda-algebras/using1110091110
agda-algebras/wholesale10442855
total55474449485

What it shows, read with the guide’s § 4.3 and § 5, is as follows:

The measurements before these, each made with the fixture headers’ hints in view, are in the guide with their numbers. The first control (#162, § 4.3) put the shell ahead on the count too. Lean answers (#184) cut the size of the server’s answers and little of its cost: answer size was not what made a server arm cost more than the shell arm. A trimmed tool surface (#191) cut what a server arm reads every turn, and its cost with it, without closing the gap to the shell arm’s cost. And a hard tier of statements posed with no proof on disk (#189, § 4.5) left nothing to transcribe: every file Opus 5 left there, with the server, with a shell, and with both, type-checks with its statement kept, so that tier sits at the model’s ceiling and cannot yet tell the instruments apart. What the headers were worth is the guide’s § 4.6.

One tier whose header named the answer

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; the control’s shell arm, which has no server at all, solves eleven. Sonnet 5 solves three of them asking Agda nothing before its final check (read the file, edit it, check it); on the other nine it asked first, with two exports_of, nine search_by_name, one search_by_type, two type_of. These are standard-library lemmas, in every frontier model’s training data, and Sonnet still looked most of them up. Opus 5 asked first on eleven rows as well, but only by probing lemmas it named itself (fifteen fill_hole, three type_of): it searched for none.

Sonnet did not look them up while the file named them. Until #219 each fixture’s header carried a Needle: line naming its lemma, qualified, and a subject works on a copy of the fixture. With that line in view the same model and server solved eleven of the twelve asking Agda nothing before the final check (run agent-sonnet5-1, 2026-09-15), and an earlier version of this page read that as the model naming the needle from memory. On most rows it was the header.

The tier’s count does not tell the arms apart: every arm solves eleven of the twelve or more. What the header’s removal shows is the route: when the file does not name the lemma, Sonnet 5 reaches for the search tools, and Opus 5 names it from memory and asks Agda whether it fits.

A tier built to need several lemmas

The sessions above come from a suite most of whose rows have the library’s own proof on disk. The composition tier (#160) has none to copy: twelve agda-algebras obligations mined so that each gold strings two to four library lemmas together, the row’s needles, through a root lemma whose premises share a middle point its conclusion does not mention (the y of x ≤ y → y ≤ z → x ≤ z). The search loop solves none of the twelve in three sweeps (comp160-fixed-1, comp160-retrieval-1, comp160-retrieval-idf-1), since fill_hole refuses such a root lemma with holes for its arguments. Opus 5 and Sonnet 5 each ran the tier with a shell, with the server, and with both, one seed each, at caps of 60 turns, 1,800 seconds, and USD 6.00, on 2026-09-29; the guide reads the runs in § 4.7.

The composition tier, per arm, checked against ADR 0001 § 9’s composition table. One seed per arm.
Opus 5Sonnet 5
shellmcpbothshellmcpboth
runcomp-opus5-shell-1comp-opus5-mcp-1comp-opus5-both-1comp-sonnet5-shell-1comp-sonnet5-mcp-1comp-sonnet5-both-1
final file checks, statement kept121212121212
solved101210635
lost to the preservation gate000595
lost to the isolation gate202102
needles in the final files (of 33)202022253226
rows on the gold's whole route5567118
turns130128148157206179
USD (list)4.144.054.872.603.553.39

What it shows is as follows:

Where each needle first appeared, per arm: the first event of each session that names it, by the rule in scripts/python/demo/needles.py (the sweeps skill’s needle-source.py, which the guide cites, ported). Read from the transcripts when this page was built; the ADR has no table of it to be checked against.
first named byOpus 5 shellOpus 5 mcpOpus 5 bothSonnet 5 shellSonnet 5 mcpSonnet 5 both
an answer of the server’s tools0840231
a Read of a file219218722
a shell command’s output280251007
the session’s own words or call, before any answer010020
never named in the session352513
needles333333333333

normal-of-smaller-congruence Sonnet 5 solved

The gold's route. One `exports_of` on the module the fixture opens lists both needles. The first file writes both arguments of `≤ⁿ-trans` and still comes back with unsolved metas at it: `_≤ⁿ_` unfolds to a function space, which fixes no normal subgroup. With the three it relates named, as the gold names them, the file checks.

