From 9b8695dda7fa9f56cd8ed9dc9690eb6b3eec2ebc Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 22 Apr 2026 16:36:58 +0000 Subject: [PATCH 1/4] Delete duplicate broken orphan modules MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit These three modules were committed in 8b8fc03 ("further proofing bridges") but never compiled and duplicated functionality already covered by in-suite modules: * EchoCategory.agda (58 sigs, 0 data) — duplicate of EchoCategorical.agda; had self-shadowing `_→_` and `_∘_` definitions that parse-errored. * EchoCNO.agda (8 sigs) — duplicate of EchoCNOBridge.agda; used invalid `Σ (x : A) (λ _ → ...)` binder syntax and had a dangling top-level `where` clause. * EchoJanusSimple.agda (9 sigs) — duplicate of EchoJanusBridge.agda; same invalid `Σ (x : A) (...)` binder syntax. Nothing in the verified suite (All.agda) imported any of them. Deleting them removes silently-broken Agda from the tree without changing what compiles. https://claude.ai/code/session_01JRLz84fAaWvRBKyXuc4tyK --- proofs/agda/EchoCNO.agda | 66 ------- proofs/agda/EchoCategory.agda | 323 ------------------------------- proofs/agda/EchoJanusSimple.agda | 52 ----- 3 files changed, 441 deletions(-) delete mode 100644 proofs/agda/EchoCNO.agda delete mode 100644 proofs/agda/EchoCategory.agda delete mode 100644 proofs/agda/EchoJanusSimple.agda diff --git a/proofs/agda/EchoCNO.agda b/proofs/agda/EchoCNO.agda deleted file mode 100644 index 90fe0be..0000000 --- a/proofs/agda/EchoCNO.agda +++ /dev/null @@ -1,66 +0,0 @@ -{-# OPTIONS --safe --without-K #-} - -module EchoCNO where - -open import Level using (Level; _⊔_) -open import Function.Base using (_∘_; id) -open import Data.Product.Base using (Σ; _,_; proj₁; proj₂) -open import Relation.Binary.PropositionalEquality using (_≡_; refl; trans; cong) -open import Echo - --- Import CNO concepts (simplified for bridging) --- We'll model CNOs as programs that preserve state (identity function) - --- Program state as a simple model -ProgramState : Set -ProgramState = ℕ → ℕ -- Simplified memory model - --- Identity program (CNO) -id-program : ProgramState → ProgramState -id-program = id - --- Echo type for CNO: fiber over identity function --- Echo id-program s = Σ (s' : ProgramState) (λ _ → id-program s' ≡ s) --- This simplifies to: Σ (s' : ProgramState) (λ _ → s' ≡ s) --- Which is essentially the singleton fiber containing only s - --- CNO as echo: the fiber contains exactly the original state -CNO-Echo : (s : ProgramState) → Echo id-program s -CNO-Echo s = s , refl - --- CNO preservation: mapping over identity preserves the echo -cno-preservation : ∀ {s : ProgramState} → Echo id-program s → Echo id-program s -cno-preservation e = e - --- CNO composition: composing identity with itself preserves echoes -cno-composition : ∀ {s : ProgramState} (e : Echo id-program s) → - map-over (id , (λ x → refl)) e ≡ e -cno-composition (s , p) = refl - --- Bridge theorem: CNOs are echoes over identity functions --- This shows that CNOs can be formalized as a special case of echo types --- where the computation is the identity function (no state change) -cno-as-echo-theorem : ∀ (s : ProgramState) → - Echo id-program s → - Σ (s' : ProgramState) (λ _ → s' ≡ s) -cno-as-echo-theorem s (s' , p) = s' , p - --- Reverse direction: echoes over identity are CNOs -echo-as-cno-theorem : ∀ (s : ProgramState) → - Σ (s' : ProgramState) (λ _ → s' ≡ s) → - Echo id-program s -echo-as-cno-theorem s (s' , p) = s' , p - --- Equivalence between CNOs and identity echoes -cno-echo-equivalence : ∀ (s : ProgramState) → - Echo id-program s ≃ (Σ (s' : ProgramState) (λ _ → s' ≡ s)) -cno-echo-equivalence s = - (λ e → cno-as-echo-theorem s e) , - (λ e' → echo-as-cno-theorem s e') , - (λ e → refl) , - (λ e' → refl) - -where - open import Function.Equivalence using (_≃_; module Equivalence) - open import Data.Product using (∃-syntax) - open import Relation.Binary.PropositionalEquality.TrustMe \ No newline at end of file diff --git a/proofs/agda/EchoCategory.agda b/proofs/agda/EchoCategory.agda deleted file mode 100644 index e91dc11..0000000 --- a/proofs/agda/EchoCategory.agda +++ /dev/null @@ -1,323 +0,0 @@ -{-# OPTIONS --safe --without-K #-} - -module EchoCategory where - -open import Level using (Level; _⊔_; suc) -open import Function.Base using (_∘_; id; _∘_) -open import Data.Product.Base using (Σ; _,_; proj₁; proj₂) -open import Relation.Binary.PropositionalEquality using (_≡_; refl; trans; cong; sym) -open import Data.Unit using (⊤; tt) -open import Echo -open import EchoCNOBridge - --- Categorical formalization of echo types --- Bridging echo type theory with category theory for universal CNO definitions - ----------------------------------------------------------------------------- --- Section 1: Basic Category Theory Definitions ----------------------------------------------------------------------------- - --- Objects in our category are sets -Ob : Set₁ -Ob = Set - --- Morphisms (functions between sets) -_→_ : Ob → Ob → Set -A → B = A → B -- Using function type for morphisms - --- Identity morphism -id-morph : ∀ {A : Ob} → A → A -id-morph = id - --- Composition of morphisms -_∘_ : ∀ {A B C : Ob} → (B → C) → (A → B) → (A → C) -_∘_ = _∘_ - --- Category laws (proof assertions included) -left-identity : ∀ {A B : Ob} (f : A → B) → (id ∘ f) ≡ f -left-identity f = refl - -right-identity : ∀ {A B : Ob} (f : A → B) → (f ∘ id) ≡ f -right-identity f = refl - -associativity : ∀ {A B C D : Ob} (f : C → D) (g : B → C) (h : A → B) → - f ∘ (g ∘ h) ≡ (f ∘ g) ∘ h -associativity f g h = refl - --- Proof assertion: Category laws hold -category-laws-proof : ∀ {A B C D : Ob} (f : C → D) (g : B → C) (h : A → B) → - (id ∘ f ≡ f) × ((f ∘ id) ≡ f) × (f ∘ (g ∘ h) ≡ (f ∘ g) ∘ h) -category-laws-proof f g h = left-identity f , right-identity f , associativity f g h - ----------------------------------------------------------------------------- --- Section 2: Echo Types as Fibrations ----------------------------------------------------------------------------- - --- Fibration structure: for each morphism f : A → B, we have a fiber over each y : B -Fibration : Set₁ -Fibration = ∀ {A B : Ob} (f : A → B) → B → Ob - --- Echo types form a fibration -EchoFibration : Fibration -EchoFibration {A} {B} f y = Echo f y - --- Fiber over a morphism f at point y -Fiber : Fibration → ∀ {A B : Ob} (f : A → B) → B → Ob -Fiber F f y = F f y - --- Echo fiber (specific instance) -EchoFiber : ∀ {A B : Ob} (f : A → B) → B → Ob -EchoFiber = Fiber EchoFibration - --- Proof assertion: Echo types satisfy fibration properties -echo-fibration-proof : ∀ {A B : Ob} (f : A → B) (y : B) → - EchoFiber f y ≡ Echo f y -echo-fibration-proof f y = refl - ----------------------------------------------------------------------------- --- Section 3: Echo Category (Category of Echo Types) ----------------------------------------------------------------------------- - --- Objects in Echo category are echo types -EchoOb : Set₁ -EchoOb = ∀ {A B : Ob} (f : A → B) (y : B) → Ob - --- Morphisms in Echo category are maps between echo types -EchoMorph : EchoOb → EchoOb → Set -EchoMorph E1 E2 = ∀ {A B A' B'} {f : A → B} {f' : A' → B'} {y : B} {y' : B'} → - E1 f y → E2 f' y' → Set - --- Identity morphism in Echo category -echo-id-morph : ∀ {A B : Ob} {f : A → B} {y : B} → Echo f y → Echo f y -echo-id-morph e = e - --- Composition in Echo category -echo-comp : ∀ {A B A' B' A'' B'' : Ob} {f : A → B} {f' : A' → B'} {f'' : A'' → B''} - {y : B} {y' : B'} {y'' : B''} → - (Echo f y → Echo f' y') → (Echo f' y' → Echo f'' y'') → - (Echo f y → Echo f'' y'') -echo-comp g h e = h (g e) - --- Proof assertion: Echo category satisfies category laws -echo-category-laws : ∀ {A B A' B' A'' B'' : Ob} {f : A → B} {f' : A' → B'} {f'' : A'' → B''} - {y : B} {y' : B'} {y'' : B''} - (g : Echo f y → Echo f' y') (h : Echo f' y' → Echo f'' y'') - (e : Echo f y) → - (echo-comp (echo-id-morph {f = f} {y = y}) g e ≡ g e) × - (echo-comp g (echo-id-morph {f = f'} {y = y'}) e ≡ g e) × - (echo-comp h (echo-comp g h) e ≡ echo-comp (echo-comp h g) h e) -echo-category-laws g h e = refl , refl , refl - ----------------------------------------------------------------------------- --- Section 4: CNOs as Identity Morphisms ----------------------------------------------------------------------------- - --- CNOs as identity morphisms in the echo category -CNO-Identity : ∀ {A : Ob} (s : A) → Echo id-morph s → Echo id-morph s -CNO-Identity s e = e - --- Proof assertion: CNOs satisfy identity morphism properties -cno-identity-proof : ∀ {A : Ob} (s : A) (e : Echo id-morph s) → - CNO-Identity s e ≡ e -cno-identity-proof s e = refl - --- Universal CNO definition: CNOs are identity morphisms that preserve echo structure -UniversalCNO : ∀ {A : Ob} (p : A → A) → Set -UniversalCNO p = ∀ (s : A) → Echo p s ≃ Echo id-morph s - --- Proof assertion: Identity function is a universal CNO -identity-universal-cno : UniversalCNO id-morph -identity-universal-cno s = - (λ e → (proj₁ e , trans (cong id-morph (proj₂ e)) (proj₂ e))) , - (λ e' → (proj₁ e' , trans (sym (cong id-morph (proj₂ e'))) (proj₂ e'))) , - (λ e → refl) , - (λ e' → refl) - --- Proof assertion: All CNOs are universal CNOs -all-cnos-universal : ∀ {A : Ob} (p : A → A) → (∀ s → p s ≡ s) → UniversalCNO p -all-cnos-universal p p-id s = - (λ e → (proj₁ e , trans (p-id (proj₁ e)) (proj₂ e))) , - (λ e' → (proj₁ e' , trans (sym (p-id (proj₁ e'))) (proj₂ e'))) , - (λ e → refl) , - (λ e' → refl) - ----------------------------------------------------------------------------- --- Section 5: Functorial Properties of Echo Maps ----------------------------------------------------------------------------- - --- Echo maps preserve identity -map-preserves-identity : ∀ {A B : Ob} {f : A → B} {y : B} → - map-over (id , (λ x → refl)) ≡ echo-id-morph -map-preserves-identity = refl - --- Echo maps preserve composition (functoriality) -map-preserves-composition : ∀ {A B A' B' A'' B'' : Ob} - {f : A → B} {f' : A' → B'} {f'' : A'' → B''} - (u1 : A → A') (c1 : ∀ x → f' (u1 x) ≡ f x) - (u2 : A' → A'') (c2 : ∀ x → f'' (u2 x) ≡ f' x) - {y : B} (e : Echo f y) → - map-over (u2 ∘ u1 , (λ x → trans (c2 (u1 x)) (c1 x))) e ≡ - map-over (u2 , c2) (map-over (u1 , c1) e) -map-preserves-composition u1 c1 u2 c2 (x , p) = refl - --- Proof assertion: Echo maps form a functor -echo-functor-proof : ∀ {A B A' B' A'' B'' : Ob} - {f : A → B} {f' : A' → B'} {f'' : A'' → B''} - (u1 : A → A') (c1 : ∀ x → f' (u1 x) ≡ f x) - (u2 : A' → A'') (c2 : ∀ x → f'' (u2 x) ≡ f' x) - {y : B} (e : Echo f y) → - (map-over (id , (λ x → refl)) e ≡ e) × - (map-over (u2 ∘ u1 , (λ x → trans (c2 (u1 x)) (c1 x))) e ≡ - map-over (u2 , c2) (map-over (u1 , c1) e)) -echo-functor-proof u1 c1 u2 c2 e = - map-preserves-identity , map-preserves-composition u1 c1 u2 c2 e - ----------------------------------------------------------------------------- --- Section 6: Model Independence and Universal Properties ----------------------------------------------------------------------------- - --- Model independence: Echo types are preserved across isomorphic models -ModelIndependence : ∀ {A B A' B' : Ob} (f : A → B) (f' : A' → B') - (isoA : A ≃ A') (isoB : B ≃ B') - (commute : ∀ a → f' (proj₁ (isoA a)) ≡ proj₁ (isoB (f a))) - {y : B} → - Echo f y ≃ Echo f' (proj₁ (isoB y)) -ModelIndependence f f' isoA isoB commute {y} = - (λ e → (proj₁ (isoA (proj₁ e)) , trans (commute (proj₁ e)) (cong (proj₁ ∘ isoB) (proj₂ e)))) , - (λ e' → (proj₁ (sym (isoA) (proj₁ e')) , trans (sym (commute (proj₁ (sym (isoA) (proj₁ e'))))) (cong (sym (proj₁ ∘ isoB)) (proj₂ e')))) , - (λ e → refl) , - (λ e' → refl) - --- Proof assertion: Model independence preserves echo structure -model-independence-proof : ∀ {A B A' B' : Ob} (f : A → B) (f' : A' → B') - (isoA : A ≃ A') (isoB : B ≃ B') - (commute : ∀ a → f' (proj₁ (isoA a)) ≡ proj₁ (isoB (f a))) - {y : B} (e : Echo f y) → - proj₁ (ModelIndependence f f' isoA isoB commute e) ≡ - proj₁ (isoA (proj₁ e)) -model-independence-proof f f' isoA isoB commute e = refl - --- Universal property: Echo types are the universal way to capture fibers -UniversalProperty : ∀ {A B : Ob} (f : A → B) (y : B) → - (∀ {X : Ob} (g : X → Echo f y) → - ∃ λ (h : X → A) → ∀ x → g x ≡ (h x , refl)) → - Set -UniversalProperty f y prop = ⊤ -- Simplified: echo types satisfy universal property - --- Proof assertion: Echo types satisfy universal property -universal-property-proof : ∀ {A B : Ob} (f : A → B) (y : B) → - UniversalProperty f y (λ g → tt , (λ x → refl)) -universal-property-proof f y = tt - ----------------------------------------------------------------------------- --- Section 7: Connection to Absolute Zero's Category Theory Goals ----------------------------------------------------------------------------- - --- Universal CNO definition matching Absolute Zero's goals -AbsoluteZeroUniversalCNO : ∀ {A : Ob} (p : A → A) → Set -AbsoluteZeroUniversalCNO p = - (∀ s → p s ≡ s) × -- State preservation - (∀ s → Echo p s ≃ Echo id s) × -- Echo equivalence - (∀ s → fiber-energy p s T ≡ zero) -- Zero energy (from EchoThermodynamics) - -where - open import EchoThermodynamics - T : Temperature - T = 1 - --- Proof assertion: Identity function satisfies Absolute Zero's universal CNO definition -identity-satisfies-absolute-zero : AbsoluteZeroUniversalCNO id-morph -identity-satisfies-absolute-zero = - (λ s → refl) , -- State preservation - (λ s → identity-universal-cno s) , -- Echo equivalence - (λ s → cno-zero-energy s T) -- Zero energy - --- Categorical equivalence with Absolute Zero's CNO category -CNO-Category-Equivalence : ∀ {A : Ob} (p : A → A) → - UniversalCNO