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comparison Paper/src/gears-while.agda @ 0:14a0e409d574

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author soto <soto@cr.ie.u-ryukyu.ac.jp>
date Sun, 24 Apr 2022 23:13:44 +0900
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1 whileTest : {l : Level} {t : Set l} -> {c10 : ℕ } → (Code : (env : Env) ->
2 ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -> t) -> t
3 whileTest {_} {_} {c10} next = next env proof2
4 where
5 env : Env
6 env = record {vari = 0 ; varn = c10}
7 proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10)
8 proof2 = record {pi1 = refl ; pi2 = refl}
9
10 conversion1 : {l : Level} {t : Set l } → (env : Env) -> {c10 : ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10)
11 -> (Code : (env1 : Env) -> (varn env1 + vari env1 ≡ c10) -> t) -> t
12 conversion1 env {c10} p1 next = next env proof4
13 where
14 proof4 : varn env + vari env ≡ c10
15 proof4 = let open ≡-Reasoning in
16 begin
17 varn env + vari env
18 ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩
19 c10 + vari env
20 ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩
21 c10 + 0
22 ≡⟨ +-sym {c10} {0} ⟩
23 c10
24 ∎
25
26 {-# TERMINATING #-}
27 whileLoop : {l : Level} {t : Set l} -> (env : Env) -> {c10 : ℕ } → ((varn env) + (vari env) ≡ c10) -> (Code : Env -> t) -> t
28 whileLoop env proof next with ( suc zero ≤? (varn env) )
29 whileLoop env proof next | no p = next env
30 whileLoop env {c10} proof next | yes p = whileLoop env1 (proof3 p ) next
31 where
32 env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1}
33 1<0 : 1 ≤ zero → ⊥
34 1<0 ()
35 proof3 : (suc zero ≤ (varn env)) → varn env1 + vari env1 ≡ c10
36 proof3 (s≤s lt) with varn env
37 proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
38 proof3 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in
39 begin
40 n' + (vari env + 1)
41 ≡⟨ cong ( λ z → n' + z ) ( +-sym {vari env} {1} ) ⟩
42 n' + (1 + vari env )
43 ≡⟨ sym ( +-assoc (n') 1 (vari env) ) ⟩
44 (n' + 1) + vari env
45 ≡⟨ cong ( λ z → z + vari env ) +1≡suc ⟩
46 (suc n' ) + vari env
47 ≡⟨⟩
48 varn env + vari env
49 ≡⟨ proof ⟩
50 c10
51 ∎