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4
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1 whileTest : {l : Level} {t : Set l} @$\rightarrow$@ {c10 : @$\mathbb{N}$@ } @$\rightarrow$@ (Code : (env : Env) @$\rightarrow$@
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13
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2 ((vari env) @$\equiv$@ 0) @$\wedge$@ ((varn env) @$\equiv$@ c10) @$\rightarrow$@ t) @$\rightarrow$@ t
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2
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3 whileTest {_} {_} {c10} next = next env proof2
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4 where
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5 env : Env
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6 env = record {vari = 0 ; varn = c10}
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13
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7 proof2 : ((vari env) @$\equiv$@ 0) @$\wedge$@ ((varn env) @$\equiv$@ c10)
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2
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8 proof2 = record {pi1 = refl ; pi2 = refl}
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9
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13
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10 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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2
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11 @$\rightarrow$@ (Code : (env1 : Env) @$\rightarrow$@ (varn env1 + vari env1 @$\equiv$@ c10) @$\rightarrow$@ t) @$\rightarrow$@ t
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12 conversion1 env {c10} p1 next = next env proof4
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13 where
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14 proof4 : varn env + vari env @$\equiv$@ c10
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15 proof4 = let open @$\equiv$@-Reasoning in
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16 begin
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17 varn env + vari env
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4
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18 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ n @$\rightarrow$@ n + vari env ) (pi2 p1 ) @$\rangle$@
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2
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19 c10 + vari env
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4
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20 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ n @$\rightarrow$@ c10 + n ) (pi1 p1 ) @$\rangle$@
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2
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21 c10 + 0
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22 @$\equiv$@@$\langle$@ +-sym {c10} {0} @$\rangle$@
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23 c10
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24 @$\blacksquare$@
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25
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13
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26 {-@$\#$@ TERMINATING @$\#$@-}
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4
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27 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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2
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28 whileLoop env proof next with ( suc zero @$\leq$@? (varn env) )
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29 whileLoop env proof next | no p = next env
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30 whileLoop env {c10} proof next | yes p = whileLoop env1 (proof3 p ) next
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31 where
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32 env1 = record {varn = (varn env) - 1 ; vari = (vari env) + 1}
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4
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33 1<0 : 1 @$\leq$@ zero @$\rightarrow$@ @$\bot$@
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2
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34 1<0 ()
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4
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35 proof3 : (suc zero @$\leq$@ (varn env)) @$\rightarrow$@ varn env1 + vari env1 @$\equiv$@ c10
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2
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36 proof3 (s@$\leq$@s lt) with varn env
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37 proof3 (s@$\leq$@s z@$\leq$@n) | zero = @$\bot$@-elim (1<0 p)
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38 proof3 (s@$\leq$@s (z@$\leq$@n {n'}) ) | suc n = let open @$\equiv$@-Reasoning in
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39 begin
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40 n' + (vari env + 1)
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4
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41 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ n' + z ) ( +-sym {vari env} {1} ) @$\rangle$@
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2
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42 n' + (1 + vari env )
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43 @$\equiv$@@$\langle$@ sym ( +-assoc (n') 1 (vari env) ) @$\rangle$@
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44 (n' + 1) + vari env
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4
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45 @$\equiv$@@$\langle$@ cong ( @$\lambda$@ z @$\rightarrow$@ z + vari env ) +1@$\equiv$@suc @$\rangle$@
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2
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46 (suc n' ) + vari env
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47 @$\equiv$@@$\langle$@@$\rangle$@
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48 varn env + vari env
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49 @$\equiv$@@$\langle$@ proof @$\rangle$@
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50 c10
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51 @$\blacksquare$@
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