p ≃ AbsoluteZeroUniversalCNO p -CNO-Category-Equivalence p = - (λ u → (proj₁ (u tt)) , (proj₂ (u tt)) , (λ s → cno-zero-energy s T)) , - (λ az → λ _ → (az.1 , az.2)) , - (λ u → refl) , - (λ az → refl) - --- Proof assertion: Categorical equivalence holds -categorical-equivalence-proof : ∀ {A : Ob} (p : A → A) (u : UniversalCNO p) → - proj₁ (CNO-Category-Equivalence p u) ≡ proj₁ (u tt) -categorical-equivalence-proof p u = refl - ----------------------------------------------------------------------------- --- Section 8: Practical Applications ----------------------------------------------------------------------------- - --- CNO detection via categorical properties -categorical-cno-detection : ∀ {A : Ob} (p : A → A) → Bool -categorical-cno-detection p = - if (UniversalCNO p) then true else false - --- Proof assertion: Categorical detection works for identity -categorical-detection-works : categorical-cno-detection id-morph ≡ true -categorical-detection-works = refl - --- Category-theoretic optimization -categorical-optimization : ∀ {A : Ob} (p : A → A) → - UniversalCNO p → - ∀ s → Echo p s ≃ Echo id s -categorical-optimization p u s = proj₂ (u s) - --- Proof assertion: Categorical optimization preserves echo structure -categorical-optimization-proof : ∀ {A : Ob} (p : A → A) (u : UniversalCNO p) s → - categorical-optimization p u s ≡ proj₂ (u s) -categorical-optimization-proof p u s = refl - ----------------------------------------------------------------------------- --- Export ----------------------------------------------------------------------------- - -module EchoCategory where - -- Basic category theory - Ob = Ob - _→_ = _→_ - id-morph = id-morph - _∘_ = _∘_ - category-laws-proof = category-laws-proof - - -- Echo fibration - EchoFibration = EchoFibration - EchoFiber = EchoFiber - echo-fibration-proof = echo-fibration-proof - - -- Echo category - EchoOb = EchoOb - EchoMorph = EchoMorph - echo-id-morph = echo-id-morph - echo-comp = echo-comp - echo-category-laws = echo-category-laws - - -- CNO categorical properties - CNO-Identity = CNO-Identity - cno-identity-proof = cno-identity-proof - UniversalCNO = UniversalCNO - identity-universal-cno = identity-universal-cno - all-cnos-universal = all-cnos-universal - - -- Functorial properties - map-preserves-identity = map-preserves-identity - map-preserves-composition = map-preserves-composition - echo-functor-proof = echo-functor-proof - - -- Model independence - ModelIndependence = ModelIndependence - model-independence-proof = model-independence-proof - - -- Absolute Zero connection - AbsoluteZeroUniversalCNO = AbsoluteZeroUniversalCNO - identity-satisfies-absolute-zero = identity-satisfies-absolute-zero - CNO-Category-Equivalence = CNO-Category-Equivalence - categorical-equivalence-proof = categorical-equivalence-proof - - -- Practical applications - categorical-cno-detection = categorical-cno-detection - categorical-detection-works = categorical-detection-works - categorical-optimization = categorical-optimization - categorical-optimization-proof = categorical-optimization-proof \ No newline at end of file diff --git a/proofs/agda/EchoJanusSimple.agda b/proofs/agda/EchoJanusSimple.agda deleted file mode 100644 index d6fcdd8..0000000 --- a/proofs/agda/EchoJanusSimple.agda +++ /dev/null @@ -1,52 +0,0 @@ -{-# OPTIONS --safe --without-K #-} - -module EchoJanusSimple where - -open import Level using (Level; _⊔_) -open import Function.Base using (_∘_; id) -open import Data.Product.Base using (Σ; _,_) -open import Relation.Binary.PropositionalEquality using (_≡_; refl; trans; cong) -open import Data.Maybe using (Maybe; just; nothing) -open import Data.Bool.Base using (Bool; true; false) -open import Data.String using (_≡ᵇ_) -open import Echo - --- Simple bridge between Echo Types and JanusKey's Reversible File Operations - --- File system state: mapping from paths to content -FilePath : Set -FilePath = String - -FileContent : Set -FileContent = String - -FileSystem : Set -FileSystem = FilePath → Maybe FileContent - --- Initial empty filesystem -empty-fs : FileSystem -empty-fs = λ _ → nothing - --- Delete file -delete-file : FileSystem → FilePath → FileSystem -delete-file fs path = λ p → if p ≡ᵇ path then nothing else fs p - --- JanusKey delete operation as a function -delete-op : FilePath → FileSystem → FileSystem -delete-op path = delete-file - --- Echo type for delete operation -EchoDelete : FilePath → FileSystem → Set -EchoDelete path fs' = Echo (delete-op path) fs' - --- Theorem: Delete operations are reversible via echo types -delete-reversible : ∀ {path : FilePath} {fs' : FileSystem} → - EchoDelete path fs' → - Sig (fs : FileSystem) (delete-op path fs ≡ fs') -delete-reversible {fs = fs} (fs , p) = fs , p - where - Sig = Σ - --- Example: Empty filesystem delete -empty-delete-example : EchoDelete "test.txt" empty-fs -empty-delete-example = empty-fs , refl \ No newline at end of file From c94e38ad94e39acacd00c1f9e7d7f4f0e16dcefb Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 22 Apr 2026 16:39:46 +0000 Subject: [PATCH 2/4] Fix DyadicEchoBridge universe error and wire in Dyadic MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Three defects kept this file from typechecking: 1. Lines 29-32, 42: the body `Echo (λ _ → ⊤) ⊤` conflated the unit *type* (⊤ : Set) with the unit *value* (tt : ⊤). Echo expects a function A → B and a value y : B, not a function into Set. Changed to `Echo {A = ⊤} (λ _ → tt) tt` — pinning A explicitly because the inferred source type is not constrained anywhere else. 2. ProtocolProvenance pattern-matched on a SessionEcho input whose type depends on the Session constructor. The End case has type ⊤, not Echo, so the single-clause catch-all "ProtocolProvenance echo = echo" could not typecheck. Split into five clauses, synthesising a canonical Echo in the End case via echo-intro. 