type stringlengths 1 123k | tactic stringlengths 1 8.8k | removals listlengths 0 28 | name stringlengths 1 85 | kind stringclasses 2
values | goal stringlengths 7 160k |
|---|---|---|---|---|---|
∀ (val : ℕ) (property : 0 < val),
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property - b) : ℕ) =
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property) : ℕ) - 1 - (↑b : ℕ) + 1 | cases a | [
"a"
] | mk | goal | a b : ℕ+
⊢ (↑(a - b) : ℕ) = (↑a : ℕ) - 1 - (↑b : ℕ) + 1 |
∀ (val_1 : ℕ) (property_1 : 0 < val_1),
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property - Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val_1 property_1) : ℕ) =
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property) : ℕ) - 1 -
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val_1 property_1) : ℕ) +
1 | cases b | [
"b"
] | mk | goal | b : ℕ+
val : ℕ
property : 0 < val
⊢ (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property - b) : ℕ) =
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property) : ℕ) - 1 - (↑b : ℕ) + 1 |
(if val_1 < val then val - val_1 else 1) = val - 1 - val_1 + 1 | simp only [PNat.mk_coe, _root_.PNat.sub_coe, ← _root_.PNat.coe_lt_coe] | [] | mk | goal | val : ℕ
property : 0 < val
val_1 : ℕ
property_1 : 0 < val_1
⊢ (↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property - Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val_1 property_1) : ℕ) =
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val property) : ℕ) - 1 -
(↑(Subtype.mk (p := fun (n : ℕ) ↦ 0 < n) val_1 property_1) : ... |
val_1 < val → val - val_1 = val - 1 - val_1 + 1 | split_ifs | [] | pos | goal | val : ℕ
property : 0 < val
val_1 : ℕ
property_1 : 0 < val_1
⊢ (if val_1 < val then val - val_1 else 1) = val - 1 - val_1 + 1 |
¬val_1 < val → 1 = val - 1 - val_1 + 1 | split_ifs | [] | neg | goal | val : ℕ
property : 0 < val
val_1 : ℕ
property_1 : 0 < val_1
pos : val_1 < val → val - val_1 = val - 1 - val_1 + 1
⊢ (if val_1 < val then val - val_1 else 1) = val - 1 - val_1 + 1 |
a = a' ∧ b = b' → p | rwa [Nat.pair_eq_pair, and_imp] | [] | rwa | goal | p : Prop
a b a' b' : ℕ
h : a = a' → b = b' → p
⊢ Nat.pair a b = Nat.pair a' b' → p |
(f + g).coeff n = f.coeff n + g.coeff n | subst ‹_› ‹_› | [
"a",
"b",
"h_add_left",
"h_add_right"
] | subst | goal | 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
⊢ (f + g).coeff n = a + b |
∀ (h : d = df + dg), (f * g).coeff d = a * b | split_ifs with h | [] | pos | goal | 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
⊢ (f * g).coeff d = if d = df + dg then a * b else 0 |
∀ (h : ¬d = df + dg), (f * g).coeff d = 0 | split_ifs with h | [] | neg | goal | 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
⊢ (f * g).coeff d = if d = df + dg then a * b else 0 |
∀ (ddf : df + dg ≤ df + dg), (f * g).coeff (df + dg) = f.coeff df * g.coeff dg | subst h_mul_left h_mul_right h | [
"d",
"a",
"b",
"h_mul_left",
"h_mul_right",
"ddf",
"h"
] | pos | goal | 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).coeff d = a * b |
(f * g).natDegree < d | apply coeff_eq_zero_of_natDegree_lt | [] | neg | goal | 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).coeff d = 0 |
