tactic stringlengths 1 8.8k | premises listlengths 0 85 | goal stringlengths 3 286k |
|---|---|---|
tauto | [] | p q r : Prop
h : r → p → q
inl : ∀ (h : ¬r), r → p → r ∧ q
inr inr₁ : ∀ (h : p → q), r → p → r ∧ q
inr₂ : ∀ (h : p → q), r → p → r
inr₃ : ∀ (h : p → q), r → p → q
inr₄ : r → p → ∀ (h : ¬p), q
inr₅ inr₆ inr₇ : r → p → ∀ (h : q), q
⊢ r ∧ p → r ∧ q |
Filter.Eventually.mp hp (Filter.Eventually.of_forall hq) | [
"Filter.Eventually.of_forall",
"Filter.Eventually.mp"
] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), p x → q x
hp : ∀ᶠ (x : α) in f, p x
⊢ ∀ᶠ (x : α) in f, q x |
Filter.Frequently.mp hp (Filter.Eventually.of_forall hq) | [
"Filter.Eventually.of_forall",
"Filter.Frequently.mp"
] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), p x → q x
hp : ∃ᶠ (x : α) in f, p x
⊢ ∃ᶠ (x : α) in f, q x |
congr! 2 | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq a : ∀ (x : α), p x ↔ q x
⊢ (∀ᶠ (x : α) in f, p x) ↔ ∀ᶠ (x : α) in f, q x |
hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), p x ↔ q x
x : α
⊢ p x ↔ q x |
exact hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), p x ↔ q x
x : α
⊢ p x ↔ q x |
congr! 2 | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq a : ∀ (x : α), p x ↔ q x
⊢ (∃ᶠ (x : α) in f, p x) ↔ ∃ᶠ (x : α) in f, q x |
hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), p x ↔ q x
x : α
⊢ p x ↔ q x |
exact hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), p x ↔ q x
x : α
⊢ p x ↔ q x |
(natDegree_C a).le | [
"Polynomial.natDegree_C",
"Eq.le"
] | R : Type u_1
inst : Semiring R
a : R
⊢ natDegree.{u_1} (R := R) ((C : (a : R) → R[X]) a) ≤ 0 |
(natDegree_natCast _).le | [
"Eq.le",
"Polynomial.natDegree_natCast"
] | R : Type u_1
inst : Semiring R
n : ℕ
⊢ natDegree.{u_1} (R := R) (↑n : R[X]) ≤ 0 |
natDegree_zero.le | [
"Eq.le",
"Polynomial.natDegree_zero"
] | R : Type u_1
inst : Semiring R
⊢ natDegree.{u_1} (R := R) 0 ≤ 0 |
natDegree_one.le | [
"Eq.le",
"Polynomial.natDegree_one"
] | R : Type u_1
inst : Semiring R
⊢ natDegree.{u_1} (R := R) 1 ≤ 0 |
subst ‹_› ‹_› | [] | R : Type u_1
inst : Semiring R
n : ℕ
a b : R
f g : R[X]
h_add_left : f.coeff n = a
h_add_right : g.coeff n = b
subst : (f + g).coeff n = f.coeff n + g.coeff n
⊢ (f + g).coeff n = a + b |
apply coeff_add | [
"Polynomial.coeff_add"
] | R : Type u_1
inst : Semiring R
n : ℕ
f g : R[X]
⊢ (f + g).coeff n = f.coeff n + g.coeff n |
split_ifs with h | [] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
pos : ∀ (h : d = df + dg), (f * g).coeff d = a * b
