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disjointify.lean
import Mathlib.Tactic
inductive E
| lit : Bool → E
| var : Nat → E
| ite : E → E → E → E
deriving DecidableEq
def E.hasNestedIf : E → Bool
| lit _ => false
| var _ => false
| ite (ite _ _ _) _ _ => true
| ite _ t e => t.hasNestedIf || e.hasNestedIf
def E.hasConstantIf : E → Bool
| lit _ => false
| var _ => false
| ite (lit _) _ _ => true
| ite i t e => i.hasConstantIf || t.hasConstantIf || e.hasConstantIf
def E.hasRedundantIf : E → Bool
| lit _ => false
| var _ => false
| ite i t e => t == e || i.hasRedundantIf || t.hasRedundantIf || e.hasRedundantIf
def E.vars : E → List Nat
| lit _ => []
| var i => [i]
| ite i t e => i.vars ++ t.vars ++ e.vars
def List.disjoint : List Nat → List Nat → Bool
| [], _ => true
| (x::xs), ys => x ∉ ys && xs.disjoint ys
def E.disjoint : E → Bool
| lit _ => true
| var _ => true
| ite i t e =>
i.vars.disjoint t.vars && i.vars.disjoint e.vars && i.disjoint && t.disjoint && e.disjoint
def E.normalized (e : E) : Bool :=
!e.hasNestedIf && !e.hasConstantIf && !e.hasRedundantIf && e.disjoint
def E.eval (f : Nat → Bool) : E → Bool
| lit b => b
| var i => f i
| ite i t e => bif i.eval f then t.eval f else e.eval f
open E
def E.occs : E → List ℕ
| lit _ => []
| var v => List.ofFn (fun i : Fin (v+1) => if i = Fin.last _ then 1 else 0)
| ite i t e => List.zipWith (· + ·) (List.zipWith (· + ·) i.occs t.occs) e.occs
def E.elimVar (v : ℕ) : E → (E × E)
| lit b => (lit b, lit b)
| var w => if v = w then (lit true, lit false) else (var w, var w)
| ite i t e =>
let (i₁, i₂) := i.elimVar v
let (t₁, t₂) := t.elimVar v
let (e₁, e₂) := e.elimVar v
(ite i₁ t₁ e₁, ite i₂ t₂ e₂)
theorem E.eval_elimVar_fst (v : ℕ) : ∀ (e : E) (f : ℕ → Bool),
eval f (e.elimVar v).1 = eval (fun w => if v = w then true else f w) e
| lit b, _ => by simp [elimVar, eval]
| var w, f => by
simp [elimVar, eval]
split_ifs <;>
simp_all [eval, beq_iff_eq]
| ite i t e, f => by
simp [elimVar, eval]
simp only [eval_elimVar_fst]
theorem E.eval_elimVar_snd (v : ℕ) : ∀ (e : E) (f : ℕ → Bool),
eval f (e.elimVar v).2 = eval (fun w => if v = w then false else f w) e
| lit b, _ => by simp [elimVar, eval]
| var w, f => by
simp [elimVar, eval]
split_ifs <;>
simp_all [eval, beq_iff_eq]
| ite i t e, f => by
simp [elimVar, eval]
simp only [eval_elimVar_snd]
theorem E.vars_elimVar_fst (v : ℕ) : ∀ (e : E),
(e.elimVar v).1.vars = e.vars.filter (v ≠ ·)
| lit _ => by simp [elimVar, vars]
| var w => by
simp [elimVar, vars]
split_ifs <;>
simp_all [vars, beq_iff_eq, List.filter]
| ite i t e => by
simp [elimVar, vars]
simp only [vars_elimVar_fst, decide_not]
theorem E.vars_elimVar_snd (v : ℕ) : ∀ (e : E),
(e.elimVar v).2.vars = e.vars.filter (v ≠ ·)
| lit _ => by simp [elimVar, vars]
