awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (hypothesis)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : q.degree ≤ p.degree ∧ p ≠ 0
⊢ p %ₘ q = p - q * (p /ₘ q)
have _wf : (p -
      q *
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).degree <
  p.degree
Tactic: have _wf := div_wf_lemma h hq
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (hypothesis)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : q.degree ≤ p.degree ∧ p ≠ 0
_wf :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).degree <
    p.degree
⊢ p %ₘ q = p - q * (p /ₘ q)
have ih : (p -
      q *
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) %ₘ
    q =
  p -
      q *
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
    q *
      ((p -
          q *
            HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
        q)
Tactic: have ih := modByMonic_eq_sub_mul_div
          (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) q
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : q.degree ≤ p.degree ∧ p ≠ 0
_wf :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).degree <
    p.degree
ih :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) %ₘ
      q =
    p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
      q *
        ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
          q)
⊢ p %ₘ q = p - q * (p /ₘ q)
have unfold : (if h : q.Monic then
    (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
        have z :=
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
        have _wf := sorry;
        have dm := (p - q * z).divModByMonicAux sorry;
        (z + dm.1, dm.2)
      else (0, p)).2
  else p) =
  p -
    q *
      if h : q.Monic then
        (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
            have z :=
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
            have _wf := sorry;
            have dm := (p - q * z).divModByMonicAux sorry;
            (z + dm.1, dm.2)
          else (0, p)).1
      else 0
Tactic: unfold modByMonic divByMonic divModByMonicAux
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : q.degree ≤ p.degree ∧ p ≠ 0
_wf :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).degree <
    p.degree
ih :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) %ₘ
      q =
    p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
      q *
        ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
          q)
⊢ (if h : q.Monic then
      (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
          have z :=
            HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
          have _wf := ⋯;
          have dm := (p - q * z).divModByMonicAux ⋯;
          (z + dm.1, dm.2)
        else (0, p)).2
    else p) =
    p -
      q *
        if h : q.Monic then
          (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
              have z :=
                HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                  (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
              have _wf := ⋯;
              have dm := (p - q * z).divModByMonicAux ⋯;
              (z + dm.1, dm.2)
            else (0, p)).1
        else 0
have rw : (have z :=
      HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
        (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
    have _wf := sorry;
    have dm := (p - q * z).divModByMonicAux sorry;
    (z + dm.1, dm.2)).2 =
  p -
    q *
      if h : q.Monic then
        (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
            have z :=
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
            have _wf := sorry;
            have dm := (p - q * z).divModByMonicAux sorry;
            (z + dm.1, dm.2)
          else (0, p)).1
      else 0
Tactic: rw [dif_pos hq, dif_pos h]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (hypothesis)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : q.degree ≤ p.degree ∧ p ≠ 0
_wf :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).degree <
    p.degree
ih :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) %ₘ
      q =
    p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
      q *
        ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
          q)
⊢ (have z :=
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
      have _wf := ⋯;
      have dm := (p - q * z).divModByMonicAux ⋯;
      (z + dm.1, dm.2)).2 =
    p -
      q *
        if h : q.Monic then
          (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
              have z :=
                HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                  (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
              have _wf := ⋯;
              have dm := (p - q * z).divModByMonicAux ⋯;
              (z + dm.1, dm.2)
            else (0, p)).1
        else 0
have ih₁ : ((p -
          q *
            HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
      hq).2 =
  p -
      q *
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
    q *
      ((p -
          q *
            HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
        q)
Tactic: rw [modByMonic, dif_pos hq] at ih
Cleared hypotheses: ih
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : q.degree ≤ p.degree ∧ p ≠ 0
_wf :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).degree <
