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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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44
lean/Spa/Language/Properties.lean
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@@ -4,7 +4,7 @@ namespace Spa
open Graph
theorem Fin.castAdd_ne_natAdd {n m : ℕ} (i : Fin n) (j : Fin m) :
lemma Fin.castAdd_ne_natAdd {n m : ℕ} (i : Fin n) (j : Fin m) :
Fin.castAdd m i ≠ Fin.natAdd n j := by
intro h
have := congrArg Fin.val h
@@ -17,7 +17,7 @@ section Embeddings
variable {g₁ g₂ : Graph} {ρ₁ ρ₂ : Env}
theorem Trace.comp_left {idx₁ idx₂ : g₁.Index}
lemma Trace.comp_left {idx₁ idx₂ : g₁.Index}
(tr : Trace g₁ idx₁ idx₂ ρ₁ ρ₂) :
Trace (g₁ ∙ g₂) (idx₁.castAdd g₂.size) (idx₂.castAdd g₂.size) ρ₁ ρ₂ := by
induction tr with
@@ -29,7 +29,7 @@ theorem Trace.comp_left {idx₁ idx₂ : g₁.Index}
· rwa [show (g₁ ∙ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl, Fin.append_left]
· exact List.mem_append_left _ (List.mem_map_of_mem _ he)
theorem Trace.comp_right {idx₁ idx₂ : g₂.Index}
lemma Trace.comp_right {idx₁ idx₂ : g₂.Index}
(tr : Trace g₂ idx₁ idx₂ ρ₁ ρ₂) :
Trace (g₁ ∙ g₂) (idx₁.natAdd g₁.size) (idx₂.natAdd g₁.size) ρ₁ ρ₂ := by
induction tr with
@@ -41,7 +41,7 @@ theorem Trace.comp_right {idx₁ idx₂ : g₂.Index}
· rwa [show (g₁ ∙ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl, Fin.append_right]
· exact List.mem_append_right _ (List.mem_map_of_mem _ he)
theorem Trace.link_left {idx₁ idx₂ : g₁.Index}
lemma Trace.link_left {idx₁ idx₂ : g₁.Index}
(tr : Trace g₁ idx₁ idx₂ ρ₁ ρ₂) :
Trace (g₁ ⤳ g₂) (idx₁.castAdd g₂.size) (idx₂.castAdd g₂.size) ρ₁ ρ₂ := by
induction tr with
@@ -53,7 +53,7 @@ theorem Trace.link_left {idx₁ idx₂ : g₁.Index}
· rwa [show (g₁ ⤳ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl, Fin.append_left]
· exact List.mem_append_left _ (List.mem_append_left _ (List.mem_map_of_mem _ he))
theorem Trace.link_right {idx₁ idx₂ : g₂.Index}
lemma Trace.link_right {idx₁ idx₂ : g₂.Index}
(tr : Trace g₂ idx₁ idx₂ ρ₁ ρ₂) :
Trace (g₁ ⤳ g₂) (idx₁.natAdd g₁.size) (idx₂.natAdd g₁.size) ρ₁ ρ₂ := by
induction tr with
@@ -66,21 +66,21 @@ theorem Trace.link_right {idx₁ idx₂ : g₂.Index}
· exact List.mem_append_left _
(List.mem_append_right _ (List.mem_map_of_mem _ he))
theorem EndToEndTrace.comp_left (etr : EndToEndTrace g₁ ρ₁ ρ₂) :
lemma EndToEndTrace.comp_left (etr : EndToEndTrace g₁ ρ₁ ρ₂) :
EndToEndTrace (g₁ ∙ g₂) ρ₁ ρ₂ := by
obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := etr
exact ⟨i₁.castAdd g₂.size, List.mem_append_left _ (List.mem_map_of_mem _ h₁),