neg : ∀ (h : ¬d = df + dg), (f * g).coeff d = 0
⊢ (f * g).coeff d =... |
subst h_mul_left h_mul_right h | [] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : d = df + dg
pos : ∀ (ddf : df + dg ≤ df + dg), (f * g).coeff (df + dg) = f.coeff df * g.coeff dg
⊢ (f * g).coeff d... |
coeff_mul_add_eq_of_natDegree_le ‹_› ‹_› | [
"Polynomial.coeff_mul_add_eq_of_natDegree_le"
] | R : Type u_1
inst : Semiring R
df dg : ℕ
f g : R[X]
h_mul_left : f.natDegree ≤ df
h_mul_right : g.natDegree ≤ dg
ddf : df + dg ≤ df + dg
⊢ (f * g).coeff (df + dg) = f.coeff df * g.coeff dg |
exact coeff_mul_add_eq_of_natDegree_le ‹_› ‹_› | [
"Polynomial.coeff_mul_add_eq_of_natDegree_le"
] | R : Type u_1
inst : Semiring R
df dg : ℕ
f g : R[X]
h_mul_left : f.natDegree ≤ df
h_mul_right : g.natDegree ≤ dg
ddf : df + dg ≤ df + dg
⊢ (f * g).coeff (df + dg) = f.coeff df * g.coeff dg |
apply coeff_eq_zero_of_natDegree_lt | [
"Polynomial.coeff_eq_zero_of_natDegree_lt"
] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : ¬d = df + dg
neg : (f * g).natDegree < d
⊢ (f * g).coeff d = 0 |
apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_) | [
"lt_of_le_of_lt",
"lt_of_le_of_ne"
] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : ¬d = df + dg
apply : (f * g).natDegree ≤ df + dg
apply₁ : df + dg ≠ d
⊢ (f * g).natDegree < d |
natDegree_mul_le_of_le ‹_› ‹_› | [
"Polynomial.natDegree_mul_le_of_le"
] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : ¬d = df + dg
⊢ (f * g).natDegree ≤ df + dg |
exact natDegree_mul_le_of_le ‹_› ‹_› | [
"Polynomial.natDegree_mul_le_of_le"
] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : ¬d = df + dg
⊢ (f * g).natDegree ≤ df + dg |
ne_comm.mp h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : ¬d = df + dg
⊢ df + dg ≠ d |
exact ne_comm.mp h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left_1 : f.natDegree ≤ df
h_mul_right_1 : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : ¬d = df + dg
⊢ df + dg ≠ d |
split_ifs with h | [] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
pos : ∀ (h : o = m * n), (p ^ m).coeff o = a ^ m
neg : ∀ (h : ¬o = m * n), (p ^ m).coeff o = 0
⊢ (p ^ m).coeff o = if o = m * n then a ^ m else 0 |
subst h h_pow_bas | [] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
h : o = m * n
pos : ∀ (h_exp : m * n ≤ m * n), (p ^ m).coeff (m * n) = p.coeff n ^ m
⊢ (p ^ m).coeff o = a ^ m |
coeff_pow_of_natDegree_le ‹_› | [
"Polynomial.coeff_pow_of_natDegree_le"
] | R : Type u_1
inst : Semiring R
m n : ℕ
