234 lines
10 KiB
Lean4
234 lines
10 KiB
Lean4
import Spa.Language.Traces
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namespace Spa
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open Graph
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theorem Fin.castAdd_ne_natAdd {n m : ℕ} (i : Fin n) (j : Fin m) :
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Fin.castAdd m i ≠ Fin.natAdd n j := by
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intro h
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have := congrArg Fin.val h
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simp only [Fin.coe_castAdd, Fin.coe_natAdd] at this
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omega
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/-! ### Trace embeddings -/
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section Embeddings
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variable {g₁ g₂ : Graph} {ρ₁ ρ₂ : Env}
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theorem Trace.comp_left {idx₁ idx₂ : g₁.Index}
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(tr : Trace g₁ idx₁ idx₂ ρ₁ ρ₂) :
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Trace (g₁ ∙ g₂) (idx₁.castAdd g₂.size) (idx₂.castAdd g₂.size) ρ₁ ρ₂ := by
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induction tr with
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| single hbs =>
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exact Trace.single (by rwa [show (g₁ ∙ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl,
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Fin.append_left])
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| edge hbs he _ ih =>
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refine Trace.edge ?_ ?_ ih
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· rwa [show (g₁ ∙ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl, Fin.append_left]
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· exact List.mem_append_left _ (List.mem_map_of_mem _ he)
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theorem Trace.comp_right {idx₁ idx₂ : g₂.Index}
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(tr : Trace g₂ idx₁ idx₂ ρ₁ ρ₂) :
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Trace (g₁ ∙ g₂) (idx₁.natAdd g₁.size) (idx₂.natAdd g₁.size) ρ₁ ρ₂ := by
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induction tr with
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| single hbs =>
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exact Trace.single (by rwa [show (g₁ ∙ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl,
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Fin.append_right])
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| edge hbs he _ ih =>
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refine Trace.edge ?_ ?_ ih
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· rwa [show (g₁ ∙ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl, Fin.append_right]
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· exact List.mem_append_right _ (List.mem_map_of_mem _ he)
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theorem Trace.link_left {idx₁ idx₂ : g₁.Index}
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(tr : Trace g₁ idx₁ idx₂ ρ₁ ρ₂) :
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Trace (g₁ ⤳ g₂) (idx₁.castAdd g₂.size) (idx₂.castAdd g₂.size) ρ₁ ρ₂ := by
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induction tr with
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| single hbs =>
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exact Trace.single (by rwa [show (g₁ ⤳ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl,
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Fin.append_left])
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| edge hbs he _ ih =>
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refine Trace.edge ?_ ?_ ih
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· rwa [show (g₁ ⤳ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl, Fin.append_left]
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· exact List.mem_append_left _ (List.mem_append_left _ (List.mem_map_of_mem _ he))
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theorem Trace.link_right {idx₁ idx₂ : g₂.Index}
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(tr : Trace g₂ idx₁ idx₂ ρ₁ ρ₂) :
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Trace (g₁ ⤳ g₂) (idx₁.natAdd g₁.size) (idx₂.natAdd g₁.size) ρ₁ ρ₂ := by
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induction tr with
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| single hbs =>
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exact Trace.single (by rwa [show (g₁ ⤳ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl,
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Fin.append_right])
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| edge hbs he _ ih =>
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refine Trace.edge ?_ ?_ ih
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· rwa [show (g₁ ⤳ g₂).nodes = Fin.append g₁.nodes g₂.nodes from rfl, Fin.append_right]
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· exact List.mem_append_left _
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(List.mem_append_right _ (List.mem_map_of_mem _ he))
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theorem EndToEndTrace.comp_left (etr : EndToEndTrace g₁ ρ₁ ρ₂) :
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EndToEndTrace (g₁ ∙ g₂) ρ₁ ρ₂ := by
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obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := etr
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exact ⟨i₁.castAdd g₂.size, List.mem_append_left _ (List.mem_map_of_mem _ h₁),
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i₂.castAdd g₂.size, List.mem_append_left _ (List.mem_map_of_mem _ h₂),
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tr.comp_left⟩
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theorem EndToEndTrace.comp_right (etr : EndToEndTrace g₂ ρ₁ ρ₂) :
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EndToEndTrace (g₁ ∙ g₂) ρ₁ ρ₂ := by
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obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := etr
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exact ⟨i₁.natAdd g₁.size, List.mem_append_right _ (List.mem_map_of_mem _ h₁),
