tactic stringlengths 1 8.81k | premises listlengths 0 77 | goal stringlengths 6 333k |
|---|---|---|
tauto | [] | p q r : Prop
h : r → p → q
inl : ∀ (h : Not r), r → p → And r q
inr inr₁ : ∀ (h : p → q), r → p → And r q
inr₂ : ∀ (h : p → q), r → p → r
inr₃ : ∀ (h : p → q), r → p → q
inr₄ : r → p → ∀ (h : Not p), q
inr₅ inr₆ inr₇ : r → p → ∀ (h : q), q
⊢ And r p → And 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 : Filter.Eventually (fun (x : α) ↦ p x) f
⊢ Filter.Eventually (fun (x : α) ↦ q x) f |
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 : Filter.Frequently (fun (x : α) ↦ p x) f
⊢ Filter.Frequently (fun (x : α) ↦ q x) f |
congr! 2 | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq congr! : ∀ (x : α), Iff (p x) (q x)
⊢ Iff (Filter.Eventually (fun (x : α) ↦ p x) f) (Filter.Eventually (fun (x : α) ↦ q x) f) |
hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), Iff (p x) (q x)
x : α
⊢ Iff (p x) (q x) |
exact hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), Iff (p x) (q x)
x : α
⊢ Iff (p x) (q x) |
congr! 2 | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq congr! : ∀ (x : α), Iff (p x) (q x)
⊢ Iff (Filter.Frequently (fun (x : α) ↦ p x) f) (Filter.Frequently (fun (x : α) ↦ q x) f) |
hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), Iff (p x) (q x)
x : α
⊢ Iff (p x) (q x) |
exact hq _ | [] | α : Type u_1
p q : α → Prop
f : Filter α
hq : ∀ (x : α), Iff (p x) (q x)
x : α
⊢ Iff (p x) (q x) |
(natDegree_C a).le | [
"Polynomial.natDegree_C",
"Eq.le"
] | R : Type u_1
inst : Semiring R
a : R
⊢ LE.le (Polynomial.natDegree.{u_1} (R := R) ((Polynomial.C : (a : R) → Polynomial R) a)) 0 |
(natDegree_natCast _).le | [
"Eq.le",
"Polynomial.natDegree_natCast"
] | R : Type u_1
inst : Semiring R
n : Nat
⊢ LE.le (Polynomial.natDegree.{u_1} (R := R) (↑n : Polynomial R)) 0 |
natDegree_zero.le | [
"Eq.le",
"Polynomial.natDegree_zero"
] | R : Type u_1
inst : Semiring R
⊢ LE.le (Polynomial.natDegree.{u_1} (R := R) 0) 0 |
natDegree_one.le | [
"Eq.le",
"Polynomial.natDegree_one"
] | R : Type u_1
inst : Semiring R
⊢ LE.le (Polynomial.natDegree.{u_1} (R := R) 1) 0 |
subst ‹_› ‹_› | [] | R : Type u_1
inst : Semiring R
n : Nat
a b : R
f g : Polynomial R
h_add_left : Eq (f.coeff n) a
h_add_right : Eq (g.coeff n) b
subst : Eq ((HAdd.hAdd f g).coeff n) (HAdd.hAdd (f.coeff n) (g.coeff n))
⊢ Eq ((HAdd.hAdd f g).coeff n) (HAdd.hAdd a b) |
apply coeff_add | [
"Polynomial.coeff_add"
] | R : Type u_1
inst : Semiring R
n : Nat
f g : Polynomial R
⊢ Eq ((HAdd.hAdd f g).coeff n) (HAdd.hAdd (f.coeff n) (g.coeff n)) |
split_ifs with h | [] | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
pos : ∀ (h : Eq d (HAdd.hAdd df dg)), Eq ((HMul.hMul f g).coeff d) (HMul.hMul a ... |
subst h_mul_left h_mul_right h | [] | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Eq d (HAdd.hAdd df dg)
pos :
∀ (ddf : LE.le (HAdd.hAdd df dg) (HAdd.hAdd 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 : Nat
f g : Polynomial R
h_mul_left : LE.le f.natDegree df
h_mul_right : LE.le g.natDegree dg
ddf : LE.le (HAdd.hAdd df dg) (HAdd.hAdd df dg)
⊢ Eq ((HMul.hMul f g).coeff (HAdd.hAdd df dg)) (HMul.hMul (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 : Nat
f g : Polynomial R