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ m * n
⊢ (p ^ m).coeff (m * n) = p.coeff n ^ m |
exact coeff_pow_of_natDegree_le ‹_› | [
"Polynomial.coeff_pow_of_natDegree_le"
] | R : Type u_1
inst : Semiring R
m n : ℕ
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ m * n
⊢ (p ^ m).coeff (m * n) = p.coeff n ^ m |
apply coeff_eq_zero_of_natDegree_lt | [
"Polynomial.coeff_eq_zero_of_natDegree_lt"
] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
h : ¬o = m * n
neg : (p ^ m).natDegree < o
⊢ (p ^ m).coeff o = 0 |
apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_) | [
"lt_of_le_of_lt",
"lt_of_le_of_ne"
] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
h : ¬o = m * n
apply : (p ^ m).natDegree ≤ m * n
apply₁ : m * n ≠ o
⊢ (p ^ m).natDegree < o |
natDegree_pow_le_of_le m ‹_› | [
"Polynomial.natDegree_pow_le_of_le"
] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
h : ¬o = m * n
⊢ (p ^ m).natDegree ≤ m * n |
exact natDegree_pow_le_of_le m ‹_› | [
"Polynomial.natDegree_pow_le_of_le"
] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
h : ¬o = m * n
⊢ (p ^ m).natDegree ≤ m * n |
Iff.mp ne_comm h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
h : ¬o = m * n
⊢ m * n ≠ o |
exact Iff.mp ne_comm h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
m n o : ℕ
a : R
p : R[X]
h_pow : p.natDegree ≤ n
h_exp : m * n ≤ o
h_pow_bas : p.coeff n = a
h : ¬o = m * n
⊢ m * n ≠ o |
(natDegree_smul_le a f).trans hf | [
"Polynomial.natDegree_smul_le",
"LE.le.trans"
] | R : Type u_1
inst : Semiring R
S : Type u_2
inst_1 : SMulZeroClass S R
n : ℕ
a : S
f : R[X]
hf : f.natDegree ≤ n
⊢ (a • f).natDegree ≤ n |
(degree_smul_le a f).trans hf | [
"Polynomial.degree_smul_le",
"LE.le.trans"
] | R : Type u_1
inst : Semiring R
S : Type u_2
inst_1 : SMulZeroClass S R
n : ℕ
a : S
f : R[X]
hf : f.degree ≤ (↑n : WithBot ℕ)
⊢ (a • f).degree ≤ (↑n : WithBot ℕ) |
rfl | [
"Polynomial.coeff",
"rfl"
] | R : Type u_1
inst : Semiring R
S : Type u_2
inst_1 : SMulZeroClass S R
n : ℕ
a : S
f : R[X]
⊢ (a • f).coeff n = a • f.coeff n |
subst coeff_eq deg_eq_deg coeff_eq_deg | [] | R : Type u_1
inst : Semiring R
deg m o : ℕ
c : R
p : R[X]
h_natDeg_le : p.natDegree ≤ m
coeff_eq : p.coeff o = c
coeff_ne_zero : c ≠ 0
deg_eq_deg : m = deg
coeff_eq_deg : o = deg
subst : ∀ (coeff_ne_zero : p.coeff o ≠ 0) (h_natDeg_le : p.natDegree ≤ o), p.natDegree = o
⊢ p.natDegree = deg |
natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_› | [
"Polynomial.natDegree_eq_of_le_of_coeff_ne_zero"
] | R : Type u_1
inst : Semiring R