| var w => by
simp [elimVar, vars]
split_ifs <;>
simp_all [vars, beq_iff_eq, List.filter]
| ite i t e => by
simp [elimVar, vars]
simp only [vars_elimVar_snd, decide_not]
def E.disjointLE (v : ℕ) : E → Bool
| lit _ => true
| var _ => true
| ite i t e =>
(i.vars.filter (· ≤ v)).disjoint t.vars &&
(i.vars.filter (· ≤ v)).disjoint e.vars &&
i.disjointLE v && t.disjointLE v && e.disjointLE v
theorem List.disjoint_iff : ∀ {xs ys : List ℕ},
xs.disjoint ys = (∀ x ∈ xs, x ∉ ys : Bool)
| [], _=> by simp [List.disjoint]
| x::xs, ys => by
simp [List.disjoint, @List.disjoint_iff xs]
theorem E.disjointLE_of_le {v w : ℕ} (h : v ≤ w) :
∀ {e : E}, e.disjointLE w → e.disjointLE v
| lit _ => by simp [disjointLE]
| var _ => by simp [disjointLE]
| ite i t e => by
simp only [disjointLE, List.disjoint_iff, List.mem_filter, decide_eq_true_eq, and_imp,
Bool.and_eq_true, and_assoc]
intro h₁ h₂ hi ht he
exact ⟨fun x hx hxv => h₁ x hx (le_trans hxv h),
fun x hx hxv => h₂ x hx (le_trans hxv h),
E.disjointLE_of_le h hi,
E.disjointLE_of_le h ht,
E.disjointLE_of_le h he⟩
theorem E.disjointLE_elimVar_fst {v : ℕ} :
∀ (e : E), (∀ w < v, disjointLE w e) ↔ disjointLE v (e.elimVar v).1
| lit _ => by simp [elimVar, disjointLE]
| var w => by
simp [elimVar, disjointLE]
split_ifs
· simp [disjointLE]
· simp [disjointLE]
| ite i t e => by
simp only [disjointLE, Bool.and_eq_true, forall_and,
vars_elimVar_fst, vars_elimVar_snd,
i.disjointLE_elimVar_fst,
t.disjointLE_elimVar_fst,
e.disjointLE_elimVar_fst,
← and_assoc, and_congr_left_iff]
intros _ _ _
simp [List.disjoint_iff, List.mem_filter, decide_eq_true]
refine and_congr ?_ ?_
. refine ⟨fun h x hxi hxv hvx hxt =>
(h x (lt_of_le_of_ne hxv (Ne.symm hvx)) x hxi (le_refl x) hxt).elim,
fun h x hxv y hyi hyx hyt => ne_of_lt (lt_of_le_of_lt hyx hxv)
(Eq.symm (h y hyi (le_trans hyx (le_of_lt hxv))
(Ne.symm (ne_of_lt (lt_of_le_of_lt hyx hxv))) hyt))⟩
· refine ⟨fun h x hxi hxv hvx hxt =>
(h x (lt_of_le_of_ne hxv (Ne.symm hvx)) x hxi (le_refl x) hxt).elim,
fun h x hxv y hyi hyx hyt => ne_of_lt (lt_of_le_of_lt hyx hxv)
(Eq.symm (h y hyi (le_trans hyx (le_of_lt hxv))
(Ne.symm (ne_of_lt (lt_of_le_of_lt hyx hxv))) hyt))⟩
theorem E.disjointLE_elimVar_snd {v : ℕ} :
∀ (e : E), (∀ w < v, disjointLE w e) ↔ disjointLE v (e.elimVar v).2
| lit _ => by simp [elimVar, disjointLE]
| var w => by
simp [elimVar, disjointLE]
split_ifs
· simp [disjointLE]
· simp [disjointLE]
| ite i t e => by
simp only [disjointLE, Bool.and_eq_true, forall_and,
vars_elimVar_snd, vars_elimVar_snd,
i.disjointLE_elimVar_snd,
t.disjointLE_elimVar_snd,
e.disjointLE_elimVar_snd,
← and_assoc, and_congr_left_iff]
intros _ _ _
simp [List.disjoint_iff, List.mem_filter, decide_eq_true]
refine and_congr ?_ ?_
. refine ⟨fun h x hxi hxv hvx hxt =>