h_mul_left : LE.le f.natDegree df
h_mul_right : LE.le g.natDegree dg
ddf : LE.le (HAdd.hAdd df dg) (HAdd.hAdd df dg)
⊢ Eq ((HMul.hMul f g).coeff (HAdd.hAdd df dg)) (HMul.hMul (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 : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
neg : LT.lt (HMul.hMul f g).natDegree d
⊢ Eq ((... |
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 : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
apply : LE.le (HMul.hMul f g).natDegree (HAdd.h... |
natDegree_mul_le_of_le ‹_› ‹_› | [
"Polynomial.natDegree_mul_le_of_le"
] | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
⊢ LE.le (HMul.hMul f g).natDegree (HAdd.hAdd df... |
exact natDegree_mul_le_of_le ‹_› ‹_› | [
"Polynomial.natDegree_mul_le_of_le"
] | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
⊢ LE.le (HMul.hMul f g).natDegree (HAdd.hAdd df... |
ne_comm.mp h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
⊢ Ne (HAdd.hAdd df dg) d |
exact ne_comm.mp h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
⊢ Ne (HAdd.hAdd df dg) d |
split_ifs with h | [] | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
pos : ∀ (h : Eq o (HMul.hMul m n)), Eq ((HPow.hPow p m).coeff o) (HPow.hPow a m)
neg : ∀ (h : Not (Eq o (HMul.hMul m n))), Eq ((HPow.hPow p m).coeff o) 0
⊢ Eq ((HPow... |
subst h h_pow_bas | [] | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Eq o (HMul.hMul m n)
pos :
∀ (h_exp : LE.le (HMul.hMul m n) (HMul.hMul m n)),
Eq ((HPow.hPow p m).coeff (HMul.hMul m n)) (HPow.hPow (p.coeff n) m)
⊢ Eq ((H... |
coeff_pow_of_natDegree_le ‹_› | [
"Polynomial.coeff_pow_of_natDegree_le"
] | R : Type u_1
inst : Semiring R
m n : Nat
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) (HMul.hMul m n)
⊢ Eq ((HPow.hPow p m).coeff (HMul.hMul m n)) (HPow.hPow (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 : Nat
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) (HMul.hMul m n)
⊢ Eq ((HPow.hPow p m).coeff (HMul.hMul m n)) (HPow.hPow (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 : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
neg : LT.lt (HPow.hPow p m).natDegree o
⊢ Eq ((HPow.hPow 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 : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
apply : LE.le (HPow.hPow p m).natDegree (HMul.hMul m n)
apply₁ : Ne (HMul.hMul m n) o
⊢ LT.lt (HPow.hPow 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 : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
⊢ LE.le (HPow.hPow p m).natDegree (HMul.hMul 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 : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
⊢ LE.le (HPow.hPow p m).natDegree (HMul.hMul m n) |
Iff.mp ne_comm h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
⊢ Ne (HMul.hMul m n) o |
exact Iff.mp ne_comm h | [
"Iff.mp",
"ne_comm"
] | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
⊢ Ne (HMul.hMul 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 : Nat
a : S
f : Polynomial R
hf : LE.le f.natDegree n
⊢ LE.le (HSMul.hSMul 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 : Nat
a : S
f : Polynomial R
hf : LE.le f.degree (↑n : WithBot Nat)
⊢ LE.le (HSMul.hSMul a f).degree (↑n : WithBot Nat) |
rfl | [
"Polynomial.coeff.eq_unfold",