3. ProvenancePreservation returned the left echo verbatim, but its type was SessionEcho at (S1 >>= S2) rather than at S1. Pattern- match on S1 and synthesise a fresh canonical echo at the concatenated head; pass the right-hand echo through when S1 = End. Also wired Dyadic + DyadicEchoBridge into All.agda so they join the verified suite and cannot silently regress again. https://claude.ai/code/session_01JRLz84fAaWvRBKyXuc4tyK --- proofs/agda/All.agda | 2 ++ proofs/agda/DyadicEchoBridge.agda | 39 ++++++++++++++++++++----------- 2 files changed, 28 insertions(+), 13 deletions(-) diff --git a/proofs/agda/All.agda b/proofs/agda/All.agda index cbb5ff8..a26eb26 100644 --- a/proofs/agda/All.agda +++ b/proofs/agda/All.agda @@ -22,6 +22,8 @@ open import EchoCategorical open import EchoScope open import EchoOrdinal open import EchoJanusBridge +open import Dyadic +open import DyadicEchoBridge open import Ordinal.Base open import Ordinal.Closure diff --git a/proofs/agda/DyadicEchoBridge.agda b/proofs/agda/DyadicEchoBridge.agda index 5d89843..fffda9e 100644 --- a/proofs/agda/DyadicEchoBridge.agda +++ b/proofs/agda/DyadicEchoBridge.agda @@ -26,19 +26,25 @@ SessionAsFunction {p} (Select S1 S2) q = SessionAsFunction S1 q ⊎ SessionAsFun -- Simplified echo over session types (avoiding universe issues) SessionEcho : ∀ {p} → (q : Party) → Session p → Set SessionEcho q End = ⊤ -SessionEcho q (Send A S) = Echo (λ _ → ⊤) ⊤ -SessionEcho q (Recv A S) = Echo (λ _ → ⊤) ⊤ -SessionEcho q (Choice S1 S2) = Echo (λ _ → ⊤) ⊤ -SessionEcho q (Select S1 S2) = Echo (λ _ → ⊤) ⊤ +SessionEcho q (Send A S) = Echo {A = ⊤} (λ _ → tt) tt +SessionEcho q (Recv A S) = Echo {A = ⊤} (λ _ → tt) tt +SessionEcho q (Choice S1 S2) = Echo {A = ⊤} (λ _ → tt) tt +SessionEcho q (Select S1 S2) = Echo {A = ⊤} (λ _ → tt) tt -- Example: Echo over Alice's send protocol AliceSendEcho : SessionEcho Alice AliceSendsUnit -AliceSendEcho = echo-intro (λ _ → tt) tt - --- Protocol provenance: echo retains session structure -ProtocolProvenance : ∀ {p} {S : Session p} {q : Party} → - SessionEcho q S → Echo (λ _ → ⊤) ⊤ -ProtocolProvenance echo = echo +AliceSendEcho = echo-intro {A = ⊤} (λ _ → tt) tt + +-- Protocol provenance: echo retains session structure. For non-End +-- sessions SessionEcho q S is already Echo {A = ⊤} (λ _ → tt) tt; for End it +-- is ⊤ and we synthesize the canonical echo. +ProtocolProvenance : ∀ {p} {S : Session p} {q : Party} → + SessionEcho q S → Echo {A = ⊤} (λ _ → tt) tt +ProtocolProvenance {S = End} _ = echo-intro {A = ⊤} (λ _ → tt) tt +ProtocolProvenance {S = Send _ _} echo = echo +ProtocolProvenance {S = Recv _ _} echo = echo +ProtocolProvenance {S = Choice _ _} echo = echo +ProtocolProvenance {S = Select _ _} echo = echo -- Dyadic echo: session with echo-indexed provenance DyadicEcho : ∀ {p} → Session p → Set @@ -50,12 +56,19 @@ AliceSendWithProvenance = Alice , AliceSendEcho -- Bob's receive protocol with provenance BobReceiveWithProvenance : DyadicEcho BobReceivesUnit -BobReceiveWithProvenance = Bob , echo-intro (λ _ → tt) tt +BobReceiveWithProvenance = Bob , echo-intro {A = ⊤} (λ _ → tt) tt --- Provenance preservation under session concatenation +-- Provenance preservation under session concatenation. We synthesize +-- a fresh echo at the concatenated session, since SessionEcho at a +-- Send/Recv/Choice/Select head is always Echo {A = ⊤} (λ _ → tt) tt and at +-- End it is ⊤. ProvenancePreservation : ∀ {p} {S1 S2 : Session p} → DyadicEcho S1 → DyadicEcho S2 → DyadicEcho (S1 >>= S2) -ProvenancePreservation (q1 , echo1) (q2 , echo2) = q1 , echo1 +ProvenancePreservation {S1 = End} _ (q2 , echo2) = q2 , echo2 +ProvenancePreservation {S1 = Send _ _} (q1 , _) _ = q1 , echo-intro {A = ⊤} (λ _ → tt) tt +ProvenancePreservation {S1 = Recv _ _} (q1 , _) _ = q1 , echo-intro {A = ⊤} (λ _ → tt) tt +ProvenancePreservation {S1 = Choice _ _} (q1 , _) _ = q1 , echo-intro {A = ⊤} (λ _ → tt) tt +ProvenancePreservation {S1 = Select _ _} (q1 , _) _ = q1 , echo-intro {A = ⊤} (λ _ → tt) tt -- Echo-safe dyadic protocols: protocols where echoes preserve safety EchoSafe : ∀ {p} → Session p → Set From e35e9ccc4acbebd1c4e077e5ae15f363f72c6bc2 Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 22 Apr 2026 16:41:03 +0000 Subject: [PATCH 3/4] Rewrite EchoThermodynamics to minimum viable compile MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The previous file failed to typecheck in ~15 places: missing imports (∃, Fin, _≤_, _≃_, proj₁, proj₂, sym, _-_, if-then-else), detached top-level `where` clauses, chained `≤` that doesn't parse as written, reversible-echo-preservation using _≃_ without importing it, a self-assigning nested export module, and theorem bodies such as "echo-to-cno-thermodynamic-mapping" that branched on a proof value as if it were a Bool. Rather than salvage each aspirational theorem, this commit strips the module to the claims it can actually discharge: * Thermodynamic types (Temperature, Energy, Information, Entropy) as ℕ. * Landauer's bit-erasure energy `landauer-energy T = k * T`. * A simplified `FiberSize` (always 1) with matching `fiber-energy` and `echo-energy-cost`. * A concrete CNO model `cno-identity = id` on ProgramState. * Three zero-cost lemmas at temperature zero: - `cno-fiber-size s ≡ 1` - `fiber-energy cno-identity s 0 ≡ 0` - `echo-energy-cost f x 0 ≡ 0` Dropped but documented in-file as not-yet-discharged: CNO zero energy at non-zero T (needs real fiber enumeration), involution-based reversible-echo preservation (needs _≃_ plumbing), energy hierarchy (needs a `≤` relation and non-trivial FiberSize), CNO-detection predicates (need decidable equality on energies). Wired into All.agda. https://claude.ai/code/session_01JRLz84fAaWvRBKyXuc4tyK --- proofs/agda/All.agda | 1 + proofs/agda/EchoThermodynamics.agda | 211 +++++++--------------------- 2 files changed, 53 insertions(+), 159 deletions(-) diff --git a/proofs/agda/All.agda b/proofs/agda/All.agda index a26eb26..c8c17d0 100644 --- a/proofs/agda/All.agda +++ b/proofs/agda/All.agda @@ -24,6 +24,7 @@ open import EchoOrdinal open import EchoJanusBridge open import Dyadic open import DyadicEchoBridge +open import EchoThermodynamics open import Ordinal.Base open import Ordinal.Closure diff --git a/proofs/agda/EchoThermodynamics.agda b/proofs/agda/EchoThermodynamics.agda index 9498f05..79b8c27 100644 --- a/proofs/agda/EchoThermodynamics.agda +++ b/proofs/agda/EchoThermodynamics.agda @@ -1,196 +1,89 @@ {-# OPTIONS --safe --without-K #-} +-- Echo thermodynamics: minimum viable compile. +-- +-- Bridge module expressing Landauer-style bit-erasure cost in terms +-- of a (simplified, singleton) fiber size. This is pedagogical, not a +-- quantitative physics claim: FiberSize is always 1 and fiber-energy +-- of identity functions is zero. The load-bearing content is just +-- that CNOs (identity-preserving programs) have fiber-energy = 0. + module EchoThermodynamics where -open import Level using (Level; _⊔_) -open import Function.Base using (_∘_; id) -open import Data.Product.Base using (Σ; _,_) -open import Relation.Binary.PropositionalEquality using (_≡_; refl; trans; cong) -open import Data.Nat using (ℕ; zero; suc; _+_; _*_) -open import Data.Bool using (Bool; true; false) -open import Echo -open import EchoCNOBridge +open import Data.Nat.Base using (ℕ; zero; suc; _+_; _*_) +open import Relation.Binary.PropositionalEquality using (_≡_; refl) --- Thermodynamic formalization of echo types --- Bridging echo type theory with Landauer's principle and information thermodynamics +open import Echo using (Echo; echo-intro) ---------------------------------------------------------------------------- --- Section 1: Thermodynamic Foundations +-- Thermodynamic types (simplified units) ---------------------------------------------------------------------------- --- Boltzmann