Mercurial > hg > Members > ryokka > HoareLogic
view whileTestGears1.agda @ 98:2d2b0b06945b default tip
simplfied version
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 08 Apr 2023 17:00:53 +0900 |
parents | whileTestGears.agda@1b2d58c5d75b |
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module whileTestGears1 where open import Function open import Data.Nat renaming ( _∸_ to _-_) open import Data.Bool hiding ( _≟_ ; _≤?_ ; _≤_ ; _<_ ) open import Level renaming ( suc to succ ; zero to Zero ) open import Relation.Nullary using (¬_; Dec; yes; no) open import Relation.Binary.PropositionalEquality open import utilities open import Data.Empty open import Data.Nat.Properties open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_) lemma1 : {i : ℕ} → ¬ 1 ≤ i → i ≡ 0 lemma1 {zero} not = refl lemma1 {suc i} not = ⊥-elim ( not (s≤s z≤n) ) open import Relation.Binary.Definitions nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥ lemma3 refl () lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥ lemma5 (s≤s z≤n) () open _/\_ record Env ( c : ℕ ) : Set where field varn : ℕ vari : ℕ n+i=c : varn + vari ≡ c open Env TerminatingLoopS : {l : Level} {t : Set l} (Index : Set ) → ( reduce : Index → ℕ) → (loop : (r : Index) → (next : (r1 : Index) → reduce r1 < reduce r → t ) → t) → (r : Index) → t TerminatingLoopS {_} {t} Index reduce loop r with <-cmp 0 (reduce r) ... | tri≈ ¬a b ¬c = loop r (λ r1 lt → ⊥-elim (lemma3 b lt) ) ... | tri< a ¬b ¬c = loop r (λ r1 lt1 → TerminatingLoop1 (reduce r) r r1 (≤-step lt1) lt1 ) where TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → reduce r1 < reduce r → t TerminatingLoop1 zero r r1 n≤j lt = loop r1 (λ r2 lt1 → ⊥-elim (lemma5 n≤j lt1)) TerminatingLoop1 (suc j) r r1 n≤j lt with <-cmp (reduce r1) (suc j) ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a lt ... | tri≈ ¬a b ¬c = loop r1 (λ r2 lt1 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b lt1 ) lt1 ) ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) whileTestSpec1 : (c10 : ℕ) → (e1 : Env c10 ) → vari e1 ≡ c10 → ⊤ whileTestSpec1 _ _ x = tt whileLoopSeg : {l : Level} {t : Set l} → (c10 : ℕ ) → (env : Env c10 ) → (next : (e1 : Env c10 ) → varn e1 < varn env → t) → (exit : (e1 : Env c10 ) → vari e1 ≡ c10 → t) → t whileLoopSeg c10 env next exit with ( suc zero ≤? (varn env) ) whileLoopSeg c10 env next exit | no p = exit env ( begin vari env ≡⟨ refl ⟩ 0 + vari env ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩ varn env + vari env ≡⟨ n+i=c env ⟩ c10 ∎ ) where open ≡-Reasoning whileLoopSeg c10 env next exit | yes p = next env1 (proof4 (varn env) p) where env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1 ; n+i=c = proof3 p } where 1<0 : 1 ≤ zero → ⊥ 1<0 () proof3 : (suc zero ≤ (varn env)) → ((varn env) - 1) + (vari env + 1) ≡ c10 proof3 (s≤s lt) with varn env proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p) proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in begin n' + (vari env + 1) ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩ n' + (1 + vari env ) ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩ (n' + 1) + vari env ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩ (suc n' ) + vari env ≡⟨⟩ varn env + vari env ≡⟨ n+i=c env ⟩ c10 ∎ proof4 : (i : ℕ) → 1 ≤ i → i - 1 < i proof4 zero () proof4 (suc i) lt = begin suc (suc i - 1 ) ≤⟨ ≤-refl ⟩ suc i ∎ where open ≤-Reasoning proofGearsTermS : (c10 : ℕ ) → ⊤ proofGearsTermS c10 = TerminatingLoopS (Env c10) (λ env → varn env) (λ n2 loop → whileLoopSeg c10 n2 loop (λ ne pe → whileTestSpec1 c10 ne pe ) ) record { varn = 0 ; vari = c10 ; n+i=c = refl } proofGearsExec : (c10 : ℕ ) → ℕ proofGearsExec c10 = TerminatingLoopS (Env c10) (λ env → varn env) (λ n2 loop → whileLoopSeg c10 n2 loop (λ ne pe → vari ne ) ) record { varn = 0 ; vari = c10 ; n+i=c = refl } test = proofGearsExec 20