i₂.castAdd g₂.size, List.mem_append_left _ (List.mem_map_of_mem _ h₂),
tr.comp_left⟩
theorem EndToEndTrace.comp_right (etr : EndToEndTrace g₂ ρ₁ ρ₂) :
lemma EndToEndTrace.comp_right (etr : EndToEndTrace g₂ ρ₁ ρ₂) :
EndToEndTrace (g₁ ∙ g₂) ρ₁ ρ₂ := by
obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := etr
exact ⟨i₁.natAdd g₁.size, List.mem_append_right _ (List.mem_map_of_mem _ h₁),
i₂.natAdd g₁.size, List.mem_append_right _ (List.mem_map_of_mem _ h₂),
tr.comp_right⟩
theorem EndToEndTrace.concat {ρ₃ : Env} (etr₁ : EndToEndTrace g₁ ρ₁ ρ₂)
lemma EndToEndTrace.concat {ρ₃ : Env} (etr₁ : EndToEndTrace g₁ ρ₁ ρ₂)
(etr₂ : EndToEndTrace g₂ ρ₂ ρ₃) : EndToEndTrace (g₁ ⤳ g₂) ρ₁ ρ₃ := by
obtain ⟨i₁, h₁, i₂, h₂, tr₁⟩ := etr₁
obtain ⟨j₁, k₁, j₂, k₂, tr₂⟩ := etr₂
@@ -98,7 +98,7 @@ section Loop
variable {g : Graph} {ρ₁ ρ₂ ρ₃ : Env}
theorem Trace.loop {idx₁ idx₂ : g.Index} (tr : Trace g idx₁ idx₂ ρ₁ ρ₂) :
lemma Trace.loop {idx₁ idx₂ : g.Index} (tr : Trace g idx₁ idx₂ ρ₁ ρ₂) :
Trace (Graph.loop g) (idx₁.natAdd 2) (idx₂.natAdd 2) ρ₁ ρ₂ := by
induction tr with
| single hbs =>
@@ -112,15 +112,15 @@ theorem Trace.loop {idx₁ idx₂ : g.Index} (tr : Trace g idx₁ idx₂ ρ₁
· exact List.mem_append_left _ (List.mem_append_left _
(List.mem_append_left _ (List.mem_map_of_mem _ he)))
private theorem loop_nodes_at_in :
private lemma loop_nodes_at_in :
(Graph.loop g).nodes g.loopIn = [] :=
Fin.append_left (fun _ : Fin 2 => []) g.nodes 0
private theorem loop_nodes_at_out :
private lemma loop_nodes_at_out :
(Graph.loop g).nodes g.loopOut = [] :=
Fin.append_left (fun _ : Fin 2 => []) g.nodes 1
theorem EndToEndTrace.loop (etr : EndToEndTrace g ρ₁ ρ₂) :
lemma EndToEndTrace.loop (etr : EndToEndTrace g ρ₁ ρ₂) :
EndToEndTrace (Graph.loop g) ρ₁ ρ₂ := by
obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := etr
-- the edge in → (2 ↑ʳ i₁), reached through the second edge group
@@ -135,12 +135,12 @@ theorem EndToEndTrace.loop (etr : EndToEndTrace g ρ₁ ρ₂) :
exact Trace.concat (Trace.single (loop_nodes_at_in ▸ EvalBasicStmts.nil)) hin
(Trace.concat tr.loop hout (Trace.single (loop_nodes_at_out ▸ EvalBasicStmts.nil)))
private theorem loop_edge_out_in :
private lemma loop_edge_out_in :
((g.loopOut, g.loopIn) : (Graph.loop g).Edge) ∈ (Graph.loop g).edges := by
refine List.mem_append_right _ ?_
exact List.mem_cons_self _ _
theorem EndToEndTrace.loop_concat (etr₁ : EndToEndTrace (Graph.loop g) ρ₁ ρ₂)
lemma EndToEndTrace.loop_concat (etr₁ : EndToEndTrace (Graph.loop g) ρ₁ ρ₂)
(etr₂ : EndToEndTrace (Graph.loop g) ρ₂ ρ₃) :
EndToEndTrace (Graph.loop g) ρ₁ ρ₃ := by
obtain ⟨i₁, h₁, i₂, h₂, tr₁⟩ := etr₁
