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1 whileLoopPwP!$\prime$! : {l : Level} {t : Set l} !$\rightarrow$! (n : !$\mathbb{N}$!) !$\rightarrow$! (env : Envc )
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2 !$\rightarrow$! (n !$\equiv$! varn env) !$\rightarrow$! whileTestStateP s2 env
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3 !$\rightarrow$! (next : (env : Envc ) !$\rightarrow$! (pred n !$\equiv$! varn env) !$\rightarrow$! whileTestStateP s2 env !$\rightarrow$! t)
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4 !$\rightarrow$! (exit : (env : Envc ) !$\rightarrow$! whileTestStateP sf env !$\rightarrow$! t) !$\rightarrow$! t
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5 whileLoopPwP!$\prime$! zero env refl refl _ exit = exit env refl
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6 whileLoopPwP!$\prime$! (suc n) env refl refl next _ =
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7 next (record env {varn = pred (varn env) ; vari = suc (vari env) }) refl (+-suc n (vari env))
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8
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9
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10 loopPwP!$\prime$! : {l : Level} {t : Set l} !$\rightarrow$! (n : !$\mathbb{N}$!) !$\rightarrow$! (env : Envc )
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11 !$\rightarrow$! (n !$\equiv$! varn env) !$\rightarrow$! whileTestStateP s2 env
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12 !$\rightarrow$! (exit : (env : Envc ) !$\rightarrow$! whileTestStateP sf env !$\rightarrow$! t) !$\rightarrow$! t
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13 loopPwP!$\prime$! zero env refl refl exit = exit env refl
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14 loopPwP!$\prime$! (suc n) env refl refl exit
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15 = whileLoopPwP!$\prime$! (suc n) env refl refl (!$\lambda$! env x y !$\rightarrow$! loopPwP!$\prime$! n env x y exit) exit
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