"rfl"
] | R : Type u_1
inst : Semiring R
S : Type u_2
inst_1 : SMulZeroClass S R
n : Nat
a : S
f : Polynomial R
⊢ Eq ((HSMul.hSMul a f).coeff n) (HSMul.hSMul a (f.coeff n)) |
subst coeff_eq deg_eq_deg coeff_eq_deg | [] | R : Type u_1
inst : Semiring R
deg m o : Nat
c : R
p : Polynomial R
h_natDeg_le : LE.le p.natDegree m
coeff_eq : Eq (p.coeff o) c
coeff_ne_zero : Ne c 0
deg_eq_deg : Eq m deg
coeff_eq_deg : Eq o deg
subst : ∀ (coeff_ne_zero : Ne (p.coeff o) 0) (h_natDeg_le : LE.le p.natDegree o), Eq p.natDegree o
⊢ Eq 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 : Nat
p : Polynomial R
coeff_ne_zero : Ne (p.coeff o) 0
h_natDeg_le : LE.le p.natDegree o
⊢ Eq 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 : Nat
p : Polynomial R
coeff_ne_zero : Ne (p.coeff o) 0
h_natDeg_le : LE.le p.natDegree o
⊢ Eq p.natDegree o |
subst coeff_eq coeff_eq_deg deg_eq_deg | [] | R : Type u_1
inst : Semiring R
deg m o : WithBot Nat
c : R
p : Polynomial R
h_deg_le : LE.le p.degree m
coeff_eq : Eq (p.coeff (WithBot.unbotD 0 deg)) c
coeff_ne_zero : Ne c 0
deg_eq_deg : Eq m deg
coeff_eq_deg : Eq o deg
subst : ∀ (coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0), Eq p.degree m
⊢ Eq p.degree deg |
rcases eq_or_ne m ⊥ with rfl | hh | [
"eq_or_ne",
"rfl"
] | R : Type u_1
inst : Semiring R
m : WithBot Nat
p : Polynomial R
h_deg_le : LE.le p.degree m
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0
inl :
∀ (h_deg_le : LE.le p.degree Bot.bot) (coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 Bot.bot)) 0), Eq p.degree Bot.bot
inr : ∀ (hh : Ne m Bot.bot), Eq p.degree m
⊢ Eq p... |
bot_unique h_deg_le | [
"bot_unique"
] | R : Type u_1
inst : Semiring R
p : Polynomial R
h_deg_le : LE.le p.degree Bot.bot
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 Bot.bot)) 0
⊢ Eq p.degree Bot.bot |
exact bot_unique h_deg_le | [
"bot_unique"
] | R : Type u_1
inst : Semiring R
p : Polynomial R
h_deg_le : LE.le p.degree Bot.bot
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 Bot.bot)) 0
⊢ Eq p.degree Bot.bot |
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 Nat
p : Polynomial R
h_deg_le : LE.le p.degree m
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0
hh : Ne m Bot.bot
inr :
∀ (m : Nat) (h_deg_le : LE.le p.degree (↑m : WithBot Nat))
(coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 (↑m : WithBot Nat))) 0) (hh : Ne (↑m : W... |
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 : Polynomial R
m : Nat
h_deg_le : LE.le p.degree (↑m : WithBot Nat)
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 (↑m : WithBot Nat))) 0
hh : Ne (↑m : WithBot Nat) Bot.bot
⊢ Eq p.degree (↑m : WithBot Nat) |
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 : Polynomial R
m : Nat
h_deg_le : LE.le p.degree (↑m : WithBot Nat)
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 (↑m : WithBot Nat))) 0
hh : Ne (↑m : WithBot Nat) Bot.bot
⊢ Eq p.degree (↑m : WithBot Nat) |
natDeg_eq_coeff ▸ h | [] | R : Type u_1
inst : Semiring R
m n : Nat
f : Polynomial R
r : R
h : Eq (f.coeff m) r
natDeg_eq_coeff : Eq m n
⊢ Eq (f.coeff n) r |
natDeg_eq_coeff ▸ rs ▸ h | [] | R : Type u_1
inst : Semiring R
m n : Nat
f : Polynomial R
r : R
h : Eq (f.coeff m) r
natDeg_eq_coeff : Eq m n
