Mercurial > hg > Members > ryokka > HoareLogic

comparison whileTestGears1.agda @ 98:2d2b0b06945b default tip

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
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
comparison
equal deleted inserted replaced
97:1b2d58c5d75b 98:2d2b0b06945b
1 module whileTestGears1 where
2
3 open import Function
4 open import Data.Nat renaming ( _∸_ to _-_)
5 open import Data.Bool hiding ( _≟_ ; _≤?_ ; _≤_ ; _<_ )
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
9 open import utilities
10 open import Data.Empty
11 open import Data.Nat.Properties
12 open import Data.Unit hiding ( _≟_ ; _≤?_ ; _≤_)
13
14 lemma1 : {i : ℕ} → ¬ 1 ≤ i → i ≡ 0
15 lemma1 {zero} not = refl
16 lemma1 {suc i} not = ⊥-elim ( not (s≤s z≤n) )
17
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
22 lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥
23 lemma3 refl ()
24 lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥
25 lemma5 (s≤s z≤n) ()
26
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 )
47 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c n≤j )
48
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
93
94
95
96
97
98
99
100
101
102
103
104
105
106