obligation
comp-group-normal-of-smaller-congruence
tier
compositional
stratum
agda-algebras/composition
the needles
Classical.Structures.Group.Congruences.GroupCongruences.≤ⁿ-trans, Classical.Structures.Group.Congruences.GroupCongruences.normalOf-mono
model
claude-sonnet-5
turns
14
tool calls
13
wall
135 s
cost, list price
USD 0.28
the hole’s goal, as get_goal reported it
{θ φ : Con (proj₁ 𝒢) ℓ} {𝑵 : NormalSubgroup ℓ} → θ ⊆ φ → normalOf φ ≤ⁿ 𝑵 → normalOf θ ≤ⁿ 𝑵
the session, 14 turns and 13 tool calls
  1. 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.
  2. the work directory
    Read()
    answered
    1 -- Group-normal-of-smaller-congruence.agda
    the whole answer, as the agent saw it (1,278 characters of text)
    1	-- Group-normal-of-smaller-congruence.agda
2	--
3	-- File: data/benchmarks/agda-algebras-composition-v0/obligations/Group-normal-of-smaller-congruence.agda
4	--
5	-- Benchmark obligation: comp-group-normal-of-smaller-congruence
6	-- Difficulty: compositional
7	-- Import stratum: composition
8	--
9	-- If θ ⊆ φ and the normal subgroup of φ lies in N, the normal subgroup of θ lies in N.
10	--
11	module Group-normal-of-smaller-congruence where
12	
13	open import AgdaDojang.Debug
14	
15	open import Data.Product                            using ( proj₁ )
16	open import Level                                   using ( Level )
17	open import Classical.Structures.Group.Basic        using ( Group )
18	open import Classical.Structures.Group.Congruences  using ( module GroupCongruences )
19	open import Setoid.Congruences.Basic                using ( Con )
20	open import Setoid.Congruences.Lattice              using ( _⊆_ ; ⊆-refl )
21	
22	module _ {α ρ ℓ : Level} (𝒢 : Group α ρ) where
23	
24	  open GroupCongruences 𝒢  using ( NormalSubgroup ; normalOf ; _≤ⁿ_ ; ≤ⁿ-refl )
25	
26	  normal-of-smaller-congruence : {θ φ : Con (proj₁ 𝒢) ℓ} {𝑵 : NormalSubgroup ℓ}
27	    → θ ⊆ φ → normalOf φ ≤ⁿ 𝑵 → normalOf θ ≤ⁿ 𝑵
28	  normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N = {!!}
29
  3. 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.
  4. agda-mcp
    exports_of(module: GroupCongruences, limit: 0)
    answered
    elapsedMs: 5825exports: 44modules: 3
    the whole answer, as the agent saw it (11,630 characters of JSON)
    {"checkedFromSource":true,"elapsedMs":5825,"exports":[{"name":"_≈ⁿ_","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nGroupCongruences.NormalSubgroup 𝒢 ℓ →\nGroupCongruences.NormalSubgroup 𝒢 ℓ → Set (α Level.⊔ ℓ)"},{"name":"_≤ⁿ_","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nGroupCongruences.NormalSubgroup 𝒢 ℓ →\nGroupCongruences.NormalSubgroup 𝒢 ℓ → Set (α Level.⊔ ℓ)"},{"name":"BelowTrivial","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nRelation.Unary.Pred\n(Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ →\nSet (α Level.⊔ ρ Level.⊔ ℓ)"},{"name":"Nontrivialᴺ","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nRelation.Unary.Pred\n(Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ →\nSet (α Level.⊔ ρ Level.⊔ ℓ)"},{"name":"NormalCongruenceIso","type":"{α ρ : Level} →\nGroup α ρ → (ℓ : Level) → Set (α Level.⊔ ρ Level.⊔ Level.suc ℓ)"},{"name":"NormalCongruenceIso⁻¹","type":"{α ρ : Level} →\nGroup α ρ → (ℓ : Level) → Set (α Level.⊔ ρ Level.⊔ Level.suc ℓ)"},{"name":"NormalRel","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nRelation.Unary.Pred\n(Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ →\nRelation.Binary.Core.Rel\n(Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ"},{"name":"NormalSubgroup","type":"{α ρ : Level} →\nGroup α ρ → (ℓ : Level) → Set (α Level.⊔ ρ Level.⊔ Level.suc ℓ)"},{"name":"NormalSubgroup-Poset","type":"{α ρ : Level} →\nGroup α ρ →\n(ℓ : Level) →\nRelation.Binary.Bundles.Poset (α Level.⊔ ρ Level.⊔ Level.suc ℓ)\n(α Level.⊔ ℓ) (α Level.⊔ ℓ)"},{"name":"below-diagonal→below-trivial","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\nSetoid.Congruences.Monolith.BelowDiagonal\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n(GroupCongruences.congruenceOf 𝒢 𝑵) →\nGroupCongruences.BelowTrivial 𝒢 (GroupCongruences.set 𝒢 𝑵)"},{"name":"below-trivial→below-diagonal","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\nGroupCongruences.BelowTrivial 𝒢 (GroupCongruences.set 𝒢 𝑵) →\nSetoid.Congruences.Monolith.BelowDiagonal\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n(GroupCongruences.congruenceOf 𝒢 𝑵)"},{"name":"con-below-diagonal→below-trivial","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(θ\n : Con (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n   ℓ) →\nSetoid.Congruences.Monolith.BelowDiagonal\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢) θ →\nGroupCongruences.BelowTrivial 𝒢\n(GroupCongruences.set 𝒢 (GroupCongruences.normalOf 𝒢 θ))"},{"name":"con-below-trivial→below-diagonal","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(θ\n : Con (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n   ℓ) →\nGroupCongruences.BelowTrivial 𝒢\n(GroupCongruences.set 𝒢 (GroupCongruences.normalOf 𝒢 θ)) →\nSetoid.Congruences.Monolith.BelowDiagonal\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢) θ"},{"name":"con-nontrivial→nonzero","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(θ\n : Con (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n   ℓ) →\nGroupCongruences.Nontrivialᴺ 𝒢\n(GroupCongruences.set 𝒢 (GroupCongruences.normalOf 𝒢 θ)) →\nSetoid.Congruences.Monolith.Nonzero\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢) θ"},{"name":"con-nonzero→nontrivial","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(θ\n : Con (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n   ℓ) →\nSetoid.Congruences.Monolith.Nonzero\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢) θ →\nGroupCongruences.Nontrivialᴺ 𝒢\n(GroupCongruences.set 𝒢 (GroupCongruences.normalOf 𝒢 θ))"},{"name":"congruenceOf","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nGroupCongruences.NormalSubgroup 𝒢 ℓ →\nCon (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢) ℓ"},{"name":"congruenceOf-cong","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑴 𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\n(𝒢 GroupCongruences.≈ⁿ 𝑴) 𝑵 →\nGroupCongruences.congruenceOf 𝒢 𝑴 Setoid.Congruences.Lattice.≑\nGroupCongruences.congruenceOf 𝒢 𝑵"},{"name":"congruenceOf-mono","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑴 𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\n(𝒢 GroupCongruences.≤ⁿ 𝑴) 𝑵 →\nGroupCongruences.congruenceOf 𝒢 𝑴 ⊆\nGroupCongruences.congruenceOf 𝒢 𝑵"},{"name":"congruenceOf∘normalOf","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(θ\n : Con (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n   ℓ) →\nGroupCongruences.congruenceOf 𝒢 (GroupCongruences.normalOf 𝒢 θ)\nSetoid.Congruences.Lattice.≑ θ"},{"name":"congruence→normal","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(N\n : Relation.Unary.Pred\n   (Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ) →\nClassical.Structures.Group.Subgroups.IsSubgroup 𝒢 N →\nSetoid.Congruences.Basic.IsCongruence\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n(GroupCongruences.NormalRel 𝒢 N) →\nClassical.Structures.Group.Conjugation.Conjugate.IsNormal 𝒢 N"},{"name":"nontrivial→nonzero","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\nGroupCongruences.Nontrivialᴺ 𝒢 (GroupCongruences.set 𝒢 𝑵) →\nSetoid.Congruences.Monolith.Nonzero\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n(GroupCongruences.congruenceOf 𝒢 𝑵)"},{"name":"nonzero→nontrivial","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\nSetoid.Congruences.Monolith.Nonzero\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n(GroupCongruences.congruenceOf 𝒢 𝑵) →\nGroupCongruences.Nontrivialᴺ 𝒢 (GroupCongruences.set 𝒢 𝑵)"},{"name":"normal-congruence-iso","type":"{α ρ : Level} (𝒢 : Group α ρ) (ℓ : Level) →\nGroupCongruences.NormalCongruenceIso 𝒢 ℓ"},{"name":"normal-congruence-iso⁻¹","type":"{α ρ : Level} (𝒢 : Group α ρ) (ℓ : Level) →\nGroupCongruences.NormalCongruenceIso⁻¹ 𝒢 ℓ"},{"name":"normalOf","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nCon (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\nℓ →\nGroupCongruences.NormalSubgroup 𝒢 ℓ"},{"name":"normalOf-cong","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(θ φ\n : Con (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n   ℓ) →\nθ Setoid.Congruences.Lattice.≑ φ →\n(𝒢 GroupCongruences.≈ⁿ GroupCongruences.normalOf 𝒢 θ)\n(GroupCongruences.normalOf 𝒢 φ)"},{"name":"normalOf-mono","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(θ φ\n : Con (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n   ℓ) →\nθ ⊆ φ →\n(𝒢 GroupCongruences.≤ⁿ GroupCongruences.normalOf 𝒢 θ)\n(GroupCongruences.normalOf 𝒢 φ)"},{"name":"normalOf∘congruenceOf","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\n(𝒢 GroupCongruences.≈ⁿ\n GroupCongruences.normalOf 𝒢 (GroupCongruences.congruenceOf 𝒢 𝑵))\n𝑵"},{"name":"set","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nGroupCongruences.NormalSubgroup 𝒢 ℓ →\nRelation.Unary.Pred\n(Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ"},{"name":"set-isSubgroup","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\nClassical.Structures.Group.Subgroups.IsSubgroup 𝒢\n(GroupCongruences.set 𝒢 𝑵)"},{"name":"set-normal","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n(𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ) →\nClassical.Structures.Group.Conjugation.Conjugate.IsNormal 𝒢\n(GroupCongruences.set 𝒢 𝑵)"},{"name":"simpleAlgebra→simple","type":"{α ρ : Level} (𝒢 : Group α ρ) (ℓ₀ : Level) →\nSetoid.Congruences.Simple.IsSimple\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n(α Level.⊔ ρ Level.⊔ ℓ₀) →\nClassical.Structures.Group.Simple.Simple.IsSimple 𝒢 ℓ₀"},{"name":"simple→simpleAlgebra","type":"{α ρ : Level} (𝒢 : Group α ρ) (ℓ₀ : Level) →\nClassical.Structures.Group.Simple.Simple.IsSimple 𝒢 ℓ₀ →\nSetoid.Congruences.Simple.IsSimple\n(Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢)\n(α Level.⊔ ρ Level.⊔ ℓ₀)"},{"name":"∙ε⁻¹","type":"{α ρ : Level} (𝒢 : Group α ρ)\n(x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain (𝒢 .proj₁))) →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝒢 Classical.Structures.Group.Basic.Group-Op.∙ x)\n ((𝒢 Classical.Structures.Group.Basic.Group-Op.⁻¹)\n  (Classical.Structures.Group.Basic.Group-Op.ε 𝒢)))\nx"},{"name":"∙⁻¹∙","type":"{α ρ : Level} (𝒢 : Group α ρ)\n(x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain (𝒢 .proj₁))) →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝒢 Classical.Structures.Group.Basic.Group-Op.∙\n  (𝒢 Classical.Structures.Group.Basic.Group-Op.∙ x)\n  ((𝒢 Classical.Structures.Group.Basic.Group-Op.⁻¹) y))\n y)\nx"},{"name":"∙⁻¹≈ε→≈","type":"{α ρ : Level} (𝒢 : Group α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain (𝒢 .proj₁))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝒢 Classical.Structures.Group.Basic.Group-Op.∙ x)\n ((𝒢 Classical.Structures.Group.Basic.Group-Op.⁻¹) y))\n(Classical.Structures.Group.Basic.Group-Op.ε 𝒢) →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢 ]\n Relation.Binary.Bundles.Setoid.≈ x)\ny"},{"name":"≈ⁿ-isEquivalence","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nRelation.Binary.Structures.IsEquivalence (GroupCongruences._≈ⁿ_ 𝒢)"},{"name":"≈ⁿ-refl","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n{𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ} →\n(𝒢 GroupCongruences.≈ⁿ 𝑵) 𝑵"},{"name":"≈ⁿ-sym","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n{𝑴 𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ} →\n(𝒢 GroupCongruences.≈ⁿ 𝑴) 𝑵 → (𝒢 GroupCongruences.≈ⁿ 𝑵) 𝑴"},{"name":"≈ⁿ-trans","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n{𝑳 𝑴 𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ} →\n(𝒢 GroupCongruences.≈ⁿ 𝑳) 𝑴 →\n(𝒢 GroupCongruences.≈ⁿ 𝑴) 𝑵 → (𝒢 GroupCongruences.≈ⁿ 𝑳) 𝑵"},{"name":"≈→∙⁻¹≈ε","type":"{α ρ : Level} (𝒢 : Group α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢 ]\n Relation.Binary.Bundles.Setoid.≈ x)\ny →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Group.Congruences.GroupCongruences.𝑮 𝒢 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝒢 Classical.Structures.Group.Basic.Group-Op.∙ x)\n ((𝒢 Classical.Structures.Group.Basic.Group-Op.⁻¹) y))\n(Classical.Structures.Group.Basic.Group-Op.ε 𝒢)"},{"name":"≤ⁿ-isPartialOrder","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level} →\nRelation.Binary.Structures.IsPartialOrder (GroupCongruences._≈ⁿ_ 𝒢)\n(GroupCongruences._≤ⁿ_ 𝒢)"},{"name":"≤ⁿ-refl","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n{𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ} →\n(𝒢 GroupCongruences.≤ⁿ 𝑵) 𝑵"},{"name":"≤ⁿ-trans","type":"{α ρ : Level} (𝒢 : Group α ρ) {ℓ : Level}\n{𝑳 𝑴 𝑵 : GroupCongruences.NormalSubgroup 𝒢 ℓ} →\n(𝒢 GroupCongruences.≤ⁿ 𝑳) 𝑴 →\n(𝒢 GroupCongruences.≤ⁿ 𝑴) 𝑵 → (𝒢 GroupCongruences.≤ⁿ 𝑳) 𝑵"}],"lane":{"load":"first","loadElapsedMs":5780},"module":"GroupCongruences","modules":["ConNormal","NormalCon","SubgroupRel"],"project":{"root":"<work>","rootSource":"server-config"},"total":44,"truncated":false}
