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1 module whileTestGears where
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2
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3 open import Function
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4 open import Data.Nat
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5 open import Data.Bool hiding ( _@$\stackrel{?}{=}$@_ ; _@$\leq$@?_ ; _@$\leq$@_ ; _<_)
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6 open import Data.Product
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7 open import Level renaming ( suc to succ ; zero to Zero )
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8 open import Relation.Nullary using (@$\neg$@_; Dec; yes; no)
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9 open import Relation.Binary.PropositionalEquality
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10 open import Agda.Builtin.Unit
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11
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12 open import utilities
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13 open _@$\wedge$@_
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14
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15 -- original codeGear (with non terminatinng )
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16
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17 record Env : Set (succ Zero) where
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18 field
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19 varn : @$\mathbb{N}$@
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20 vari : @$\mathbb{N}$@
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21 open Env
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22
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23 whileTest : {l : Level} {t : Set l} @$\rightarrow$@ (c10 : @$\mathbb{N}$@) @$\rightarrow$@ (Code : Env @$\rightarrow$@ t) @$\rightarrow$@ t
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24 whileTest c10 next = next (record {varn = c10 ; vari = 0 } )
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25
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26 {-@$\#$@ TERMINATING @$\#$@-}
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27 whileLoop : {l : Level} {t : Set l} @$\rightarrow$@ Env @$\rightarrow$@ (Code : Env @$\rightarrow$@ t) @$\rightarrow$@ t
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28 whileLoop env next with lt 0 (varn env)
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29 whileLoop env next | false = next env
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30 whileLoop env next | true =
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31 whileLoop (record env {varn = (varn env) - 1 ; vari = (vari env) + 1}) next
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32
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33 test1 : Env
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34 test1 = whileTest 10 (@$\lambda$@ env @$\rightarrow$@ whileLoop env (@$\lambda$@ env1 @$\rightarrow$@ env1 ))
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35
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36 proof1 : whileTest 10 (@$\lambda$@ env @$\rightarrow$@ whileLoop env (@$\lambda$@ e @$\rightarrow$@ (vari e) @$\equiv$@ 10 ))
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37 proof1 = refl
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38
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39 -- codeGear with pre-condtion and post-condition
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40 --
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41 -- ↓PostCondition
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42 whileTest' : {l : Level} {t : Set l} @$\rightarrow$@ {c10 : @$\mathbb{N}$@ } @$\rightarrow$@ (Code : (env : Env ) @$\rightarrow$@ ((vari env) @$\equiv$@ 0) @$\wedge$@ ((varn env) @$\equiv$@ c10) @$\rightarrow$@ t) @$\rightarrow$@ t
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43 whileTest' {_} {_} {c10} next = next env proof2
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44 where
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45 env : Env
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46 env = record {vari = 0 ; varn = c10 }
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47 proof2 : ((vari env) @$\equiv$@ 0) @$\wedge$@ ((varn env) @$\equiv$@ c10) -- PostCondition
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48 proof2 = record {pi1 = refl ; pi2 = refl}
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49
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50
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51 open import Data.Empty
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52 open import Data.Nat.Properties
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53
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54
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55 {-@$\#$@ TERMINATING @$\#$@-} -- ↓PreCondition(Invaliant)
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56 whileLoop' : {l : Level} {t : Set l} @$\rightarrow$@ (env : Env ) @$\rightarrow$@ {c10 : @$\mathbb{N}$@ } @$\rightarrow$@ ((varn env) + (vari env) @$\equiv$@ c10) @$\rightarrow$@ (Code : Env @$\rightarrow$@ t) @$\rightarrow$@ t
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57 whileLoop' env proof next with ( suc zero @$\leq$@? (varn env) )
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58 whileLoop' env proof next | no p = next env
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59 whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next
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60 where
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61 env1 = record env {varn = (varn env) - 1 ; vari = (vari env) + 1}
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62 1<0 : 1 @$\leq$@ zero @$\rightarrow$@ @$\bot$@
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63 1<0 ()
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64 proof3 : (suc zero @$\leq$@ (varn env)) @$\rightarrow$@ varn env1 + vari env1 @$\equiv$@ c10
