154 lines
6.0 KiB
Lean4
154 lines
6.0 KiB
Lean4
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/-
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Port of `Analysis/Forward/Lattices.agda`.
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The Agda module instantiates `Lattice.FiniteMap` twice (variables ↦ abstract
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values, states ↦ variable maps) and re-exports everything with ᵛ/ᵐ suffixes.
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In Lean the two instantiations are `abbrev`s and the FiniteMap API is used
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directly; the module parameters (the finite-height lattice `L`, the program)
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become section variables.
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Correspondence:
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VariableValues, StateVariables ↦ VariableValues, StateVariables
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isLatticeᵛ/isLatticeᵐ, ⊔ᵛ, ≼ᵛ … ↦ (the FiniteMap Lattice instances)
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fixedHeightᵛ ↦ varsFixedHeight
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⊥ᵛ, ⊥ᵛ-contains-bottoms ↦ botV, FiniteMap.bot_contains_bots
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states-in-Map ↦ states_memKey
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variablesAt ↦ variablesAt
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variablesAt-∈ ↦ variablesAt_mem
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variablesAt-≈ ↦ (congruence, trivial with `=`)
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joinForKey, joinForKey-Mono ↦ joinForKey, joinForKey_mono
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joinAll, joinAll-Mono,
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joinAll-k∈ks-≡ ↦ joinAll, joinAll_mono, joinAll_mem_eq
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variablesAt-joinAll ↦ variablesAt_joinAll
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⟦_⟧ᵛ ↦ interpV
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⟦⊥ᵛ⟧ᵛ∅ ↦ interpV_botV_nil
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⟦⟧ᵛ-respects-≈ᵛ ↦ (trivial with `=`)
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⟦⟧ᵛ-⊔ᵛ-∨ ↦ interpV_sup
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⟦⟧ᵛ-foldr ↦ interpV_foldr
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-/
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import Spa.Language
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import Spa.Lattice.FiniteMap
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namespace Spa
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variable (L : Type) [Lattice L] (prog : Program)
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/-- Agda: `VariableValues`. -/
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abbrev VariableValues : Type := FiniteMap String L prog.vars
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/-- Agda: `StateVariables`. -/
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abbrev StateVariables : Type := FiniteMap prog.State (VariableValues L prog) prog.states
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variable {h : ℕ}
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/-- Agda: `fixedHeightᵛ`. -/
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def varsFixedHeight (fhL : FixedHeight L h) :
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FixedHeight (VariableValues L prog) (prog.vars.length * h) :=
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FiniteMap.fixedHeight fhL prog.vars
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/-- Agda: `⊥ᵛ`. -/
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def botV (fhL : FixedHeight L h) : VariableValues L prog :=
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(varsFixedHeight L prog fhL).bot
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/-- Agda: `fixedHeight` on `StateVariables` (assembled in `Forward.agda`'s
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fixpoint call; named here for reuse). -/
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def statesFixedHeight (fhL : FixedHeight L h) :
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FixedHeight (StateVariables L prog) (prog.states.length * (prog.vars.length * h)) :=
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FiniteMap.fixedHeight (varsFixedHeight L prog fhL) prog.states
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variable {L prog}
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omit [Lattice L] in
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/-- Agda: `states-in-Map`. -/
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theorem states_memKey (s : prog.State) (sv : StateVariables L prog) :
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FiniteMap.MemKey s sv :=
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FiniteMap.memKey_iff.mpr (prog.states_complete s)
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/-- Agda: `variablesAt`. -/
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def variablesAt (s : prog.State) (sv : StateVariables L prog) :
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VariableValues L prog :=
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(FiniteMap.locate (states_memKey s sv)).1
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omit [Lattice L] in
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/-- Agda: `variablesAt-∈`. -/
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theorem variablesAt_mem (s : prog.State) (sv : StateVariables L prog) :
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(s, variablesAt s sv) ∈ sv :=
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(FiniteMap.locate (states_memKey s sv)).2
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/-- Agda: `m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ`, specialized the way `Forward.agda` uses it. -/