  5. 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.
  6. the work directory
    Edit(replace_all: false, old_string: open GroupCongruences 𝒢 using ( NormalSubgroup ; normalOf ; _≤ⁿ_ ; ≤ⁿ-refl ) normal-of-smaller-congruence : {θ φ : Con (proj₁ 𝒢) ℓ} {𝑵 : NormalSubgroup ℓ}…, new_string: open GroupCongruences 𝒢 using ( NormalSubgroup ; normalOf ; _≤ⁿ_ ; ≤ⁿ-refl ) open GroupCongruences 𝒢 using ( normalOf-mono ; ≤ⁿ-trans ) normal-of-small…)
    answered
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your co…
    the whole answer, as the agent saw it (152 characters of text)
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  7. agda-mcp
    check_file()
    answered
    success: falseholesCount: 0diagnosticsTotal: 2verdict.exitCode: 42elapsedMs: 5829diagnostics: 2holes: 0
    the whole answer, as the agent saw it (1,683 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[{"code":"UnsolvedConstraints","involved":{"metaTypes":["_x_43 (𝒢 = 𝒢) (θ⊆φ = θ⊆φ) (φ≤N = φ≤N) = x",": Relation.Binary.Bundles.Setoid.Carrier","(Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))","(blocked on _x_43)","x = _x_43",": Relation.Binary.Bundles.Setoid.Carrier","(Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))","(blocked on _x_43)"]},"message":"Failed to solve the following constraints:\n  _x_43 (𝒢 = 𝒢) (θ⊆φ = θ⊆φ) (φ≤N = φ≤N) = x\n    : Relation.Binary.Bundles.Setoid.Carrier\n      (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))\n    (blocked on _x_43)\n  x = _x_43\n    : Relation.Binary.Bundles.Setoid.Carrier\n      (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))\n    (blocked on _x_43)","severity":"error"},{"code":"UnsolvedMetaVariables","col":5,"file":"<work>/Group-normal-of-smaller-congruence.agda","involved":{"metaTypes":["<work>/Group-normal-of-smaller-congruence.agda:30.5-13","<work>/Group-normal-of-smaller-congruence.agda:30.5-41","<work>/Group-normal-of-smaller-congruence.agda:30.5-41","<work>/Group-normal-of-smaller-congruence.agda:30.15-36"]},"line":30,"message":"Unsolved metas at the following locations:\n  <work>/Group-normal-of-smaller-congruence.agda:30.5-13\n  <work>/Group-normal-of-smaller-congruence.agda:30.5-41\n  <work>/Group-normal-of-smaller-congruence.agda:30.5-41\n  <work>/Group-normal-of-smaller-congruence.agda:30.15-36","range":{"endCol":41,"endLine":30,"startCol":5,"startLine":30},"severity":"error"}],"diagnosticsTotal":2,"elapsedMs":5829,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":false,"timedOut":false,"verdict":{"exitCode":42}}
  8. 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.
  9. agda-mcp
    type_of(expr: normalOf-mono θ φ θ⊆φ, line: 30, column: 36)
    answered
    error.code: NotInScopeelapsedMs: 448
    the whole answer, as the agent saw it (674 characters of JSON)
    {"checkedFromSource":true,"elapsedMs":448,"error":{"code":"NotInScope","message":"1.1-14: error: [NotInScope]\nNot in scope:\n  normalOf-mono at 1.1-14\n    (did you mean\n       'Classical.Structures.Group.Congruences.GroupCongruences.normalOf-cong' or\n       'Classical.Structures.Group.Congruences.GroupCongruences.normalOf-mono' or\n       'GroupCongruences.normalOf-cong' or\n       'GroupCongruences.normalOf-mono'?)\nwhen scope checking normalOf-mono","stage":"expression"},"expr":"normalOf-mono θ φ θ⊆φ","lane":{"load":"changed","loadElapsedMs":441},"project":{"root":"<work>","rootSource":"server-config"},"scope":"toplevel (line 30, column 36 is inside no goal)"}
  10. 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.
  11. agda-mcp
    definition_of(name: GroupCongruences.normalOf-mono)
    answered
    elapsedMs: 2definitions: 1
    the whole answer, as the agent saw it (445 characters of JSON)
    {"checkedFromSource":false,"definitions":[{"col":3,"endCol":16,"endLine":596,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Classical/Structures/Group/Congruences.lagda.md","line":596,"qualified":"Classical.Structures.Group.Congruences.GroupCongruences.normalOf-mono"}],"elapsedMs":2,"lane":{"load":"reused"},"name":"GroupCongruences.normalOf-mono","project":{"root":"<work>","rootSource":"server-config"},"scope":"toplevel","unlocated":[]}
  12. the work directory
    Read(offset: 560, limit: 70)
    answered
    560 ```
    the whole answer, as the agent saw it (3,281 characters of text)
    560	```
561	
562	#### Normality is necessary, not merely sufficient
563	
564	`NormalCon`{.AgdaModule} shows normality *suffices* for `NormalRel N`{.AgdaFunction} to
565	be a congruence.  The converse holds too, and is worth proving rather than asserting:
566	were it left as prose, the claim "the correspondence is with the *normal* subgroups"
567	would be doing work the formal development had not done, and nothing would rule out the
568	relation of some non-normal subgroup slipping into `Con 𝑮 ℓ`{.AgdaFunction}.
569	
570	The proof needs no new group arithmetic.  If `NormalRel N`{.AgdaFunction} is a
571	congruence, its identity class is normal by
572	`IdentityClass-normal`{.AgdaFunction} — and that class is
573	`{ x ∣ x ∙ ε ⁻¹ ∈ N }`, which `respects`{.AgdaField} identifies with
574	`N`{.AgdaBound} itself.
575	
576	```agda
577	  -- If the relation of an equality-respecting subgroup is a congruence, the subgroup
578	  -- is normal.  With `NormalCon.∼-isCongruence` this makes normality *equivalent* to
579	  -- compatibility, so `NormalSubgroup ℓ` is exactly the source of the correspondence.
580	  congruence→normal : {ℓ : Level} (N : Pred G ℓ) → IsSubgroup 𝒢 N
581	    →  IsCongruence 𝑮 (NormalRel N) → IsNormal N
582	  congruence→normal N N-sg isCon g {x} x∈N =
583	    respects  (∙ε⁻¹ (x ^ g))
584	              (ConNormal.IdentityClass-normal (NormalRel N , isCon) g
585	                (respects (≈sym (∙ε⁻¹ x)) x∈N))
586	    where open IsSubgroup N-sg using ( respects )
587	```
588	
589	#### Monotonicity
590	
591	Both maps act by (co)restriction of the underlying predicates and relations, so
592	monotonicity is immediate in each direction.
593	
594	```agda
595	  -- The congruence-to-subgroup map is monotone.
596	  normalOf-mono : (θ φ : Con 𝑮 ℓ) → θ ⊑ φ → normalOf θ ≤ⁿ normalOf φ
597	  normalOf-mono _ _ θ⊑φ p = θ⊑φ p
598	
599	  -- The subgroup-to-congruence map is monotone.
600	  congruenceOf-mono : (𝑴 𝑵 : NormalSubgroup ℓ) → 𝑴 ≤ⁿ 𝑵 → congruenceOf 𝑴 ⊑ congruenceOf 𝑵
601	  congruenceOf-mono _ _ 𝑴≤𝑵 p = 𝑴≤𝑵 p
602	```
603	
604	`OrderIso`{.AgdaRecord} asks only for monotonicity, not for the maps to be well defined
605	on equivalence classes, so that latter property — which "isomorphism of posets" is
606	usually taken to include — is recorded here rather than left implicit.  It costs
607	nothing: each equivalence *is* mutual containment, so applying the matching
608	monotonicity twice suffices.
609	
610	```agda
611	  -- Both maps respect the equivalences of the two sides.
612	  normalOf-cong : (θ φ : Con 𝑮 ℓ) → θ ≑ φ → normalOf θ ≈ⁿ normalOf φ
613	  normalOf-cong  θ φ (θ⊑φ , φ⊑θ) =
614	    normalOf-mono θ φ θ⊑φ , normalOf-mono φ θ φ⊑θ
615	
616	  congruenceOf-cong : (𝑴 𝑵 : NormalSubgroup ℓ)
617	    →  𝑴 ≈ⁿ 𝑵 → congruenceOf 𝑴 ≑ congruenceOf 𝑵
618	  congruenceOf-cong 𝑴 𝑵 (𝑴≤𝑵 , 𝑵≤𝑴) =
619	    congruenceOf-mono 𝑴 𝑵 𝑴≤𝑵 , congruenceOf-mono 𝑵 𝑴 𝑵≤𝑴
620	```
621	
622	#### Mutual inverseness
623	
624	On congruences, `θ_{N_θ} ≑ θ`: the relation `θ_{N_θ}` holds at `(x , y)` when
625	`(x ∙ y ⁻¹) θ ε`, and multiplying on the right by `y`{.AgdaBound} converts that to
626	`x θ y` through `∙⁻¹∙`{.AgdaFunction} and the unit law — while multiplying `x θ y` on
627	the right by `y ⁻¹` converts it back through `invʳ-law`{.AgdaFunction}.
628	
629	```agda
  13. 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.
  14. agda-mcp
    definition_of(name: GroupCongruences.≤ⁿ-trans)
    answered
    elapsedMs: 3definitions: 1
    the whole answer, as the agent saw it (435 characters of JSON)
    {"checkedFromSource":false,"definitions":[{"col":3,"endCol":11,"endLine":284,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Classical/Structures/Group/Congruences.lagda.md","line":284,"qualified":"Classical.Structures.Group.Congruences.GroupCongruences.≤ⁿ-trans"}],"elapsedMs":3,"lane":{"load":"reused"},"name":"GroupCongruences.≤ⁿ-trans","project":{"root":"<work>","rootSource":"server-config"},"scope":"toplevel","unlocated":[]}
  15. the work directory
    Read(offset: 250, limit: 50)
    answered
    250 set (N , _ , _) = N
    the whole answer, as the agent saw it (2,089 characters of text)
    250	  set (N , _ , _) = N
251	
252	  -- ... and its two proof components.
253	  set-isSubgroup : (𝑵 : NormalSubgroup ℓ) → IsSubgroup 𝒢 (set 𝑵)
254	  set-isSubgroup (_ , isSubgroup , _) = isSubgroup
255	
256	  set-normal : (𝑵 : NormalSubgroup ℓ) → IsNormal (set 𝑵)
257	  set-normal (_ , _ , isNormal) = isNormal
258	
259	  infix 4 _≤ⁿ_ _≈ⁿ_
260	
261	  -- The inclusion order on normal subgroups ...
262	  _≤ⁿ_ : NormalSubgroup ℓ → NormalSubgroup ℓ → Type (α ⊔ ℓ)
263	  𝑴 ≤ⁿ 𝑵 = set 𝑴 ⊆ set 𝑵
264	
265	  -- ... and the equivalence of mutual inclusion it is antisymmetric over.
266	  _≈ⁿ_ : NormalSubgroup ℓ → NormalSubgroup ℓ → Type (α ⊔ ℓ)
267	  𝑴 ≈ⁿ 𝑵 = 𝑴 ≤ⁿ 𝑵 × 𝑵 ≤ⁿ 𝑴
268	```
269	
270	Calling the correspondence an *order* isomorphism presupposes that both sides really
271	are ordered, so the partial-order laws are proved rather than assumed.  They are the
272	laws of `_⊆_`{.AgdaFunction} on predicates, and antisymmetry holds by construction:
273	`_≈ⁿ_`{.AgdaFunction} *is* mutual inclusion.  This mirrors
274	`⊆-isPartialOrder`{.AgdaFunction} and `Con-Poset`{.AgdaFunction} on the congruence
275	side, down to the same implicit-argument discipline — `_≤ⁿ_`{.AgdaFunction} is a
276	defined relation, not an injective type former, so Agda cannot recover the endpoint
277	arguments of the helper lemmas from the expected field types and they are forwarded by
278	hand.
279	
280	```agda
281	  ≤ⁿ-refl : {𝑵 : NormalSubgroup ℓ} → 𝑵 ≤ⁿ 𝑵
282	  ≤ⁿ-refl p = p
283	
284	  ≤ⁿ-trans : {𝑳 𝑴 𝑵 : NormalSubgroup ℓ} → 𝑳 ≤ⁿ 𝑴 → 𝑴 ≤ⁿ 𝑵 → 𝑳 ≤ⁿ 𝑵
285	  ≤ⁿ-trans 𝑳≤𝑴 𝑴≤𝑵 p = 𝑴≤𝑵 (𝑳≤𝑴 p)
286	
287	  ≈ⁿ-refl : {𝑵 : NormalSubgroup ℓ} → 𝑵 ≈ⁿ 𝑵
288	  ≈ⁿ-refl = (λ p → p) , (λ p → p)
289	
290	  ≈ⁿ-sym : {𝑴 𝑵 : NormalSubgroup ℓ} → 𝑴 ≈ⁿ 𝑵 → 𝑵 ≈ⁿ 𝑴
291	  ≈ⁿ-sym = swap
292	
293	  ≈ⁿ-trans : {𝑳 𝑴 𝑵 : NormalSubgroup ℓ} → 𝑳 ≈ⁿ 𝑴 → 𝑴 ≈ⁿ 𝑵 → 𝑳 ≈ⁿ 𝑵
294	  ≈ⁿ-trans (𝑳≤𝑴 , 𝑴≤𝑳) (𝑴≤𝑵 , 𝑵≤𝑴) = (λ p → 𝑴≤𝑵 (𝑳≤𝑴 p)) , (λ p → 𝑴≤𝑳 (𝑵≤𝑴 p))
295	
296	  ≈ⁿ-isEquivalence : IsEquivalence (_≈ⁿ_ {ℓ})
297	  ≈ⁿ-isEquivalence {ℓ} = record
298	    { refl   = λ {𝑵} → ≈ⁿ-refl {ℓ} {𝑵}
299	    ; sym    = λ {𝑴} {𝑵} → ≈ⁿ-sym {ℓ} {𝑴} {𝑵}
  16. 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.
  17. the work directory
    Edit(replace_all: false, old_string: normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N = ≤ⁿ-trans (normalOf-mono θ φ θ⊆φ) φ≤N, new_string: normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N = ≤ⁿ-trans {normalOf θ} {normalOf φ} {𝑵} (normalOf-mono θ φ θ⊆φ) φ≤N)
    answered
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your co…
    the whole answer, as the agent saw it (152 characters of text)
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  18. agda-mcp
    check_file()
    answered
    success: falseholesCount: 0diagnosticsTotal: 1verdict.exitCode: 42elapsedMs: 5678diagnostics: 1holes: 0
    the whole answer, as the agent saw it (1,019 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[{"code":"UnequalTerms","col":15,"file":"<work>/Group-normal-of-smaller-congruence.agda","involved":{"actual":"Data.Product.Σ (Relation.Unary.Pred (Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ) (λ N → Classical.Structures.Group.Subgroups.IsSubgroup 𝒢 N Data.Product.× Classical.Structures.Group.Conjugation.Conjugate.IsNormal 𝒢 N)","expected":"Level"},"line":30,"message":"Data.Product.Σ\n(Relation.Unary.Pred\n (Classical.Structures.Group.Congruences.GroupCongruences.G 𝒢) ℓ)\n(λ N →\n   Classical.Structures.Group.Subgroups.IsSubgroup 𝒢 N Data.Product.×\n   Classical.Structures.Group.Conjugation.Conjugate.IsNormal 𝒢 N)\n!=< Level\nwhen checking that the expression normalOf θ has type Level","range":{"endCol":25,"endLine":30,"startCol":15,"startLine":30},"severity":"error"}],"diagnosticsTotal":1,"elapsedMs":5678,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":false,"timedOut":false,"verdict":{"exitCode":42}}
  19. 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.
  20. the work directory
    Edit(replace_all: false, old_string: ≤ⁿ-trans {normalOf θ} {normalOf φ} {𝑵} (normalOf-mono θ φ θ⊆φ) φ≤N, new_string: ≤ⁿ-trans {ℓ} {normalOf θ} {normalOf φ} {𝑵} (normalOf-mono θ φ θ⊆φ) φ≤N)
    answered
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your co…
    the whole answer, as the agent saw it (152 characters of text)
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  21. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5643diagnostics: 0holes: 0
    the whole answer, as the agent saw it (221 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5643,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  22. the model
    check_file reports success on the final file (exit code 0, no diagnostics, no holes).