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65 proof3 (s@$\leq$@s lt) with varn env
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66 proof3 (s@$\leq$@s z@$\leq$@n) | zero = @$\bot$@-elim (1<0 p)
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67 proof3 (s@$\leq$@s (z@$\leq$@n {n'}) ) | suc n = let open @$\equiv$@-Reasoning in
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68 begin
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69 n' + (vari env + 1)
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70 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ n' + z ) ( +-sym {vari env} {1} ) @$\rangle$@
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71 n' + (1 + vari env )
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72 @$\equiv$@@$\langle$@ sym ( +-assoc (n') 1 (vari env) ) @$\rangle$@
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73 (n' + 1) + vari env
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74 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ z + vari env ) +1@$\equiv$@suc @$\rangle$@
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75 (suc n' ) + vari env
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76 @$\equiv$@@$\langle$@@$\rangle$@
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77 varn env + vari env
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78 @$\equiv$@@$\langle$@ proof @$\rangle$@
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79 c10
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80 @$\blacksquare$@
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81
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82 -- Condition to Invariant
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83 conversion1 : {l : Level} {t : Set l } @$\rightarrow$@ (env : Env ) @$\rightarrow$@ {c10 : @$\mathbb{N}$@ } @$\rightarrow$@ ((vari env) @$\equiv$@ 0) @$\wedge$@ ((varn env) @$\equiv$@ c10)
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84 @$\rightarrow$@ (Code : (env1 : Env ) @$\rightarrow$@ (varn env1 + vari env1 @$\equiv$@ c10) @$\rightarrow$@ t) @$\rightarrow$@ t
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85 conversion1 env {c10} p1 next = next env proof4
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86 where
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87 proof4 : varn env + vari env @$\equiv$@ c10
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88 proof4 = let open @$\equiv$@-Reasoning in
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89 begin
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90 varn env + vari env
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91 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ n @$\rightarrow$@ n + vari env ) (pi2 p1 ) @$\rangle$@
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92 c10 + vari env
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93 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ n @$\rightarrow$@ c10 + n ) (pi1 p1 ) @$\rangle$@
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94 c10 + 0
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95 @$\equiv$@@$\langle$@ +-sym {c10} {0} @$\rangle$@
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96 c10
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97 @$\blacksquare$@
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98
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99 -- all proofs are connected
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100 proofGears : {c10 : @$\mathbb{N}$@ } @$\rightarrow$@ Set
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101 proofGears {c10} = whileTest' {_} {_} {c10} (@$\lambda$@ n p1 @$\rightarrow$@ conversion1 n p1 (@$\lambda$@ n1 p2 @$\rightarrow$@ whileLoop' n1 p2 (@$\lambda$@ n2 @$\rightarrow$@ ( vari n2 @$\equiv$@ c10 ))))
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102
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103 --
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104 -- codeGear with loop step and closed environment
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105 --
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106
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107 open import Relation.Binary
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108
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109 record Envc : Set (succ Zero) where
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110 field
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111 c10 : @$\mathbb{N}$@
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112 varn : @$\mathbb{N}$@
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113 vari : @$\mathbb{N}$@
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114 open Envc
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115
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116 whileTestP : {l : Level} {t : Set l} @$\rightarrow$@ (c10 : @$\mathbb{N}$@) @$\rightarrow$@ (Code : Envc @$\rightarrow$@ t) @$\rightarrow$@ t
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117 whileTestP c10 next = next (record {varn = c10 ; vari = 0 ; c10 = c10 } )
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118
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119 whileLoopP : {l : Level} {t : Set l} @$\rightarrow$@ Envc @$\rightarrow$@ (next : Envc @$\rightarrow$@ t) @$\rightarrow$@ (exit : Envc @$\rightarrow$@ t) @$\rightarrow$@ t
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120 whileLoopP env next exit with <-cmp 0 (varn env)
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121 whileLoopP env next exit | tri@$\thickapprox$@ @$\neg$@a b @$\neg$@c = exit env
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122 whileLoopP env next exit | tri< a @$\neg$@b @$\neg$@c =
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123 next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 })
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124
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125 -- equivalent of whileLoopP but it looks like an induction on varn
