internlm/Lean-Github huggingface dataset:
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : x₀ = x
⊢ coords ys x₀ = 0
Tactic:
let ceq := coords_well_defined x₀ _ _ eqnStep
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : x₀ = x
ceq : coords ((coords t x₀, x₀) :: ys) x₀ = coords t x₀ := coords_well_defined x₀ ((coords t x₀, x₀) :: ys) t eqnStep
⊢ coords ys x₀ = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : x₀ = x
ceq : coords ((coords t x₀, x₀) :: ys) x₀ = coords t x₀ := coords_well_defined x₀ ((coords t x₀, x₀) :: ys) t eqnStep
⊢ coords ys x₀ = 0
Tactic:
simp [coords, monomCoeff] at ceq
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : x₀ = x
ceq : coords ys x₀ = 0
⊢ coords ys x₀ = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : x₀ = x
ceq : coords ys x₀ = 0
⊢ coords ys x₀ = 0
Tactic:
assumption
Proof state after:
no goals
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
⊢ coords ys x = 0
Tactic:
let hx := hyp x
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx : coords ((a₀, x₀) :: t) x = 0 := hyp x
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx : coords ((a₀, x₀) :: t) x = 0 := hyp x
⊢ coords ys x = 0
Tactic:
simp [coords, monomCoeff] at hx
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx : (match x₀ == x with
    | true => a₀
    | false => 0) +
    coords t x =
  0 :=
  hyp x
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx : (match x₀ == x with
    | true => a₀
    | false => 0) +
    coords t x =
  0 :=
  hyp x
⊢ coords ys x = 0
Tactic:
have lf : (x₀ == x) = false := decide_eq_false c
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx : (match x₀ == x with
    | true => a₀
    | false => 0) +
    coords t x =
  0 :=
  hyp x
lf : (x₀ == x) = false
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx : (match x₀ == x with
    | true => a₀
    | false => 0) +
    coords t x =
  0 :=
  hyp x
lf : (x₀ == x) = false
⊢ coords ys x = 0
Tactic:
rw [lf] at hx
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx :
  (match false with
      | true => a₀
      | false => 0) +
      coords t x =
    0
lf : (x₀ == x) = false
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
hx :
  (match false with
      | true => a₀
      | false => 0) +
      coords t x =
    0
lf : (x₀ == x) = false
⊢ coords ys x = 0
Tactic:
simp [zero_add] at hx
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
⊢ coords ys x = 0
Tactic:
let ceq := coords_well_defined x _ _ eqnStep
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
ceq : coords ((coords t x₀, x₀) :: ys) x = coords t x := coords_well_defined x ((coords t x₀, x₀) :: ys) t eqnStep
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
ceq : coords ((coords t x₀, x₀) :: ys) x = coords t x := coords_well_defined x ((coords t x₀, x₀) :: ys) t eqnStep
⊢ coords ys x = 0
Tactic:
simp [coords, monomCoeff, lf] at ceq
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
ceq : coords ys x = coords t x
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
ceq : coords ys x = coords t x
⊢ coords ys x = 0
Tactic:
rw [hx] at ceq
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
ceq : coords ys x = 0
⊢ coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
x : X
c : ¬x₀ = x
lf : (x₀ == x) = false
hx : coords t x = 0
ceq : coords ys x = 0
⊢ coords ys x = 0
Tactic:
exact ceq
Proof state after:
no goals
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
⊢ List.length ys < List.length (h :: t)
Tactic:
simp [List.length_cons]
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
⊢ List.length ys < Nat.succ (List.length t)
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
⊢ List.length ys < Nat.succ (List.length t)
Tactic:
apply Nat.le_trans lIneqStep
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
⊢ List.length t ≤ Nat.succ (List.length t)
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
⊢ List.length t ≤ Nat.succ (List.length t)
Tactic:
apply Nat.le_succ
Proof state after:
no goals
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
x✝ : List.length ys < List.length (h :: t)
⊢ ys ≃ []
Tactic:
apply equiv_e_of_zero_coeffs
Proof state after:
case hyp
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
x✝ : List.length ys < List.length (h :: t)
⊢ ∀ (x : X), coords ys x = 0
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
case hyp
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
x✝ : List.length ys < List.length (h :: t)
⊢ ∀ (x : X), coords ys x = 0
Tactic:
exact tail_coeffs
Proof state after:
no goals
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