(h x (lt_of_le_of_ne hxv (Ne.symm hvx)) x hxi (le_refl x) hxt).elim,
fun h x hxv y hyi hyx hyt => ne_of_lt (lt_of_le_of_lt hyx hxv)
(Eq.symm (h y hyi (le_trans hyx (le_of_lt hxv))
(Ne.symm (ne_of_lt (lt_of_le_of_lt hyx hxv))) hyt))⟩
· refine ⟨fun h x hxi hxv hvx hxt =>
(h x (lt_of_le_of_ne hxv (Ne.symm hvx)) x hxi (le_refl x) hxt).elim,
fun h x hxv y hyi hyx hyt => ne_of_lt (lt_of_le_of_lt hyx hxv)
(Eq.symm (h y hyi (le_trans hyx (le_of_lt hxv))
(Ne.symm (ne_of_lt (lt_of_le_of_lt hyx hxv))) hyt))⟩
@[simp]
theorem E.hasNestedIf_elimVar_fst (v : ℕ) :(e : E) →
hasNestedIf (elimVar v e).1 = hasNestedIf e
| lit _ => by simp [elimVar, hasNestedIf]
| var _ => by
simp [elimVar, hasNestedIf]
split_ifs <;> simp [hasNestedIf]
| .ite (.ite _ _ _) _ _ => by
simp [hasNestedIf, elimVar]
| .ite (lit _) t e => by
simp [hasNestedIf, elimVar, t.hasNestedIf_elimVar_fst,
e.hasNestedIf_elimVar_fst]
| .ite (var _) t e => by
simp [hasNestedIf, elimVar, t.hasNestedIf_elimVar_fst,
e.hasNestedIf_elimVar_fst]
split_ifs <;>
simp [hasNestedIf, elimVar, t.hasNestedIf_elimVar_fst,
e.hasNestedIf_elimVar_fst]
@[simp]
theorem E.hasNestedIf_elimVar_snd (v : ℕ) :(e : E) →
hasNestedIf (elimVar v e).2 = hasNestedIf e
| lit _ => by simp [elimVar, hasNestedIf]
| var _ => by
simp [elimVar, hasNestedIf]
split_ifs <;> simp [hasNestedIf]
| .ite (.ite _ _ _) _ _ => by
simp [hasNestedIf, elimVar]
| .ite (lit _) t e => by
simp [hasNestedIf, elimVar, t.hasNestedIf_elimVar_snd,
e.hasNestedIf_elimVar_snd]
| .ite (var _) t e => by
simp [hasNestedIf, elimVar, t.hasNestedIf_elimVar_snd,
e.hasNestedIf_elimVar_snd]
split_ifs <;>
simp [hasNestedIf, elimVar, t.hasNestedIf_elimVar_snd,
e.hasNestedIf_elimVar_snd]
def E.dedupLE (v : ℕ) : (e : E) → (h : ∀ w < v, e.disjointLE w) →
{ e' : E // e'.eval = e.eval ∧ e'.disjointLE v ∧
(∀ w, v < w → w ∈ e'.vars → w ∈ e.vars)
∧ (!e.hasNestedIf → !e'.hasNestedIf) }
| lit b, _ => ⟨lit b, by simp [disjointLE, eval]⟩
| var i, _ => ⟨var i, by simp [disjointLE, eval]⟩
| ite i t e, h =>
let i' := i.elimVar v
let t' := t.elimVar v
let e' := e.elimVar v
⟨ite (var v) (ite i'.1 t'.1 e'.1) (ite i'.2 t'.2 e'.2), by
ext f
simp [eval, eval_elimVar_fst, eval_elimVar_snd]
by_cases hfv : f v = true
· simp [hfv]
congr <;>
ext <;> split_ifs <;> simp_all
· simp [eq_false_of_ne_true hfv]
congr <;>
ext <;> split_ifs <;> simp_all, by
simp
[disjointLE, E.vars_elimVar_snd, E.disjoint, E.vars, and_assoc,
E.vars_elimVar_fst, List.disjoint_iff, List.mem_filter] at h ⊢
have h₁ : ∀ (x : ℕ), x ∈ vars i → x ≤ v → ¬v = x → x ∈ vars t → v = x := by
intro x hxi hxv hvx hxt
exact ((h x (lt_of_le_of_ne hxv (Ne.symm hvx))).1 x hxi (le_refl x) hxt).elim
have h₂ : ∀ (x : ℕ), x ∈ vars i → x ≤ v → ¬v = x → x ∈ vars e → v = x := by
intro x hxi hxv hvx hxe