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i₂.natAdd g₁.size, List.mem_append_right _ (List.mem_map_of_mem _ h₂),
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tr.comp_right⟩
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theorem EndToEndTrace.concat {ρ₃ : Env} (etr₁ : EndToEndTrace g₁ ρ₁ ρ₂)
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(etr₂ : EndToEndTrace g₂ ρ₂ ρ₃) : EndToEndTrace (g₁ ⤳ g₂) ρ₁ ρ₃ := by
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obtain ⟨i₁, h₁, i₂, h₂, tr₁⟩ := etr₁
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obtain ⟨j₁, k₁, j₂, k₂, tr₂⟩ := etr₂
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refine ⟨i₁.castAdd g₂.size, List.mem_map_of_mem _ h₁,
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j₂.natAdd g₁.size, List.mem_map_of_mem _ k₂,
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Trace.concat tr₁.link_left ?_ tr₂.link_right⟩
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exact List.mem_append_right _
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(List.mem_product.mpr ⟨List.mem_map_of_mem _ h₂, List.mem_map_of_mem _ k₁⟩)
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end Embeddings
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/-! ### Loops -/
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section Loop
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variable {g : Graph} {ρ₁ ρ₂ ρ₃ : Env}
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theorem Trace.loop {idx₁ idx₂ : g.Index} (tr : Trace g idx₁ idx₂ ρ₁ ρ₂) :
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Trace (Graph.loop g) (idx₁.natAdd 2) (idx₂.natAdd 2) ρ₁ ρ₂ := by
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induction tr with
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| single hbs =>
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exact Trace.single (by
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rwa [show (Graph.loop g).nodes = Fin.append (fun _ : Fin 2 => []) g.nodes from rfl,
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Fin.append_right])
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| edge hbs he _ ih =>
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refine Trace.edge ?_ ?_ ih
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· rwa [show (Graph.loop g).nodes = Fin.append (fun _ : Fin 2 => []) g.nodes from rfl,
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Fin.append_right]
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· exact List.mem_append_left _ (List.mem_append_left _
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(List.mem_append_left _ (List.mem_map_of_mem _ he)))
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private theorem loop_nodes_at_in :
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(Graph.loop g).nodes g.loopIn = [] :=
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Fin.append_left (fun _ : Fin 2 => []) g.nodes 0
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private theorem loop_nodes_at_out :
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(Graph.loop g).nodes g.loopOut = [] :=
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Fin.append_left (fun _ : Fin 2 => []) g.nodes 1
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theorem EndToEndTrace.loop (etr : EndToEndTrace g ρ₁ ρ₂) :
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EndToEndTrace (Graph.loop g) ρ₁ ρ₂ := by
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obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := etr
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-- the edge in → (2 ↑ʳ i₁), reached through the second edge group
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have hin : (g.loopIn, i₁.natAdd 2) ∈ (Graph.loop g).edges := by
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refine List.mem_append_left _ (List.mem_append_left _ (List.mem_append_right _ ?_))
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exact List.mem_map_of_mem _ (List.mem_map_of_mem _ h₁)
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-- the edge (2 ↑ʳ i₂) → out, reached through the third edge group
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have hout : (i₂.natAdd 2, g.loopOut) ∈ (Graph.loop g).edges := by
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refine List.mem_append_left _ (List.mem_append_right _ ?_)
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exact List.mem_map_of_mem _ (List.mem_map_of_mem _ h₂)
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refine ⟨g.loopIn, List.mem_singleton_self _, g.loopOut, List.mem_singleton_self _, ?_⟩
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exact Trace.concat (Trace.single (loop_nodes_at_in ▸ EvalBasicStmts.nil)) hin
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(Trace.concat tr.loop hout (Trace.single (loop_nodes_at_out ▸ EvalBasicStmts.nil)))
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private theorem loop_edge_out_in :
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((g.loopOut, g.loopIn) : (Graph.loop g).Edge) ∈ (Graph.loop g).edges := by
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refine List.mem_append_right _ ?_
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exact List.mem_cons_self _ _
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theorem EndToEndTrace.loop_concat (etr₁ : EndToEndTrace (Graph.loop g) ρ₁ ρ₂)
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(etr₂ : EndToEndTrace (Graph.loop g) ρ₂ ρ₃) :
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EndToEndTrace (Graph.loop g) ρ₁ ρ₃ := by
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obtain ⟨i₁, h₁, i₂, h₂, tr₁⟩ := etr₁
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obtain ⟨j₁, k₁, j₂, k₂, tr₂⟩ := etr₂
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simp only [Graph.loop_inputs, Graph.loop_outputs, List.mem_singleton] at h₁ h₂ k₁ k₂
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subst h₁; subst h₂; subst k₁; subst k₂
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exact ⟨g.loopIn, List.mem_singleton_self _, g.loopOut, List.mem_singleton_self _,
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Trace.concat tr₁ loop_edge_out_in tr₂⟩
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theorem EndToEndTrace.loop_empty {ρ : Env} : EndToEndTrace (Graph.loop g) ρ ρ := by
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have hedge : ((g.loopIn, g.loopOut) : (Graph.loop g).Edge) ∈ (Graph.loop g).edges :=