o : ℕ
p : R[X]
coeff_ne_zero : p.coeff o ≠ 0
h_natDeg_le : p.natDegree ≤ o
⊢ p.natDegree = o |
exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_› | [
"Polynomial.natDegree_eq_of_le_of_coeff_ne_zero"
] | R : Type u_1
inst : Semiring R
o : ℕ
p : R[X]
coeff_ne_zero : p.coeff o ≠ 0
h_natDeg_le : p.natDegree ≤ o
⊢ p.natDegree = o |
subst coeff_eq coeff_eq_deg deg_eq_deg | [] | R : Type u_1
inst : Semiring R
deg m o : WithBot ℕ
c : R
p : R[X]
h_deg_le : p.degree ≤ m
coeff_eq : p.coeff (WithBot.unbotD 0 deg) = c
coeff_ne_zero : c ≠ 0
deg_eq_deg : m = deg
coeff_eq_deg : o = deg
subst : ∀ (coeff_ne_zero : p.coeff (WithBot.unbotD 0 m) ≠ 0), p.degree = m
⊢ p.degree = deg |
rcases eq_or_ne m ⊥ with rfl | hh | [
"Bot.bot",
"eq_or_ne",
"rfl"
] | R : Type u_1
inst : Semiring R
m : WithBot ℕ
p : R[X]
h_deg_le : p.degree ≤ m
coeff_ne_zero : p.coeff (WithBot.unbotD 0 m) ≠ 0
inl : ∀ (h_deg_le : p.degree ≤ ⊥) (coeff_ne_zero : p.coeff (WithBot.unbotD 0 ⊥) ≠ 0), p.degree = ⊥
inr : ∀ (hh : m ≠ ⊥), p.degree = m
⊢ p.degree = m |
bot_unique h_deg_le | [
"bot_unique"
] | R : Type u_1
inst : Semiring R
p : R[X]
h_deg_le : p.degree ≤ ⊥
coeff_ne_zero : p.coeff (WithBot.unbotD 0 ⊥) ≠ 0
⊢ p.degree = ⊥ |
exact bot_unique h_deg_le | [
"bot_unique"
] | R : Type u_1
inst : Semiring R
p : R[X]
h_deg_le : p.degree ≤ ⊥
coeff_ne_zero : p.coeff (WithBot.unbotD 0 ⊥) ≠ 0
⊢ p.degree = ⊥ |
obtain ⟨m, rfl⟩ := WithBot.ne_bot_iff_exists.mp hh | [
"WithBot.ne_bot_iff_exists",
"Iff.mp",
"rfl"
] | R : Type u_1
inst : Semiring R
m : WithBot ℕ
p : R[X]
h_deg_le : p.degree ≤ m
coeff_ne_zero : p.coeff (WithBot.unbotD 0 m) ≠ 0
hh : m ≠ ⊥
inr :
∀ (m : ℕ) (h_deg_le : p.degree ≤ (↑m : WithBot ℕ)) (coeff_ne_zero : p.coeff (WithBot.unbotD 0 (↑m : WithBot ℕ)) ≠ 0)
(hh : (↑m : WithBot ℕ) ≠ ⊥), p.degree = (↑m : WithBot... |
degree_eq_of_le_of_coeff_ne_zero ‹_› ‹_› | [
"Polynomial.degree_eq_of_le_of_coeff_ne_zero"
] | R : Type u_1
inst : Semiring R
p : R[X]
m : ℕ
h_deg_le : p.degree ≤ (↑m : WithBot ℕ)
coeff_ne_zero : p.coeff (WithBot.unbotD 0 (↑m : WithBot ℕ)) ≠ 0
hh : (↑m : WithBot ℕ) ≠ ⊥
⊢ p.degree = (↑m : WithBot ℕ) |
exact degree_eq_of_le_of_coeff_ne_zero ‹_› ‹_› | [
"Polynomial.degree_eq_of_le_of_coeff_ne_zero"
] | R : Type u_1
inst : Semiring R
p : R[X]
m : ℕ
h_deg_le : p.degree ≤ (↑m : WithBot ℕ)
coeff_ne_zero : p.coeff (WithBot.unbotD 0 (↑m : WithBot ℕ)) ≠ 0
hh : (↑m : WithBot ℕ) ≠ ⊥
⊢ p.degree = (↑m : WithBot ℕ) |
natDeg_eq_coeff ▸ h | [
"Eq.rec"
] | R : Type u_1
inst : Semiring R
m n : ℕ
f : R[X]
r : R
