Mercurial > hg > Papers > 2020 > ryokka-master
comparison paper/src/gears-while.agda.replaced @ 2:c7acb9211784
add code, figure. and paper fix content
author | ryokka |
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date | Mon, 27 Jan 2020 20:41:36 +0900 |
parents | |
children | b5fffa8ae875 |
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1:ee44dbda6bd3 | 2:c7acb9211784 |
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1 whileTest : {l : Level} {t : Set l} @$\rightarrow$@ {c10 : @$\mathbb{N}$@ } → (Code : (env : Env) @$\rightarrow$@ | |
2 ((vari env) @$\equiv$@ 0) /\ ((varn env) @$\equiv$@ c10) @$\rightarrow$@ t) @$\rightarrow$@ t | |
3 whileTest {_} {_} {c10} next = next env proof2 | |
4 where | |
5 env : Env | |
6 env = record {vari = 0 ; varn = c10} | |
7 proof2 : ((vari env) @$\equiv$@ 0) /\ ((varn env) @$\equiv$@ c10) | |
8 proof2 = record {pi1 = refl ; pi2 = refl} | |
9 | |
10 conversion1 : {l : Level} {t : Set l } → (env : Env) @$\rightarrow$@ {c10 : @$\mathbb{N}$@ } → ((vari env) @$\equiv$@ 0) /\ ((varn env) @$\equiv$@ c10) | |
11 @$\rightarrow$@ (Code : (env1 : Env) @$\rightarrow$@ (varn env1 + vari env1 @$\equiv$@ c10) @$\rightarrow$@ t) @$\rightarrow$@ t | |
12 conversion1 env {c10} p1 next = next env proof4 | |
13 where | |
14 proof4 : varn env + vari env @$\equiv$@ c10 | |
15 proof4 = let open @$\equiv$@-Reasoning in | |
16 begin | |
17 varn env + vari env | |
18 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ n → n + vari env ) (pi2 p1 ) @$\rangle$@ | |
19 c10 + vari env | |
20 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ n → c10 + n ) (pi1 p1 ) @$\rangle$@ | |
21 c10 + 0 | |
22 @$\equiv$@@$\langle$@ +-sym {c10} {0} @$\rangle$@ | |
23 c10 | |
24 @$\blacksquare$@ | |
25 | |
26 {-# TERMINATING #-} | |
27 whileLoop : {l : Level} {t : Set l} @$\rightarrow$@ (env : Env) @$\rightarrow$@ {c10 : @$\mathbb{N}$@ } → ((varn env) + (vari env) @$\equiv$@ c10) @$\rightarrow$@ (Code : Env @$\rightarrow$@ t) @$\rightarrow$@ t | |
28 whileLoop env proof next with ( suc zero @$\leq$@? (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 @$\leq$@ zero → @$\bot$@ | |
34 1<0 () | |
35 proof3 : (suc zero @$\leq$@ (varn env)) → varn env1 + vari env1 @$\equiv$@ c10 | |
36 proof3 (s@$\leq$@s lt) with varn env | |
37 proof3 (s@$\leq$@s z@$\leq$@n) | zero = @$\bot$@-elim (1<0 p) | |
38 proof3 (s@$\leq$@s (z@$\leq$@n {n'}) ) | suc n = let open @$\equiv$@-Reasoning in | |
39 begin | |
40 n' + (vari env + 1) | |
41 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z → n' + z ) ( +-sym {vari env} {1} ) @$\rangle$@ | |
42 n' + (1 + vari env ) | |
43 @$\equiv$@@$\langle$@ sym ( +-assoc (n') 1 (vari env) ) @$\rangle$@ | |
44 (n' + 1) + vari env | |
45 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z → z + vari env ) +1@$\equiv$@suc @$\rangle$@ | |
46 (suc n' ) + vari env | |
47 @$\equiv$@@$\langle$@@$\rangle$@ | |
48 varn env + vari env | |
49 @$\equiv$@@$\langle$@ proof @$\rangle$@ | |
50 c10 | |
51 @$\blacksquare$@ |