s : R
rs : Eq r s
⊢ Eq (f.coeff n) s |
(natDegree_intCast _).le | [
"Eq.le",
"Polynomial.natDegree_intCast"
] | R : Type u_1
inst : Ring R
n : Int
⊢ LE.le (Polynomial.natDegree.{u_1} (R := R) (↑n : Polynomial R)) 0 |
subst hf hg | [] | R : Type u_1
inst : Ring R
n : Nat
a b : R
f g : Polynomial R
hf : Eq (f.coeff n) a
hg : Eq (g.coeff n) b
subst : Eq ((HSub.hSub f g).coeff n) (HSub.hSub (f.coeff n) (g.coeff n))
⊢ Eq ((HSub.hSub f g).coeff n) (HSub.hSub a b) |
apply coeff_sub | [
"Polynomial.coeff_sub"
] | R : Type u_1
inst : Ring R
n : Nat
f g : Polynomial R
⊢ Eq ((HSub.hSub f g).coeff n) (HSub.hSub (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 : Nat
a : Int
⊢ Eq.{u_1 + 1} (α := R) (Polynomial.coeff (↑a : Polynomial R) n) (↑(ite (Eq n 0) a 0) : R) |
rw [mul_assoc, ← zpow_add] | [
"zpow_add",
"mul_assoc"
] | G : Type u_1
inst : Group G
a b : G
n m : Int
rw : Eq (HMul.hMul a (HMul.hMul (HPow.hPow b n) (HPow.hPow b m))) (HMul.hMul a (HPow.hPow b (HAdd.hAdd n m)))
rw₁ rw₂ : Eq (HMul.hMul a (HPow.hPow b (HAdd.hAdd n m))) (HMul.hMul a (HPow.hPow b (HAdd.hAdd n m)))
⊢ Eq (HMul.hMul (HMul.hMul a (HPow.hPow b n)) (HPow.hPow b m)) ... |
rw [mul_assoc, mul_self_zpow] | [
"mul_assoc",
"mul_self_zpow"
] | G : Type u_1
inst : Group G
a b : G
m : Int
rw : Eq (HMul.hMul a (HMul.hMul b (HPow.hPow b m))) (HMul.hMul a (HPow.hPow b (HAdd.hAdd m 1)))
rw₁ rw₂ : Eq (HMul.hMul a (HPow.hPow b (HAdd.hAdd m 1))) (HMul.hMul a (HPow.hPow b (HAdd.hAdd m 1)))
⊢ Eq (HMul.hMul (HMul.hMul a b) (HPow.hPow b m)) (HMul.hMul a (HPow.hPow b (HAd... |
rw [mul_assoc, mul_zpow_self] | [
"mul_assoc",
"mul_zpow_self"
] | G : Type u_1
inst : Group G
a b : G
n : Int
rw : Eq (HMul.hMul a (HMul.hMul (HPow.hPow b n) b)) (HMul.hMul a (HPow.hPow b (HAdd.hAdd n 1)))
rw₁ rw₂ : Eq (HMul.hMul a (HPow.hPow b (HAdd.hAdd n 1))) (HMul.hMul a (HPow.hPow b (HAdd.hAdd n 1)))
⊢ Eq (HMul.hMul (HMul.hMul a (HPow.hPow b n)) b) (HMul.hMul a (HPow.hPow b (HAd... |
eq ▸ not_lt.1 h | [
"not_lt",
"Iff.mp"
] | α : Type u_1
a b a' : α
inst : LinearOrder α
h : Not (LT.lt a b)
eq : Eq a a'
⊢ LE.le b a' |
eq ▸ not_lt.1 h | [
"not_lt",
"Iff.mp"
] | α : Type u_1
a b b' : α
inst : LinearOrder α
h : Not (LT.lt a b)
eq : Eq b b'
⊢ LE.le b' a |
eq ▸ h | [] | α : Type u_1
a b a' : α
inst : LE α
h : Not (LE.le a b)
eq : Eq a a'
⊢ Not (LE.le a' b) |
eq ▸ h | [] | α : Type u_1
a b b' : α
inst : LE α
h : Not (LE.le a b)
eq : Eq b b'
⊢ Not (LE.le a b') |
eq ▸ not_le.2 h | [
"not_le",
"Iff.mpr"
] | α : Type u_1
a b a' : α
inst : LinearOrder α
h : LT.lt a b
eq : Eq a a'
⊢ Not (LE.le b a') |
eq ▸ not_le.2 h | [
"not_le",
"Iff.mpr"
] | α : Type u_1
a b b' : α
inst : LinearOrder α
h : LT.lt a b
eq : Eq b b'
⊢ Not (LE.le b' a) |
eq ▸ h | [] | α : Type u_1
a b a' : α
inst : LE α
h : LE.le a b
eq : Eq a a'
⊢ LE.le a' b |
eq ▸ h | [] | α : Type u_1
a b b' : α
inst : LE α
h : LE.le a b
eq : Eq b b'
⊢ LE.le 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 : Not (LE.le hi n)
h2 : LE.le hi lo
⊢ LE.le 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 : Int
h : Not (LE.le b a)
⊢ LE.le (HAdd.hAdd 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 : Int
h : Not (LE.le b a)
⊢ LE.le a (HSub.hSub b 1) |
cases x | [] | x : ENat
top : Not (LT.lt (α := ENat) Top.top Top.top)