h : f.coeff m = r
natDeg_eq_coeff : m = n
⊢ f.coeff n = r |
natDeg_eq_coeff ▸ rs ▸ h | [
"Eq.rec"
] | R : Type u_1
inst : Semiring R
m n : ℕ
f : R[X]
r : R
h : f.coeff m = r
natDeg_eq_coeff : m = n
s : R
rs : r = s
⊢ f.coeff n = s |
(natDegree_intCast _).le | [
"Eq.le",
"Polynomial.natDegree_intCast"
] | R : Type u_1
inst : Ring R
n : ℤ
⊢ natDegree.{u_1} (R := R) (↑n : R[X]) ≤ 0 |
subst hf hg | [] | R : Type u_1
inst : Ring R
n : ℕ
a b : R
f g : R[X]
hf : f.coeff n = a
hg : g.coeff n = b
subst : (f - g).coeff n = f.coeff n - g.coeff n
⊢ (f - g).coeff n = a - b |
apply coeff_sub | [
"Polynomial.coeff_sub"
] | R : Type u_1
inst : Ring R
n : ℕ
f g : R[X]
⊢ (f - g).coeff n = f.coeff n - g.coeff n |
simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero] | [
"Int.cast_zero",
"Polynomial.C_eq_intCast",
"Polynomial.coeff_C",
"Int.cast_ite"
] | R : Type u_1
inst : Ring R
n : ℕ
a : ℤ
⊢ Eq.{u_1 + 1} (α := R) (coeff (↑a : R[X]) n) (↑(if n = 0 then a else 0) : R) |
rw [mul_assoc, ← zpow_add] | [
"zpow_add",
"mul_assoc"
] | G : Type u_1
inst : Group G
a b : G
n m : ℤ
rw : a * (b ^ n * b ^ m) = a * b ^ (n + m)
rw₁ rw₂ : a * b ^ (n + m) = a * b ^ (n + m)
⊢ a * b ^ n * b ^ m = a * b ^ (n + m) |
rw [mul_assoc, mul_self_zpow] | [
"mul_assoc",
"mul_self_zpow"
] | G : Type u_1
inst : Group G
a b : G
m : ℤ
rw : a * (b * b ^ m) = a * b ^ (m + 1)
rw₁ rw₂ : a * b ^ (m + 1) = a * b ^ (m + 1)
⊢ a * b * b ^ m = a * b ^ (m + 1) |
rw [mul_assoc, mul_zpow_self] | [
"mul_assoc",
"mul_zpow_self"
] | G : Type u_1
inst : Group G
a b : G
n : ℤ
rw : a * (b ^ n * b) = a * b ^ (n + 1)
rw₁ rw₂ : a * b ^ (n + 1) = a * b ^ (n + 1)
⊢ a * b ^ n * b = a * b ^ (n + 1) |
eq ▸ not_lt.1 h | [
"Eq.rec",
"not_lt",
"Iff.mp"
] | α : Type u_1
a b a' : α
inst : LinearOrder α
h : ¬a < b
eq : a = a'
⊢ b ≤ a' |
eq ▸ not_lt.1 h | [
"Eq.rec",
"not_lt",
"Iff.mp"
] | α : Type u_1
a b b' : α
inst : LinearOrder α
h : ¬a < b
eq : b = b'
⊢ b' ≤ a |
eq ▸ h | [
"Eq.rec"
] | α : Type u_1
a b a' : α
inst : LE α
h : ¬a ≤ b
eq : a = a'
⊢ ¬a' ≤ b |
eq ▸ h | [
"Eq.rec"
] | α : Type u_1
a b b' : α
inst : LE α
h : ¬a ≤ b
eq : b = b'
⊢ ¬a ≤ b' |
eq ▸ not_le.2 h | [
"not_le",
"Iff.mpr",
"Eq.rec"
] | α : Type u_1
a b a' : α
inst : LinearOrder α
h : a < b
eq : a = a'
⊢ ¬b ≤ a' |
eq ▸ not_le.2 h | [
"not_le",
"Iff.mpr",
"Eq.rec"
] | α : Type u_1
a b b' : α
inst : LinearOrder α
h : a < b
eq : b = b'
⊢ ¬b' ≤ a |
eq ▸ h | [
"Eq.rec"
] | α : Type u_1
a b a' : α
inst : LE α
h : a ≤ b
eq : a = a'
⊢ a' ≤ b |
eq ▸ h | [
"Eq.rec"
] | α : Type u_1
