312 lines
11 KiB
Lean4
312 lines
11 KiB
Lean4
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/-
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Port of `Analysis/Constant.agda`.
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Correspondence:
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showable, ≡-equiv, ≡-Decidable-ℤ ↦ (mathlib/derived instances)
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ConstLattice (AboveBelow ℤ) ↦ ConstLattice
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AB.Plain (+ 0) ↦ constFixedHeight
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plus, minus ↦ plus, minus
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plus-Monoˡ/ʳ, minus-Monoˡ/ʳ (postulates in Agda!)
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↦ plus_mono_left/right, minus_mono_left/right
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— now actually proved, via AboveBelow.le_cases
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plus-Mono₂, minus-Mono₂ ↦ plus_mono₂, minus_mono₂
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⟦_⟧ᶜ ↦ interpConst
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⟦⟧ᶜ-respects-≈ᶜ ↦ (trivial with `=`)
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⟦⟧ᶜ-⊔ᶜ-∨, ⟦⟧ᶜ-⊓ᶜ-∧ ↦ interpConst_sup, interpConst_inf
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s₁≢s₂⇒¬s₁∧s₂ ↦ interpConst_mk_disjoint
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latticeInterpretationᶜ ↦ constInterpretation
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WithProg.eval, eval-Monoʳ ↦ ConstAnalysis.eval, eval_mono
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ConstEval ↦ ConstAnalysis.exprEvaluator
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plus-valid, minus-valid ↦ plus_valid, minus_valid
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eval-valid, ConstEvalValid ↦ eval_valid
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output ↦ ConstAnalysis.output
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analyze-correct ↦ ConstAnalysis.analyze_correct
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-/
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import Spa.Analysis.Forward
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import Spa.Analysis.Utils
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import Spa.Showable
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namespace Spa
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abbrev ConstLattice : Type := AboveBelow ℤ
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/-- Agda: `AB.Plain (+ 0)`'s `fixedHeight`. -/
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def constFixedHeight : FixedHeight ConstLattice 2 :=
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AboveBelow.plainFixedHeight (0 : ℤ)
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namespace ConstAnalysis
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open AboveBelow in
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/-- Agda: `plus`. -/
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def plus : ConstLattice → ConstLattice → ConstLattice
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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 z₁, mk z₂ => mk (z₁ + z₂)
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open AboveBelow in
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/-- Agda: `minus`. -/
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def minus : ConstLattice → ConstLattice → ConstLattice
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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 z₁, mk z₂ => mk (z₁ - z₂)
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/-- Agda: `plus-Monoˡ` — a postulate there, a theorem here. -/
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theorem plus_mono_left (s₂ : ConstLattice) : Monotone (plus · s₂) := by
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intro a b h
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rcases AboveBelow.le_cases h with rfl | rfl | rfl
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· rcases s₂ with _ | _ | y <;> rcases b with _ | _ | x <;>
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simp only [plus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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| exact AboveBelow.bot_le' _
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· rcases s₂ with _ | _ | y <;> rcases a with _ | _ | x <;>
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simp only [plus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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· exact le_refl _
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/-- Agda: `plus-Monoʳ` — a postulate there, a theorem here. -/
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theorem plus_mono_right (s₁ : ConstLattice) : Monotone (plus s₁) := by
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intro a b h
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rcases AboveBelow.le_cases h with rfl | rfl | rfl
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· rcases s₁ with _ | _ | x <;> rcases b with _ | _ | y <;>
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simp only [plus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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| exact AboveBelow.bot_le' _
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· rcases s₁ with _ | _ | x <;> rcases a with _ | _ | y <;>
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simp only [plus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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· exact le_refl _
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/-- Agda: `plus-Mono₂`. -/
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theorem plus_mono₂ : Monotone₂ plus :=
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⟨plus_mono_left, plus_mono_right⟩
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/-- Agda: `minus-Monoˡ` — a postulate there, a theorem here. -/
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theorem minus_mono_left (s₂ : ConstLattice) : Monotone (minus · s₂) := by
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intro a b h
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rcases AboveBelow.le_cases h with rfl | rfl | rfl
