Mercurial > hg > Papers > 2020 > soto-midterm
comparison src/whileLoopPSem.agda.replaced @ 1:73127e0ab57c
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author | soto@cr.ie.u-ryukyu.ac.jp |
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date | Tue, 08 Sep 2020 18:38:08 +0900 |
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0:b919985837a3 | 1:73127e0ab57c |
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1 whileLoopPSem : {l : Level} {t : Set l} @$\rightarrow$@ (input : Envc ) @$\rightarrow$@ (vari input) + (varn input) @$\equiv$@ (c10 input) | |
2 @$\rightarrow$@ (next : (output : Envc ) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output) + (varn output) @$\equiv$@ (c10 output)) @$\rightarrow$@ t) | |
3 @$\rightarrow$@ (exit : (output : Envc ) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output @$\equiv$@ c10 output)) @$\rightarrow$@ t) @$\rightarrow$@ t | |
4 whileLoopPSem env s next exit with varn env | s | |
5 ... | zero | _ = exit env (proof (@$\lambda$@ z @$\rightarrow$@ z)) | |
6 ... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof @$\lambda$@ x @$\rightarrow$@ +-suc varn (vari env) ) | |
7 | |
8 | |
9 loopPPSem : (input output : Envc ) @$\rightarrow$@ output @$\equiv$@ loopPP (varn input) input refl | |
10 @$\rightarrow$@ (vari input) + (varn input) @$\equiv$@ (c10 input) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output @$\equiv$@ c10 output)) | |
11 loopPPSem input output refl s2p = loopPPSemInduct (varn input) input refl refl s2p | |
12 where | |
13 lem : (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc) @$\rightarrow$@ n + suc (vari env) @$\equiv$@ suc (n + vari env) | |
14 lem n env = +-suc (n) (vari env) | |
15 loopPPSemInduct : (n : @$\mathbb{N}$@) @$\rightarrow$@ (current : Envc) @$\rightarrow$@ (eq : n @$\equiv$@ varn current) @$\rightarrow$@ (loopeq : output @$\equiv$@ loopPP n current eq) | |
16 @$\rightarrow$@ ((vari current) + (varn current) @$\equiv$@ (c10 current) ) @$\rightarrow$@ ((vari current) + (varn current) @$\equiv$@ (c10 current) ) implies ((vari output @$\equiv$@ c10 output)) | |
17 loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (@$\lambda$@ x @$\rightarrow$@ refl) | |
18 loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) = | |
19 whileLoopPSem current refl | |
20 (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl) | |
21 (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl) |