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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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26
lean/Spa/Analysis/Forward.lean
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@@ -13,7 +13,7 @@ def updateVariablesForState (s : prog.State) (sv : StateVariables L prog) :
VariableValues L prog :=
(prog.code s).foldl (fun vs bs => E.eval s bs vs) (variablesAt s sv)
theorem updateVariablesForState_mono (s : prog.State) :
lemma updateVariablesForState_mono (s : prog.State) :
Monotone (updateVariablesForState (L := L) s) := fun _ _ hle =>
foldl_mono' (prog.code s) _ (E.eval_mono s ·) (variablesAt_le hle s)
@@ -21,15 +21,15 @@ def updateAll (sv : StateVariables L prog) : StateVariables L prog :=
FiniteMap.generalizedUpdate id updateVariablesForState
prog.states sv
theorem updateAll_mono : Monotone (updateAll (L := L) (prog := prog)) :=
lemma updateAll_mono : Monotone (updateAll (L := L) (prog := prog)) :=
FiniteMap.generalizedUpdate_monotone monotone_id updateVariablesForState_mono
theorem updateAll_mem_eq {s : prog.State} {vs : VariableValues L prog}
lemma updateAll_mem_eq {s : prog.State} {vs : VariableValues L prog}
{sv : StateVariables L prog} (hmem : (s, vs) ∈ updateAll sv) :
vs = updateVariablesForState s sv :=
FiniteMap.generalizedUpdate_mem_eq (prog.states_complete s) hmem
theorem variablesAt_updateAll (s : prog.State) (sv : StateVariables L prog) :
lemma variablesAt_updateAll (s : prog.State) (sv : StateVariables L prog) :
variablesAt s (updateAll sv) = updateVariablesForState s sv :=
updateAll_mem_eq (variablesAt_mem s (updateAll sv))
@@ -38,7 +38,7 @@ variable [FiniteHeightLattice L]
def analyze (sv : StateVariables L prog) : StateVariables L prog :=
updateAll (joinAll sv)
theorem analyze_mono : Monotone (analyze (L := L) (prog := prog)) := fun _ _ hle =>
lemma analyze_mono : Monotone (analyze (L := L) (prog := prog)) := fun _ _ hle =>
updateAll_mono (joinAll_mono hle)
variable [DecidableEq L]
@@ -48,10 +48,10 @@ def result : StateVariables L prog :=
Fixedpoint.aFix analyze analyze_mono
variable (L prog) in
theorem result_eq : result L prog = analyze (result L prog) :=
lemma result_eq : result L prog = analyze (result L prog) :=
Fixedpoint.aFix_eq analyze analyze_mono
theorem joinForKey_initialState :
lemma joinForKey_initialState :
joinForKey prog.initialState (result L prog) = botV L prog := by
rw [joinForKey, prog.incoming_initialState_eq_nil]
rfl
@@ -59,7 +59,7 @@ theorem joinForKey_initialState :
variable [I : LatticeInterpretation L] [V : ValidStmtEvaluator L prog]
omit [FiniteHeightLattice L] [DecidableEq L] in
theorem eval_fold_valid {s : prog.State} {bss : List BasicStmt}
lemma eval_fold_valid {s : prog.State} {bss : List BasicStmt}
{vs : VariableValues L prog} {ρ₁ ρ₂ : Env}
(hbss : EvalBasicStmts ρ₁ bss ρ₂) (hvs : ⟦ vs ⟧ ρ₁) :
⟦ bss.foldl (fun vs bs => E.eval s bs vs) vs ⟧ ρ₂ := by
@@ -68,7 +68,7 @@ theorem eval_fold_valid {s : prog.State} {bss : List BasicStmt}
| cons hbs _ ih => exact ih (ValidStmtEvaluator.valid hbs hvs)
omit [FiniteHeightLattice L] [DecidableEq L] in
theorem updateVariablesForState_matches {s : prog.State}
lemma updateVariablesForState_matches {s : prog.State}
{sv : StateVariables L prog} {ρ₁ ρ₂ : Env}
(hbss : EvalBasicStmts ρ₁ (prog.code s) ρ₂)
(hvs : ⟦ variablesAt s sv ⟧ ρ₁) :
@@ -76,14 +76,14 @@ theorem updateVariablesForState_matches {s : prog.State}
eval_fold_valid hbss hvs
omit [FiniteHeightLattice L] [DecidableEq L] in
theorem updateAll_matches {s : prog.State} {sv : StateVariables L prog}
lemma updateAll_matches {s : prog.State} {sv : StateVariables L prog}
{ρ₁ ρ₂ : Env} (hbss : EvalBasicStmts ρ₁ (prog.code s) ρ₂)
(hvs : ⟦ variablesAt s sv ⟧ ρ₁) :
⟦ variablesAt s (updateAll sv) ⟧ ρ₂ := by
rw [variablesAt_updateAll]
exact updateVariablesForState_matches hbss hvs
theorem stepTrace {s₁ : prog.State} {ρ₁ ρ₂ : Env}
lemma stepTrace {s₁ : prog.State} {ρ₁ ρ₂ : Env}
(hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁)
(hbss : EvalBasicStmts ρ₁ (prog.code s₁) ρ₂) :
⟦ variablesAt s₁ (result L prog) ⟧ ρ₂ := by
@@ -92,7 +92,7 @@ theorem stepTrace {s₁ : prog.State} {ρ₁ ρ₂ : Env}
rw [variablesAt_joinAll]
exact hjoin
theorem walkTrace {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
lemma walkTrace {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
(hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁)
(tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) :
⟦ variablesAt s₂ (result L prog) ⟧ ρ₂ := by
@@ -108,7 +108,7 @@ theorem walkTrace {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
exact ih (interp_foldr hstep hmem)
omit V in
theorem interp_joinForKey_initialState :
lemma interp_joinForKey_initialState :
⟦ joinForKey prog.initialState (result L prog) ⟧ [] := by
rw [joinForKey_initialState]
exact interp_botV_nil