The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the server’s own, in full. The client billed 5,671 thinking tokens over the session; their text is not in the archive.

solvedthe judge, in outcome.json

Every gate passed, so the row counts as a solve. Its index entry names no library original, so the judge looks for no citation of one.

Agda’s exit code 0
statement preserved yes
turns 14

Where each needle came from: the first event of the session that names it.

  • ≤ⁿ-trans: first shown by the answer to call 2, exports_of on GroupCongruences; the final file uses it.
  • normalOf-mono: first shown by the answer to call 2, exports_of on GroupCongruences; the final file uses it.
the file the judge read, marked against the obligation it started as
-- Group-normal-of-smaller-congruence.agda
--
-- File: data/benchmarks/agda-algebras-composition-v0/obligations/Group-normal-of-smaller-congruence.agda
--
-- Benchmark obligation: comp-group-normal-of-smaller-congruence
-- Difficulty: compositional
-- Import stratum: composition
--
-- If θ ⊆ φ and the normal subgroup of φ lies in N, the normal subgroup of θ lies in N.
--
module Group-normal-of-smaller-congruence where

open import AgdaDojang.Debug

open import Data.Product                            using ( proj₁ )
open import Level                                   using ( Level )
open import Classical.Structures.Group.Basic        using ( Group )
open import Classical.Structures.Group.Congruences  using ( module GroupCongruences )
open import Setoid.Congruences.Basic                using ( Con )
open import Setoid.Congruences.Lattice              using ( _⊆_ ; ⊆-refl )

module _ {α ρ ℓ : Level} (𝒢 : Group α ρ) where

  open GroupCongruences 𝒢  using ( NormalSubgroup ; normalOf ; _≤ⁿ_ ; ≤ⁿ-refl )
  open GroupCongruences 𝒢  using ( normalOf-mono ; ≤ⁿ-trans )

  normal-of-smaller-congruence : {θ φ : Con (proj₁ 𝒢) ℓ} {𝑵 : NormalSubgroup ℓ}
    → θ ⊆ φ → normalOf φ ≤ⁿ 𝑵 → normalOf θ ≤ⁿ 𝑵
  normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N = {!!}
  normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N =
    ≤ⁿ-trans {ℓ} {normalOf θ} {normalOf φ} {𝑵} (normalOf-mono θ φ θ⊆φ) φ≤N

reports/agent-bench/comp-sonnet5-mcp-1/subjects/comp-group-normal-of-smaller-congruence/final/Group-normal-of-smaller-congruence.agda

normal-of-smaller-congruence Opus 5 solved

The same row, and no needle in the proof. `get_goal` prints the goal unfolded, a membership to a membership; a Read of the file `definition_of` located has `normalOf-mono` in it; and the session writes `λ p → φ≤N (θ⊆φ p)`, probes it with `fill_hole`, and checks it.