a b b' : α
inst : LE α
h : a ≤ b
eq : b = b'
⊢ a ≤ b' |
le_trans (le_of_not_ge h1) h2 | [
"le_of_not_ge",
"le_trans"
] | α : Type u_1
hi n lo : α
inst : LinearOrder α
h1 : ¬hi ≤ n
h2 : hi ≤ lo
⊢ n ≤ lo |
Int.add_one_le_iff.2 (Int.not_le.1 h) | [
"Iff.mpr",
"Int.add_one_le_iff",
"Iff.mp",
"Int.not_le"
] | a b : ℤ
h : ¬b ≤ a
⊢ a + 1 ≤ b |
Int.le_sub_one_iff.2 (Int.not_le.1 h) | [
"Iff.mpr",
"Int.le_sub_one_iff",
"Iff.mp",
"Int.not_le"
] | a b : ℤ
h : ¬b ≤ a
⊢ a ≤ b - 1 |
cases x | [] | x : ℕ∞
top : ¬LT.lt (α := ℕ∞) ⊤ ⊤
coe : ∀ (a : ℕ), ¬LT.lt (α := ℕ∞) ⊤ (↑a : ℕ∞)
⊢ ¬⊤ < x |
simp | [] | ⊢ ¬LT.lt (α := ℕ∞) ⊤ ⊤ |
simp | [] | a : ℕ
⊢ ¬LT.lt (α := ℕ∞) ⊤ (↑a : ℕ∞) |
rfl | [
"WithTop.map₂",
"WithTop.some",
"rfl",
"Option.map₂"
] | m n : ℕ
⊢ HAdd.hAdd (α := ℕ∞) (↑m : ℕ∞) (↑n : ℕ∞) = (↑(m + n) : ℕ∞) |
rfl | [
"WithTop.some",
"WithTop.sub",
"rfl"
] | m n : ℕ
⊢ HSub.hSub (α := ℕ∞) (↑m : ℕ∞) (↑n : ℕ∞) = (↑(m - n) : ℕ∞) |
rfl | [
"WithTop.some",
"rfl"
] | m n : ℕ
⊢ HMul.hMul (α := ℕ∞) (↑m : ℕ∞) (↑n : ℕ∞) = (↑(m * n) : ℕ∞) |
rfl | [
"rfl"
] | n : ℕ
inst : n.AtLeastTwo
⊢ Eq (α := ℕ∞) (OfNat.ofNat n) (↑(OfNat.ofNat n) : ℕ∞) |
rfl | [
"WithTop.some",
"rfl"
] | ⊢ Eq (α := ℕ∞) 0 (↑0 : ℕ∞) |
rfl | [
"WithTop.some",
"rfl"
] | ⊢ Eq (α := ℕ∞) 1 (↑1 : ℕ∞) |
rfl | [] | R : Type u_2
M : Type u_3
inst : AddMonoid M
inst_1 : SMul R M
p : R × M
l : NF R M
⊢ (p ::ᵣ l).eval = p.1 • p.2 + l.eval |
simp [eval] | [
"Mathlib.Tactic.Module.NF.eval"
] | M : Type u_3
inst : AddMonoid M
x : M
⊢ x = eval (R := ℕ) [(1, x)] |
rfl | [
"List.sum",
"Mathlib.Tactic.Module.NF.eval",
"List.foldr",
"rfl"
] | M : Type u_3
inst : AddMonoid M
⊢ Eq.{u_3 + 1} (α := M) 0 (eval (R := ℕ) []) |
simp only [eval_cons, ← h, add_assoc] | [
"add_assoc",
"Mathlib.Tactic.Module.NF.eval_cons"
] | R : Type u_2
M : Type u_3
inst : AddMonoid M
inst_1 : SMul R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : l₁.eval + (a₂ ::ᵣ l₂).eval = l.eval
⊢ (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval |
simp only [← h, eval_cons, add_smul, add_assoc] | [
"add_assoc",
"add_smul",
"Mathlib.Tactic.Module.NF.eval_cons"
] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval + l₂.eval = l.eval
simp : r₁ • x + (l₁.eval + (r₂ • x + l₂.eval)) = r₁ • x + (r₂ • x + (l₁.eval + l₂.eval))
⊢ ((r₁, x) ::ᵣ l₁).eval + ((r₂, x) ::ᵣ l₂).eval = ((r₁ + r₂, x) ::ᵣ l).eval |
congr! 1 | [] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval + l₂.eval = l.eval
h₁ : l₁.eval + (r₂ • x + l₂.eval) = r₂ • x + (l₁.eval + l₂.eval)
