223 lines
7.3 KiB
Lean4
223 lines
7.3 KiB
Lean4
import Spa.Analysis.Forward
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import Spa.Analysis.Utils
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import Spa.Interp
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import Spa.Showable
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namespace Spa
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inductive Sign where
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| plus
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| minus
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| zero
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deriving DecidableEq
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instance : Showable Sign :=
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⟨fun
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| .plus => "+"
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| .minus => "-"
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| .zero => "0"⟩
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instance : Inhabited Sign := ⟨.zero⟩
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abbrev SignLattice : Type := AboveBelow Sign
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open AboveBelow in
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def plus : SignLattice → SignLattice → SignLattice
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| bot, _ => bot
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| _, bot => bot
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| top, _ => top
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| _, top => top
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| mk .plus, mk .plus => mk .plus
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| mk .plus, mk .minus => top
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| mk .plus, mk .zero => mk .plus
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| mk .minus, mk .plus => top
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| mk .minus, mk .minus => mk .minus
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| mk .minus, mk .zero => mk .minus
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| mk .zero, mk .plus => mk .plus
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| mk .zero, mk .minus => mk .minus
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| mk .zero, mk .zero => mk .zero
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open AboveBelow in
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def minus : SignLattice → SignLattice → SignLattice
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| bot, _ => bot
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| _, bot => bot
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| top, _ => top
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| _, top => top
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| mk .plus, mk .plus => top
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| mk .plus, mk .minus => mk .plus
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| mk .plus, mk .zero => mk .plus
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| mk .minus, mk .plus => mk .minus
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| mk .minus, mk .minus => top
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| mk .minus, mk .zero => mk .minus
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| mk .zero, mk .plus => mk .minus
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| mk .zero, mk .minus => mk .plus
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| mk .zero, mk .zero => mk .zero
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theorem plus_mono₂ : Monotone₂ plus :=
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AboveBelow.monotone₂_of_strict plus
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(fun y => by cases y <;> rfl)
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(fun x => by rcases x with _ | _ | s <;> first | rfl | (cases s <;> rfl))
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(fun y hy => by cases y <;> first | exact absurd rfl hy | rfl)
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(fun x hx => by
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rcases x with _ | _ | s <;>
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first | exact absurd rfl hx | rfl | (cases s <;> rfl))
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theorem minus_mono₂ : Monotone₂ minus :=
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AboveBelow.monotone₂_of_strict minus
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(fun y => by cases y <;> rfl)
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(fun x => by rcases x with _ | _ | s <;> first | rfl | (cases s <;> rfl))
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(fun y hy => by cases y <;> first | exact absurd rfl hy | rfl)
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(fun x hx => by
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rcases x with _ | _ | s <;>
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first | exact absurd rfl hx | rfl | (cases s <;> rfl))
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def interpSign : SignLattice → Value → Prop
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| .bot, _ => False
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| .top, _ => True
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| .mk .plus, v => ∃ n : ℕ, v = .int (n + 1)
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| .mk .zero, v => v = .int 0
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| .mk .minus, v => ∃ n : ℕ, v = .int (-(n + 1))
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theorem interpSign_mk_disjoint {s₁ s₂ : Sign} (hne : s₁ ≠ s₂) {v : Value} :
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¬(interpSign (.mk s₁) v ∧ interpSign (.mk s₂) v) := by
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rintro ⟨h₁, h₂⟩
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rcases s₁ <;> rcases s₂ <;> try exact hne rfl
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all_goals simp only [interpSign] at h₁ h₂
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· obtain ⟨n₁, rfl⟩ := h₁
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obtain ⟨n₂, hv⟩ := h₂
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injection hv with hz
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omega
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· obtain ⟨n₁, rfl⟩ := h₁
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injection h₂ with hz
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omega
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· obtain ⟨n₁, rfl⟩ := h₁
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obtain ⟨n₂, hv⟩ := h₂
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injection hv with hz
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omega
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· obtain ⟨n₁, rfl⟩ := h₁
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injection h₂ with hz
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omega
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· subst h₁
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obtain ⟨n₂, hv⟩ := h₂
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injection hv with hz
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omega
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· subst h₁
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obtain ⟨n₂, hv⟩ := h₂
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injection hv with hz
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omega
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instance signInterpretation : LatticeInterpretation SignLattice where
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interp := interpSign
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interp_sup := fun v h => AboveBelow.interp_sup_of (fun _ h => h) (fun _ => trivial) v h
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interp_inf := fun v h => AboveBelow.interp_inf_of (fun hne _ => interpSign_mk_disjoint hne) v h
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namespace SignAnalysis
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variable (prog : Program)