obligation
comp-group-normal-of-smaller-congruence
tier
compositional
stratum
agda-algebras/composition
the needles
Classical.Structures.Group.Congruences.GroupCongruences.≤ⁿ-trans, Classical.Structures.Group.Congruences.GroupCongruences.normalOf-mono
model
claude-opus-5
turns
9
tool calls
8
wall
54 s
cost, list price
USD 0.26
the hole’s goal, as get_goal reported it
{θ φ : Con (proj₁ 𝒢) ℓ} {𝑵 : NormalSubgroup ℓ} → θ ⊆ φ → normalOf φ ≤ⁿ 𝑵 → normalOf θ ≤ⁿ 𝑵
the session, 9 turns and 8 tool calls
  1. the model
    I'll start by reading the file.
  2. the work directory
    Read()
    answered
    1 -- Group-normal-of-smaller-congruence.agda
    the whole answer, as the agent saw it (1,278 characters of text)
    1	-- Group-normal-of-smaller-congruence.agda
2	--
3	-- File: data/benchmarks/agda-algebras-composition-v0/obligations/Group-normal-of-smaller-congruence.agda
4	--
5	-- Benchmark obligation: comp-group-normal-of-smaller-congruence
6	-- Difficulty: compositional
7	-- Import stratum: composition
8	--
9	-- If θ ⊆ φ and the normal subgroup of φ lies in N, the normal subgroup of θ lies in N.
10	--
11	module Group-normal-of-smaller-congruence where
12	
13	open import AgdaDojang.Debug
14	
15	open import Data.Product                            using ( proj₁ )
16	open import Level                                   using ( Level )
17	open import Classical.Structures.Group.Basic        using ( Group )
18	open import Classical.Structures.Group.Congruences  using ( module GroupCongruences )
19	open import Setoid.Congruences.Basic                using ( Con )
20	open import Setoid.Congruences.Lattice              using ( _⊆_ ; ⊆-refl )
21	
22	module _ {α ρ ℓ : Level} (𝒢 : Group α ρ) where
23	
24	  open GroupCongruences 𝒢  using ( NormalSubgroup ; normalOf ; _≤ⁿ_ ; ≤ⁿ-refl )
25	
26	  normal-of-smaller-congruence : {θ φ : Con (proj₁ 𝒢) ℓ} {𝑵 : NormalSubgroup ℓ}
27	    → θ ⊆ φ → normalOf φ ≤ⁿ 𝑵 → normalOf θ ≤ⁿ 𝑵
28	  normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N = {!!}
29
  3. 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.
  4. agda-mcp
    definition_of(name: normalOf)
    one of 2 calls the model issued in this turn
    answered
    elapsedMs: 5913definitions: 4
    the whole answer, as the agent saw it (1,164 characters of JSON)
    {"checkedFromSource":true,"definitions":[{"col":5,"endCol":11,"endLine":409,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Classical/Structures/Group/Congruences.lagda.md","line":409,"qualified":"Classical.Structures.Group.Congruences.GroupCongruences.NormalCon.normal"},{"col":3,"endCol":11,"endLine":556,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Classical/Structures/Group/Congruences.lagda.md","line":556,"qualified":"Classical.Structures.Group.Congruences.GroupCongruences.normalOf"},{"col":5,"endCol":11,"endLine":409,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Classical/Structures/Group/Congruences.lagda.md","line":409,"qualified":"Classical.Structures.Group.Congruences.GroupCongruences.NormalCon.normal"},{"col":3,"endCol":11,"endLine":556,"file":"<nix>/agda-algebras-unstable-2026-09-07/src/Classical/Structures/Group/Congruences.lagda.md","line":556,"qualified":"Classical.Structures.Group.Congruences.GroupCongruences.normalOf"}],"elapsedMs":5913,"lane":{"load":"first","loadElapsedMs":5865},"name":"normalOf","project":{"root":"<work>","rootSource":"server-config"},"recovered":"did-you-mean","scope":"toplevel","unlocated":[]}
  5. agda-mcp
    get_goal(line: 28, column: 48)
    one of 2 calls the model issued in this turn
    refused
    No hole at line 28, column 48 in <work>/Group-normal-of-smaller-congruence.agda (a position addresses the hole whose spa…
    the whole answer, as the agent saw it (525 characters of text)
    No hole at line 28, column 48 in <work>/Group-normal-of-smaller-congruence.agda (a position addresses the hole whose span contains it; starting at it counts).
  nearest holes (1 in the file):
    index 0 at line 28, column 54
  Address a hole by the (line, column) its own listing reports — get_diagnostics.holes, check_file.holes, or the holes list every fill_hole response carries.
  Take that listing from the latest response: a fill above a hole moves it when the candidate is a different length, and renumbers it always.
  6. agda-mcp
    get_goal(line: 28, column: 54)
    answered
    goal: θ .proj₁ x (Function.Bundles.Func.to (Setoid.Algebras.Basic.Algebra.Interp (𝒢 .proj₁)) (Classical.Signatures.Group.Op-Group.ε-Op Data.Product., (λ ()))) → 𝑵 .proj₁ xelapsedMs: 8context: 10
    the whole answer, as the agent saw it (3,441 characters of JSON)
    {"checkedFromSource":false,"context":[{"name":"α","type":"Level"},{"name":"ρ","type":"Level"},{"name":"ℓ","type":"Level"},{"name":"𝒢","type":"Data.Product.Σ (Setoid.Algebras.Basic.Algebra α ρ)\n(λ 𝑨 →\n   (i : Classical.Theories.Group.Eq-Group)\n   (ρ₁\n    : Data.Fin.Base.Fin 3 →\n      Relation.Binary.Bundles.Setoid.Carrier\n      (Setoid.Algebras.Basic.Algebra.Domain 𝑨)) →\n   (Setoid.Algebras.Basic.Algebra.Domain 𝑨\n    Relation.Binary.Bundles.Setoid.≈\n    Function.Bundles.Func.to\n    (Setoid.Terms.Basic.Environment.⟦ 𝑨 ⟧\n     (proj₁ (Classical.Theories.Group.Th-Group i)))\n    ρ₁)\n   (Function.Bundles.Func.to\n    (Setoid.Terms.Basic.Environment.⟦ 𝑨 ⟧\n     (Data.Product.proj₂ (Classical.Theories.Group.Th-Group i)))\n    ρ₁))"},{"name":"θ","type":"Data.Product.Σ\n(Relation.Binary.Bundles.Setoid.Carrier\n (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢)) →\n Relation.Binary.Bundles.Setoid.Carrier\n (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢)) →\n Set ℓ)\n(Setoid.Congruences.Basic.IsCongruence (proj₁ 𝒢))"},{"name":"φ","type":"Data.Product.Σ\n(Relation.Binary.Bundles.Setoid.Carrier\n (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢)) →\n Relation.Binary.Bundles.Setoid.Carrier\n (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢)) →\n Set ℓ)\n(Setoid.Congruences.Basic.IsCongruence (proj₁ 𝒢))"},{"name":"𝑵","type":"Data.Product.Σ\n(Relation.Binary.Bundles.Setoid.Carrier\n (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢)) →\n Set ℓ)\n(λ N →\n   Data.Product.Σ\n   (Classical.Structures.Group.Subgroups.IsSubgroup 𝒢 N)\n   (λ x₁ →\n      (g\n       : Relation.Binary.Bundles.Setoid.Carrier\n         (Setoid.Algebras.Basic.Algebra.Domain (𝒢 .proj₁)))\n      {x = x₂\n       : Relation.Binary.Bundles.Setoid.Carrier\n         (Setoid.Algebras.Basic.Algebra.Domain (𝒢 .proj₁))} →\n      N x₂ →\n      N\n      (Function.Bundles.Func.to\n       (Setoid.Algebras.Basic.Algebra.Interp (𝒢 .proj₁))\n       (Classical.Signatures.Group.Op-Group.∙-Op Data.Product.,\n        Classical.Operations.pair\n        (Function.Bundles.Func.to\n         (Setoid.Algebras.Basic.Algebra.Interp (𝒢 .proj₁))\n         (Classical.Signatures.Group.Op-Group.∙-Op Data.Product.,\n          Classical.Operations.pair g x₂))\n        (Function.Bundles.Func.to\n         (Setoid.Algebras.Basic.Algebra.Interp (𝒢 .proj₁))\n         (Classical.Signatures.Group.Op-Group.⁻¹-Op Data.Product.,\n          (λ _ → g)))))))"},{"name":"θ⊆φ","type":"{x = x₁\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))}\n{y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))} →\nproj₁ θ x₁ y → proj₁ φ x₁ y"},{"name":"φ≤N","type":"{x = x₁\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))} →\nφ .proj₁ x₁\n(Function.Bundles.Func.to\n (Setoid.Algebras.Basic.Algebra.Interp (𝒢 .proj₁))\n (Classical.Signatures.Group.Op-Group.ε-Op Data.Product., (λ ()))) →\n𝑵 .proj₁ x₁"},{"name":"x","type":"Relation.Binary.Bundles.Setoid.Carrier\n(Setoid.Algebras.Basic.Algebra.Domain (proj₁ 𝒢))"}],"elapsedMs":8,"goal":"θ .proj₁ x\n(Function.Bundles.Func.to\n (Setoid.Algebras.Basic.Algebra.Interp (𝒢 .proj₁))\n (Classical.Signatures.Group.Op-Group.ε-Op Data.Product., (λ ()))) →\n𝑵 .proj₁ x","lane":{"load":"reused"},"module":"Group-normal-of-smaller-congruence","project":{"root":"<work>","rootSource":"server-config"},"source":"interaction-lane"}
  7. the work directory
    Read(offset: 380, limit: 220)
    answered
    380 ≈⇒∼ x≈y = respects (≈sym (≈→∙⁻¹≈ε x≈y)) ε-closed
    the whole answer, as the agent saw it (10,770 characters of text)
    380	    ≈⇒∼ x≈y = respects (≈sym (≈→∙⁻¹≈ε x≈y)) ε-closed
381	
382	    -- Consequently the relation may be transported along ≈ in either argument.
383	    ∼-resp : ∀ {x x' y y'} → x ≈ x' → y ≈ y' → x ∼ y → x' ∼ y'
384	    ∼-resp x≈x' y≈y' p = ∼-trans (∼-trans (≈⇒∼ (≈sym x≈x')) p) (≈⇒∼ y≈y')
385	```
386	
387	#### From a normal subgroup to a congruence
388	
389	Adding normality makes the relation compatible with the three operations of
390	`Sig-Group`{.AgdaFunction}, hence a congruence.
391	
392	Compatibility with `∙-Op`{.AgdaInductiveConstructor} is the substantive clause and the
393	only one that uses normality in an essential way: from `x ∙ y ⁻¹ ∈ N` and
394	`u ∙ v ⁻¹ ∈ N` we must produce `(x ∙ u) ∙ (y ∙ v) ⁻¹ ∈ N`, and the two given elements do
395	not multiply to it — one of them must first be *moved past* `x`{.AgdaBound}, which is
396	precisely conjugation.  Compatibility with `⁻¹-Op`{.AgdaInductiveConstructor} is one
397	conjugation as well; with `ε-Op`{.AgdaInductiveConstructor} it is reflexivity.
398	
399	```agda
400	  module NormalCon {ℓ : Level} (𝑵 : NormalSubgroup ℓ) where
401	
402	    private
403	      N : Pred G ℓ
404	      N = set 𝑵
405	
406	    open IsSubgroup (set-isSubgroup 𝑵)  using ( respects ; ∙-closed )
407	    open SubgroupRel N (set-isSubgroup 𝑵) public
408	
409	    normal : IsNormal N
410	    normal = set-normal 𝑵
411	
412	    -- Compatibility with the curried multiplication.  The element x ∙ y ⁻¹ is on the
413	    -- wrong side of x ∙ u, so u ∙ v ⁻¹ is first conjugated by x — the one step that
414	    -- normality supplies and that fails for a non-normal subgroup.
415	    ∼-∙ : ∀ {x y u v} → x ∼ y → u ∼ v → (x ∙ u) ∼ (y ∙ v)
416	    ∼-∙ {x} {y} {u} {v} p q = respects step (∙-closed (normal x q) p)
417	      where
418	      step : x ∙ (u ∙ v ⁻¹) ∙ x ⁻¹ ∙ (x ∙ y ⁻¹) ≈ x ∙ u ∙ (y ∙ v) ⁻¹
419	      step = begin
420	        x ∙ (u ∙ v ⁻¹) ∙ x ⁻¹ ∙ (x ∙ y ⁻¹)    ≈⟨ assoc-law (x ∙ (u ∙ v ⁻¹)) (x ⁻¹) (x ∙ y ⁻¹) ⟩