constant (simplified for formalization) -k : ℕ -k = 1 -- Simplified constant for formal analysis - --- Temperature in arbitrary units Temperature : Set Temperature = ℕ --- Energy in arbitrary units (Joules) Energy : Set Energy = ℕ --- Landauer's principle: minimum energy to erase one bit -landauer-energy : Temperature → Energy -landauer-energy T = k * T - --- Information content (number of bits) Information : Set Information = ℕ --- Entropy of a system (simplified) Entropy : Set Entropy = ℕ +-- Boltzmann constant in arbitrary units. +k : ℕ +k = 1 + +-- Landauer's principle: minimum energy to erase one bit at temperature T. +landauer-energy : Temperature → Energy +landauer-energy T = k * T + ---------------------------------------------------------------------------- --- Section 2: Echo Type Thermodynamics +-- Simplified fiber size ---------------------------------------------------------------------------- --- Fiber size represents information content -fiber-size : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (y : B) → Set -fiber-size {A = A} f y = ∃ λ n → (vec : Fin n → A) → (∀ i → f (vec i) ≡ y) - --- Simplified fiber size (count of inhabitants) +-- Fiber size as a natural number. Deliberately simplified to 1 at this +-- stage; a real count would require decidable equality on the codomain +-- plus an enumeration of preimages, which this module does not yet have. FiberSize : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (y : B) → ℕ -FiberSize {A = A} f y = 1 -- Simplified for formalization +FiberSize f y = 1 --- Energy dissipation based on fiber analysis --- Larger fibers = more information loss = more energy dissipation +-- Energy dissipation for a single-bit erasure at given temperature, +-- scaled by the simplified fiber size (= 1). fiber-energy : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (y : B) → Temperature → Energy -fiber-energy {A = A} f y T = landauer-energy T * (FiberSize f y) +fiber-energy f y T = landauer-energy T * FiberSize f y --- Thermodynamic cost of computation +-- Thermodynamic cost of computing f on input x at temperature T. echo-energy-cost : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → Temperature → Energy echo-energy-cost f x T = fiber-energy f (f x) T ---------------------------------------------------------------------------- --- Section 3: CNO Thermodynamic Properties ----------------------------------------------------------------------------- - --- CNOs have minimal fiber size (singleton) -cno-fiber-size : ∀ (s : ProgramState) → FiberSize cno-identity s ≡ 1 -cno-fiber-size s = refl - --- CNOs dissipate zero energy (Landauer's principle) -cno-zero-energy : ∀ (s : ProgramState) (T : Temperature) → - fiber-energy cno-identity s T ≡ zero -cno-zero-energy s T = refl - --- CNOs preserve information (no information loss) -cno-information-preservation : ∀ (s : ProgramState) → - echo-energy-cost cno-identity s T ≡ zero -cno-information-preservation s = refl - -where - T : Temperature - T = 1 -- Default temperature for analysis - ----------------------------------------------------------------------------- --- Section 4: Information Loss Analysis ----------------------------------------------------------------------------- - --- Information loss in general echoes --- Measures difference between input and output information content -echo-information-loss : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → Information - echo-information-loss f x = 1 -- Simplified: 1 bit lost per computation - --- CNOs have zero information loss -cno-zero-information-loss : ∀ (s : ProgramState) → echo-information-loss cno-identity s ≡ zero -cno-zero-information-loss s = refl - --- Information loss theorem: larger fibers = more information loss -information-loss-theorem : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → - echo-information-loss f x ≡ (FiberSize f (f x) - 1) -information-loss-theorem f x = refl - ----------------------------------------------------------------------------- --- Section 5: Reversible Computing Bridge ----------------------------------------------------------------------------- - --- Reversible computation preserves echo structure -reversible-echo-preservation : ∀ {a} {A : Set a} (f : A → A) → - (∀ x → f (f x) ≡ x) → -- f is involution - ∀ x → Echo f (f x) ≃ Echo id x -reversible-echo-preservation f inv x = - (λ e → (proj₁ e , trans (inv (proj₁ e)) (proj₂ e))) , - (λ e' → (proj₁ e' , trans (sym (inv (proj₁ e'))) (proj₂ e'))) , - (λ e → refl) , - (λ e' → refl) - --- CNOs are reversible (trivial case) -cno-reversibility : ∀ (s : ProgramState) → - reversible-echo-preservation cno-identity (λ x → refl) s -cno-reversibility s = reversible-echo-preservation cno-identity (λ x → refl) s - ----------------------------------------------------------------------------- --- Section 6: Thermodynamic Equivalence +-- CNO (identity-preserving) thermodynamic properties ---------------------------------------------------------------------------- --- Thermodynamic equivalence: CNOs minimize energy dissipation -thermodynamic-optimality : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) (T : Temperature) → - fiber-energy cno-identity x T ≤ fiber-energy f (f x) T -thermodynamic-optimality f x T = refl -- Simplified: CNOs have minimal energy - --- Energy hierarchy: CNOs < Reversible < Irreversible -energy-hierarchy : ∀ {a} {A : Set a} (f : A → A) (x : A) (T : Temperature) → - let - cno-energy = fiber-energy cno-identity x T - rev-energy = fiber-energy f (f x) T - irrev-energy = fiber-energy (λ x → x) (x) T -- Simplified irreversible - in cno-energy ≤ rev-energy ≤ irrev-energy -energy-hierarchy f x T = refl - ----------------------------------------------------------------------------- --- Section 7: Practical Applications ----------------------------------------------------------------------------- +-- Abstract program state. +ProgramState : Set +ProgramState = ℕ → ℕ --- Energy-efficient computation analysis -energy-efficient-computation : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) (T : Temperature) → - Bool -energy-efficient-computation f x T = - if (fiber-energy f (f x) T ≡ zero) then true else false +-- CNO modelled as identity on ProgramState. +cno-identity : ProgramState → ProgramState +cno-identity s = s --- CNO detection via thermodynamic analysis -cno-thermodynamic-detection : ∀ (p : ProgramState → ProgramState) (s : ProgramState) (T : Temperature) → - Bool -cno-thermodynamic-detection p s T = - if (fiber-energy p s T ≡ zero) then true else false - --- Energy optimization theorem -energy-optimization-theorem : ∀ (p : ProgramState → ProgramState) (s : ProgramState) (T : Temperature) → - cno-thermodynamic-detection p s T ≡ true → - fiber-energy p s T ≡ zero -energy-optimization-theorem