exact ((h x (lt_of_le_of_ne hxv (Ne.symm hvx))).2.1 x hxi (le_refl x) hxe).elim
simp only [← disjointLE_elimVar_fst, ← disjointLE_elimVar_snd]
simp only [forall_and] at h
refine ⟨h₁, h₂, ?_, ?_, ?_, h₁, h₂, ?_⟩ <;> tauto, by
simp [vars, vars_elimVar_fst, vars_elimVar_snd,
List.mem_filter, or_imp]
intro w hvw
simp [ne_of_gt hvw, ne_of_lt hvw]
tauto, by
simp [hasNestedIf]
cases i
· simp (config := {contextual := true}) [hasNestedIf]
· simp [hasNestedIf, elimVar]
split_ifs <;> simp (config := {contextual := true}) [hasNestedIf]
· simp [hasNestedIf, elimVar]⟩
theorem E.lt_disjointLE_iff_disjointLE_of_not_mem_vars {v : ℕ} : ∀ {e : E},
(he : v ∉ e.vars) → (∀ w < v, e.disjointLE w) ↔ e.disjointLE v
| lit _ => by simp [disjointLE]
| var w => by simp [disjointLE]
| ite i t e => by
simp only [disjointLE, vars, List.mem_append, not_or,
and_imp, Bool.and_eq_true, List.disjoint_iff, List.mem_filter,
decide_eq_true_iff]
intro hvi hvt hve
simp only [forall_and, ← and_assoc,
← i.lt_disjointLE_iff_disjointLE_of_not_mem_vars hvi,
← t.lt_disjointLE_iff_disjointLE_of_not_mem_vars hvt,
← e.lt_disjointLE_iff_disjointLE_of_not_mem_vars hve]
simp only [and_congr_left_iff]
intros h₁ h₂ h₃
refine and_congr ?_ ?_
. exact ⟨fun h x hxi hxv hxt =>
h x (lt_of_le_of_ne hxv (by rintro rfl; simp_all))
x hxi (le_refl _) hxt, fun h x hxv y hyi hyx hyt =>
h y hyi (le_trans hyx (le_of_lt hxv)) hyt⟩
. exact ⟨fun h x hxi hxv hxt =>
h x (lt_of_le_of_ne hxv (by rintro rfl; simp_all))
x hxi (le_refl _) hxt, fun h x hxv y hyi hyx hyt =>
h y hyi (le_trans hyx (le_of_lt hxv)) hyt⟩
theorem E.lt_disjointLE_iff_disjointLE_of_not_mem_vars' : ∀ {v : ℕ} {e : E},
(he : v ∉ e.vars) → (∀ w < v, w ∈ e.vars → e.disjointLE w) ↔ e.disjointLE v
| v, e, he => by
conv_rhs => rw [← E.lt_disjointLE_iff_disjointLE_of_not_mem_vars he]
refine ⟨?_, ?_⟩
· rintro h w hwv
by_cases hw : w ∈ e.vars
· exact h w hwv hw
· rw [← E.lt_disjointLE_iff_disjointLE_of_not_mem_vars' hw]
intro x hxw hxe
exact h x (lt_of_le_of_lt (le_of_lt hxw) hwv) hxe
· intro h w hwv _
exact h w hwv
theorem E.disjoint_iff_disjointLE : ∀ {e : E}, e.disjoint ↔ ∀ v, e.disjointLE v
| lit _ => by simp [disjoint, disjointLE]
| var _ => by simp [disjoint, disjointLE]
| E.ite i t e => by
simp only [disjoint, disjointLE, i.disjoint_iff_disjointLE,
t.disjoint_iff_disjointLE, e.disjoint_iff_disjointLE,
List.disjoint_iff, decide_eq_true_iff, Bool.and_eq_true,
E.vars, List.mem_append, or_assoc, List.mem_filter,
and_imp, and_assoc]
refine ⟨?_, ?_⟩
· intro h v
tauto
· intro h
simp only [← forall_and, and_assoc]
intro v
have := h v
have h₁ := this.1 v
have h₂ := this.2.1 v
simp at h₁ h₂
tauto
def E.dedupAllLE : (v : ℕ) → (e : E) →
{ e' : E // e'.eval = e.eval ∧ e'.disjointLE v ∧
(∀ w, v < w → w ∈ e'.vars → w ∈ e.vars) ∧