421	        x ∙ (u ∙ v ⁻¹) ∙ (x ⁻¹ ∙ (x ∙ y ⁻¹))  ≈⟨ ∙-cong ≈refl (\\-leftDividesʳ x (y ⁻¹)) ⟩
422	        x ∙ (u ∙ v ⁻¹) ∙ y ⁻¹                 ≈˘⟨ ∙-cong (assoc-law x u (v ⁻¹)) ≈refl ⟩
423	        x ∙ u ∙ v ⁻¹ ∙ y ⁻¹                   ≈⟨ assoc-law (x ∙ u) (v ⁻¹) (y ⁻¹) ⟩
424	        x ∙ u ∙ (v ⁻¹ ∙ y ⁻¹)                 ≈˘⟨ ∙-cong ≈refl (⁻¹-anti-homo-∙ y v) ⟩
425	        x ∙ u ∙ (y ∙ v) ⁻¹                    ∎
426	
427	    -- Compatibility with the curried inverse: conjugating x ∙ y ⁻¹ by x ⁻¹ produces
428	    -- y ⁻¹ ∙ (x ⁻¹) ⁻¹, which is the relation the other way round.
429	    ∼-⁻¹ : ∀ {x y} → x ∼ y → x ⁻¹ ∼ y ⁻¹
430	    ∼-⁻¹ {x} {y} p = ∼-sym (respects step (normal (x ⁻¹) p))
431	      where
432	      step : x ⁻¹ ∙ (x ∙ y ⁻¹) ∙ (x ⁻¹) ⁻¹ ≈ y ⁻¹ ∙ (x ⁻¹) ⁻¹
433	      step = ∙-cong (\\-leftDividesʳ x (y ⁻¹)) ≈refl
434	
435	    -- Compatibility with every operation symbol.  Each clause is the corresponding
436	    -- curried fact, transported across the tuple-vs-curried interpretation bridges of
437	    -- [Classical.Structures.Group.Subgroups].
438	    ∼-compatible : 𝑮 ∣≈ _∼_
439	    ∼-compatible ∙-Op   {u} {v} p = ∼-resp  (≈sym (interp-tuple-∙ 𝒢 u))
440	                                            (≈sym (interp-tuple-∙ 𝒢 v))
441	                                            (∼-∙ (p 0F) (p 1F))
442	    ∼-compatible ε-Op   {u} {v} p = ∼-resp  (≈sym (interp-tuple-ε 𝒢 u))
443	                                            (≈sym (interp-tuple-ε 𝒢 v))
444	                                            ∼-refl
445	    ∼-compatible ⁻¹-Op  {u} {v} p = ∼-resp  (≈sym (interp-tuple-⁻¹ 𝒢 u))
446	                                            (≈sym (interp-tuple-⁻¹ 𝒢 v))
447	                                            (∼-⁻¹ (p 0F))
448	
449	    -- The relation of a normal subgroup is a congruence of the group algebra.
450	    ∼-isCongruence : IsCongruence 𝑮 _∼_
451	    ∼-isCongruence = mkcon ≈⇒∼ ∼-isEquivalence ∼-compatible
452	```
453	
454	For a normal subgroup the right-handed relation used here agrees with the left-handed
455	coset relation `Coset._∼_`{.AgdaFunction} of [Classical.Structures.Group.Cosets][],
456	which is the one [FLRP.Bridge][] builds on.  Both directions are one application of
457	`∼-⁻¹`{.AgdaFunction} followed by an involutivity rewrite, so nothing about the
458	development depends on which of the two presentations a consumer prefers.
459	
460	```agda
461	    -- The right-coset relation of a normal subgroup is its left-coset relation ...
462	    rel→coset : ∀ {x y} → x ∼ y → Coset._∼_ 𝒢 N (set-isSubgroup 𝑵) x y
463	    rel→coset {x} {y} p = respects (∙-cong ≈refl (⁻¹-involutive y)) (∼-⁻¹ p)
464	
465	    -- ... and conversely.
466	    coset→rel : ∀ {x y} → Coset._∼_ 𝒢 N (set-isSubgroup 𝑵) x y → x ∼ y
467	    coset→rel {x} {y} p = ∼-resp  (⁻¹-involutive x) (⁻¹-involutive y)
468	                                  (∼-⁻¹ (respects (∙-cong ≈refl (≈sym (⁻¹-involutive y))) p))
469	```
470	
471	The forward map of the correspondence packages the relation with its congruence proof.
472	
473	```agda
474	  -- N ↦ θ_N : a normal subgroup gives a congruence of the group algebra.
475	  congruenceOf : NormalSubgroup ℓ → Con 𝑮 ℓ
476	  congruenceOf 𝑵 = NormalRel (set 𝑵) , NormalCon.∼-isCongruence 𝑵
477	```
478	
479	#### From a congruence to a normal subgroup
480	
481	In the other direction the ingredients are the curried consequences of a congruence's
482	compatibility: it is preserved by multiplication, by inversion, and hence by
483	conjugation.  These are read off from `is-compatible`{.AgdaFunction} at the canonical
484	tuples, with no interpretation bridge needed — the curried accessors of
485	`Group-Op`{.AgdaModule} are *defined* by applying the interpreted symbol to exactly
486	those tuples.
487	
488	```agda
489	  module ConNormal {ℓ : Level} ((_θ_ , θcon) : Con 𝑮 ℓ) where
490	
491	    -- The equivalence laws of θ, read off the record.
492	    θ-refl : ∀ {x} → x θ x
493	    θ-refl = IsEquivalence.refl (is-equivalence θcon)
494	
495	    θ-sym : ∀ {x y} → x θ y → y θ x
496	    θ-sym = IsEquivalence.sym (is-equivalence θcon)
497	
498	    θ-trans : ∀ {x y z} → x θ y → y θ z → x θ z
499	    θ-trans = IsEquivalence.trans (is-equivalence θcon)
500	
501	    -- A congruence relates ≈-equal elements.
502	    θ-reflexive : ∀ {x y} → x ≈ y → x θ y
503	    θ-reflexive = reflexive θcon
504	
505	    -- Compatibility with the curried multiplication ...
506	    θ-∙ : ∀ {x y u v} → x θ y → u θ v → (x ∙ u) θ (y ∙ v)
507	    θ-∙ {x} {y} {u} {v} p q = is-compatible θcon ∙-Op λ { 0F → p ; 1F → q }
508	
509	    -- ... and with the curried inverse.
510	    θ-⁻¹ : ∀ {x y} → x θ y → (x ⁻¹) θ (y ⁻¹)
511	    θ-⁻¹ p = is-compatible θcon ⁻¹-Op (λ _ → p)
512	
513	    -- Hence conjugation by any element preserves the congruence.
514	    θ-conj : ∀ g {x y} → x θ y → x ^ g θ y ^ g
515	    θ-conj g {x} {y} p = θ-∙ (θ-∙ (θ-refl {x = g}) p) (θ-refl {x = g ⁻¹})
516	```
517	
518	The class of the identity is an equality-respecting subgroup, and it is normal.
519	Membership is `x θ ε`, so each subgroup law is one application of the corresponding
520	compatibility fact followed by a `≈`-step that renormalizes the right-hand side back to
521	`ε`{.AgdaFunction}: `ε ∙ ε ≈ ε`, `ε ⁻¹ ≈ ε`, and `conj g ε ≈ ε`.
522	
523	```agda
524	    -- The θ-class of the identity.
525	    IdentityClass : Pred G ℓ
526	    IdentityClass x = x θ ε
527	
528	    -- It respects the setoid equality, because a congruence contains it.
529	    IdentityClass-respects : IdentityClass Respects _≈_
530	    IdentityClass-respects x≈y p = θ-trans (θ-reflexive (≈sym x≈y)) p
531	
532	    IdentityClass-ε : ε ∈ IdentityClass
533	    IdentityClass-ε = θ-refl
534	
535	    IdentityClass-∙ : ∀ {x y} → x ∈ IdentityClass → y ∈ IdentityClass
536	      → x ∙ y ∈ IdentityClass
537	    IdentityClass-∙ p q = θ-trans (θ-∙ p q) (θ-reflexive (idˡ-law ε))
538	
539	    IdentityClass-⁻¹ : ∀ {x} → x ∈ IdentityClass → x ⁻¹ ∈ IdentityClass
540	    IdentityClass-⁻¹ p = θ-trans (θ-⁻¹ p) (θ-reflexive ε⁻¹≈ε)
541	
542	    -- The identity class is an equality-respecting subgroup ...
543	    IdentityClass-isSubgroup : IsSubgroup 𝒢 IdentityClass
544	    IdentityClass-isSubgroup = mkIsSubgroup 𝒢  IdentityClass-respects IdentityClass-∙
545	                                               IdentityClass-ε IdentityClass-⁻¹
546	
547	    -- ... and it is normal, since conjugation fixes the identity.
548	    IdentityClass-normal : IsNormal IdentityClass
549	    IdentityClass-normal g p = θ-trans (θ-conj g p) (θ-reflexive (conj-ε g))
550	```
551	
552	The backward map of the correspondence packages the class with its two proofs.
553	
554	```agda
555	  -- θ ↦ N_θ : a congruence gives a normal subgroup of the group.
556	  normalOf : Con 𝑮 ℓ → NormalSubgroup ℓ
557	  normalOf θ =  ConNormal.IdentityClass θ
558	             ,  ConNormal.IdentityClass-isSubgroup θ
559	             ,  ConNormal.IdentityClass-normal θ
560	```
561	
562	#### Normality is necessary, not merely sufficient
563	
564	`NormalCon`{.AgdaModule} shows normality *suffices* for `NormalRel N`{.AgdaFunction} to
565	be a congruence.  The converse holds too, and is worth proving rather than asserting:
566	were it left as prose, the claim "the correspondence is with the *normal* subgroups"
567	would be doing work the formal development had not done, and nothing would rule out the
568	relation of some non-normal subgroup slipping into `Con 𝑮 ℓ`{.AgdaFunction}.
569	
570	The proof needs no new group arithmetic.  If `NormalRel N`{.AgdaFunction} is a
571	congruence, its identity class is normal by
572	`IdentityClass-normal`{.AgdaFunction} — and that class is
573	`{ x ∣ x ∙ ε ⁻¹ ∈ N }`, which `respects`{.AgdaField} identifies with
574	`N`{.AgdaBound} itself.
575	
576	```agda
577	  -- If the relation of an equality-respecting subgroup is a congruence, the subgroup
578	  -- is normal.  With `NormalCon.∼-isCongruence` this makes normality *equivalent* to
579	  -- compatibility, so `NormalSubgroup ℓ` is exactly the source of the correspondence.
580	  congruence→normal : {ℓ : Level} (N : Pred G ℓ) → IsSubgroup 𝒢 N
581	    →  IsCongruence 𝑮 (NormalRel N) → IsNormal N
582	  congruence→normal N N-sg isCon g {x} x∈N =
583	    respects  (∙ε⁻¹ (x ^ g))
584	              (ConNormal.IdentityClass-normal (NormalRel N , isCon) g
585	                (respects (≈sym (∙ε⁻¹ x)) x∈N))
586	    where open IsSubgroup N-sg using ( respects )
587	```
588	
589	#### Monotonicity
590	
591	Both maps act by (co)restriction of the underlying predicates and relations, so
592	monotonicity is immediate in each direction.
593	
594	```agda
595	  -- The congruence-to-subgroup map is monotone.
596	  normalOf-mono : (θ φ : Con 𝑮 ℓ) → θ ⊑ φ → normalOf θ ≤ⁿ normalOf φ
597	  normalOf-mono _ _ θ⊑φ p = θ⊑φ p
598	
599	  -- The subgroup-to-congruence map is monotone.
  8. 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.
  9. agda-mcp
    fill_hole(line: 28, column: 54, candidate: λ p → φ≤N (θ⊆φ p))
    answered
    status: okremainingHoles: 0verdict.exitCode: 0elapsedMs: 5381holes: 0
    the whole answer, as the agent saw it (201 characters of JSON)
    {"candidate":"λ p → φ≤N (θ⊆φ p)","checkedFromSource":true,"elapsedMs":5381,"holes":[],"project":{"root":"<work>","rootSource":"server-config"},"remainingHoles":0,"status":"ok","verdict":{"exitCode":0}}
  10. the work directory
    Edit(replace_all: false, old_string: θ⊆φ φ≤N = {!!}, new_string: θ⊆φ φ≤N = λ p → φ≤N (θ⊆φ p))
    answered
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your co…
    the whole answer, as the agent saw it (152 characters of text)
    The file <work>/Group-normal-of-smaller-congruence.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  11. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5282diagnostics: 0holes: 0
    the whole answer, as the agent saw it (222 characters of JSON)
    {"checkedFromSource":false,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5282,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  12. the model
    check_file reports success (exit 0, no diagnostics, no holes) on the final file, with the hole filled by `λ p → φ≤N (θ⊆φ p)`.

