253 lines
9.8 KiB
Text
253 lines
9.8 KiB
Text
abbrev Var := Nat
|
||
|
||
inductive Expr where
|
||
| num (v : Nat)
|
||
| var (i : Var)
|
||
| add (a b : Expr)
|
||
| mulL (k : Nat) (a : Expr)
|
||
| mulR (a : Expr) (k : Nat)
|
||
deriving Inhabited, Repr
|
||
|
||
structure Context where
|
||
vars : List Nat
|
||
|
||
def List.getIdx : List α → Var → α → α
|
||
| [], i, u => u
|
||
| a::as, 0, u => a
|
||
| a::as, i+1, u => getIdx as i u
|
||
|
||
def Var.denote (ctx : Context) (v : Var) : Nat :=
|
||
ctx.vars.getIdx v 0
|
||
|
||
def Expr.denote (ctx : Context) : Expr → Nat
|
||
| Expr.add a b => Nat.add (denote ctx a) (denote ctx b)
|
||
| Expr.num k => k
|
||
| Expr.var v => v.denote ctx
|
||
| Expr.mulL k e => k * denote ctx e
|
||
| Expr.mulR e k => denote ctx e * k
|
||
|
||
abbrev Monomials := List (Nat × Var)
|
||
|
||
def Monomials.denote (ctx : Context) (m : Monomials) : Nat :=
|
||
match m with
|
||
| [] => 0
|
||
| (k, v) :: m => k * v.denote ctx + denote ctx m
|
||
|
||
def Monomials.addM (m : Monomials) (k : Nat) (v : Var) : Monomials :=
|
||
match m with
|
||
| [] => [(k, v)]
|
||
| (k', v') :: m => if v = v' then (k' + k, v) :: m else (k', v') :: addM m k v
|
||
|
||
attribute [local simp] Nat.add_comm Nat.add_assoc Nat.add_left_comm Nat.right_distrib Nat.left_distrib Nat.mul_assoc Nat.mul_comm
|
||
|
||
@[simp] theorem Monomials.denote_addM (ctx : Context) (m : Monomials) (k : Nat) (v : Var) : (m.addM k v).denote ctx = m.denote ctx + k * v.denote ctx := by
|
||
induction m with
|
||
| nil => simp [denote]
|
||
| cons kv m ih => cases kv with | _ k' v' =>
|
||
by_cases h : v = v'
|
||
. simp [h, denote, addM]
|
||
. simp [h, denote, addM, ih]
|
||
|
||
def Monomials.add (m₁ m₂ : Monomials) : Monomials :=
|
||
match m₂ with
|
||
| [] => m₁
|
||
| (k, v) :: m₂ => add (m₁.addM k v) m₂
|
||
|
||
@[simp] theorem Monomials.denote_add (ctx : Context) (m₁ m₂ : Monomials) : (m₁.add m₂).denote ctx = m₁.denote ctx + m₂.denote ctx := by
|
||
induction m₂ generalizing m₁ with
|
||
| nil => simp [add, denote]
|
||
| cons kv m₂ ih => cases kv with | _ k v =>
|
||
simp [add, denote, ih]
|
||
|
||
def Monomials.mul (k : Nat) (m : Monomials) : Monomials :=
|
||
if k = 0 then
|
||
[]
|
||
else
|
||
go m
|
||
where
|
||
go : Monomials → Monomials
|
||
| [] => []
|
||
| (k', v) :: m => (k*k', v) :: go m
|
||
|
||
@[simp] theorem Monomials.denote_mul (ctx : Context) (k : Nat) (m : Monomials) : (m.mul k).denote ctx = k * m.denote ctx := by
|
||
simp [mul]
|
||
by_cases h : k = 0
|
||
. simp [denote, h]
|
||
. simp [denote, h]
|
||
induction m with
|
||
| nil => simp [mul.go, denote]
|
||
| cons kv m ih => cases kv with | _ k' v => simp [mul.go, denote, ih]
|
||
|
||
def Monomials.insertSorted (k : Nat) (v : Var) (m : Monomials) : Monomials :=
|
||
match m with
|
||
| [] => [(k, v)]
|
||
| (k', v') :: m => if v < v' then (k, v) :: (k', v') :: m else (k', v') :: insertSorted k v m
|
||
|
||
@[simp] theorem Monomials.denote_insertSorted (ctx : Context) (k : Nat) (v : Var) (m : Monomials) : (m.insertSorted k v).denote ctx = m.denote ctx + k * v.denote ctx := by
|
||
induction m with
|
||
| nil => simp [insertSorted, denote]
|
||
| cons kv m ih => cases kv with | _ k' v' => by_cases h : v < v' <;> simp [h, insertSorted, denote, ih]
|
||
|
||
def Monomials.sort (m : Monomials) : Monomials :=
|
||
let rec go (m : Monomials) (r : Monomials) : Monomials :=
