91 lines
3.0 KiB
Lean4
91 lines
3.0 KiB
Lean4
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/-
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Port of `Language.agda` (the `Program` record and re-exports).
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Correspondence:
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Program record ↦ structure Program (defs in the `Program` namespace)
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graph ↦ Program.graph
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State ↦ Program.State
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initialState ↦ Program.initialState
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finalState ↦ Program.finalState
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trace ↦ Program.trace
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vars, vars-Unique ↦ Program.vars, Program.vars_nodup
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(Finset.toList + Finset.nodup_toList replace
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`to-Listˢ` and the intrinsic MapSet uniqueness)
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states, states-complete, states-Unique
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↦ Program.states, .states_complete, .states_nodup
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code ↦ Program.code
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_≟_, _≟ᵉ_ ↦ (instances, automatic for Fin/products)
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incoming ↦ Program.incoming
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initialState-pred-∅ ↦ Program.incoming_initialState_eq_nil
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edge⇒incoming ↦ Program.mem_incoming_of_edge
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-/
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import Spa.Language.Base
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import Spa.Language.Semantics
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import Spa.Language.Graphs
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import Spa.Language.Traces
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import Spa.Language.Properties
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import Mathlib.Data.Finset.Sort
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import Mathlib.Data.String.Basic
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namespace Spa
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structure Program where
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rootStmt : Stmt
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namespace Program
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variable (p : Program)
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def graph : Graph := Graph.wrap (buildCfg p.rootStmt)
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abbrev State : Type := p.graph.Index
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def initialState : p.State := (buildCfg p.rootStmt).wrapInput
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def finalState : p.State := (buildCfg p.rootStmt).wrapOutput
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/-- Agda: `Program.trace`. -/
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theorem trace {ρ : Env} (h : EvalStmt [] p.rootStmt ρ) :
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Trace p.graph p.initialState p.finalState [] ρ := by
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obtain ⟨i₁, h₁, i₂, h₂, tr⟩ := EndToEndTrace.wrap (buildCfg_sufficient h)
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rw [Graph.wrap_inputs, List.mem_singleton] at h₁
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rw [Graph.wrap_outputs, List.mem_singleton] at h₂
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subst h₁; subst h₂
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exact tr
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/-- Agda: `vars` (via `vars-Set = Stmt-vars rootStmt`). `Finset.toList` is
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noncomputable, so the variables are listed in sorted order instead — this is
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the computable stand-in for MapSet's `to-List`. -/
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def vars : List String := p.rootStmt.vars.sort (· ≤ ·)
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/-- Agda: `vars-Unique`. -/
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theorem vars_nodup : p.vars.Nodup := Finset.sort_nodup _ _
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def states : List p.State := p.graph.indices
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/-- Agda: `states-complete`. -/
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theorem states_complete (s : p.State) : s ∈ p.states := p.graph.mem_indices s
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/-- Agda: `states-Unique`. -/
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theorem states_nodup : p.states.Nodup := p.graph.nodup_indices
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/-- Agda: `code`. -/
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def code (st : p.State) : List BasicStmt := p.graph.nodes st
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/-- Agda: `incoming`. -/
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def incoming (s : p.State) : List p.State := p.graph.predecessors s
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/-- Agda: `initialState-pred-∅`. -/
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theorem incoming_initialState_eq_nil : p.incoming p.initialState = [] :=
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Graph.wrap_predecessors_eq_nil (buildCfg p.rootStmt) p.initialState
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(by rw [Graph.wrap_inputs]; exact List.mem_singleton_self _)
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/-- Agda: `edge⇒incoming`. -/
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theorem mem_incoming_of_edge {s₁ s₂ : p.State}
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(h : (s₁, s₂) ∈ p.graph.edges) : s₁ ∈ p.incoming s₂ :=
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p.graph.mem_predecessors_of_edge h
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end Program
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end Spa
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