@@ -150,7 +150,7 @@ theorem EndToEndTrace.loop_concat (etr₁ : EndToEndTrace (Graph.loop g) ρ₁
exact ⟨g.loopIn, List.mem_singleton_self _, g.loopOut, List.mem_singleton_self _,
Trace.concat tr₁ loop_edge_out_in tr₂⟩
theorem EndToEndTrace.loop_empty {ρ : Env} : EndToEndTrace (Graph.loop g) ρ ρ := by
lemma EndToEndTrace.loop_empty {ρ : Env} : EndToEndTrace (Graph.loop g) ρ ρ := by
have hedge : ((g.loopIn, g.loopOut) : (Graph.loop g).Edge) ∈ (Graph.loop g).edges :=
List.mem_append_right _ (List.mem_cons_of_mem _ (List.mem_cons_self _ _))
exact ⟨g.loopIn, List.mem_singleton_self _, g.loopOut, List.mem_singleton_self _,
@@ -161,16 +161,16 @@ end Loop
/-! ### Singletons, wrap, and the main result -/
theorem EndToEndTrace.singleton {bss : List BasicStmt} {ρ₁ ρ₂ : Env}
lemma EndToEndTrace.singleton {bss : List BasicStmt} {ρ₁ ρ₂ : Env}
(h : EvalBasicStmts ρ₁ bss ρ₂) : EndToEndTrace (Graph.singleton bss) ρ₁ ρ₂ :=
⟨(0 : Fin 1), List.mem_singleton_self _, (0 : Fin 1), List.mem_singleton_self _,
Trace.single h⟩
theorem EndToEndTrace.singleton_nil (ρ : Env) :
lemma EndToEndTrace.singleton_nil (ρ : Env) :
EndToEndTrace (Graph.singleton []) ρ ρ :=
EndToEndTrace.singleton EvalBasicStmts.nil
theorem EndToEndTrace.wrap {g : Graph} {ρ₁ ρ₂ : Env}
lemma EndToEndTrace.wrap {g : Graph} {ρ₁ ρ₂ : Env}
(etr : EndToEndTrace g ρ₁ ρ₂) : EndToEndTrace (Graph.wrap g) ρ₁ ρ₂ :=
(EndToEndTrace.singleton_nil ρ₁).concat (etr.concat (EndToEndTrace.singleton_nil ρ₂))
@@ -198,13 +198,13 @@ def Graph.wrapInput (g : Graph) : (Graph.wrap g).Index :=
def Graph.wrapOutput (g : Graph) : (Graph.wrap g).Index :=
Fin.natAdd 1 ((Fin.natAdd g.size (0 : Fin 1)))
theorem Graph.wrap_inputs (g : Graph) :
lemma Graph.wrap_inputs (g : Graph) :
(Graph.wrap g).inputs = [g.wrapInput] := rfl
theorem Graph.wrap_outputs (g : Graph) :
lemma Graph.wrap_outputs (g : Graph) :
(Graph.wrap g).outputs = [g.wrapOutput] := rfl
private theorem not_mem_edges_castAdd_link {g₂ : Graph} (i : Fin 1)
private lemma not_mem_edges_castAdd_link {g₂ : Graph} (i : Fin 1)
(idx : (Graph.singleton [] ⤳ g₂).Index) :
((idx, i.castAdd g₂.size) : (Graph.singleton [] ⤳ g₂).Edge)
∉ (Graph.singleton [] ⤳ g₂).edges := by
@@ -221,7 +221,7 @@ private theorem not_mem_edges_castAdd_link {g₂ : Graph} (i : Fin 1)
obtain ⟨j, -, heq⟩ := List.mem_map.mp hb
exact Fin.castAdd_ne_natAdd i j heq.symm
theorem Graph.wrap_predecessors_eq_nil (g : Graph) (idx : (Graph.wrap g).Index)
lemma Graph.wrap_predecessors_eq_nil (g : Graph) (idx : (Graph.wrap g).Index)
(h : idx ∈ (Graph.wrap g).inputs) :
(Graph.wrap g).predecessors idx = [] := by
rw [Graph.wrap_inputs, List.mem_singleton] at h