Delete more LLM-generated comments from the migration

lean/Spa/Analysis/Forward/Lattices.lean

@@ -1,32 +1,3 @@
-/-
-Port of `Analysis/Forward/Lattices.agda`.
-
-The Agda module instantiates `Lattice.FiniteMap` twice (variables ↦ abstract
-values, states ↦ variable maps) and re-exports everything with ᵛ/ᵐ suffixes.
-In Lean the two instantiations are `abbrev`s and the FiniteMap API is used
-directly; the module parameters (the finite-height lattice `L`, the program)
-become section variables, with the finite-height structure and the lattice
-interpretation arriving by instance resolution as in Agda.
-
-Correspondence:
-  VariableValues, StateVariables  ↦ VariableValues, StateVariables
-  isLatticeᵛ/isLatticeᵐ, ⊔ᵛ, ≼ᵛ … ↦ (the FiniteMap Lattice instances)
-  fixedHeightᵛ, fixedHeightᵐ      ↦ (the FiniteMap FiniteHeightLattice instance)
-  ⊥ᵛ, ⊥ᵛ-contains-bottoms         ↦ botV, FiniteMap.bot_contains_bots
-  states-in-Map                   ↦ states_memKey
-  variablesAt                     ↦ variablesAt
-  variablesAt-∈                   ↦ variablesAt_mem
-  variablesAt-≈                   ↦ (congruence, trivial with `=`)
-  joinForKey, joinForKey-Mono     ↦ joinForKey, joinForKey_mono
-  joinAll, joinAll-Mono,
-  joinAll-k∈ks-≡                  ↦ joinAll, joinAll_mono, joinAll_mem_eq
-  variablesAt-joinAll             ↦ variablesAt_joinAll
-  ⟦_⟧ᵛ                            ↦ interpV
-  ⟦⊥ᵛ⟧ᵛ∅                          ↦ interpV_botV_nil
-  ⟦⟧ᵛ-respects-≈ᵛ                 ↦ (trivial with `=`)
-  ⟦⟧ᵛ-⊔ᵛ-∨                        ↦ interpV_sup
-  ⟦⟧ᵛ-foldr                       ↦ interpV_foldr
--/
 import Spa.Language
 import Spa.Lattice.FiniteMap
 
@@ -34,36 +5,29 @@ namespace Spa
 
 variable (L : Type) [Lattice L] (prog : Program)
 
-/-- Agda: `VariableValues`. -/
 abbrev VariableValues : Type := FiniteMap String L prog.vars
 
-/-- Agda: `StateVariables`. -/
 abbrev StateVariables : Type := FiniteMap prog.State (VariableValues L prog) prog.states
 
-/-- Agda: `⊥ᵛ` (the bottom of `fixedHeightᵛ`, now found by instance search). -/
 def botV [FiniteHeightLattice L] : VariableValues L prog :=
   (⊥ : VariableValues L prog)
 
 variable {L prog}
 
 omit [Lattice L] in
-/-- Agda: `states-in-Map`. -/
 theorem states_memKey (s : prog.State) (sv : StateVariables L prog) :
     FiniteMap.MemKey s sv :=
   FiniteMap.memKey_iff.mpr (prog.states_complete s)
 
-/-- Agda: `variablesAt`. -/
 def variablesAt (s : prog.State) (sv : StateVariables L prog) :
     VariableValues L prog :=
   (FiniteMap.locate (states_memKey s sv)).1
 
 omit [Lattice L] in
-/-- Agda: `variablesAt-∈`. -/
 theorem variablesAt_mem (s : prog.State) (sv : StateVariables L prog) :
     (s, variablesAt s sv) ∈ sv :=
   (FiniteMap.locate (states_memKey s sv)).2
 