⊢ r₁ • x + (l₁.eval + (r₂ • x + l₂.eval)) = r₁ • x + (r₂ • x + (l₁.eval + l₂.eval)) |
simp only [← add_assoc] | [
"add_assoc"
] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval + l₂.eval = l.eval
h₁ : l₁.eval + r₂ • x + l₂.eval = r₂ • x + l₁.eval + l₂.eval
⊢ l₁.eval + (r₂ • x + l₂.eval) = r₂ • x + (l₁.eval + l₂.eval) |
congr! 1 | [] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval + l₂.eval = l.eval
h₁ : l₁.eval + r₂ • x = r₂ • x + l₁.eval
⊢ l₁.eval + r₂ • x + l₂.eval = r₂ • x + l₁.eval + l₂.eval |
rw [add_comm] | [
"add_comm"
] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval + l₂.eval = l.eval
h₁ h₂ : r₂ • x + l₁.eval = r₂ • x + l₁.eval
⊢ l₁.eval + r₂ • x = r₂ • x + l₁.eval |
simp only [eval_cons, ← h] | [
"Mathlib.Tactic.Module.NF.eval_cons"
] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : (a₁ ::ᵣ l₁).eval + l₂.eval = l.eval
simp : a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) = a₂.1 • a₂.2 + (a₁.1 • a₁.2 + l₁.eval + l₂.eval)
⊢ (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₂ ::ᵣ l).eval |
nth_rw 4 [add_comm] | [
"add_comm"
] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : (a₁ ::ᵣ l₁).eval + l₂.eval = l.eval
nth_rw : a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) = a₁.1 • a₁.2 + l₁.eval + l₂.eval + a₂.1 • a₂.2
⊢ a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) = a₂.... |
simp only [add_assoc] | [
"add_assoc"
] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : (a₁ ::ᵣ l₁).eval + l₂.eval = l.eval
simp : a₁.1 • a₁.2 + (l₁.eval + (a₂.1 • a₂.2 + l₂.eval)) = a₁.1 • a₁.2 + (l₁.eval + (l₂.eval + a₂.1 • a₂.2))
⊢ a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) =... |
congr! 2 | [] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : (a₁ ::ᵣ l₁).eval + l₂.eval = l.eval
h₁ : a₂.1 • a₂.2 + l₂.eval = l₂.eval + a₂.1 • a₂.2
⊢ a₁.1 • a₁.2 + (l₁.eval + (a₂.1 • a₂.2 + l₂.eval)) = a₁.1 • a₁.2 + (l₁.eval + (l₂.eval + a₂.1 • a₂.2)) |
rw [add_comm] | [
"add_comm"
] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : (a₁ ::ᵣ l₁).eval + l₂.eval = l.eval
h₁ h₂ : l₂.eval + a₂.1 • a₂.2 = l₂.eval + a₂.1 • a₂.2
⊢ a₂.1 • a₂.2 + l₂.eval = l₂.eval + a₂.1 • a₂.2 |
rw [hx₁, hx₂, ← h₁, ← h₂, h] | [] | R : Type u_2
M : Type u_3
R₁ : Type u_4
R₂ : Type u_5
inst : AddCommMonoid M
inst_1 : Semiring R
inst_2 : Module R M
inst_3 : Semiring R₁
inst_4 : Module R₁ M
inst_5 : Semiring R₂
inst_6 : Module R₂ M
l₁ l₂ l : NF R M
l₁' : NF R₁ M
l₂' : NF R₂ M
x₁ x₂ : M
hx₁ : x₁ = l₁'.eval
hx₂ : x₂ = l₂'.eval
h₁ : l₁.eval = l₁'.eval
... |