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 411 thinking tokens over the session; their text is not in the archive.

solvedthe judge, in outcome.json

Every gate passed, so the row counts as a solve. Its index entry names no library original, so the judge looks for no citation of one.

Agda’s exit code 0
statement preserved yes
turns 9

Where each needle came from: the first event of the session that names it.

  • ≤ⁿ-trans: never named in the session; the final file does not use it.
  • normalOf-mono: first shown by the answer to call 5, Read of Congruences.lagda.md, a file an earlier definition_of answer had named; the final file does not use it.
the file the judge read, marked against the obligation it started as
-- Group-normal-of-smaller-congruence.agda
--
-- File: data/benchmarks/agda-algebras-composition-v0/obligations/Group-normal-of-smaller-congruence.agda
--
-- Benchmark obligation: comp-group-normal-of-smaller-congruence
-- Difficulty: compositional
-- Import stratum: composition
--
-- If θ ⊆ φ and the normal subgroup of φ lies in N, the normal subgroup of θ lies in N.
--
module Group-normal-of-smaller-congruence where

open import AgdaDojang.Debug

open import Data.Product                            using ( proj₁ )
open import Level                                   using ( Level )
open import Classical.Structures.Group.Basic        using ( Group )
open import Classical.Structures.Group.Congruences  using ( module GroupCongruences )
open import Setoid.Congruences.Basic                using ( Con )
open import Setoid.Congruences.Lattice              using ( _⊆_ ; ⊆-refl )

module _ {α ρ ℓ : Level} (𝒢 : Group α ρ) where

  open GroupCongruences 𝒢  using ( NormalSubgroup ; normalOf ; _≤ⁿ_ ; ≤ⁿ-refl )

  normal-of-smaller-congruence : {θ φ : Con (proj₁ 𝒢) ℓ} {𝑵 : NormalSubgroup ℓ}
    → θ ⊆ φ → normalOf φ ≤ⁿ 𝑵 → normalOf θ ≤ⁿ 𝑵
  normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N = {!!}
  normal-of-smaller-congruence {θ} {φ} {𝑵} θ⊆φ φ≤N = λ p → φ≤N (θ⊆φ p)

reports/agent-bench/comp-opus5-mcp-1/subjects/comp-group-normal-of-smaller-congruence/final/Group-normal-of-smaller-congruence.agda

below-join-bound Sonnet 5 refused

A file that checks and is not a solve. One `exports_of` lists both needles and the session writes the gold's proof, `≤-trans x≤y∨z (∨-least y≤w z≤w)`, supplying the middle point through the second needle's conclusion. To bring the two into scope it appended them to the fixture's own `using` list, and the preservation gate refused the file. `≤-trans` is the root lemma the loop reached for on this row, and `fill_hole` refused it with holes for its arguments.