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126 whileLoopP' : {l : Level} {t : Set l} @$\rightarrow$@ (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc) @$\rightarrow$@ (n @$\equiv$@ varn env) @$\rightarrow$@ (next : Envc @$\rightarrow$@ t) @$\rightarrow$@ (exit : Envc @$\rightarrow$@ t) @$\rightarrow$@ t
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127 whileLoopP' zero env refl _ exit = exit env
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128 whileLoopP' (suc n) env refl next _ = next (record {c10 = (c10 env) ; varn = varn env ; vari = suc (vari env) })
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129
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130 -- normal loop without termination
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131 {-@$\#$@ TERMINATING @$\#$@-}
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132 loopP : {l : Level} {t : Set l} @$\rightarrow$@ Envc @$\rightarrow$@ (exit : Envc @$\rightarrow$@ t) @$\rightarrow$@ t
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133 loopP env exit = whileLoopP env (@$\lambda$@ env @$\rightarrow$@ loopP env exit ) exit
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134
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135 whileTestPCall : (c10 : @$\mathbb{N}$@ ) @$\rightarrow$@ Envc
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136 whileTestPCall c10 = whileTestP {_} {_} c10 (@$\lambda$@ env @$\rightarrow$@ loopP env (@$\lambda$@ env @$\rightarrow$@ env))
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137
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138 --
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139 -- codeGears with states of condition
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140 --
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141 data whileTestState : Set where
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142 s1 : whileTestState
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143 s2 : whileTestState
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144 sf : whileTestState
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145
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146 whileTestStateP : whileTestState @$\rightarrow$@ Envc @$\rightarrow$@ Set
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147 whileTestStateP s1 env = (vari env @$\equiv$@ 0) @$\wedge$@ (varn env @$\equiv$@ c10 env)
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148 whileTestStateP s2 env = (varn env + vari env @$\equiv$@ c10 env)
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149 whileTestStateP sf env = (vari env @$\equiv$@ c10 env)
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150
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151 whileTestPwP : {l : Level} {t : Set l} @$\rightarrow$@ (c10 : @$\mathbb{N}$@) @$\rightarrow$@ ((env : Envc ) @$\rightarrow$@ whileTestStateP s1 env @$\rightarrow$@ t) @$\rightarrow$@ t
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152 whileTestPwP c10 next = next env record { pi1 = refl ; pi2 = refl } where
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153 env : Envc
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154 env = whileTestP c10 ( @$\lambda$@ env @$\rightarrow$@ env )
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155
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156 whileLoopPwP : {l : Level} {t : Set l} @$\rightarrow$@ (env : Envc ) @$\rightarrow$@ whileTestStateP s2 env
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157 @$\rightarrow$@ (next : (env : Envc ) @$\rightarrow$@ whileTestStateP s2 env @$\rightarrow$@ t)
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158 @$\rightarrow$@ (exit : (env : Envc ) @$\rightarrow$@ whileTestStateP sf env @$\rightarrow$@ t) @$\rightarrow$@ t
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159 whileLoopPwP env s next exit with <-cmp 0 (varn env)
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160 whileLoopPwP env s next exit | tri@$\thickapprox$@ @$\neg$@a b @$\neg$@c = exit env (lem (sym b) s)
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161 where
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162 lem : (varn env @$\equiv$@ 0) @$\rightarrow$@ (varn env + vari env @$\equiv$@ c10 env) @$\rightarrow$@ vari env @$\equiv$@ c10 env
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163 lem refl refl = refl
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164 whileLoopPwP env s next exit | tri< a @$\neg$@b @$\neg$@c = next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) (proof5 a)
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165 where
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166 1<0 : 1 @$\leq$@ zero @$\rightarrow$@ @$\bot$@
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167 1<0 ()
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168 proof5 : (suc zero @$\leq$@ (varn env)) @$\rightarrow$@ (varn env - 1) + (vari env + 1) @$\equiv$@ c10 env
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169 proof5 (s@$\leq$@s lt) with varn env
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170 proof5 (s@$\leq$@s z@$\leq$@n) | zero = @$\bot$@-elim (1<0 a)
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171 proof5 (s@$\leq$@s (z@$\leq$@n {n'}) ) | suc n = let open @$\equiv$@-Reasoning in
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172 begin
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173 n' + (vari env + 1)
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174 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ n' + z ) ( +-sym {vari env} {1} ) @$\rangle$@
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175 n' + (1 + vari env )
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176 @$\equiv$@@$\langle$@ sym ( +-assoc (n') 1 (vari env) ) @$\rangle$@
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177 (n' + 1) + vari env
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178 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ z + vari env ) +1@$\equiv$@suc @$\rangle$@
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179 (suc n' ) + vari env