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def eval : Expr → VariableValues SignLattice prog → SignLattice
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| .add e₁ e₂, vs => plus (eval e₁ vs) (eval e₂ vs)
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| .sub e₁ e₂, vs => minus (eval e₁ vs) (eval e₂ vs)
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| .var k, vs =>
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if h : FiniteMap.MemKey k vs then (FiniteMap.locate h).1 else .top
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| .num 0, _ => .mk .zero
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| .num (_ + 1), _ => .mk .plus
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theorem eval_mono (e : Expr) : Monotone (eval prog e) := by
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induction e with
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| add e₁ e₂ ih₁ ih₂ =>
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intro vs₁ vs₂ h
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exact eval_combine₂ plus_mono₂ (ih₁ h) (ih₂ h)
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| sub e₁ e₂ ih₁ ih₂ =>
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intro vs₁ vs₂ h
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exact eval_combine₂ minus_mono₂ (ih₁ h) (ih₂ h)
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| var k =>
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intro vs₁ vs₂ h
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simp only [eval]
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by_cases hk : k ∈ prog.vars
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· rw [dif_pos (FiniteMap.memKey_iff.mpr hk),
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dif_pos (FiniteMap.memKey_iff.mpr hk)]
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exact FiniteMap.le_of_mem_mem prog.vars_nodup h
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(FiniteMap.locate _).2 (FiniteMap.locate _).2
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· rw [dif_neg (fun hm => hk (FiniteMap.memKey_iff.mp hm)),
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dif_neg (fun hm => hk (FiniteMap.memKey_iff.mp hm))]
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| num n =>
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intro vs₁ vs₂ _
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cases n <;> exact le_refl _
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instance exprEvaluator : ExprEvaluator SignLattice prog :=
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⟨eval prog, eval_mono prog⟩
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def output : String :=
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show' (result SignLattice prog)
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/-- A nonneg-shifted interpretation `∃ n : ℕ, z = n + 1` just means `z` is positive. -/
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private theorem int_pos_iff (z : ℤ) : (∃ n : ℕ, z = (n : ℤ) + 1) ↔ 0 < z := by
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constructor
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· rintro ⟨n, rfl⟩; omega
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· intro h; exact ⟨(z - 1).toNat, by omega⟩
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/-- Dually, `∃ n : ℕ, z = -(n + 1)` just means `z` is negative. -/
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private theorem int_neg_iff (z : ℤ) : (∃ n : ℕ, z = -((n : ℤ) + 1)) ↔ z < 0 := by
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constructor
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· rintro ⟨n, rfl⟩; omega
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· intro h; exact ⟨(-z - 1).toNat, by omega⟩
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theorem plus_valid {g₁ g₂ : SignLattice} {z₁ z₂ : ℤ}
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(h₁ : ⟦g₁⟧ (.int z₁)) (h₂ : ⟦g₂⟧ (.int z₂)) :
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⟦plus g₁ g₂⟧ (.int (z₁ + z₂)) := by
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rcases g₁ with _ | _ | s₁ <;> rcases g₂ with _ | _ | s₂ <;>
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(try rcases s₁) <;> (try rcases s₂) <;>
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simp only [plus, signInterpretation, interpSign, Value.int.injEq, int_pos_iff, int_neg_iff]
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at h₁ h₂ ⊢ <;>
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omega
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theorem minus_valid {g₁ g₂ : SignLattice} {z₁ z₂ : ℤ}
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(h₁ : ⟦g₁⟧ (.int z₁)) (h₂ : ⟦g₂⟧ (.int z₂)) :
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⟦minus g₁ g₂⟧ (.int (z₁ - z₂)) := by
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rcases g₁ with _ | _ | s₁ <;> rcases g₂ with _ | _ | s₂ <;>
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(try rcases s₁) <;> (try rcases s₂) <;>
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simp only [minus, signInterpretation, interpSign, Value.int.injEq, int_pos_iff, int_neg_iff]
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at h₁ h₂ ⊢ <;>
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omega
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instance eval_valid : ValidExprEvaluator SignLattice prog := by
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constructor
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intro vs ρ e v hev
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induction hev with
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| num n =>
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intro _
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show ⟦eval prog (.num n) vs⟧ (.int n)
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cases n with
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| zero => rfl
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| succ n' => exact ⟨n', congrArg Value.int (by norm_cast)⟩
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| var x v hxv =>
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intro hvs
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show ⟦eval prog (.var x) vs⟧ v
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simp only [eval]
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by_cases hk : FiniteMap.MemKey x vs
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· rw [dif_pos hk]
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exact hvs _ _ (FiniteMap.locate hk).2 _ hxv
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· rw [dif_neg hk]
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exact trivial
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| add e₁ e₂ z₁ z₂ _ _ ih₁ ih₂ =>
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intro hvs
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have h₁ : ⟦eval prog e₁ vs⟧ (.int z₁) := ih₁ hvs
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have h₂ : ⟦eval prog e₂ vs⟧ (.int z₂) := ih₂ hvs
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show ⟦eval prog (.add e₁ e₂) vs⟧ (.int (z₁ + z₂))
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exact plus_valid h₁ h₂
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| sub e₁ e₂ z₁ z₂ _ _ ih₁ ih₂ =>
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intro hvs
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have h₁ : ⟦eval prog e₁ vs⟧ (.int z₁) := ih₁ hvs
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have h₂ : ⟦eval prog e₂ vs⟧ (.int z₂) := ih₂ hvs
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show ⟦eval prog (.sub e₁ e₂) vs⟧ (.int (z₁ - z₂))
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exact minus_valid h₁ h₂
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theorem analyze_correct {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
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⟦ variablesAt prog.finalState (result SignLattice prog) ⟧ ρ :=
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Spa.analyze_correct SignLattice prog hrun
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end SignAnalysis
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end Spa
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