mirror of https://github.com/nmvdw/HITs-Examples
33 lines
1.8 KiB
Plaintext
33 lines
1.8 KiB
Plaintext
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{-# OPTIONS --without-K --rewriting #-}
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open import HoTT
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open import Syntax
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module Language where
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data Program : Set where
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fail : Program
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exec : Syntax -> State -> Program
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postulate
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assignp : (z : State) (x n : Nat) -> exec (x := n) z == exec skip ( z [ x :== n ])
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comp₁ : (z : State) (S : Syntax) -> exec (conc skip S) z == exec S z
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comp₂ : (z z' : State) (S₁ S₂ S₁' : Syntax) -> exec S₁ z == exec S₁' z' -> exec (conc S₁ S₂) z == exec (conc S₁' S₂) z'
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while₁ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> equals z x n -> exec (while x == n do S) z == exec (conc S (while x == n do S)) z
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while₂ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> unequals z x n -> exec (while x == n do S) z == exec skip z
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while₃ : (z : State) (x n : Nat) (S : Syntax) -> undefined z x -> exec (while x == n do S) z == fail
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Program-elim :
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(Y : Set)
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-> (failY : Y)
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-> (execY : Syntax -> State -> Y)
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-> (assignY : (z : State) (x n : Nat) -> execY (x := n) z == execY skip ( z [ x :== n ]) )
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-> (compY₁ : (z : State) (S : Syntax) -> execY (conc skip S) z == execY S z )
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-> (compY₂ : (z z' : State) (S₁ S₂ S₁' : Syntax) -> execY S₁ z == execY S₁' z' -> execY (conc S₁ S₂) z == execY (conc S₁' S₂) z')
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-> (whileY₁ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> equals z x n -> execY (while x == n do S) z == execY (conc S (while x == n do S)) z)
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-> (whileY₂ : (z : State) (x n : Nat) (S : Syntax) -> defined z x -> unequals z x n -> execY (while x == n do S) z == execY skip z)
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-> (whileY₃ : (z : State) (x n : Nat) (S : Syntax) -> undefined z x -> execY (while x == n do S) z == failY)
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-> Program -> Y
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Program-elim _ failY _ _ _ _ _ _ _ fail = failY
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Program-elim _ _ execY _ _ _ _ _ _ (exec s z) = execY s z
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