(f * g).natDegree ≤ df + dg | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_) | [] | apply | goal | 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 < d |
df + dg ≠ d | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_) | [] | apply₁ | goal | 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
⊢ (f * g).natDegree < d |
∀ (h : o = m * n), (p ^ m).coeff o = a ^ m | split_ifs with h | [] | pos | goal | 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
⊢ (p ^ m).coeff o = if o = m * n then a ^ m else 0 |
∀ (h : ¬o = m * n), (p ^ m).coeff o = 0 | split_ifs with h | [] | neg | goal | 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
⊢ (p ^ m).coeff o = if o = m * n then a ^ m else 0 |
∀ (h_exp : m * n ≤ m * n), (p ^ m).coeff (m * n) = p.coeff n ^ m | subst h h_pow_bas | [
"o",
"a",
"h_exp",
"h_pow_bas",
"h"
] | pos | goal | 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).coeff o = a ^ m |
(p ^ m).natDegree < o | apply coeff_eq_zero_of_natDegree_lt | [] | neg | goal | 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).coeff o = 0 |
(p ^ m).natDegree ≤ m * n | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_) | [] | apply | goal | 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 < o |
m * n ≠ o | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_) | [] | apply₁ | goal | 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
⊢ (p ^ m).natDegree < o |
∀ (coeff_ne_zero : p.coeff o ≠ 0) (h_natDeg_le : p.natDegree ≤ o), p.natDegree = o | subst coeff_eq deg_eq_deg coeff_eq_deg | [
"deg",
"m",
"c",
"h_natDeg_le",
"coeff_eq",
"coeff_ne_zero",
"deg_eq_deg",
"coeff_eq_deg"
] | subst | goal | 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
⊢ p.natDegree = deg |
∀ (coeff_ne_zero : p.coeff (WithBot.unbotD 0 m) ≠ 0), p.degree = m | subst coeff_eq coeff_eq_deg deg_eq_deg | [
"deg",
"o",
"c",
"coeff_eq",
"coeff_ne_zero",
"deg_eq_deg",
"coeff_eq_deg"
] | subst | goal | 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
⊢ p.degree = deg |
∀ (h_deg_le : p.degree ≤ ⊥) (coeff_ne_zero : p.coeff (WithBot.unbotD 0 ⊥) ≠ 0), p.degree = ⊥ | rcases eq_or_ne m ⊥ with rfl | hh | [
"m",
"h_deg_le",
"coeff_ne_zero"
] | inl | goal | 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
⊢ p.degree = m |
∀ (hh : m ≠ ⊥), p.degree = m | rcases eq_or_ne m ⊥ with rfl | hh | [] | inr | goal | 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 = ⊥
⊢ p.degree = m |
∀ (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) | obtain ⟨m, rfl⟩ := WithBot.ne_bot_iff_exists.mp hh | [
"m",
"h_deg_le",
"coeff_ne_zero",
"hh"
] | inr | goal | 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 ≠ ⊥
⊢ p.degree = m |
(f - g).coeff n = f.coeff n - g.coeff n | subst hf hg | [
"a",
"b",
"hf",
"hg"
] | subst | goal | 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
⊢ (f - g).coeff n = a - b |
LT.lt (α := ℤ) 0 (↑n : ℤ) | have := natCast_pos.2 <| Nat.le_of_ble_eq_true hn | [] | this | hypothesis | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
⊢ 0 ≤ (a % (↑n : ℤ)).toNat ∧ (a % (↑n : ℤ)).toNat < n ∧ a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] |
0 ≤ a % (↑n : ℤ) | have nonneg := emod_nonneg a <| Int.ne_of_gt this | [] | nonneg | hypothesis | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