|
||
match m with
|
||
| [] => r
|
||
| (k, v) :: m => go m (r.insertSorted k v)
|
||
go m []
|
||
|
||
@[simp] theorem Monomials.denote_sort_go (ctx : Context) (m : Monomials) (r : Monomials) : (sort.go m r).denote ctx = m.denote ctx + r.denote ctx := by
|
||
induction m generalizing r with
|
||
| nil => simp [sort.go, denote]; done
|
||
| cons kv m ih => cases kv with | _ k v => simp [sort.go, denote, ih]
|
||
|
||
@[simp] theorem Monomials.denote_sort (ctx : Context) (m : Monomials) : m.sort.denote ctx = m.denote ctx := by
|
||
simp [sort, denote]
|
||
|
||
@[simp] theorem Monomials.denote_append (ctx : Context) (m₁ m₂ : Monomials) : (m₁ ++ m₂).denote ctx = m₁.denote ctx + m₂.denote ctx := by
|
||
match m₁ with
|
||
| [] => simp [denote]
|
||
| (k, v) :: m₁ => simp [denote, denote_append ctx m₁ m₂]
|
||
|
||
@[simp] theorem Monomials.denote_cons (ctx : Context) (k : Nat) (v : Var) (m : Monomials) : denote ctx ((k, v) :: m) = k * v.denote ctx + m.denote ctx := by
|
||
match m with
|
||
| [] => simp [denote]
|
||
| _ :: m => simp [denote, denote_cons ctx k v m]
|
||
|
||
@[simp] theorem Monomials.denote_reverseAux (ctx : Context) (m₁ m₂ : Monomials) : denote ctx (List.reverseAux m₁ m₂) = denote ctx (m₁ ++ m₂) := by
|
||
match m₁ with
|
||
| [] => simp [denote, List.reverseAux]
|
||
| (k, v) :: m₁ => simp [denote, List.reverseAux, denote_reverseAux ctx m₁]
|
||
|
||
@[simp] theorem Monomials.denote_reverse (ctx : Context) (m : Monomials) : denote ctx (List.reverse m) = denote ctx m := by
|
||
simp [List.reverse]
|
||
|
||
def Monomials.cancelAux (fuel : Nat) (m₁ m₂ r₁ r₂ : Monomials) : Monomials × Monomials :=
|
||
match fuel with
|
||
| 0 => (r₁.reverse ++ m₁, r₂.reverse ++ m₂)
|
||
| fuel + 1 =>
|
||
match m₁, m₂ with
|
||
| m₁, [] => (r₁.reverse ++ m₁, r₂.reverse)
|
||
| [], m₂ => (r₁.reverse, r₂.reverse ++ m₂)
|
||
| (k₁, v₁) :: m₁, (k₂, v₂) :: m₂ =>
|
||
if v₁ < v₂ then
|
||
cancelAux fuel m₁ ((k₂, v₂) :: m₂) ((k₁, v₁) :: r₁) r₂
|
||
else if v₁ > v₂ then
|
||
cancelAux fuel ((k₁, v₁) :: m₁) m₂ r₁ ((k₂, v₂) :: r₂)
|
||
else if k₁ < k₂ then
|
||
cancelAux fuel m₁ m₂ r₁ ((k₂ - k₁, v₁) :: r₂)
|
||
else if k₁ > k₂ then
|
||
cancelAux fuel m₁ m₂ ((k₁ - k₂, v₁) :: r₁) r₂
|
||
else
|
||
cancelAux fuel m₁ m₂ r₁ r₂
|
||
|
||
def Monomials.denote_eq (ctx : Context) (mp : Monomials × Monomials) : Prop := mp.1.denote ctx = mp.2.denote ctx
|
||
|
||
-- TODO : cleanup proof
|
||
theorem Monomials.denote_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Monomials)
|
||
(h : denote_eq ctx (r₁.reverse ++ m₁, r₂.reverse ++ m₂)) : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) := by
|
||
induction fuel generalizing m₁ m₂ r₁ r₂ with
|
||
| zero => assumption
|
||
| succ fuel ih =>
|
||
simp [cancelAux]
|
||
split
|
||
. simp_all
|
||
. simp_all
|
||
. rename_i k₁ v₁ m₁ k₂ v₂ m₂
|
||
by_cases hltv : v₁ < v₂
|
||
. simp [hltv]; apply ih; simp [denote_eq, denote] at h |-; assumption
|
||
. by_cases hgtv : v₁ > v₂
|
||
. simp [hltv, hgtv]; apply ih; simp [denote_eq, denote] at h |-; assumption
|
||
. simp [hltv, hgtv]
|
||
have heqv : v₁ = v₂ := Nat.le_antisymm (Nat.ge_of_not_lt hgtv) (Nat.ge_of_not_lt hltv)
|
||
subst heqv
|
||
by_cases hltk : k₁ < k₂
|
||
. simp [hltk]; apply ih; simp [denote_eq, denote] at h |-
|
||
rw [Nat.mul_sub_right_distrib, ← Nat.add_assoc, ← Nat.add_sub_assoc]
|
||
apply Eq.symm
|
||
apply Nat.sub_eq_of_eq_add
|
||
