48 lines
1.8 KiB
Text
48 lines
1.8 KiB
Text
module
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@[expose] public section
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abbrev Variable := String
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def State := Variable → Nat
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inductive Stmt : Type where
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| skip : Stmt
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| assign : Variable → (State → Nat) → Stmt
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| seq : Stmt → Stmt → Stmt
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| ifThenElse : (State → Prop) → Stmt → Stmt → Stmt
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| whileDo : (State → Prop) → Stmt → Stmt
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infix:60 ";; " => Stmt.seq
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export Stmt (skip assign seq ifThenElse whileDo)
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set_option quotPrecheck false in
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notation s:70 "[" x:70 "↦" n:70 "]" => (fun v ↦ if v = x then n else s v)
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inductive BigStep : Stmt → State → State → Prop where
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| skip (s : State) : BigStep skip s s
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| assign (x : Variable) (a : State → Nat) (s : State) : BigStep (assign x a) s (s[x ↦ a s])
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| seq {S T : Stmt} {s t u : State} (hS : BigStep S s t) (hT : BigStep T t u) :
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BigStep (S;; T) s u
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| if_true {B : State → Prop} {s t : State} (hcond : B s) (S T : Stmt) (hbody : BigStep S s t) :
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BigStep (ifThenElse B S T) s t
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| if_false {B : State → Prop} {s t : State} (hcond : ¬ B s) (S T : Stmt) (hbody : BigStep T s t) :
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BigStep (ifThenElse B S T) s t
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| while_true {B S s t u} (hcond : B s) (hbody : BigStep S s t) (hrest : BigStep (whileDo B S) t u) :
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BigStep (whileDo B S) s u
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| while_false {B S s} (hcond : ¬ B s) : BigStep (whileDo B S) s s
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notation:55 "(" S:55 "," s:55 ")" " ==> " t:55 => BigStep S s t
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example {B S T s t} (hcond : B s) : (ifThenElse B S T, s) ==> t → (S, s) ==> t := by
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grind [BigStep]
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attribute [grind] BigStep
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theorem cases_if_of_true {B S T s t} (hcond : B s) : (ifThenElse B S T, s) ==> t → (S, s) ==> t := by
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grind
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theorem cases_if_of_false {B S T s t} (hcond : ¬ B s) : (ifThenElse B S T, s) ==> t → (T, s) ==> t := by
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grind
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theorem if_iff {B S T s t} : (ifThenElse B S T, s) ==> t ↔ (B s ∧ (S, s) ==> t) ∨ (¬ B s ∧ (T, s) ==> t) := by
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grind
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