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· rcases s₂ with _ | _ | y <;> rcases b with _ | _ | x <;>
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simp only [minus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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| exact AboveBelow.bot_le' _
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· rcases s₂ with _ | _ | y <;> rcases a with _ | _ | x <;>
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simp only [minus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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· exact le_refl _
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/-- Agda: `minus-Monoʳ` — a postulate there, a theorem here. -/
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theorem minus_mono_right (s₁ : ConstLattice) : Monotone (minus s₁) := by
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intro a b h
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rcases AboveBelow.le_cases h with rfl | rfl | rfl
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· rcases s₁ with _ | _ | x <;> rcases b with _ | _ | y <;>
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simp only [minus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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| exact AboveBelow.bot_le' _
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· rcases s₁ with _ | _ | x <;> rcases a with _ | _ | y <;>
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simp only [minus] <;>
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first
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| exact le_refl _
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| exact AboveBelow.le_top' _
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· exact le_refl _
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/-- Agda: `minus-Mono₂`. -/
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theorem minus_mono₂ : Monotone₂ minus :=
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⟨minus_mono_left, minus_mono_right⟩
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/-- Agda: `⟦_⟧ᶜ`. -/
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def interpConst : ConstLattice → Value → Prop
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| .bot, _ => False
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| .top, _ => True
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| .mk z, v => v = .int z
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/-- Agda: `s₁≢s₂⇒¬s₁∧s₂`. -/
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theorem interpConst_mk_disjoint {z₁ z₂ : ℤ} (hne : z₁ ≠ z₂) {v : Value} :
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¬(interpConst (.mk z₁) v ∧ interpConst (.mk z₂) v) := by
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rintro ⟨h₁, h₂⟩
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rw [h₁] at h₂
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injection h₂ with hz
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exact hne hz
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/-- Agda: `⟦⟧ᶜ-⊔ᶜ-∨`. -/
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theorem interpConst_sup {s₁ s₂ : ConstLattice} (v : Value)
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(h : interpConst s₁ v ∨ interpConst s₂ v) : interpConst (s₁ ⊔ s₂) v := by
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rcases s₁ with _ | _ | z₁
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· rcases h with h | h
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· exact h.elim
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· rw [AboveBelow.bot_sup]
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exact h
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· exact trivial
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· rcases s₂ with _ | _ | z₂
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· rcases h with h | h
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· rw [AboveBelow.sup_bot]
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exact h
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· exact h.elim
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· rw [AboveBelow.sup_top]
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exact trivial
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· by_cases hz : z₁ = z₂
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· subst hz
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rw [AboveBelow.mk_sup_mk, if_pos rfl]
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rcases h with h | h <;> exact h
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· rw [AboveBelow.mk_sup_mk, if_neg hz]
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exact trivial
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/-- Agda: `⟦⟧ᶜ-⊓ᶜ-∧`. -/
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theorem interpConst_inf {s₁ s₂ : ConstLattice} (v : Value)
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(h : interpConst s₁ v ∧ interpConst s₂ v) : interpConst (s₁ ⊓ s₂) v := by
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rcases s₁ with _ | _ | z₁
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· exact h.1
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· rw [AboveBelow.top_inf]
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exact h.2
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· rcases s₂ with _ | _ | z₂
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· exact h.2
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· rw [AboveBelow.inf_top]
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exact h.1
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· by_cases hz : z₁ = z₂
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· subst hz
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rw [AboveBelow.mk_inf_mk, if_pos rfl]
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exact h.1
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· exact absurd h (interpConst_mk_disjoint hz)
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/-- Agda: `latticeInterpretationᶜ`. -/
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def constInterpretation : LatticeInterpretation ConstLattice where
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interp := interpConst
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interp_sup := fun {l₁ l₂} v h => interpConst_sup (s₁ := l₁) (s₂ := l₂) v h