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180 @$\equiv$@@$\langle$@@$\rangle$@
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181 varn env + vari env
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182 @$\equiv$@@$\langle$@ s @$\rangle$@
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183 c10 env
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184 @$\blacksquare$@
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185
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186
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187 whileLoopPwP' : {l : Level} {t : Set l} @$\rightarrow$@ (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc ) @$\rightarrow$@ (n @$\equiv$@ varn env) @$\rightarrow$@ whileTestStateP s2 env
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188 @$\rightarrow$@ (next : (env : Envc ) @$\rightarrow$@ (pred n @$\equiv$@ varn env) @$\rightarrow$@ whileTestStateP s2 env @$\rightarrow$@ t)
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189 @$\rightarrow$@ (exit : (env : Envc ) @$\rightarrow$@ whileTestStateP sf env @$\rightarrow$@ t) @$\rightarrow$@ t
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190 whileLoopPwP' zero env refl refl next exit = exit env refl
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191 whileLoopPwP' (suc n) env refl refl next exit = next (record env {varn = pred (varn env) ; vari = suc (vari env) }) refl (+-suc n (vari env))
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192
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193
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194 {-@$\#$@ TERMINATING @$\#$@-}
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195 loopPwP : {l : Level} {t : Set l} @$\rightarrow$@ (env : Envc ) @$\rightarrow$@ whileTestStateP s2 env @$\rightarrow$@ (exit : (env : Envc ) @$\rightarrow$@ whileTestStateP sf env @$\rightarrow$@ t) @$\rightarrow$@ t
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196 loopPwP env s exit = whileLoopPwP env s (@$\lambda$@ env s @$\rightarrow$@ loopPwP env s exit ) exit
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197
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198
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199 loopPwP' : {l : Level} {t : Set l} @$\rightarrow$@ (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc ) @$\rightarrow$@ (n @$\equiv$@ varn env) @$\rightarrow$@ whileTestStateP s2 env @$\rightarrow$@ (exit : (env : Envc ) @$\rightarrow$@ whileTestStateP sf env @$\rightarrow$@ t) @$\rightarrow$@ t
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200 loopPwP' zero env refl refl exit = exit env refl
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201 loopPwP' (suc n) env refl refl exit = whileLoopPwP' (suc n) env refl refl (@$\lambda$@ env x y @$\rightarrow$@ loopPwP' n env x y exit) exit
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202
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203
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204 loopHelper : (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc ) @$\rightarrow$@ (eq : varn env @$\equiv$@ n) @$\rightarrow$@ (seq : whileTestStateP s2 env) @$\rightarrow$@ loopPwP' n env (sym eq) seq @$\lambda$@ env@$\_{1}$@ x @$\rightarrow$@ (vari env@$\_{1}$@ @$\equiv$@ c10 env@$\_{1}$@)
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205 loopHelper zero env eq refl rewrite eq = refl
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206 loopHelper (suc n) env eq refl rewrite eq = loopHelper n (record { c10 = suc (n + vari env) ; varn = n ; vari = suc (vari env) }) refl (+-suc n (vari env))
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207
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208
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209 -- all codtions are correctly connected and required condtion is proved in the continuation
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210 -- use required condition as t in (env @$\rightarrow$@ t) @$\rightarrow$@ t
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211 --
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212 whileTestPCallwP : (c : @$\mathbb{N}$@ ) @$\rightarrow$@ Set
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213 whileTestPCallwP c = whileTestPwP {_} {_} c ( @$\lambda$@ env s @$\rightarrow$@ loopPwP env (conv env s) ( @$\lambda$@ env s @$\rightarrow$@ vari env @$\equiv$@ c10 env ) ) where
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214 conv : (env : Envc ) @$\rightarrow$@ (vari env @$\equiv$@ 0) @$\wedge$@ (varn env @$\equiv$@ c10 env) @$\rightarrow$@ varn env + vari env @$\equiv$@ c10 env
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215 conv e record { pi1 = refl ; pi2 = refl } = +zero
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216
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217
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218 whileTestPCallwP' : (c : @$\mathbb{N}$@ ) @$\rightarrow$@ Set
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219 whileTestPCallwP' c = whileTestPwP {_} {_} c (@$\lambda$@ env s @$\rightarrow$@ loopPwP' (varn env) env refl (conv env s) ( @$\lambda$@ env s @$\rightarrow$@ vari env @$\equiv$@ c10 env ) ) where
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220 conv : (env : Envc ) @$\rightarrow$@ (vari env @$\equiv$@ 0) @$\wedge$@ (varn env @$\equiv$@ c10 env) @$\rightarrow$@ varn env + vari env @$\equiv$@ c10 env
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221 conv e record { pi1 = refl ; pi2 = refl } = +zero
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222
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223 helperCallwP : (c : @$\mathbb{N}$@) @$\rightarrow$@ whileTestPCallwP' c
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224 helperCallwP c = whileTestPwP {_} {_} c (@$\lambda$@ env s @$\rightarrow$@ loopHelper c (record { c10 = c ; varn = c ; vari = zero }) refl +zero)
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225
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226 --
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227 -- Using imply relation to make soundness explicit