(!e.hasNestedIf → !e'.hasNestedIf) }
| 0, e => e.dedupLE 0 (by simp)
| v+1, e =>
let e' := E.dedupAllLE v e
let e'' := @E.dedupLE (v+1) e' (fun w hw =>
E.disjointLE_of_le (Nat.le_of_lt_succ hw) e'.2.2.1)
⟨e''.1, by rw [e''.2.1, e'.2.1], e''.2.2.1,
fun w hvw hwe => by
have := e''.2.2.2.1 w hvw hwe
exact e'.2.2.2.1 w (Nat.lt_of_succ_lt hvw) this, by
intro h
apply e''.2.2.2.2
exact e'.2.2.2.2 h⟩
def List.leastGE : (l : List ℕ) → {n : ℕ // ∀ m ∈ l, m ≤ n}
| [] => ⟨0, by simp⟩
| a::l =>
let m := List.leastGE l
if ham : a ≤ m then ⟨m.1, by
intro k
rw [List.mem_cons]
rintro (rfl | hkl)
· exact ham
· exact m.2 _ hkl⟩
else ⟨a, by
intro k
rw [List.mem_cons]
rintro (rfl | hkl)
· exact le_refl _
· exact le_trans (m.2 _ hkl) (le_of_not_ge ham)⟩
def E.dedup (e : E) : { e' : E // e'.eval = e.eval ∧ e'.disjoint ∧
(!e.hasNestedIf → !e'.hasNestedIf) } :=
let v := List.leastGE e.vars
let e' := E.dedupAllLE v e
⟨e'.1, e'.2.1, by
rw [E.disjoint_iff_disjointLE]
intro w
by_cases hw : w ≤ v.1
· exact disjointLE_of_le hw e'.2.2.1
· have hw : w ∉ e'.1.vars := by
intro hwe
have := e'.2.2.2.1 w (lt_of_not_ge hw) hwe
exact hw (v.2 _ this)
rw [← E.lt_disjointLE_iff_disjointLE_of_not_mem_vars' hw]
intro x _ hxe'
refine disjointLE_of_le ?_ e'.2.2.1
refine le_of_not_lt ?_
intro hvx
have := e'.2.2.2.1 _ hvx hxe'
have := v.2 _ this
exact not_le_of_gt hvx this, e'.2.2.2.2⟩
@[simp]
def E.denestSize : E → ℕ
| lit _ => 0
| var _ => 0
| .ite i t e =>
2 * E.denestSize i +
max (E.denestSize t) (E.denestSize e) + 1
def E.denest : (e : E) → { e' : E // e'.eval = e.eval ∧ !e'.hasNestedIf }
| lit b => ⟨lit b, by simp [eval, hasNestedIf]⟩
| var i => ⟨var i, by simp [eval, hasNestedIf]⟩
| .ite (.ite a b c) d e =>
have : 2 * denestSize a +
max (2 * denestSize b + max (denestSize d) (denestSize e) + 1)
(2 * denestSize c + max (denestSize d) (denestSize e) + 1) <
2 * (2 * denestSize a + max (denestSize b) (denestSize c) + 1) +
max (denestSize d) (denestSize e) := by
simp only [denestSize, add_assoc, add_lt_add_iff_left]
simp only [← add_assoc, max_add_add_right]
simp only [← two_mul, max_mul_mul_left]
linarith
have ⟨t', ht'⟩ := E.denest (.ite a (.ite b d e) (.ite c d e))
⟨t', by
simp only [Function.funext_iff] at *
simp [eval, ht'.1, ht'.2, hasNestedIf]
intro f
cases eval f a <;> simp⟩
| .ite (lit true) t e =>
have : denestSize t < 2 + max (denestSize t) (denestSize e) + 1 :=
calc denestSize t < max (denestSize t) (denestSize e) + 3 :=
Nat.lt_succ_of_le (le_add_right (le_max_left _ _))
_ = _ := by ring
⟨E.denest t, by simp [eval, hasNestedIf, (E.denest t).2]⟩
| .ite (lit false) t e =>
have : denestSize e < 2 + max (denestSize t) (denestSize e) + 1 :=
calc denestSize e < max (denestSize t) (denestSize e) + 3 :=
Nat.lt_succ_of_le (le_add_right (le_max_right _ _))
