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1 whileLoopPSem : {l : Level} {t : Set l} @$\rightarrow$@ (input : Envc ) @$\rightarrow$@ (vari input) + (varn input) @$\equiv$@ (c10 input)
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2 @$\rightarrow$@ (next : (output : Envc ) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output) + (varn output) @$\equiv$@ (c10 output)) @$\rightarrow$@ t)
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3 @$\rightarrow$@ (exit : (output : Envc ) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output @$\equiv$@ c10 output)) @$\rightarrow$@ t) @$\rightarrow$@ t
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4 whileLoopPSem env s next exit with varn env | s
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5 ... | zero | _ = exit env (proof (@$\lambda$@ z @$\rightarrow$@ z))
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6 ... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof @$\lambda$@ x @$\rightarrow$@ +-suc varn (vari env) )
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7
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8
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9 loopPPSem : (input output : Envc ) @$\rightarrow$@ output @$\equiv$@ loopPP (varn input) input refl
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10 @$\rightarrow$@ (vari input) + (varn input) @$\equiv$@ (c10 input) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output @$\equiv$@ c10 output))
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11 loopPPSem input output refl s2p = loopPPSemInduct (varn input) input refl refl s2p
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12 where
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13 lem : (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc) @$\rightarrow$@ n + suc (vari env) @$\equiv$@ suc (n + vari env)
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14 lem n env = +-suc (n) (vari env)
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15 loopPPSemInduct : (n : @$\mathbb{N}$@) @$\rightarrow$@ (current : Envc) @$\rightarrow$@ (eq : n @$\equiv$@ varn current) @$\rightarrow$@ (loopeq : output @$\equiv$@ loopPP n current eq)
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16 @$\rightarrow$@ ((vari current) + (varn current) @$\equiv$@ (c10 current) ) @$\rightarrow$@ ((vari current) + (varn current) @$\equiv$@ (c10 current) ) implies ((vari output @$\equiv$@ c10 output))
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17 loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (@$\lambda$@ x @$\rightarrow$@ refl)
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18 loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) =
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19 whileLoopPSem current refl
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20 (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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21 (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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