simp only [eval_cons, ← h, sub_eq_add_neg, add_assoc] | [
"add_assoc",
"sub_eq_add_neg",
"Mathlib.Tactic.Module.NF.eval_cons"
] | R : Type u_2
M : Type u_3
inst : SMul R M
inst_1 : AddGroup M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : l₁.eval - (a₂ ::ᵣ l₂).eval = l.eval
⊢ (a₁ ::ᵣ l₁).eval - (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval |
simp only [← h, eval_cons, sub_eq_add_neg, neg_add, add_smul, neg_smul, add_assoc] | [
"neg_smul",
"add_assoc",
"sub_eq_add_neg",
"neg_add",
"add_smul",
"Mathlib.Tactic.Module.NF.eval_cons"
] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval - l₂.eval = l.eval
simp : r₁ • x + (l₁.eval + (-(r₂ • x) + -l₂.eval)) = r₁ • x + (-(r₂ • x) + (l₁.eval + -l₂.eval))
⊢ ((r₁, x) ::ᵣ l₁).eval - ((r₂, x) ::ᵣ l₂).eval = ((r₁ - r₂, x) ::ᵣ l).eval |
congr! 1 | [] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval - l₂.eval = l.eval
h₁ : l₁.eval + (-(r₂ • x) + -l₂.eval) = -(r₂ • x) + (l₁.eval + -l₂.eval)
⊢ r₁ • x + (l₁.eval + (-(r₂ • x) + -l₂.eval)) = r₁ • x + (-(r₂ • x) + (l₁.eval + -l₂.eval)) |
simp only [← add_assoc] | [
"add_assoc"
] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval - l₂.eval = l.eval
h₁ : l₁.eval + -(r₂ • x) + -l₂.eval = -(r₂ • x) + l₁.eval + -l₂.eval
⊢ l₁.eval + (-(r₂ • x) + -l₂.eval) = -(r₂ • x) + (l₁.eval + -l₂.eval) |
congr! 1 | [] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval - l₂.eval = l.eval
h₁ : l₁.eval + -(r₂ • x) = -(r₂ • x) + l₁.eval
⊢ l₁.eval + -(r₂ • x) + -l₂.eval = -(r₂ • x) + l₁.eval + -l₂.eval |
rw [add_comm] | [
"add_comm"
] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
r₁ r₂ : R
x : M
l₁ l₂ l : NF R M
h : l₁.eval - l₂.eval = l.eval
h₁ h₂ : -(r₂ • x) + l₁.eval = -(r₂ • x) + l₁.eval
⊢ l₁.eval + -(r₂ • x) = -(r₂ • x) + l₁.eval |
simp only [eval_cons, neg_smul, neg_add, sub_eq_add_neg, ← h, ← add_assoc] | [
"neg_smul",
"add_assoc",
"sub_eq_add_neg",
"neg_add",
"Mathlib.Tactic.Module.NF.eval_cons"
] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : (a₁ ::ᵣ l₁).eval - l₂.eval = l.eval
simp : a₁.1 • a₁.2 + l₁.eval + -(a₂.1 • a₂.2) + -l₂.eval = -(a₂.1 • a₂.2) + a₁.1 • a₁.2 + l₁.eval + -l₂.eval
⊢ (a₁ ::ᵣ l₁).eval - (a₂ ::ᵣ l₂).eval = ((-a₂.1, a₂.2) :... |
congr! 1 | [] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : (a₁ ::ᵣ l₁).eval - l₂.eval = l.eval
h₁ : a₁.1 • a₁.2 + l₁.eval + -(a₂.1 • a₂.2) = -(a₂.1 • a₂.2) + a₁.1 • a₁.2 + l₁.eval
⊢ a₁.1 • a₁.2 + l₁.eval + -(a₂.1 • a₂.2) + -l₂.eval = -(a₂.1 • a₂.2) + a₁.1 • a₁... |
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