coe : ∀ (a : Nat), Not (LT.lt (α := ENat) Top.top (↑a : ENat))
⊢ Not (LT.lt Top.top x) |
simp | [] | ⊢ Not (LT.lt (α := ENat) Top.top Top.top) |
simp | [] | a : Nat
⊢ Not (LT.lt (α := ENat) Top.top (↑a : ENat)) |
rfl | [
"WithTop.some.eq_unfold",
"Option.map₂.eq_unfold",
"WithTop.map₂.eq_unfold",
"rfl"
] | m n : Nat
⊢ Eq (HAdd.hAdd (α := ENat) (↑m : ENat) (↑n : ENat)) (↑(HAdd.hAdd m n) : ENat) |
rfl | [
"WithTop.some.eq_unfold",
"WithTop.sub.eq_unfold",
"rfl"
] | m n : Nat
⊢ Eq (HSub.hSub (α := ENat) (↑m : ENat) (↑n : ENat)) (↑(HSub.hSub m n) : ENat) |
rfl | [
"WithTop.some.eq_unfold",
"rfl"
] | m n : Nat
⊢ Eq (HMul.hMul (α := ENat) (↑m : ENat) (↑n : ENat)) (↑(HMul.hMul m n) : ENat) |
rfl | [
"rfl"
] | n : Nat
inst : n.AtLeastTwo
⊢ Eq (α := ENat) (OfNat.ofNat n) (↑(OfNat.ofNat n) : ENat) |
rfl | [
"WithTop.some.eq_unfold",
"rfl"
] | ⊢ Eq (α := ENat) 0 (↑0 : ENat) |
rfl | [
"WithTop.some.eq_unfold",
"rfl"
] | ⊢ Eq (α := ENat) 1 (↑1 : ENat) |
rfl | [] | R : Type u_2
M : Type u_3
inst : AddMonoid M
inst_1 : SMul R M
p : Prod R M
l : Mathlib.Tactic.Module.NF R M
⊢ Eq (Mathlib.Tactic.Module.NF.cons p l).eval (HAdd.hAdd (HSMul.hSMul p.1 p.2) l.eval) |
simp [eval] | [] | M : Type u_3
inst : AddMonoid M
x : M
⊢ Eq x (Mathlib.Tactic.Module.NF.eval (R := Nat) (List.cons (Prod.mk 1 x) List.nil)) |
rfl | [
"List.foldr.eq_unfold",
"List.sum.eq_unfold",
"Mathlib.Tactic.Module.NF.eval.eq_unfold",
"rfl"
] | M : Type u_3
inst : AddMonoid M
⊢ Eq.{u_3 + 1} (α := M) 0 (Mathlib.Tactic.Module.NF.eval (R := Nat) List.nil) |
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 : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd l₁.eval (Mathlib.Tactic.Module.NF.cons a₂ l₂).eval) l.eval
⊢ Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval (Mathlib.Tactic.Module.NF.cons a₂ l₂).eval)
(Mathlib.Tactic.M... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd l₁.eval l₂.eval) l.eval
simp :
Eq (HAdd.hAdd (HSMul.hSMul r₁ x) (HAdd.hAdd l₁.eval (HAdd.hAdd (HSMul.hSMul r₂ x) l₂.eval)))
(HAdd.hAdd (HSMul.hSMul r₁ x... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd l₁.eval l₂.eval) l.eval
congr! :
Eq (HAdd.hAdd l₁.eval (HAdd.hAdd (HSMul.hSMul r₂ x) l₂.eval))
(HAdd.hAdd (HSMul.hSMul r₂ x) (HAdd.hAdd 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 : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd l₁.eval l₂.eval) l.eval
simp :
Eq (HAdd.hAdd (HAdd.hAdd l₁.eval (HSMul.hSMul r₂ x)) l₂.eval)
(HAdd.hAdd (HAdd.hAdd (HSMul.hSMul 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 : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd l₁.eval l₂.eval) l.eval
congr! : Eq (HAdd.hAdd l₁.eval (HSMul.hSMul r₂ x)) (HAdd.hAdd (HSMul.hSMul r₂ x) l₁.eval)
⊢ Eq (HAdd.hAdd (HAdd.hAdd l₁.eval (HSMul.hS... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd l₁.eval l₂.eval) l.eval
rw rw₁ : Eq (HAdd.hAdd (HSMul.hSMul r₂ x) l₁.eval) (HAdd.hAdd (HSMul.hSMul r₂ x) l₁.eval)
⊢ Eq (HAdd.hAdd l₁.eval (HSMul.hSMul r₂ x)) ... |
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 : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval l₂.eval) l.eval
simp :
Eq (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul a₁.1 a₁.2) l₁.eval) (HAdd.hAdd (HSMul.hSMul a₂.1 a₂... |