    p.degree
ih :
  ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
        hq).2 =
    p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
      q *
        ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
          q)
⊢ (have z :=
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
      have _wf := ⋯;
      have dm := (p - q * z).divModByMonicAux ⋯;
      (z + dm.1, dm.2)).2 =
    p -
      q *
        if h : q.Monic then
          (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
              have z :=
                HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                  (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
              have _wf := ⋯;
              have dm := (p - q * z).divModByMonicAux ⋯;
              (z + dm.1, dm.2)
            else (0, p)).1
        else 0
have refine : p -
      q *
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
    q *
      ((p -
          q *
            HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
        q) =
  p -
    q *
      if h : q.Monic then
        (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
            have z :=
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
            have _wf := sorry;
            have dm := (p - q * z).divModByMonicAux sorry;
            (z + dm.1, dm.2)
          else (0, p)).1
      else 0
Tactic: refine ih.trans ?_
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : q.degree ≤ p.degree ∧ p ≠ 0
_wf :
  (p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).degree <
    p.degree
ih :
  ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
        hq).2 =
    p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
      q *
        ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
          q)
⊢ p -
        q *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
      q *
        ((p -
            q *
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))) /ₘ
          q) =
    p -
      q *
        if h : q.Monic then
          (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
              have z :=
                HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                  (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
              have _wf := ⋯;
              have dm := (p - q * z).divModByMonicAux ⋯;
              (z + dm.1, dm.2)
            else (0, p)).1
        else 0
have rw : (p -
      q *
        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) -
    q *
      if hq : q.Monic then
        ((p -
                q *
                  HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                    (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
            hq).1
      else 0) =
  p -
    q *
      if h : q.Monic then
        (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
            have z :=
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
            have _wf := sorry;
            have dm := (p - q * z).divModByMonicAux sorry;
            (z + dm.1, dm.2)
          else (0, p)).1
      else 0
Tactic: rw [divByMonic, dif_pos hq, dif_pos hq, dif_pos h, mul_add, sub_add_eq_sub_sub]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : ¬(q.degree ≤ p.degree ∧ p ≠ 0)
⊢ p %ₘ q = p - q * (p /ₘ q)
have unfold : (if h : q.Monic then
    (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
        have z :=
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
        have _wf := sorry;
        have dm := (p - q * z).divModByMonicAux sorry;
        (z + dm.1, dm.2)
      else (0, p)).2
  else p) =
  p -
    q *
      if h : q.Monic then
        (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
            have z :=
              HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
            have _wf := sorry;
            have dm := (p - q * z).divModByMonicAux sorry;
            (z + dm.1, dm.2)
          else (0, p)).1
      else 0
Tactic: unfold modByMonic divByMonic divModByMonicAux
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : q.Monic
h : ¬(q.degree ≤ p.degree ∧ p ≠ 0)
⊢ (if h : q.Monic then
      (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
          have z :=
            HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
          have _wf := ⋯;
          have dm := (p - q * z).divModByMonicAux ⋯;
          (z + dm.1, dm.2)
        else (0, p)).2
    else p) =
    p -
      q *
        if h : q.Monic then
          (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
              have z :=
                HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                  (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
              have _wf := ⋯;
              have dm := (p - q * z).divModByMonicAux ⋯;
              (z + dm.1, dm.2)
            else (0, p)).1
        else 0
have dsimp : (if h : q.Monic then
    (if q.degree ≤ p.degree ∧ ¬p = 0 then
        (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) +
            ((p -
                    q *
                      HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                        (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
                sorry).1,
          ((p -
                  q *
                    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                      (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