p s T det = det - ----------------------------------------------------------------------------- --- Section 8: Connection to Absolute Zero ----------------------------------------------------------------------------- - --- Map echo thermodynamic properties to Absolute Zero's CNO properties -echo-to-cno-thermodynamic-mapping : ∀ (p : ProgramState → ProgramState) → - (∀ σ → fiber-energy p σ T ≡ zero) → - (∀ σ → cno-thermodynamic-detection p σ T ≡ true) -echo-to-cno-thermodynamic-mapping p zero-energy σ = - if (zero-energy σ) then refl else refl - --- Thermodynamic verification of CNO properties -thermodynamic-cno-verification : ∀ (p : ProgramState → ProgramState) → - (∀ σ → fiber-energy p σ T ≡ zero) → - (∀ σ → echo-energy-cost p σ T ≡ zero) → - (∀ σ → echo-information-loss p σ ≡ zero) → - (∀ σ → cno-zero-energy σ T) -thermodynamic-cno-verification p zero-energy zero-cost zero-loss σ = - zero-energy σ - ----------------------------------------------------------------------------- --- Export ----------------------------------------------------------------------------- +-- CNOs have unit fiber size (trivially, by FiberSize's simplification). +cno-fiber-size : ∀ (s : ProgramState) → FiberSize cno-identity s ≡ 1 +cno-fiber-size s = refl -module EchoThermodynamics where - fiber-energy = fiber-energy - cno-zero-energy = cno-zero-energy - echo-information-loss = echo-information-loss - cno-zero-information-loss = cno-zero-information-loss - reversible-echo-preservation = reversible-echo-preservation - thermodynamic-optimality = thermodynamic-optimality - cno-thermodynamic-detection = cno-thermodynamic-detection \ No newline at end of file +-- CNOs at temperature zero dissipate zero energy. This is the only +-- strong-form zero-energy statement the simplified FiberSize lets us +-- prove: landauer-energy 0 = 0 * FiberSize = 0. The stronger claim +-- "CNOs dissipate zero energy at every temperature" is not yet +-- discharged — it needs FiberSize to track actual preimages. +cno-zero-energy-at-zero : ∀ (s : ProgramState) → + fiber-energy cno-identity s 0 ≡ 0 +cno-zero-energy-at-zero s = refl + +-- Echo-cost at temperature zero is likewise zero, for any function. +echo-cost-at-zero : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → + echo-energy-cost f x 0 ≡ 0 +echo-cost-at-zero f x = refl From 095bd21c9e68dc34d7953dc08670e541005eb2bf Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 22 Apr 2026 16:43:28 +0000 Subject: [PATCH 4/4] Rewrite EchoStabilityTests against surviving modules MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The previous file cross-linked four modules. Three of them (EchoCategory, EchoCNO, EchoCNO's thermodynamic mapping) were deleted or rewritten in earlier commits on this branch, leaving roughly 75 % of the tests pointing at missing symbols. On top of that the file itself never typechecked — bad Σ-binder syntax, bogus `where` at top level, self-assigning export module. Rewrite strategy: keep only tests that lock upstream shapes via refl against modules we still ship, document the dropped sections in comments so follow-up work can repopulate them. The surviving tests are: Section 1 (Echo core) echo-intro-fst — proj₁ of echo-intro lands at the supplied x map-over-fst — map-over advances the first component by u map-over-id-check — map-over with identity is the identity Section 2 (thermodynamic core) cno-fiber-is-one — the CNO fiber proof is refl cno-zero-T-check — fiber-energy of CNO at T=0 is 0 echo-cost-zero-T-check — echo-energy-cost of any f at T=0 is 0 Dropped sections (noted in-file for future work): universal-CNO / categorical equivalence (needs EchoCategorical API parity), CNO-singleton equivalence (needs tests against EchoCNOBridge's CNOEcho/nullop-echo), thermodynamic optimality / hierarchy (needs non-trivial FiberSize and a _≤_ on energies). Wired into All.agda. https://claude.ai/code/session_01JRLz84fAaWvRBKyXuc4tyK --- proofs/agda/All.agda | 1 + proofs/agda/EchoStabilityTests.agda | 373 ++++++---------------------- 2 files changed, 76 insertions(+), 298 deletions(-) diff --git a/proofs/agda/All.agda b/proofs/agda/All.agda index c8c17d0..94e94de 100644 --- a/proofs/agda/All.agda +++ b/proofs/agda/All.agda @@ -25,6 +25,7 @@ open import EchoJanusBridge open import Dyadic open import DyadicEchoBridge open import EchoThermodynamics +open import EchoStabilityTests open import Ordinal.Base open import Ordinal.Closure diff --git a/proofs/agda/EchoStabilityTests.agda b/proofs/agda/EchoStabilityTests.agda index a54b5d8..c846673 100644 --- a/proofs/agda/EchoStabilityTests.agda +++ b/proofs/agda/EchoStabilityTests.agda @@ -1,317 +1,94 @@ {-# OPTIONS --safe --without-K #-} +-- Stability tests: a lightweight harness of propositional-equality +-- lemmas that wedge the shape of core Echo/Thermodynamics theorems +-- against unintended rewrites. Each "test" here is a theorem whose +-- body is `refl`, so it breaks immediately under a silent rename, +-- signature change, or constructor reshuffle in the upstream module. +-- +-- Scope note: the earlier version of this file tested against +-- EchoCategory, EchoCNO, and several theorems that were never proved. +-- Those modules were deleted (duplicates of EchoCategorical / +-- EchoCNOBridge), so the corresponding sections were removed rather +-- than rewritten against a different API. Reintroduce on a per- +-- section basis as new content lands. + module EchoStabilityTests where -open import Level using (Level; _⊔_) -open import Function.Base using (_∘_; id) +open import Data.Nat.Base using (ℕ; zero) open import Data.Product.Base using (Σ; _,_; proj₁; proj₂) -open import Relation.Binary.PropositionalEquality using (_≡_; refl; trans; cong) -open import Data.Nat using (ℕ) -open import Echo -open import EchoCNOBridge -open import EchoThermodynamics -open import EchoCategory - --- Local definitions needed for testing -ProgramState : Set -ProgramState = ℕ → ℕ -- Simplified memory model for testing - --- Comprehensive stability tests for echo types framework --- Phase 1: Core stability verification +open import Function.Base using (id; _∘_) +open import Relation.Binary.PropositionalEquality using (_≡_; refl) + +open import Echo using (Echo; echo-intro; MapOver; map-over; map-over-id; map-over-comp) +open import EchoThermodynamics using + ( Temperature + ; fiber-energy + ; echo-energy-cost + ; cno-identity + ; cno-fiber-size + ; cno-zero-energy-at-zero + ; echo-cost-at-zero + ) ---------------------------------------------------------------------------- --- Section 1: Core Echo Type Stability Tests +-- Section 1: Core Echo type stability ---------------------------------------------------------------------------- --- Test 1: Echo type definition is well-formed -EchoDefinitionTest : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (y : B) → Set -EchoDefinitionTest f y = Echo f y - -echo-definition-well-formed : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (y : B) → - EchoDefinitionTest f y ≡ Echo f y -echo-definition-well-formed f y = refl - --- Test 2: Echo introduction preserves function behavior -echo-intro-test : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → - Echo f (f x) -echo-intro-test f x = echo-intro f x - -echo-intro-correct : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → - proj₁ (echo-intro-test f x) ≡ x -echo-intro-correct f x = refl - --- Test 3: Map-over preserves echo structure -map-over-test : ∀ {a a' b} {A : Set a} {A' : Set a'} {B : Set b} - {f : A → B} {f' : A' → B} → - MapOver f f' → ∀ {y : B} → Echo f y → Echo f' y -map-over-test = map-over - -map-over-preserves : ∀ {a a' b} {A : Set a} {A' : Set a'} {B : Set b} - {f : A → B} {f' : A' → B} - (u , commute) (e : Echo f (f (proj₁ e))) → - proj₁ (map-over-test (u , commute) e) ≡ u (proj₁ e) -map-over-preserves (u , commute) (x , p) = refl - --- Test 4: Identity law holds -map-over-id-test : ∀ {a b} {A : Set a} {B : Set b} {f : A → B} {y : B} (e : Echo f y) → - map-over (id , (λ x → refl)) e ≡ e -map-over-id-test (x , p) = refl - --- Test 5: Composition law holds -map-over-comp-test : ∀ {a a' a'' b} - {A : Set a} {A' : Set a'} {A'' : Set a''} {B : Set b} - {f : A → B} {f' : A' → B} {f'' : A'' → B} - (u1 : A → A') (c1 : ∀ x → f' (u1 x) ≡ f x) - (u2 : A' → A'') (c2 : ∀ x → f'' (u2 x) ≡ f' x) - {y : B} (e : Echo f y) → - map-over (u2 ∘ u1 , (λ x → trans (c2 (u1 x)) (c1 x))) e ≡ - map-over (u2 , c2) (map-over (u1 , c1) e) -map-over-comp-test u1 c1 u2 c2 (x , p) = refl - --- Proof assertion: Core echo type properties are stable -core-echo-stability : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → - (echo-intro-test f x ≡ echo-intro f x) × - (map-over-id-test (echo-intro f x) ≡ refl) -core-echo-stability f x = refl , refl +-- Test 1. Echo introduction places the provided x as the first +-- component of the sigma. Breaks under a silent reshape of Echo. +echo-intro-fst : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) → + proj₁ (echo-intro f x) ≡ x +echo-intro-fst f x = refl + +-- Test 2. map-over advances the first component by the underlying +-- map u. Breaks if map-over's tuple shape changes. Uses fully +-- explicit f and f' so Agda can pin the MapOver implicit args. +map-over-fst : ∀ {a a' b} {A : Set a} {A' : Set a'} {B : Set b} + (f : A → B) (f' : A' → B) {y : B} + (u : A → A') (c : ∀ x → f' (u x) ≡ f x) + (x : A) (p : f x ≡ y) → + proj₁ (map-over {f = f} {f' = f'} (u , c) (x , p)) ≡ u x +map-over-fst f f' u c x p = refl + +-- Test 3. Identity law for map-over (restates the pinned lemma so +-- both the name and the shape are locked in by this module). +map-over-id-check : ∀ {a b} {A : Set a} {B : Set b} {f : A → B} {y : B} + (e : Echo f y) → + map-over (id , (λ _ → refl)) e ≡ e +map-over-id-check e = map-over-id e ---------------------------------------------------------------------------- --- Section 2: CNO Bridge Stability Tests +-- Section 2: Thermodynamic stability ---------------------------------------------------------------------------- --- Test 6: CNO echo equivalence -cno-echo-equiv-test : ∀ (s : ProgramState) → - Echo cno-identity s ≃ (Σ (s' : ProgramState) , (s' ≡ s)) -cno-echo-equiv-test s = cno-echo-equivalence s - -cno-echo-equiv-correct : ∀ (s : ProgramState) (e : Echo cno-identity s) → - proj₁ (proj₁ (cno-echo-equiv-test s e)) ≡ proj₁ e -cno-echo-equiv-correct s (s' , p) = refl - --- Test 7: All CNOs are echoes -all-cnos-echos-test : ∀ (p : ProgramState → ProgramState) → - (∀ σ → p σ ≡ σ) → - ∀ σ → Echo p σ ≃ Echo id σ -all-cnos-echos-test p p-id σ = all-cnos-are-echos p p-id σ - -all-cnos-echos-correct : ∀ (p : ProgramState → ProgramState) (p-id : ∀ σ → p σ ≡ σ) σ → - proj₁ (all-cnos-echos-test p p-id σ) ≡ - (λ e → (proj₁ e , trans (p-id (proj₁ e)) (proj₂ e))) -all-cnos-echos-correct p p-id σ = refl - --- Test 8: CNO composition preserves structure -cno-composition-test : ∀ (s : ProgramState) (e : Echo cno-identity s) → - map-over (id , (λ x → refl)) e ≡ e -cno-composition-test s e = cno-composition-echo s e +-- Test 4. The CNO fiber has size one. +cno-fiber-is-one : ∀ (s : ℕ → ℕ) → cno-fiber-size s ≡ refl +cno-fiber-is-one s = refl -cno-composition-correct : ∀ (s : ProgramState) (e : Echo cno-identity s) → - cno-composition-test s e ≡ refl -cno-composition-correct s e = refl +-- Test 5. At temperature zero, a CNO dissipates zero energy. +cno-zero-T-check : ∀ (s : ℕ → ℕ) → + fiber-energy cno-identity s 0 ≡ 0 +cno-zero-T-check s = cno-zero-energy-at-zero s --- Proof assertion: CNO bridge is stable -cno-bridge-stability : ∀ (s : ProgramState) → - (cno-echo-equiv-test s ≡ cno-echo-equivalence s) × - (cno-composition-test s (s , refl) ≡ refl) -cno-bridge-stability s = refl , refl +-- Test 6. At temperature zero, *any* echo-energy-cost is zero. Locks +-- the shape of the cost function against changes to FiberSize. +echo-cost-zero-T-check : ∀ {a b} {A : Set a} {B : Set b} + (f : A → B) (x : A) → + echo-energy-cost f x 0 ≡ 0 +echo-cost-zero-T-check f x = echo-cost-at-zero f x ---------------------------------------------------------------------------- --- Section 3: Thermodynamic Stability Tests +-- What is deliberately not tested here ---------------------------------------------------------------------------- --- Test 9: CNO zero energy -cno-zero-energy-test : ∀ (s : ProgramState) (T : Temperature) → - fiber-energy cno-identity s T ≡ zero -cno-zero-energy-test s T = cno-zero-energy s T - -cno-zero-energy-correct : ∀ (s : ProgramState) (T : Temperature) → - cno-zero-energy-test s T ≡ refl -cno-zero-energy-correct s T = refl - --- Test 10: CNO information preservation -cno-info-preservation-test : ∀ (s : ProgramState) → - echo-information-loss cno-identity s ≡ zero -cno-info-preservation-test s = cno-zero-information-loss s - -cno-info-preservation-correct : ∀ (s : ProgramState) → - cno-info-preservation-test s ≡ refl -cno-info-preservation-correct s = refl - --- Test 11: Thermodynamic verification -thermodynamic-verification-test : ∀ (p : ProgramState → ProgramState) → - (∀ σ → fiber-energy p σ T ≡ zero) → - (∀ σ → cno-zero-energy σ T) -thermodynamic-verification-test p zero-energy σ = thermodynamic-cno-verification p zero-energy (λ _ → refl) (λ _ → refl) σ - -where - T : Temperature - T = 1 - -thermodynamic-verification-correct : ∀ (p : ProgramState → ProgramState) (zero-energy : ∀ σ → fiber-energy p σ T ≡ zero) σ → - thermodynamic-verification-test p zero-energy σ ≡ zero-energy σ -thermodynamic-verification-correct p zero-energy σ = refl - --- Proof assertion: Thermodynamic bridge is stable -thermodynamic-stability : ∀ (s : ProgramState) (T : Temperature) → - (cno-zero-energy-test s T ≡ refl) × - (cno-info-preservation-test s ≡ refl) -thermodynamic-stability s T = refl , refl - ----------------------------------------------------------------------------- --- Section 4: Categorical Stability Tests ----------------------------------------------------------------------------- - --- Test 12: Echo fibration properties -echo-fibration-test : ∀ {A B : Ob} (f : A → B) (y : B) → - EchoFiber f y ≡ Echo f y -echo-fibration-test f y = echo-fibration-proof f y - -echo-fibration-correct : ∀ {A B : Ob} (f : A → B) (y : B) → - echo-fibration-test f y ≡ refl - echo-fibration-correct f y = refl - --- Test 13: Universal CNO definition -universal-cno-test : ∀ {A : Ob} (p : A → A) → Set -universal-cno-test p = UniversalCNO p - -universal-cno-correct : universal-cno-test id-morph ≡ UniversalCNO id-morph -universal-cno-correct = refl - --- Test 14: Categorical equivalence -categorical-equiv-test : ∀ {A : Ob} (p : A → A) → - UniversalCNO p ≃ AbsoluteZeroUniversalCNO p -categorical-equiv-test p = CNO-Category-Equivalence p - -categorical-equiv-correct : ∀ {A : Ob} (p : A → A) (u : UniversalCNO p) → - proj₁ (categorical-equiv-test p u) ≡ proj₁ (u tt) -categorical-equiv-correct p u = categorical-equivalence-proof p u - --- Test 15: Model independence -model-independence-test : ∀ {A B A' B' : Ob} (f : A → B) (f' : A' → B') - (isoA : A ≃ A') (isoB : B ≃ B') - (commute : ∀ a → f' (proj₁ (isoA a)) ≡ proj₁ (isoB (f a))) - {y : B} → - Echo f y ≃ Echo f' (proj₁ (isoB y)) -model-independence-test f f' isoA isoB commute = ModelIndependence f f' isoA isoB commute - -model-independence-correct : ∀ {A B A' B' : Ob} (f : A → B) (f' : A' → B') - (isoA : A ≃ A') (isoB : B ≃ B') - (commute : ∀ a → f' (proj₁ (isoA a)) ≡ proj₁ (isoB (f a))) - {y : B} (e : Echo f y) → - proj₁ (model-independence-test f f' isoA isoB commute e) ≡ - proj₁ (isoA (proj₁ e)) -model-independence-correct f f' isoA isoB commute e = model-independence-proof f f' isoA isoB commute e - --- Proof assertion: Categorical bridge is stable -categorical-stability : ∀ {A : Ob} (p : A → A) → - (universal-cno-test p ≡ UniversalCNO p) × - (categorical-equiv-test p ≡ CNO-Category-Equivalence p) -categorical-stability p = refl , refl - ----------------------------------------------------------------------------- --- Section 5: Integration Stability Tests ----------------------------------------------------------------------------- - --- Test 16: Cross-module consistency -cross-module-test : ∀ (s : ProgramState) → - (Echo cno-identity s ≃ ((s' : ProgramState) × (s' ≡ s))) × - (fiber-energy cno-identity s T ≡ zero) × - (UniversalCNO cno-identity s) -cross-module-test s = - cno-echo-equivalence s , - cno-zero-energy s T , - identity-universal-cno s - -where - T : Temperature - T = 1 - -cross-module-correct : ∀ (s : ProgramState) → - proj₁ (cross-module-test s) ≡ cno-echo-equivalence s -cross-module-correct s = refl - --- Test 17: Bridge coherence -bridge-coherence-test : ∀ (s : ProgramState) → - proj₂ (cross-module-test s) ≡ cno-zero-energy s T -bridge-coherence-test s = refl - --- Proof assertion: Integration is stable -integration-stability : ∀ (s : ProgramState) → - (cross-module-test s ≡ (cno-echo-equivalence s , cno-zero-energy s T , identity-universal-cno s)) × - (bridge-coherence-test s ≡ refl) -integration-stability s = refl , refl - ----------------------------------------------------------------------------- --- Section 6: Comprehensive Stability Report ----------------------------------------------------------------------------- - --- Overall stability metric -StabilityMetric : Set -StabilityMetric = ℕ - --- Calculate stability score (0-100) -calculate-stability : - (core : ℕ) → (cno : ℕ) → (thermo : ℕ) → (cat : ℕ) → (integ : ℕ) → StabilityMetric -calculate-stability core cno thermo cat integ = (core + cno + thermo + cat + integ) / 5 - --- Current stability assessment -current-stability : StabilityMetric -current-stability = calculate-stability 98 95 90 88 92 - --- Proof assertion: Overall stability is high -overall-stability-proof : current-stability ≡ 92 -overall-stability-proof = refl - --- Stability improvement targets -stability-targets : - (core : ℕ) → (cno : ℕ) → (thermo : ℕ) → (cat : ℕ) → (integ : ℕ) → StabilityMetric -stability-targets core cno thermo cat integ = calculate-stability 100 100 95 95 98 - --- Target stability -target-stability : StabilityMetric -target-stability = stability-targets 100 100 95 95 98 - --- Proof assertion: Target is achievable -target-achievable : target-stability ≡ 97 - target-achievable = refl - ----------------------------------------------------------------------------- --- Export ----------------------------------------------------------------------------- - -module EchoStabilityTests where - -- Core stability tests - echo-definition-well-formed = echo-definition-well-formed - echo-intro-correct = echo-intro-correct - map-over-preserves = map-over-preserves - map-over-id-test = map-over-id-test - map-over-comp-test = map-over-comp-test - core-echo-stability = core-echo-stability - - -- CNO bridge tests - cno-echo-equiv-correct = cno-echo-equiv-correct - all-cnos-echos-correct = all-cnos-echos-correct - cno-composition-correct = cno-composition-correct - cno-bridge-stability = cno-bridge-stability - - -- Thermodynamic tests - cno-zero-energy-correct = cno-zero-energy-correct - cno-info-preservation-correct = cno-info-preservation-correct - thermodynamic-verification-correct = thermodynamic-verification-correct - thermodynamic-stability = thermodynamic-stability - - -- Categorical tests - echo-fibration-correct = echo-fibration-correct - universal-cno-correct = universal-cno-correct - categorical-equiv-correct = categorical-equiv-correct - model-independence-correct = model-independence-correct - categorical-stability = categorical-stability - - -- Integration tests - cross-module-correct = cross-module-correct - bridge-coherence-test = bridge-coherence-test - integration-stability = integration-stability - - -- Stability metrics - current-stability = current-stability - overall-stability-proof = overall-stability-proof - target-stability = target-stability - target-achievable = target-achievable \ No newline at end of file +-- * Universal CNO / categorical equivalence: the EchoCategory module +-- was a duplicate of EchoCategorical and has been deleted. Port +-- these tests to EchoCategorical when it grows a matching API. +-- * CNO singleton equivalence under identity: the EchoCNO module +-- was a duplicate of EchoCNOBridge and has been deleted. Add +-- stability tests against EchoCNOBridge's CNOEcho / nullop-echo +-- lemmas in a follow-up pass. +-- * Thermodynamic optimality / energy hierarchy: needs a real fiber +-- count and a `≤` relation on energies, neither of which the +-- simplified EchoThermodynamics yet provides.