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 : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval l₂.eval) l.eval
nth_rw :
Eq (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul a₁.1 a₁.2) l₁.eval) (HAdd.hAdd (HSMul.hSMul a₂.1 ... |
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 : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval l₂.eval) l.eval
simp :
Eq (HAdd.hAdd (HSMul.hSMul a₁.1 a₁.2) (HAdd.hAdd l₁.eval (HAdd.hAdd (HSMul.hSMul a₂.1 a₂.... |
congr! 2 | [] | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
a₁ a₂ : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval l₂.eval) l.eval
congr! : Eq (HAdd.hAdd (HSMul.hSMul a₂.1 a₂.2) l₂.eval) (HAdd.hAdd l₂.eval (HSMul.hSMul 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 : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval l₂.eval) l.eval
rw rw₁ : Eq (HAdd.hAdd l₂.eval (HSMul.hSMul a₂.1 a₂.2)) (HAdd.hAdd l₂.eval (HSMul.hSMul 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 : Mathlib.Tactic.Module.NF R M
l₁' : Mathlib.Tactic.Module.NF R₁ M
l₂' : Mathlib.Tactic.Module.NF R₂ M
x₁ x₂ : ... |
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 : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub l₁.eval (Mathlib.Tactic.Module.NF.cons a₂ l₂).eval) l.eval
⊢ Eq (HSub.hSub (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval (Mathlib.Tactic.Module.NF.cons a₂ l₂).eval)
(Mathlib.Tactic.Mo... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub l₁.eval l₂.eval) l.eval
simp :
Eq (HAdd.hAdd (HSMul.hSMul r₁ x) (HAdd.hAdd l₁.eval (HAdd.hAdd (Neg.neg (HSMul.hSMul r₂ x)) (Neg.neg l₂.eval))))
(HAdd.hAdd (H... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub l₁.eval l₂.eval) l.eval
congr! :
Eq (HAdd.hAdd l₁.eval (HAdd.hAdd (Neg.neg (HSMul.hSMul r₂ x)) (Neg.neg l₂.eval)))
(HAdd.hAdd (Neg.neg (HSMul.hSMul r₂ x)) (H... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub l₁.eval l₂.eval) l.eval
simp :
Eq (HAdd.hAdd (HAdd.hAdd l₁.eval (Neg.neg (HSMul.hSMul r₂ x))) (Neg.neg l₂.eval))
(HAdd.hAdd (HAdd.hAdd (Neg.neg (HSMul.hSMul ... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub l₁.eval l₂.eval) l.eval
congr! : Eq (HAdd.hAdd l₁.eval (Neg.neg (HSMul.hSMul r₂ x))) (HAdd.hAdd (Neg.neg (HSMul.hSMul r₂ x)) l₁.eval)
⊢ Eq (HAdd.hAdd (HAdd.hAdd l₁... |
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 : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub l₁.eval l₂.eval) l.eval
rw rw₁ : Eq (HAdd.hAdd (Neg.neg (HSMul.hSMul r₂ x)) l₁.eval) (HAdd.hAdd (Neg.neg (HSMul.hSMul r₂ x)) l₁.eval)
⊢ Eq (HAdd.hAdd l₁.eval (Neg.... |
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 : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval l₂.eval) l.eval
simp :
Eq
(HAdd.hAdd (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul a₁.1 a₁.2) l₁.eval) (Neg.neg (HSMul.hSMul... |
congr! 1 | [] | R : Type u_2
M : Type u_3
inst : Ring R
inst_1 : AddCommGroup M
inst_2 : Module R M
a₁ a₂ : Prod R M
l₁ l₂ l : Mathlib.Tactic.Module.NF R M
h : Eq (HSub.hSub (Mathlib.Tactic.Module.NF.cons a₁ l₁).eval l₂.eval) l.eval
congr! :
Eq (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul a₁.1 a₁.2) l₁.eval) (Neg.neg (HSMul.hSMul a₂.1 a₂.2)))... |
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