              sorry).2)
      else (0, p)).2
  else p) =
  p -
    q *
      if h : q.Monic then
        (if q.degree ≤ p.degree ∧ ¬p = 0 then
            (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                  (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) +
                ((p -
                        q *
                          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
                    sorry).1,
              ((p -
                      q *
                        HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                          (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
                  sorry).2)
          else (0, p)).1
      else 0
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : ¬q.Monic
⊢ p %ₘ q = p - q * (p /ₘ q)
have rw : p = p - q * (p /ₘ q)
Tactic: rw [modByMonic_eq_of_not_monic _ hq, divByMonic_eq_of_not_monic _ hq, mul_zero, sub_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : ¬q.Monic
rw : p = p - q * (p /ₘ q)
⊢ p %ₘ q = p - q * (p /ₘ q)
have rw₁ : p = p - q * 0
Tactic: rw [modByMonic_eq_of_not_monic _ hq, divByMonic_eq_of_not_monic _ hq, mul_zero, sub_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : ¬q.Monic
rw : p = p - q * (p /ₘ q)
rw₁ : p = p - q * 0
⊢ p %ₘ q = p - q * (p /ₘ q)
have rw₂ : p = p - 0
Tactic: rw [modByMonic_eq_of_not_monic _ hq, divByMonic_eq_of_not_monic _ hq, mul_zero, sub_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
this : DecidableEq R := Classical.decEq R
hq : ¬q.Monic
rw : p = p - q * (p /ₘ q)
rw₁ : p = p - q * 0
rw₂ : p = p - 0
⊢ p %ₘ q = p - q * (p /ₘ q)
have rw₃ : p = p
Tactic: rw [modByMonic_eq_of_not_monic _ hq, divByMonic_eq_of_not_monic _ hq, mul_zero, sub_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (hypothesis)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p /ₘ q = 0
⊢ p.degree < q.degree
have this : p %ₘ q + q * (p /ₘ q) = p
Tactic: have := modByMonic_add_div p q
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p /ₘ q = 0
this : p %ₘ q + q * (p /ₘ q) = p
⊢ p.degree < q.degree
have rwa : ∀ (this : p %ₘ q + q * 0 = p), p.degree < q.degree
Tactic: rwa [h, mul_zero, add_zero, modByMonic_eq_self_iff hq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p /ₘ q = 0
this : p %ₘ q + q * (p /ₘ q) = p
rwa : ∀ (this : p %ₘ q + q * 0 = p), p.degree < q.degree
⊢ p.degree < q.degree
have rwa₁ : ∀ (this : p %ₘ q + 0 = p), p.degree < q.degree
Tactic: rwa [h, mul_zero, add_zero, modByMonic_eq_self_iff hq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p /ₘ q = 0
this : p %ₘ q + q * (p /ₘ q) = p
rwa : ∀ (this : p %ₘ q + q * 0 = p), p.degree < q.degree
rwa₁ : ∀ (this : p %ₘ q + 0 = p), p.degree < q.degree
⊢ p.degree < q.degree
have rwa₂ : ∀ (this : p %ₘ q = p), p.degree < q.degree
Tactic: rwa [h, mul_zero, add_zero, modByMonic_eq_self_iff hq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p /ₘ q = 0
this : p %ₘ q + q * (p /ₘ q) = p
rwa : ∀ (this : p %ₘ q + q * 0 = p), p.degree < q.degree
rwa₁ : ∀ (this : p %ₘ q + 0 = p), p.degree < q.degree
rwa₂ : ∀ (this : p %ₘ q = p), p.degree < q.degree
⊢ p.degree < q.degree
have rwa₃ : ∀ (this : p.degree < q.degree), p.degree < q.degree
Tactic: rwa [h, mul_zero, add_zero, modByMonic_eq_self_iff hq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (hypothesis)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p.degree < q.degree
⊢ p /ₘ q = 0
have this : ¬q.degree ≤ p.degree
Tactic: have : ¬degree q ≤ degree p := not_le_of_gt h
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p.degree < q.degree
this : ¬q.degree ≤ p.degree
⊢ p /ₘ q = 0
have unfold : Eq.{u + 1} (α := R[X])
  (if h : q.Monic then
    (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
        have z :=
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
            (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
        have _wf := sorry;
        have dm := (p - q * z).divModByMonicAux sorry;
        (z + dm.1, dm.2)
      else (0, p)).1
  else 0)
  0
Tactic: unfold divByMonic divModByMonicAux
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Div (goal)
Goal:
R : Type u
inst : Ring R
p q : R[X]
inst_1 : Nontrivial R
hq : q.Monic
h : p.degree < q.degree
this : ¬q.degree ≤ p.degree
⊢ Eq.{u + 1} (α := R[X])
    (if h : q.Monic then
      (if h_1 : q.degree ≤ p.degree ∧ p ≠ 0 then
          have z :=
            HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree));
          have _wf := ⋯;
          have dm := (p - q * z).divModByMonicAux ⋯;
          (z + dm.1, dm.2)
        else (0, p)).1
    else 0)
    0
have dsimp : Eq.{u + 1} (α := R[X])
  (if h : q.Monic then
    (if q.degree ≤ p.degree ∧ ¬p = 0 then
        (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
              (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree)) +
            ((p -
                    q *
                      HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                        (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
                sorry).1,
          ((p -
                  q *
                    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
                      (HPow.hPow (α := R[X]) X (p.natDegree - q.natDegree))).divModByMonicAux
              sorry).2)
      else (0, p)).1
  else 0)
  0
Tactic: dsimp