-/-- Agda: `m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ`, specialized the way `Forward.agda` uses it. -/
 theorem variablesAt_le {sv₁ sv₂ : StateVariables L prog} (hle : sv₁ ≤ sv₂)
     (s : prog.State) : variablesAt s sv₁ ≤ variablesAt s sv₂ :=
   FiniteMap.le_of_mem_mem prog.states_nodup hle
@@ -71,12 +35,10 @@ theorem variablesAt_le {sv₁ sv₂ : StateVariables L prog} (hle : sv₁ ≤ sv
 
 variable [FiniteHeightLattice L]
 
-/-- Agda: `joinForKey`. -/
 def joinForKey (k : prog.State) (sv : StateVariables L prog) :
     VariableValues L prog :=
   (sv.valuesAt (prog.incoming k)).foldr (· ⊔ ·) (botV L prog)
 
-/-- Agda: `joinForKey-Mono`. -/
 theorem joinForKey_mono (k : prog.State) :
     Monotone (joinForKey (L := L) k) := by
   intro sv₁ sv₂ hle
@@ -84,21 +46,17 @@ theorem joinForKey_mono (k : prog.State) :
     (fun b _ _ hab => sup_le_sup_right hab b)
     (fun a _ _ hab => sup_le_sup_left hab a)
 
-/-- Agda: `joinAll` (the "Exercise 4.26" generalized update with `f = id`). -/
 def joinAll (sv : StateVariables L prog) : StateVariables L prog :=
   FiniteMap.generalizedUpdate id joinForKey prog.states sv
 
-/-- Agda: `joinAll-Mono`. -/
 theorem joinAll_mono : Monotone (joinAll (L := L) (prog := prog)) :=
   FiniteMap.generalizedUpdate_monotone monotone_id joinForKey_mono
 
-/-- Agda: `joinAll-k∈ks-≡`. -/
 theorem joinAll_mem_eq {s : prog.State} {vs : VariableValues L prog}
     {sv : StateVariables L prog} (h : (s, vs) ∈ joinAll sv) :
     vs = joinForKey s sv :=
   FiniteMap.generalizedUpdate_mem_eq (prog.states_complete s) h
 
-/-- Agda: `variablesAt-joinAll`. -/
 theorem variablesAt_joinAll (s : prog.State) (sv : StateVariables L prog) :
     variablesAt s (joinAll sv) = joinForKey s sv :=
   joinAll_mem_eq (variablesAt_mem s (joinAll sv))
@@ -108,18 +66,15 @@ theorem variablesAt_joinAll (s : prog.State) (sv : StateVariables L prog) :
 variable [I : LatticeInterpretation L]
 
 omit [FiniteHeightLattice L] in
-/-- Agda: `⟦_⟧ᵛ`. -/
 def interpV (vs : VariableValues L prog) (ρ : Env) : Prop :=
   ∀ (k : String) (l : L), (k, l) ∈ vs →
     ∀ (v : Value), Env.Mem (k, v) ρ → I.interp l v
 
-/-- Agda: `⟦⊥ᵛ⟧ᵛ∅`. -/
 theorem interpV_botV_nil : interpV (botV L prog) [] := by
   intro k l _ v hmem
   cases hmem
 
 omit [FiniteHeightLattice L] in
-/-- Agda: `⟦⟧ᵛ-⊔ᵛ-∨`. -/
 theorem interpV_sup {vs₁ vs₂ : VariableValues L prog} {ρ : Env}
     (h : interpV vs₁ ρ ∨ interpV vs₂ ρ) : interpV (vs₁ ⊔ vs₂) ρ := by
   intro k l hmem v hv
@@ -128,7 +83,6 @@ theorem interpV_sup {vs₁ vs₂ : VariableValues L prog} {ρ : Env}
   · exact I.interp_sup v (Or.inl (h _ _ h₁ _ hv))
   · exact I.interp_sup v (Or.inr (h _ _ h₂ _ hv))
 
-/-- Agda: `⟦⟧ᵛ-foldr`. -/
 theorem interpV_foldr {vs : VariableValues L prog}
     {vss : List (VariableValues L prog)} {ρ : Env}
     (hvs : interpV vs ρ) (hmem : vs ∈ vss) :