_ = _ := by ring
⟨E.denest e, by simp [eval, hasNestedIf, (E.denest e).2]⟩
| .ite (var v) t e =>
have : denestSize t < 2 + max (denestSize t) (denestSize e) + 1 :=
calc denestSize t < max (denestSize t) (denestSize e) + 3 :=
Nat.lt_succ_of_le (le_add_right (le_max_left _ _))
_ = _ := by ring
have : denestSize e < 2 + max (denestSize t) (denestSize e) + 1 :=
calc denestSize e < max (denestSize t) (denestSize e) + 3 :=
Nat.lt_succ_of_le (le_add_right (le_max_right _ _))
_ = _ := by ring
have ⟨t', ht'⟩ := E.denest t
have ⟨e', he'⟩ := E.denest e
⟨.ite (var v) t' e', by
simp only [Function.funext_iff] at *
simp [eval, ht'.1, ht'.2, he'.1, he'.2, hasNestedIf]⟩
termination_by E.denest e => e.denestSize
def E.deConstRed : (e : E) → { e' : E //
e.disjoint → !e.hasNestedIf →
(e'.eval = e.eval ∧ !e'.hasConstantIf
∧ (!e'.hasNestedIf) ∧
( e'.disjoint) ∧
(!e'.hasRedundantIf) ∧
e'.vars ⊆ e.vars) }
| lit b => ⟨lit b, by
simp (config := {contextual := true}) [eval, hasConstantIf, hasRedundantIf]⟩
| var i => ⟨var i, by
simp (config := {contextual := true}) [eval, hasConstantIf, hasRedundantIf]⟩
| .ite (lit true) t e =>
have ⟨t', ht'⟩ := E.deConstRed t
⟨t', by
simp [Function.funext_iff, disjoint, vars, eval, hasRedundantIf,
List.disjoint, hasNestedIf, List.subset_def] at *
tauto
⟩
| .ite (lit false) t e =>
have ⟨e', he'⟩ := E.deConstRed e
⟨e', by
simp [Function.funext_iff, disjoint, vars, eval, hasRedundantIf,
List.disjoint, hasNestedIf, List.subset_def] at *
tauto
⟩
| .ite (var v) t e =>
have ⟨t', ht'⟩ := E.deConstRed t
have ⟨e', he'⟩ := E.deConstRed e
if hte' : t' = e'
then ⟨t', by
subst hte'
simp [Function.funext_iff, disjoint, vars, eval, hasRedundantIf,
List.disjoint, hasNestedIf, List.subset_def] at *
aesop
⟩
else
⟨.ite (var v) t' e', by
simp [Function.funext_iff, disjoint, vars, eval, List.subset_def,
List.disjoint, hasNestedIf, hasConstantIf, hasRedundantIf] at *
intro _ _ _ _ _ _
refine ⟨?_, ?_⟩
· intro f
cases hf : f v <;> simp_all
· aesop⟩
| .ite (.ite a b c) t e =>
⟨.ite (.ite a b c) t e, by
simp [Function.funext_iff, disjoint, vars, eval, List.subset_def,
List.disjoint, hasNestedIf, hasConstantIf] at *⟩
def E.normalize (e : E) : { e' : E // e'.normalized ∧ e'.eval = e.eval } :=
have ⟨e₁, he₁⟩ := e.denest
have ⟨e₂, he₂⟩ := e₁.dedup
have ⟨e₃, he₃⟩ := e₂.deConstRed
⟨e₃, by
refine ⟨?_, ?_⟩
· simp only [normalized, Bool.and_eq_true]
aesop
· simp only [Function.funext_iff] at *
intro f
simp_all⟩
def IfNormalization : Type := { Z : E → E // ∀ e, (Z e).normalized ∧ ∀ f, (Z e).eval f = e.eval f }
example : IfNormalization :=
⟨fun e => e.normalize, by
intro e
exact ⟨e.normalize.2.1, by
simpa [Function.funext_iff] using e.normalize.2.2⟩⟩
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