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Adopt lemma as the default keyword

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Convert every theorem to lemma (mathlib's default) except the headline results a
reader of each module seeks out: analyze_correct (Forward/Sign/Constant),
aFix_eq/aFix_le (Fixedpoint), trace (Language), and Stmt.cfg_sufficient
(Language/Properties). lemma and theorem are interchangeable keywords, so no
references change.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
Danila Fedorin
2026-06-25 13:59:08 -05:00
parent 5c9c8ac55c
commit e2df847139
20 changed files with 143 additions and 143 deletions
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16
lean/Spa/Analysis/Sign.lean
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@@ -55,7 +55,7 @@ def minus : SignLattice → SignLattice → SignLattice
| mk .zero, mk .minus => mk .plus
| mk .zero, mk .zero => mk .zero
theorem plus_mono₂ : Monotone₂ plus :=
lemma plus_mono₂ : Monotone₂ plus :=
AboveBelow.monotone₂_of_strict plus
(fun y => by cases y <;> rfl)
(fun x => by rcases x with _ | _ | s <;> first | rfl | (cases s <;> rfl))
@@ -64,7 +64,7 @@ theorem plus_mono₂ : Monotone₂ plus :=
rcases x with _ | _ | s <;>
first | exact absurd rfl hx | rfl | (cases s <;> rfl))
theorem minus_mono₂ : Monotone₂ minus :=
lemma minus_mono₂ : Monotone₂ minus :=
AboveBelow.monotone₂_of_strict minus
(fun y => by cases y <;> rfl)
(fun x => by rcases x with _ | _ | s <;> first | rfl | (cases s <;> rfl))
@@ -80,7 +80,7 @@ def interpSign : SignLattice → Value → Prop
| .mk .zero, v => v = .int 0
| .mk .minus, v => ∃ n : ℕ, v = .int (-(n + 1))
theorem interpSign_mk_disjoint {s₁ s₂ : Sign} (hne : s₁ ≠ s₂) {v : Value} :
lemma interpSign_mk_disjoint {s₁ s₂ : Sign} (hne : s₁ ≠ s₂) {v : Value} :
¬(interpSign (.mk s₁) v ∧ interpSign (.mk s₂) v) := by
rintro ⟨h₁, h₂⟩
rcases s₁ <;> rcases s₂ <;> try exact hne rfl
@@ -125,7 +125,7 @@ def eval : Expr → VariableValues SignLattice prog → SignLattice
| .num 0, _ => .mk .zero
| .num (_ + 1), _ => .mk .plus
theorem eval_mono (e : Expr) : Monotone (eval prog e) := by
lemma eval_mono (e : Expr) : Monotone (eval prog e) := by
induction e with
| add e₁ e₂ ih₁ ih₂ =>
intro vs₁ vs₂ h
@@ -154,18 +154,18 @@ def output : String :=
show' (result SignLattice prog)
/-- A nonneg-shifted interpretation `∃ n : ℕ, z = n + 1` just means `z` is positive. -/
private theorem int_pos_iff (z : ℤ) : (∃ n : ℕ, z = (n : ℤ) + 1) ↔ 0 < z := by
private lemma int_pos_iff (z : ℤ) : (∃ n : ℕ, z = (n : ℤ) + 1) ↔ 0 < z := by
constructor
· rintro ⟨n, rfl⟩; omega
· intro h; exact ⟨(z - 1).toNat, by omega⟩
/-- Dually, `∃ n : ℕ, z = -(n + 1)` just means `z` is negative. -/
private theorem int_neg_iff (z : ℤ) : (∃ n : ℕ, z = -((n : ℤ) + 1)) ↔ z < 0 := by
private lemma int_neg_iff (z : ℤ) : (∃ n : ℕ, z = -((n : ℤ) + 1)) ↔ z < 0 := by
constructor
· rintro ⟨n, rfl⟩; omega
· intro h; exact ⟨(-z - 1).toNat, by omega⟩
theorem plus_valid {g₁ g₂ : SignLattice} {z₁ z₂ : ℤ}
lemma plus_valid {g₁ g₂ : SignLattice} {z₁ z₂ : ℤ}
(h₁ : ⟦g₁⟧ (.int z₁)) (h₂ : ⟦g₂⟧ (.int z₂)) :
⟦plus g₁ g₂⟧ (.int (z₁ + z₂)) := by
rcases g₁ with _ | _ | s₁ <;> rcases g₂ with _ | _ | s₂ <;>
@@ -174,7 +174,7 @@ theorem plus_valid {g₁ g₂ : SignLattice} {z₁ z₂ : ℤ}
at h₁ h₂ ⊢ <;>
omega
theorem minus_valid {g₁ g₂ : SignLattice} {z₁ z₂ : ℤ}
lemma minus_valid {g₁ g₂ : SignLattice} {z₁ z₂ : ℤ}
(h₁ : ⟦g₁⟧ (.int z₁)) (h₂ : ⟦g₂⟧ (.int z₂)) :
⟦minus g₁ g₂⟧ (.int (z₁ - z₂)) := by
rcases g₁ with _ | _ | s₁ <;> rcases g₂ with _ | _ | s₂ <;>