Mercurial > hg > Members > ryokka > HoareLogic
annotate whileTestGears1.agda @ 98:2d2b0b06945b default tip
simplfied version
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 08 Apr 2023 17:00:53 +0900 |
| parents | whileTestGears.agda@1b2d58c5d75b |
| children |
| rev | line source |
|---|---|
| 98 | 1 module whileTestGears1 where |
| 4 | 2 |
| 3 open import Function | |
| 89 | 4 open import Data.Nat renaming ( _∸_ to _-_) |
| 88 | 5 open import Data.Bool hiding ( _≟_ ; _≤?_ ; _≤_ ; _<_ ) |
| 4 | 6 open import Level renaming ( suc to succ ; zero to Zero ) |
| 7 open import Relation.Nullary using (¬_; Dec; yes; no) | |
| 8 open import Relation.Binary.PropositionalEquality | |
| 10 | 9 open import utilities |
| 11 | 10 open import Data.Empty |
| 11 open import Data.Nat.Properties | |
| 98 | 12 open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_) |
| 11 | 13 |
| 87 | 14 lemma1 : {i : ℕ} → ¬ 1 ≤ i → i ≡ 0 |
| 15 lemma1 {zero} not = refl | |
| 16 lemma1 {suc i} not = ⊥-elim ( not (s≤s z≤n) ) | |
| 11 | 17 |
| 88 | 18 open import Relation.Binary.Definitions |
| 19 | |
| 20 nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥ | |
| 21 nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x | |
| 89 | 22 lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥ |
| 23 lemma3 refl () | |
| 90 | 24 lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥ |
| 25 lemma5 (s≤s z≤n) () | |
| 88 | 26 |
| 98 | 27 open _/\_ |
| 28 | |
| 29 record Env ( c : ℕ ) : Set where | |
| 30 field | |
| 31 varn : ℕ | |
| 32 vari : ℕ | |
| 33 n+i=c : varn + vari ≡ c | |
| 34 open Env | |
| 35 | |
| 36 TerminatingLoopS : {l : Level} {t : Set l} (Index : Set ) → ( reduce : Index → ℕ) | |
| 37 → (loop : (r : Index) → (next : (r1 : Index) → reduce r1 < reduce r → t ) → t) | |
| 38 → (r : Index) → t | |
| 39 TerminatingLoopS {_} {t} Index reduce loop r with <-cmp 0 (reduce r) | |
| 40 ... | tri≈ ¬a b ¬c = loop r (λ r1 lt → ⊥-elim (lemma3 b lt) ) | |
| 41 ... | tri< a ¬b ¬c = loop r (λ r1 lt1 → TerminatingLoop1 (reduce r) r r1 (≤-step lt1) lt1 ) where | |
| 42 TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j → reduce r1 < reduce r → t | |
| 43 TerminatingLoop1 zero r r1 n≤j lt = loop r1 (λ r2 lt1 → ⊥-elim (lemma5 n≤j lt1)) | |
| 44 TerminatingLoop1 (suc j) r r1 n≤j lt with <-cmp (reduce r1) (suc j) | |
| 45 ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a lt | |
| 46 ... | tri≈ ¬a b ¬c = loop r1 (λ r2 lt1 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b lt1 ) lt1 ) | |
| 91 | 47 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j ) |
| 88 | 48 |
| 98 | 49 whileTestSpec1 : (c10 : ℕ) → (e1 : Env c10 ) → vari e1 ≡ c10 → ⊤ |
| 50 whileTestSpec1 _ _ x = tt | |
| 51 | |
| 52 whileLoopSeg : {l : Level} {t : Set l} → (c10 : ℕ ) → (env : Env c10 ) | |
| 53 → (next : (e1 : Env c10 ) → varn e1 < varn env → t) | |
| 54 → (exit : (e1 : Env c10 ) → vari e1 ≡ c10 → t) → t | |
| 55 whileLoopSeg c10 env next exit with ( suc zero ≤? (varn env) ) | |
| 56 whileLoopSeg c10 env next exit | no p = exit env ( begin | |
| 57 vari env ≡⟨ refl ⟩ | |
| 58 0 + vari env ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩ | |
| 59 varn env + vari env ≡⟨ n+i=c env ⟩ | |
| 60 c10 ∎ ) where open ≡-Reasoning | |
| 61 whileLoopSeg c10 env next exit | yes p = next env1 (proof4 (varn env) p) where | |
| 62 env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1 ; n+i=c = proof3 p } where | |
| 63 1<0 : 1 ≤ zero → ⊥ | |
| 64 1<0 () | |
| 65 proof3 : (suc zero ≤ (varn env)) → ((varn env) - 1) + (vari env + 1) ≡ c10 | |
| 66 proof3 (s≤s lt) with varn env | |
| 67 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p) | |
| 68 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in begin | |
| 69 n' + (vari env + 1) ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩ | |
| 70 n' + (1 + vari env ) ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩ | |
| 71 (n' + 1) + vari env ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩ | |
| 72 (suc n' ) + vari env ≡⟨⟩ | |
| 73 varn env + vari env ≡⟨ n+i=c env ⟩ | |
| 74 c10 | |
| 75 ∎ | |
| 76 proof4 : (i : ℕ) → 1 ≤ i → i - 1 < i | |
| 77 proof4 zero () | |
| 78 proof4 (suc i) lt = begin | |
| 79 suc (suc i - 1 ) ≤⟨ ≤-refl ⟩ | |
| 80 suc i ∎ where open ≤-Reasoning | |
| 81 | |
| 82 proofGearsTermS : (c10 : ℕ ) → ⊤ | |
| 83 proofGearsTermS c10 = | |
| 84 TerminatingLoopS (Env c10) (λ env → varn env) (λ n2 loop → whileLoopSeg c10 n2 loop (λ ne pe → whileTestSpec1 c10 ne pe ) ) | |
| 85 record { varn = 0 ; vari = c10 ; n+i=c = refl } | |
| 86 | |
| 87 proofGearsExec : (c10 : ℕ ) → ℕ | |
| 88 proofGearsExec c10 = | |
| 89 TerminatingLoopS (Env c10) (λ env → varn env) (λ n2 loop → whileLoopSeg c10 n2 loop (λ ne pe → vari ne ) ) | |
| 90 record { varn = 0 ; vari = c10 ; n+i=c = refl } | |
| 91 | |
| 92 test = proofGearsExec 20 | |
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9c91d23c2836
Abstract Termination done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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9c91d23c2836
Abstract Termination done
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parents:
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changeset
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