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interp_inf := fun {l₁ l₂} v h => interpConst_inf (s₁ := l₁) (s₂ := l₂) v h
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variable (prog : Program)
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/-- Agda: `WithProg.eval`. -/
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def eval : Expr → VariableValues ConstLattice prog → ConstLattice
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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 n, _ => .mk n
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/-- Agda: `WithProg.eval-Monoʳ`. -/
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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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exact le_refl _
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/-- Agda: the `ConstEval` instance. -/
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def exprEvaluator : ExprEvaluator ConstLattice prog :=
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⟨eval prog, eval_mono prog⟩
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/-- Agda: `WithProg.result`/`output`. -/
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def output : String :=
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show' (result constFixedHeight (exprEvaluator prog).toStmtEvaluator)
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/-- Agda: `plus-valid`. -/
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theorem plus_valid {g₁ g₂ : ConstLattice} {z₁ z₂ : ℤ}
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(h₁ : interpConst g₁ (.int z₁)) (h₂ : interpConst g₂ (.int z₂)) :
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interpConst (plus g₁ g₂) (.int (z₁ + z₂)) := by
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rcases g₁ with _ | _ | c₁
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· exact h₁.elim
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· rcases g₂ with _ | _ | c₂
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· exact h₂.elim
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· exact trivial
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· exact trivial
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· rcases g₂ with _ | _ | c₂
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· exact h₂.elim
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· exact trivial
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· injection h₁ with hz₁
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injection h₂ with hz₂
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show Value.int (z₁ + z₂) = Value.int (c₁ + c₂)
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rw [hz₁, hz₂]
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/-- Agda: `minus-valid`. -/
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theorem minus_valid {g₁ g₂ : ConstLattice} {z₁ z₂ : ℤ}
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(h₁ : interpConst g₁ (.int z₁)) (h₂ : interpConst g₂ (.int z₂)) :
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interpConst (minus g₁ g₂) (.int (z₁ - z₂)) := by
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rcases g₁ with _ | _ | c₁
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· exact h₁.elim
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· rcases g₂ with _ | _ | c₂
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· exact h₂.elim
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· exact trivial
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· exact trivial
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· rcases g₂ with _ | _ | c₂
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· exact h₂.elim
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· exact trivial
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· injection h₁ with hz₁
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injection h₂ with hz₂
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show Value.int (z₁ - z₂) = Value.int (c₁ - c₂)
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rw [hz₁, hz₂]
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/-- Agda: `eval-valid` / `ConstEvalValid`. -/
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theorem eval_valid :
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IsValidExprEvaluator (exprEvaluator prog) constInterpretation := by
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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 interpConst (eval prog (.num n) vs) (.int n)
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rfl
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| var x v hxv =>
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intro hvs
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show interpConst (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 : interpConst (eval prog e₁ vs) (.int z₁) := ih₁ hvs
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have h₂ : interpConst (eval prog e₂ vs) (.int z₂) := ih₂ hvs
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show interpConst (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 : interpConst (eval prog e₁ vs) (.int z₁) := ih₁ hvs
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have h₂ : interpConst (eval prog e₂ vs) (.int z₂) := ih₂ hvs
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show interpConst (eval prog (.sub e₁ e₂) vs) (.int (z₁ - z₂))
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exact minus_valid h₁ h₂
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/-- Agda: `WithProg.analyze-correct`. -/
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theorem analyze_correct {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
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interpV constInterpretation
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(variablesAt prog.finalState
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(result constFixedHeight (exprEvaluator prog).toStmtEvaluator)) ρ :=
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Spa.analyze_correct constFixedHeight
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((exprEvaluator prog).toStmtEvaluator_valid (eval_valid prog)) hrun
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end ConstAnalysis
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end Spa
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