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theorem variablesAt_le {sv₁ sv₂ : StateVariables L prog} (hle : sv₁ ≤ sv₂)
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(s : prog.State) : variablesAt s sv₁ ≤ variablesAt s sv₂ :=
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FiniteMap.le_of_mem_mem prog.states_nodup hle
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(variablesAt_mem s sv₁) (variablesAt_mem s sv₂)
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variable (fhL : FixedHeight L h)
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/-- Agda: `joinForKey`. -/
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def joinForKey (k : prog.State) (sv : StateVariables L prog) :
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VariableValues L prog :=
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(sv.valuesAt (prog.incoming k)).foldr (· ⊔ ·) (botV L prog fhL)
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/-- Agda: `joinForKey-Mono`. -/
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theorem joinForKey_mono (k : prog.State) :
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Monotone (joinForKey fhL k) := by
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intro sv₁ sv₂ hle
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exact foldr_mono _ (FiniteMap.valuesAt_le hle (prog.incoming k)) (le_refl _)
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(fun b _ _ hab => sup_le_sup_right hab b)
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(fun a _ _ hab => sup_le_sup_left hab a)
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/-- Agda: `joinAll` (the "Exercise 4.26" generalized update with `f = id`). -/
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def joinAll (sv : StateVariables L prog) : StateVariables L prog :=
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FiniteMap.generalizedUpdate id (joinForKey fhL) prog.states sv
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/-- Agda: `joinAll-Mono`. -/
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theorem joinAll_mono : Monotone (joinAll (prog := prog) fhL) :=
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FiniteMap.generalizedUpdate_monotone monotone_id (joinForKey_mono fhL)
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/-- Agda: `joinAll-k∈ks-≡`. -/
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theorem joinAll_mem_eq {s : prog.State} {vs : VariableValues L prog}
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{sv : StateVariables L prog} (h : (s, vs) ∈ joinAll fhL sv) :
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vs = joinForKey fhL s sv :=
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FiniteMap.generalizedUpdate_mem_eq (prog.states_complete s) h
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/-- Agda: `variablesAt-joinAll`. -/
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theorem variablesAt_joinAll (s : prog.State) (sv : StateVariables L prog) :
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variablesAt s (joinAll fhL sv) = joinForKey fhL s sv :=
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joinAll_mem_eq fhL (variablesAt_mem s (joinAll fhL sv))
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/-! ### Lifting an interpretation to variable maps -/
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variable (I : LatticeInterpretation L)
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/-- Agda: `⟦_⟧ᵛ`. -/
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def interpV (vs : VariableValues L prog) (ρ : Env) : Prop :=
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∀ (k : String) (l : L), (k, l) ∈ vs →
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∀ (v : Value), Env.Mem (k, v) ρ → I.interp l v
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/-- Agda: `⟦⊥ᵛ⟧ᵛ∅`. -/
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theorem interpV_botV_nil : interpV I (botV L prog fhL) [] := by
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intro k l _ v hmem
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cases hmem
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/-- Agda: `⟦⟧ᵛ-⊔ᵛ-∨`. -/
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theorem interpV_sup {vs₁ vs₂ : VariableValues L prog} {ρ : Env}
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(h : interpV I vs₁ ρ ∨ interpV I vs₂ ρ) : interpV I (vs₁ ⊔ vs₂) ρ := by
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intro k l hmem v hv
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obtain ⟨l₁, l₂, rfl, h₁, h₂⟩ := FiniteMap.mem_sup hmem
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rcases h with h | h
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· exact I.interp_sup v (Or.inl (h _ _ h₁ _ hv))
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· exact I.interp_sup v (Or.inr (h _ _ h₂ _ hv))
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/-- Agda: `⟦⟧ᵛ-foldr`. -/
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theorem interpV_foldr {vs : VariableValues L prog}
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{vss : List (VariableValues L prog)} {ρ : Env}
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(hvs : interpV I vs ρ) (hmem : vs ∈ vss) :
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interpV I (vss.foldr (· ⊔ ·) (botV L prog fhL)) ρ := by
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induction vss with
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| nil => cases hmem
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| cons vs' vss' ih =>
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rcases List.mem_cons.mp hmem with rfl | hmem'
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· exact interpV_sup I (Or.inl hvs)
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· exact interpV_sup I (Or.inr (ih hmem'))
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end Spa
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