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228 -- termination is shown by induction on varn
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229 --
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230
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231 data _implies_ (A B : Set ) : Set (succ Zero) where
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232 proof : ( A @$\rightarrow$@ B ) @$\rightarrow$@ A implies B
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233
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234 whileTestPSem : (c : @$\mathbb{N}$@) @$\rightarrow$@ whileTestP c ( @$\lambda$@ env @$\rightarrow$@ @$\top$@ implies (whileTestStateP s1 env) )
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235 whileTestPSem c = proof ( @$\lambda$@ _ @$\rightarrow$@ record { pi1 = refl ; pi2 = refl } )
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236
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237 whileTestPSemSound : (c : @$\mathbb{N}$@ ) (output : Envc ) @$\rightarrow$@ output @$\equiv$@ whileTestP c (@$\lambda$@ e @$\rightarrow$@ e) @$\rightarrow$@ @$\top$@ implies ((vari output @$\equiv$@ 0) @$\wedge$@ (varn output @$\equiv$@ c))
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238 whileTestPSemSound c output refl = whileTestPSem c
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239
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240
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241 whileConvPSemSound : {l : Level} @$\rightarrow$@ (input : Envc) @$\rightarrow$@ (whileTestStateP s1 input ) implies (whileTestStateP s2 input)
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242 whileConvPSemSound input = proof @$\lambda$@ x @$\rightarrow$@ (conv input x) where
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243 conv : (env : Envc ) @$\rightarrow$@ (vari env @$\equiv$@ 0) @$\wedge$@ (varn env @$\equiv$@ c10 env) @$\rightarrow$@ varn env + vari env @$\equiv$@ c10 env
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244 conv e record { pi1 = refl ; pi2 = refl } = +zero
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245
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246 loopPP : (n : @$\mathbb{N}$@) @$\rightarrow$@ (input : Envc ) @$\rightarrow$@ (n @$\equiv$@ varn input) @$\rightarrow$@ Envc
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247 loopPP zero input refl = input
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248 loopPP (suc n) input refl =
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249 loopPP n (record input { varn = pred (varn input) ; vari = suc (vari input)}) refl
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250
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251 whileLoopPSem : {l : Level} {t : Set l} @$\rightarrow$@ (input : Envc ) @$\rightarrow$@ whileTestStateP s2 input
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252 @$\rightarrow$@ (next : (output : Envc ) @$\rightarrow$@ (whileTestStateP s2 input ) implies (whileTestStateP s2 output) @$\rightarrow$@ t)
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253 @$\rightarrow$@ (exit : (output : Envc ) @$\rightarrow$@ (whileTestStateP s2 input ) implies (whileTestStateP sf output) @$\rightarrow$@ t) @$\rightarrow$@ t
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254 whileLoopPSem env s next exit with varn env | s
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255 ... | zero | _ = exit env (proof (@$\lambda$@ z @$\rightarrow$@ z))
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256 ... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof @$\lambda$@ x @$\rightarrow$@ +-suc varn (vari env) )
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257
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258 loopPPSem : (input output : Envc ) @$\rightarrow$@ output @$\equiv$@ loopPP (varn input) input refl
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259 @$\rightarrow$@ (whileTestStateP s2 input ) @$\rightarrow$@ (whileTestStateP s2 input ) implies (whileTestStateP sf output)
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260 loopPPSem input output refl s2p = loopPPSemInduct (varn input) input refl refl s2p
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261 where
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262 lem : (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc) @$\rightarrow$@ n + suc (vari env) @$\equiv$@ suc (n + vari env)
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263 lem n env = +-suc (n) (vari env)
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264 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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265 @$\rightarrow$@ (whileTestStateP s2 current ) @$\rightarrow$@ (whileTestStateP s2 current ) implies (whileTestStateP sf output)
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266 loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (@$\lambda$@ x @$\rightarrow$@ refl)
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267 loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) =
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268 whileLoopPSem current refl
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269 (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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270 (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
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271
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272 whileLoopPSemSound : {l : Level} @$\rightarrow$@ (input output : Envc )
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273 @$\rightarrow$@ whileTestStateP s2 input
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274 @$\rightarrow$@ output @$\equiv$@ loopPP (varn input) input refl
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275 @$\rightarrow$@ (whileTestStateP s2 input ) implies ( whileTestStateP sf output )
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276 whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre
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