obligation
comp-lattice-below-join-bound
tier
compositional
stratum
agda-algebras/composition
the needles
Classical.Properties.Lattice.Lattice-Order.≤-trans, Classical.Properties.Lattice.Lattice-Order.∨-least
model
claude-sonnet-5
turns
6
tool calls
5
wall
29 s
cost, list price
USD 0.09
the hole’s goal, as get_goal reported it
∀ {x y z w : 𝕌[ proj₁ 𝑳 ]} → x ≤ (y ∨ z) → y ≤ w → z ≤ w → x ≤ w
the session, 6 turns and 5 tool calls
  1. 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.
  2. the work directory
    Read()
    answered
    1 -- Lattice-below-join-bound.agda
    the whole answer, as the agent saw it (1,109 characters of text)
    1	-- Lattice-below-join-bound.agda
2	--
3	-- File: data/benchmarks/agda-algebras-composition-v0/obligations/Lattice-below-join-bound.agda
4	--
5	-- Benchmark obligation: comp-lattice-below-join-bound
6	-- Difficulty: compositional
7	-- Import stratum: composition
8	--
9	-- In a lattice, anything below a join lies below every common upper bound of the joinands.
10	--
11	module Lattice-below-join-bound where
12	
13	open import AgdaDojang.Debug
14	
15	open import Data.Product                        using ( proj₁ )
16	open import Level                               using ( Level )
17	open import Classical.Structures.Lattice.Basic  using ( Lattice ; module Lattice-Op )
18	open import Classical.Properties.Lattice        using ( module Lattice-Order )
19	open import Setoid.Algebras.Basic               using ( 𝕌[_] )
20	
21	module _ {α ρ : Level} (𝑳 : Lattice α ρ) where
22	
23	  open Lattice-Op 𝑳     using ( _∨_ )
24	  open Lattice-Order 𝑳  using ( _≤_ ; ≤-refl )
25	
26	  below-join-bound : ∀ {x y z w : 𝕌[ proj₁ 𝑳 ]} → x ≤ (y ∨ z) → y ≤ w → z ≤ w → x ≤ w
27	  below-join-bound x≤y∨z y≤w z≤w = {!!}
28
  3. 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.
  4. agda-mcp
    exports_of(module: Lattice-Order)
    one of 2 calls the model issued in this turn
    answered
    elapsedMs: 5297exports: 19modules: 0
    the whole answer, as the agent saw it (6,183 characters of JSON)
    {"checkedFromSource":true,"elapsedMs":5297,"exports":[{"name":"_≤_","type":"{α ρ : Level} (𝑳 : Lattice α ρ) →\n𝕌[ Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ] →\n𝕌[ Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ] → Set ρ"},{"name":"IsBottom","type":"{α ρ : Level} (𝑳 : Lattice α ρ) →\n𝕌[ Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ] →\nSet (α Level.⊔ ρ)"},{"name":"IsTop","type":"{α ρ : Level} (𝑳 : Lattice α ρ) →\n𝕌[ Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ] →\nSet (α Level.⊔ ρ)"},{"name":"bot-unique","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{b b'\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\nLattice-Order.IsBottom 𝑳 b →\nLattice-Order.IsBottom 𝑳 b' →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ b)\nb'"},{"name":"top-unique","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{t t'\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\nLattice-Order.IsTop 𝑳 t →\nLattice-Order.IsTop 𝑳 t' →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ t)\nt'"},{"name":"∧-greatest","type":"{α ρ : Level} (𝑳 : 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 Lattice-Order.≤ z) x →\n(𝑳 Lattice-Order.≤ z) y →\n(𝑳 Lattice-Order.≤ z) ((𝑳 Lattice-Op.∧ x) y)"},{"name":"∧-lowerʳ","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(𝑳 Lattice-Order.≤ (𝑳 Lattice-Op.∧ x) y) y"},{"name":"∧-lowerˡ","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(𝑳 Lattice-Order.≤ (𝑳 Lattice-Op.∧ x) y) x"},{"name":"∨-least","type":"{α ρ : Level} (𝑳 : 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 Lattice-Order.≤ x) z →\n(𝑳 Lattice-Order.≤ y) z →\n(𝑳 Lattice-Order.≤ (𝑳 Lattice-Op.∨ x) y) z"},{"name":"∨-upperʳ","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))}\n{y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(𝑳 Lattice-Order.≤ y) ((𝑳 Lattice-Op.∨ x) y)"},{"name":"∨-upperˡ","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))}\n{y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(𝑳 Lattice-Order.≤ x) ((𝑳 Lattice-Op.∨ x) y)"},{"name":"≤-antisym","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(𝑳 Lattice-Order.≤ x) y →\n(𝑳 Lattice-Order.≤ y) x →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ x)\ny"},{"name":"≤-from-∨","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∨ x) y)\ny →\n(𝑳 Lattice-Order.≤ x) y"},{"name":"≤-refl","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(𝑳 Lattice-Order.≤ x) x"},{"name":"≤-reflexive","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ x)\ny →\n(𝑳 Lattice-Order.≤ x) y"},{"name":"≤-respʳ-≈","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y y'\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ y)\ny' →\n(𝑳 Lattice-Order.≤ x) y → (𝑳 Lattice-Order.≤ x) y'"},{"name":"≤-respˡ-≈","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x x' y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ x)\nx' →\n(𝑳 Lattice-Order.≤ x) y → (𝑳 Lattice-Order.≤ x') y"},{"name":"≤-trans","type":"{α ρ : Level} (𝑳 : 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 Lattice-Order.≤ x) y →\n(𝑳 Lattice-Order.≤ y) z → (𝑳 Lattice-Order.≤ x) z"},{"name":"≤-via-∨","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳))} →\n(𝑳 Lattice-Order.≤ x) y →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Properties.Lattice.Lattice-Order.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∨ x) y)\ny"}],"lane":{"load":"first","loadElapsedMs":5259},"module":"Lattice-Order","modules":[],"project":{"root":"<work>","rootSource":"server-config"},"total":19,"truncated":false}
  5. agda-mcp
    exports_of(module: Lattice-Op)
    one of 2 calls the model issued in this turn
    answered
    elapsedMs: 10exports: 15modules: 0
    the whole answer, as the agent saw it (6,864 characters of JSON)
    {"checkedFromSource":false,"elapsedMs":10,"exports":[{"name":"_∧_","type":"{α ρ : Level} (𝑳 : Lattice α ρ) →\n𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ] →\n𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ] →\n𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]"},{"name":"_∨_","type":"{α ρ : Level} (𝑳 : Lattice α ρ) →\n𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ] →\n𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ] →\n𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]"},{"name":"absorbʳ-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝑳 Lattice-Op.∨ (𝑳 Lattice-Op.∧ x) y) x)\nx"},{"name":"absorbˡ-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝑳 Lattice-Op.∧ x) ((𝑳 Lattice-Op.∨ x) y))\nx"},{"name":"equations","type":"{α ρ : Level} (𝑳 : Lattice α ρ) →\nClassical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳\nClassical.Structures.Lattice.Basic.⊨ˡᵃ\nClassical.Theories.Lattice.Th-Lattice"},{"name":"interp-node-∧","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n(s t : Overture.Terms.Basic.Term (Data.Fin.Base.Fin 3))\n{η\n : Data.Fin.Base.Fin 3 →\n   𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to\n (Setoid.Terms.Basic.Environment.⟦\n  Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ⟧\n  (Overture.Terms.node Classical.Signatures.Lattice.Op-Lattice.∧-Op\n   (Classical.Operations.pair s t)))\n η)\n((𝑳 Lattice-Op.∧\n  Function.Bundles.Func.to\n  (Setoid.Terms.Basic.Environment.⟦\n   Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ⟧\n   s)\n  η)\n (Function.Bundles.Func.to\n  (Setoid.Terms.Basic.Environment.⟦\n   Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ⟧\n   t)\n  η))"},{"name":"interp-node-∨","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n(s t : Overture.Terms.Basic.Term (Data.Fin.Base.Fin 3))\n{η\n : Data.Fin.Base.Fin 3 →\n   𝕌[ Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈\n Function.Bundles.Func.to\n (Setoid.Terms.Basic.Environment.⟦\n  Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ⟧\n  (Overture.Terms.node Classical.Signatures.Lattice.Op-Lattice.∨-Op\n   (Classical.Operations.pair s t)))\n η)\n((𝑳 Lattice-Op.∨\n  Function.Bundles.Func.to\n  (Setoid.Terms.Basic.Environment.⟦\n   Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ⟧\n   s)\n  η)\n (Function.Bundles.Func.to\n  (Setoid.Terms.Basic.Environment.⟦\n   Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ⟧\n   t)\n  η))"},{"name":"∧-assoc-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y z\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝑳 Lattice-Op.∧ (𝑳 Lattice-Op.∧ x) y) z)\n((𝑳 Lattice-Op.∧ x) ((𝑳 Lattice-Op.∧ y) z))"},{"name":"∧-comm-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∧ x) y)\n((𝑳 Lattice-Op.∧ y) x)"},{"name":"∧-cong","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y u v\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ x)\ny →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ u)\nv →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∧ x) u)\n((𝑳 Lattice-Op.∧ y) v)"},{"name":"∧-idem-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∧ x) x)\nx"},{"name":"∨-assoc-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y z\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈\n (𝑳 Lattice-Op.∨ (𝑳 Lattice-Op.∨ x) y) z)\n((𝑳 Lattice-Op.∨ x) ((𝑳 Lattice-Op.∨ y) z))"},{"name":"∨-comm-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∨ x) y)\n((𝑳 Lattice-Op.∨ y) x)"},{"name":"∨-cong","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x y u v\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ x)\ny →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ u)\nv →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∨ x) u)\n((𝑳 Lattice-Op.∨ y) v)"},{"name":"∨-idem-law","type":"{α ρ : Level} (𝑳 : Lattice α ρ)\n{x\n : Relation.Binary.Bundles.Setoid.Carrier\n   (Setoid.Algebras.Basic.Algebra.Domain\n    (Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳))} →\n(Setoid.Algebras.Basic.𝔻[\n Classical.Structures.Lattice.Basic.Lattice-Op.𝑨 𝑳 ]\n Relation.Binary.Bundles.Setoid.≈ (𝑳 Lattice-Op.∨ x) x)\nx"}],"lane":{"load":"reused"},"module":"Lattice-Op","modules":[],"project":{"root":"<work>","rootSource":"server-config"},"total":15,"truncated":false}
  6. the model
    We have `≤-trans`, `∨-least`. So `below-join-bound x≤y∨z y≤w z≤w = ≤-trans x≤y∨z (∨-least y≤w z≤w)`.
  7. the work directory
    Edit(replace_all: false, old_string: open Lattice-Op 𝑳 using ( _∨_ ) open Lattice-Order 𝑳 using ( _≤_ ; ≤-refl ) below-join-bound : ∀ {x y z w : 𝕌[ proj₁ 𝑳 ]} → x ≤ (y ∨ z) → y ≤ w → z …, new_string: open Lattice-Op 𝑳 using ( _∨_ ) open Lattice-Order 𝑳 using ( _≤_ ; ≤-refl ; ≤-trans ; ∨-least ) below-join-bound : ∀ {x y z w : 𝕌[ proj₁ 𝑳 ]} → x ≤ …)
    answered
    The file <work>/Lattice-below-join-bound.agda has been updated successfully. (file state is current in your context — no…
    the whole answer, as the agent saw it (142 characters of text)
    The file <work>/Lattice-below-join-bound.agda has been updated successfully. (file state is current in your context — no need to Read it back)
  8. 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.
  9. agda-mcp
    check_file()
    answered
    success: trueholesCount: 0diagnosticsTotal: 0verdict.exitCode: 0elapsedMs: 5292diagnostics: 0holes: 0
    the whole answer, as the agent saw it (221 characters of JSON)
    {"checkedFromSource":true,"diagnostics":[],"diagnosticsTotal":0,"elapsedMs":5292,"holes":[],"holesCount":0,"project":{"root":"<work>","rootSource":"server-config"},"success":true,"timedOut":false,"verdict":{"exitCode":0}}
  10. the model
    check_file reports success on the final file (exit code 0, no diagnostics, no holes).

The model’s words and its calls are quoted from transcript.jsonl; every answer below a call is the server’s own, in full. The client billed 100 thinking tokens over the session; their text is not in the archive.

refusedthe judge, in outcome.json

Refused at the preservation gate: a line the statement is posed in changed, moved, or is missing: open Lattice-Order 𝑳 using ( _≤_ ; ≤-refl ). Agda itself exited 0 on this file.

Agda’s exit code 0
statement preserved yes
turns 6

Where each needle came from: the first event of the session that names it.

  • ≤-trans: first shown by the answer to call 2, exports_of on Lattice-Order; the final file uses it.
  • ∨-least: first shown by the answer to call 2, exports_of on Lattice-Order; the final file uses it.
the file the judge read, marked against the obligation it started as
-- Lattice-below-join-bound.agda
--
-- File: data/benchmarks/agda-algebras-composition-v0/obligations/Lattice-below-join-bound.agda
--
-- Benchmark obligation: comp-lattice-below-join-bound
-- Difficulty: compositional
-- Import stratum: composition
--
-- In a lattice, anything below a join lies below every common upper bound of the joinands.
--
module Lattice-below-join-bound where

open import AgdaDojang.Debug

open import Data.Product                        using ( proj₁ )
open import Level                               using ( Level )
open import Classical.Structures.Lattice.Basic  using ( Lattice ; module Lattice-Op )
open import Classical.Properties.Lattice        using ( module Lattice-Order )
open import Setoid.Algebras.Basic               using ( 𝕌[_] )

module _ {α ρ : Level} (𝑳 : Lattice α ρ) where

  open Lattice-Op 𝑳     using ( _∨_ )
  open Lattice-Order 𝑳  using ( _≤_ ; ≤-refl )
  open Lattice-Order 𝑳  using ( _≤_ ; ≤-refl ; ≤-trans ; ∨-least )

  below-join-bound : ∀ {x y z w : 𝕌[ proj₁ 𝑳 ]} → x ≤ (y ∨ z) → y ≤ w → z ≤ w → x ≤ w
  below-join-bound x≤y∨z y≤w z≤w = {!!}
  below-join-bound x≤y∨z y≤w z≤w = ≤-trans x≤y∨z (∨-least y≤w z≤w)

reports/agent-bench/comp-sonnet5-mcp-1/subjects/comp-lattice-below-join-bound/final/Lattice-below-join-bound.agda

What the tier says: it is a second instrument built to sit below the ceiling, and it could not tell the tools apart. Every configuration of both models wrote a proof that checks for every row, so the arms differ in the route to it and in the gates, not in what was proved. With twelve rows and one seed per arm, a difference of a row or two would not be a finding.

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 numbers; 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 14,437 files and 117 MB, counted when this page was built, and none of it is shipped to your browser; what is here is eight sessions and the tables.

Three things the build refuses to do. It will not write a page whose three tables from ADR 0001 § 9, the agents’, the control’s, and the composition tier’s, disagree with the ADR (make demo-check runs that comparison alone, from the run reports, the composition arms’ final files, and the ADR). 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 in the following files:

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.