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1 loopPPSem : (input output : Envc ) → output ≡ loopPP (varn input) input refl
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2 → (whileTestStateP s2 input ) → (whileTestStateP s2 input ) implies (whileTestStateP sf output)
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3 loopPPSem input output refl s2p = loopPPSemInduct (varn input) input refl refl s2p
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4 where
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5 lem : (n : ℕ) → (env : Envc) → n + suc (vari env) ≡ suc (n + vari env)
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6 lem n env = +-suc (n) (vari env)
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7 loopPPSemInduct : (n : ℕ) → (current : Envc) → (eq : n ≡ varn current) → (loopeq : output ≡ loopPP n current eq)
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8 → (whileTestStateP s2 current ) → (whileTestStateP s2 current ) implies (whileTestStateP sf output)
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9 loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (λ x → refl)
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10 loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) =
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11 whileLoopPSem current refl
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12 (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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13 (λ output x → loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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14
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15
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16 whileLoopPSemSound : {l : Level} → (input output : Envc )
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17 → (varn input + vari input ≡ c10 input)
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18 → output ≡ loopPP (varn input) input refl
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19 → (varn input + vari input ≡ c10 input) implies (vari output ≡ c10 output)
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20 whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre
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