simp [h]
|
||
exact Nat.mul_le_mul_right _ (Nat.le_of_lt hltk)
|
||
. by_cases hgtk : k₁ > k₂
|
||
. simp [hltk, hgtk]
|
||
apply ih
|
||
simp [denote_eq, denote] at h |-
|
||
rw [Nat.mul_sub_right_distrib, ← Nat.add_assoc, ← Nat.add_sub_assoc]
|
||
apply Nat.sub_eq_of_eq_add
|
||
simp [h]
|
||
exact Nat.mul_le_mul_right _ (Nat.le_of_lt hgtk)
|
||
. simp [hltk, hgtk]
|
||
have heqk : k₁ = k₂ := Nat.le_antisymm (Nat.ge_of_not_lt hgtk) (Nat.ge_of_not_lt hltk)
|
||
subst heqk
|
||
apply ih
|
||
simp [denote_eq, denote] at h |-
|
||
rw [← Nat.add_assoc, ← Nat.add_assoc] at h
|
||
exact Nat.add_right_cancel h
|
||
|
||
def Monomials.cancel (m₁ m₂ : Monomials) : Monomials × Monomials :=
|
||
cancelAux (m₁.length + m₂.length) m₁ m₂ [] []
|
||
|
||
theorem Monomials.denote_cancel (ctx : Context) (m₁ m₂ : Monomials) (h : denote_eq ctx (m₁, m₂)) : denote_eq ctx (cancel m₁ m₂) := by
|
||
simp [cancel]
|
||
apply denote_cancelAux
|
||
simp [h]
|
||
|
||
structure Poly where
|
||
m : Monomials := []
|
||
k : Nat := 0
|
||
deriving Repr
|
||
|
||
def Poly.denote (ctx : Context) (p : Poly) : Nat :=
|
||
p.m.denote ctx + p.k
|
||
|
||
def Poly.addK (p : Poly) (k : Nat) : Poly :=
|
||
{ p with k := p.k + k }
|
||
|
||
def Poly.addM (p : Poly) (k : Nat) (v : Var) : Poly :=
|
||
{ p with m := p.m.addM k v }
|
||
|
||
@[simp] theorem Poly.denote_addM (ctx : Context) (p : Poly) (k : Nat) (v : Var) : (p.addM k v).denote ctx = p.denote ctx + k * v.denote ctx := by
|
||
simp [denote, addM]
|
||
|
||
def Poly.add (p q : Poly) : Poly :=
|
||
{ m := p.m.add q.m, k := p.k + q.k }
|
||
|
||
@[simp] theorem Poly.denote_add (ctx : Context) (p q : Poly) : (p.add q).denote ctx = p.denote ctx + q.denote ctx := by
|
||
simp [add, denote]
|
||
|
||
def Poly.mul (k : Nat) (p : Poly) : Poly :=
|
||
{ p with m := p.m.mul k, k := p.k * k }
|
||
|
||
@[simp] theorem Poly.denote_mul (ctx : Context) (k : Nat) (p : Poly) : (p.mul k).denote ctx = k * p.denote ctx := by
|
||
simp [denote, mul]
|
||
|
||
def Poly.sort (p : Poly) : Poly :=
|
||
{ p with m := p.m.sort }
|
||
|
||
@[simp] theorem Poly.denote_sort (ctx : Context) (p : Poly) : p.sort.denote ctx = p.denote ctx := by
|
||
simp [denote, sort]
|
||
|
||
def Expr.toPoly : Expr → Poly
|
||
| Expr.num k => { k }
|
||
| Expr.var i => { m := [(1, i)] }
|
||
| Expr.add a b => a.toPoly.add b.toPoly
|
||
| Expr.mulL k a => a.toPoly.mul k
|
||
| Expr.mulR a k => a.toPoly.mul k
|
||
|
||
@[simp] theorem Expr.denote_toPoly (ctx : Context) (e : Expr) : e.toPoly.denote ctx = e.denote ctx := by
|
||
induction e with
|
||
| num k => simp [denote, toPoly, Poly.denote, Monomials.denote]
|
||
| var i => simp [denote, toPoly, Poly.denote, Monomials.denote]
|
||
| add a b iha ihb => simp [denote, toPoly, iha, ihb]; done
|
||
| mulL k a ih => simp [denote, toPoly, ih]; done
|
||
| mulR k a ih => simp [denote, toPoly, ih]; done
|
||
|
||
theorem Expr.eq_of_toPoly_sort_eq (ctx : Context) (a b : Expr) (h : a.toPoly.sort = b.toPoly.sort) : a.denote ctx = b.denote ctx := by
|
||
have h := congrArg (Poly.denote ctx) h
|
||
simp at h
|
||
assumption
|
||
|
||
example (x₁ x₂ x₃ : Nat) : (x₁ + x₂) + (x₂ + x₃) = x₃ + 2*x₂ + x₁ :=
|
||
Expr.eq_of_toPoly_sort_eq { vars := [x₁, x₂, x₃] }
|
||
(Expr.add (Expr.add (Expr.var 0) (Expr.var 1)) (Expr.add (Expr.var 1) (Expr.var 2)))
|
||
(Expr.add (Expr.add (Expr.var 2) (Expr.mulL 2 (Expr.var 1))) (Expr.var 0))
|
||
rfl
|