x✝ : List.length ys < List.length (h :: t)
step : ys ≃ [] := equiv_e_of_zero_coeffs ys tail_coeffs
eqn₁ : (coords t x₀, x₀) :: ys ≃ [(coords t x₀, x₀)] := cons_equiv_of_equiv ys [] (coords t x₀) x₀ step
eqn₂ : t ≃ [(coords t x₀, x₀)] := eqnStep.symm.trans eqn₁
eqn₃ : (a₀, x₀) :: t ≃ [(a₀, x₀), (coords t x₀, x₀)] := cons_equiv_of_equiv t [(coords t x₀, x₀)] a₀ x₀ eqn₂
⊢ sum [(a₀, x₀), (coords t x₀, x₀)] = sum [(a₀ + coords t x₀, x₀)]
Tactic:
apply Quot.sound
Proof state after:
case a
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
x✝ : List.length ys < List.length (h :: t)
step : ys ≃ [] := equiv_e_of_zero_coeffs ys tail_coeffs
eqn₁ : (coords t x₀, x₀) :: ys ≃ [(coords t x₀, x₀)] := cons_equiv_of_equiv ys [] (coords t x₀) x₀ step
eqn₂ : t ≃ [(coords t x₀, x₀)] := eqnStep.symm.trans eqn₁
eqn₃ : (a₀, x₀) :: t ≃ [(a₀, x₀), (coords t x₀, x₀)] := cons_equiv_of_equiv t [(coords t x₀, x₀)] a₀ x₀ eqn₂
⊢ ElementaryMove R X [(a₀, x₀), (coords t x₀, x₀)] [(a₀ + coords t x₀, x₀)]
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
case a
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s : FormalSum R X
h : R × X
t : List (R × X)
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : a₀ + coords t x₀ = 0
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
x✝ : List.length ys < List.length (h :: t)
step : ys ≃ [] := equiv_e_of_zero_coeffs ys tail_coeffs
eqn₁ : (coords t x₀, x₀) :: ys ≃ [(coords t x₀, x₀)] := cons_equiv_of_equiv ys [] (coords t x₀) x₀ step
eqn₂ : t ≃ [(coords t x₀, x₀)] := eqnStep.symm.trans eqn₁
eqn₃ : (a₀, x₀) :: t ≃ [(a₀, x₀), (coords t x₀, x₀)] := cons_equiv_of_equiv t [(coords t x₀, x₀)] a₀ x₀ eqn₂
⊢ ElementaryMove R X [(a₀, x₀), (coords t x₀, x₀)] [(a₀ + coords t x₀, x₀)]
Tactic:
apply ElementaryMove.addCoeffs
Proof state after:
no goals
Theorem: FormalSum.equiv_e_of_zero_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
_x✝ : (s : FormalSum R X) ×' ∀ (x : X), coords s x = 0
a✝¹ :
  ∀ (y : (s : FormalSum R X) ×' ∀ (x : X), coords s x = 0),
    (invImage (fun a => PSigma.casesOn a fun s snd => List.length s) instWellFoundedRelation).1 y _x✝ → y.1 ≃ []
s : FormalSum R X
hyp✝¹ : ∀ (x : X), coords s x = 0
a✝ :
  ∀ (y : (s : FormalSum R X) ×' ∀ (x : X), coords s x = 0),
    (invImage (fun a => PSigma.casesOn a fun s snd => List.length s) instWellFoundedRelation).1 y
        { fst := s, snd := hyp✝¹ } →
      y.1 ≃ []
h : R × X
t : List (R × X)
hyp✝ : ∀ (x : X), coords (h :: t) x = 0
mt✝ : s = h :: t
x✝¹ :
  ∀ (y : (s : FormalSum R X) ×' ∀ (x : X), coords s x = 0),
    (invImage (fun a => PSigma.casesOn a fun s snd => List.length s) instWellFoundedRelation).1 y
        { fst := h :: t, snd := hyp✝ } →
      y.1 ≃ []
a₀ : R
x₀ : X
hyp : ∀ (x : X), coords ((a₀, x₀) :: t) x = 0
mt : s = (a₀, x₀) :: t
hyp₀ : coords ((a₀, x₀) :: t) x₀ = 0 := hyp x₀
c₀ : monomCoeff R X x₀ (a₀, x₀) = a₀
hz : ¬a₀ = 0
non_zero : 0 ≠ coords t x₀
ys : FormalSum R X
eqnStep : (coords t x₀, x₀) :: ys ≃ t
lIneqStep : List.length ys < List.length t
tail_coeffs : ∀ (x : X), coords ys x = 0
x✝ : List.length ys < List.length (h :: t)
⊢ (invImage (fun a => PSigma.casesOn a fun s snd => List.length s) instWellFoundedRelation).1
    { fst := ys, snd := tail_coeffs } { fst := h :: t, snd := hyp✝ }
Tactic:
assumption
Proof state after:
no goals
Theorem: FormalSum.equiv_of_equal_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
hyp : ∀ (x : X), coords [] x = coords s₂ x
⊢ ∀ (x : X), coords s₂ x = 0
Tactic:
intro x
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
hyp : ∀ (x : X), coords [] x = coords s₂ x
x : X
⊢ coords s₂ x = 0
Theorem: FormalSum.equiv_of_equal_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
hyp : ∀ (x : X), coords [] x = coords s₂ x
x : X
⊢ coords s₂ x = 0
Tactic:
let h := hyp x
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
hyp : ∀ (x : X), coords [] x = coords s₂ x
x : X
h : coords [] x = coords s₂ x := hyp x
⊢ coords s₂ x = 0
Theorem: FormalSum.equiv_of_equal_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
hyp : ∀ (x : X), coords [] x = coords s₂ x
x : X
h : coords [] x = coords s₂ x := hyp x
⊢ coords s₂ x = 0
Tactic:
rw [← h]
Proof state after:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
hyp : ∀ (x : X), coords [] x = coords s₂ x
x : X
h : coords [] x = coords s₂ x := hyp x
⊢ coords [] x = 0
Theorem: FormalSum.equiv_of_equal_coeffs
Proof state before:
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
hyp : ∀ (x : X), coords [] x = coords s₂ x
x : X
h : coords [] x = coords s₂ x := hyp x
⊢ coords [] x = 0
Tactic:
rfl
Proof state after:
no goals