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List.mem_append_right _ (List.mem_cons_of_mem _ (List.mem_cons_self _ _))
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exact ⟨g.loopIn, List.mem_singleton_self _, g.loopOut, List.mem_singleton_self _,
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Trace.concat (Trace.single (loop_nodes_at_in ▸ EvalBasicStmts.nil)) hedge
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(Trace.single (loop_nodes_at_out ▸ EvalBasicStmts.nil))⟩
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end Loop
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/-! ### Singletons, wrap, and the main result -/
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theorem EndToEndTrace.singleton {bss : List BasicStmt} {ρ₁ ρ₂ : Env}
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(h : EvalBasicStmts ρ₁ bss ρ₂) : EndToEndTrace (Graph.singleton bss) ρ₁ ρ₂ :=
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⟨(0 : Fin 1), List.mem_singleton_self _, (0 : Fin 1), List.mem_singleton_self _,
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Trace.single h⟩
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theorem EndToEndTrace.singleton_nil (ρ : Env) :
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EndToEndTrace (Graph.singleton []) ρ ρ :=
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EndToEndTrace.singleton EvalBasicStmts.nil
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theorem EndToEndTrace.wrap {g : Graph} {ρ₁ ρ₂ : Env}
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(etr : EndToEndTrace g ρ₁ ρ₂) : EndToEndTrace (Graph.wrap g) ρ₁ ρ₂ :=
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(EndToEndTrace.singleton_nil ρ₁).concat (etr.concat (EndToEndTrace.singleton_nil ρ₂))
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theorem buildCfg_sufficient {s : Stmt} {ρ₁ ρ₂ : Env}
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(h : EvalStmt ρ₁ s ρ₂) : EndToEndTrace (buildCfg s) ρ₁ ρ₂ := by
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induction h with
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| basic ρ₁ ρ₂ bs hbs =>
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exact EndToEndTrace.singleton (EvalBasicStmts.cons hbs EvalBasicStmts.nil)
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| andThen ρ₁ ρ₂ ρ₃ s₁ s₂ _ _ ih₁ ih₂ =>
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exact ih₁.concat ih₂
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| ifTrue ρ₁ ρ₂ e z s₁ s₂ _ _ _ ih =>
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exact ih.comp_left
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| ifFalse ρ₁ ρ₂ e s₁ s₂ _ _ ih =>
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exact ih.comp_right
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| whileTrue ρ₁ ρ₂ ρ₃ e z s _ _ _ _ ih₁ ih₂ =>
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exact (ih₁.loop).loop_concat ih₂
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| whileFalse ρ e s _ =>
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exact EndToEndTrace.loop_empty
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/-! ### The wrapped graph's entry has no predecessors (Agda's "ugly" block) -/
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def Graph.wrapInput (g : Graph) : (Graph.wrap g).Index :=
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(0 : Fin 1).castAdd ((g ⤳ Graph.singleton []).size)
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def Graph.wrapOutput (g : Graph) : (Graph.wrap g).Index :=
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Fin.natAdd 1 ((Fin.natAdd g.size (0 : Fin 1)))
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theorem Graph.wrap_inputs (g : Graph) :
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(Graph.wrap g).inputs = [g.wrapInput] := rfl
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theorem Graph.wrap_outputs (g : Graph) :
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(Graph.wrap g).outputs = [g.wrapOutput] := rfl
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private theorem not_mem_edges_castAdd_link {g₂ : Graph} (i : Fin 1)
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(idx : (Graph.singleton [] ⤳ g₂).Index) :
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((idx, i.castAdd g₂.size) : (Graph.singleton [] ⤳ g₂).Edge)
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∉ (Graph.singleton [] ⤳ g₂).edges := by
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intro h
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rcases List.mem_append.mp h with h' | h'
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· rcases List.mem_append.mp h' with h'' | h''
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· -- lifted edges of `singleton []`: there are none
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simp [Graph.singleton, List.finCastAddProd] at h''
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· -- lifted edges of g₂: targets are natAdd
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obtain ⟨e, _, heq⟩ := List.mem_map.mp h''
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exact Fin.castAdd_ne_natAdd i e.2 (congrArg Prod.snd heq).symm
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· -- product edges: targets are natAdd'd inputs of g₂
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obtain ⟨-, hb⟩ := List.mem_product.mp h'
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obtain ⟨j, -, heq⟩ := List.mem_map.mp hb
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exact Fin.castAdd_ne_natAdd i j heq.symm
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theorem Graph.wrap_predecessors_eq_nil (g : Graph) (idx : (Graph.wrap g).Index)
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(h : idx ∈ (Graph.wrap g).inputs) :
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(Graph.wrap g).predecessors idx = [] := by
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rw [Graph.wrap_inputs, List.mem_singleton] at h
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subst h
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rw [Graph.predecessors, List.filter_eq_nil_iff]
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intro idx' _
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simpa using not_mem_edges_castAdd_link (g₂ := g ⤳ Graph.singleton []) 0 idx'
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end Spa
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