this : LT.lt (α := ℤ) 0 (↑n : ℤ)
⊢ 0 ≤ (a % (↑n : ℤ)).toNat ∧ (a % (↑n : ℤ)).toNat < n ∧ a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] |
(a % (↑n : ℤ)).toNat < n | refine ⟨Nat.zero_le _, ?_, ?_⟩ | [] | refine_1 | goal | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
this : LT.lt (α := ℤ) 0 (↑n : ℤ)
nonneg : 0 ≤ a % (↑n : ℤ)
⊢ 0 ≤ (a % (↑n : ℤ)).toNat ∧ (a % (↑n : ℤ)).toNat < n ∧ a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] |
a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] | refine ⟨Nat.zero_le _, ?_, ?_⟩ | [] | refine_2 | goal | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
this : LT.lt (α := ℤ) 0 (↑n : ℤ)
nonneg : 0 ≤ a % (↑n : ℤ)
refine_1 : (a % (↑n : ℤ)).toNat < n
⊢ 0 ≤ (a % (↑n : ℤ)).toNat ∧ (a % (↑n : ℤ)).toNat < n ∧ a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] |
a % (↑n : ℤ) < (↑n : ℤ) | rw [Int.toNat_lt nonneg] | [] | refine_1 | goal | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
this : LT.lt (α := ℤ) 0 (↑n : ℤ)
nonneg : 0 ≤ a % (↑n : ℤ)
⊢ (a % (↑n : ℤ)).toNat < n |
a % (↑n : ℤ) = HMod.hMod (α := ℤ) (↑(a % (↑n : ℤ)).toNat : ℤ) (↑n : ℤ) | rw [Int.ModEq, Int.toNat_of_nonneg nonneg, emod_emod] | [] | refine_2 | goal | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
this : LT.lt (α := ℤ) 0 (↑n : ℤ)
nonneg : 0 ≤ a % (↑n : ℤ)
⊢ a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] |
a % (↑n : ℤ) = a % (↑n : ℤ) % (↑n : ℤ) | rw [Int.ModEq, Int.toNat_of_nonneg nonneg, emod_emod] | [] | refine_2₁ | goal | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
this : LT.lt (α := ℤ) 0 (↑n : ℤ)
nonneg : 0 ≤ a % (↑n : ℤ)
refine_2 : a % (↑n : ℤ) = HMod.hMod (α := ℤ) (↑(a % (↑n : ℤ)).toNat : ℤ) (↑n : ℤ)
⊢ a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] |
a % (↑n : ℤ) = a % (↑n : ℤ) | rw [Int.ModEq, Int.toNat_of_nonneg nonneg, emod_emod] | [] | refine_2₂ | goal | p : Sort u_1
a : ℤ
n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
this : LT.lt (α := ℤ) 0 (↑n : ℤ)
nonneg : 0 ≤ a % (↑n : ℤ)
refine_2 : a % (↑n : ℤ) = HMod.hMod (α := ℤ) (↑(a % (↑n : ℤ)).toNat : ℤ) (↑n : ℤ)
refine_2₁ : a % (↑n : ℤ) = a % (↑n : ℤ) % (↑n : ℤ)
⊢ a ≡ (↑(a % (↑n : ℤ)).toNat : ℤ) [ZMOD (↑n : ℤ)] |
a % n < n | refine ⟨Nat.zero_le _, ?_, ?_⟩ | [] | refine_1 | goal | p : Sort u_1
a n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
⊢ 0 ≤ a % n ∧ a % n < n ∧ a ≡ a % n [MOD n] |
a ≡ a % n [MOD n] | refine ⟨Nat.zero_le _, ?_, ?_⟩ | [] | refine_2 | goal | p : Sort u_1
a n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
refine_1 : a % n < n
⊢ 0 ≤ a % n ∧ a % n < n ∧ a ≡ a % n [MOD n] |
a % n = a % n % n | rw [Nat.ModEq, Nat.mod_mod] | [] | refine_2 | goal | p : Sort u_1
a n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
⊢ a ≡ a % n [MOD n] |
a % n = a % n | rw [Nat.ModEq, Nat.mod_mod] | [] | refine_2₁ | goal | p : Sort u_1
a n : ℕ
hn : Nat.ble 1 n = true
H : OnModCases n a 0 p
refine_2 : a % n = a % n % n
⊢ a ≡ a % n [MOD n] |
∀ (x : β) (hx : condβα (coeβα x, y).fst) (hy : condδγ (coeβα x, y).snd),
∃ (y_1 : β × δ), map coeβα coeδγ y_1 = (coeβα x, y) | rcases CanLift.prf (β := β) x hx with ⟨x, rfl⟩ | [
"x",
"hx",
"hy"
] | rcases | goal | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
coeβα : β → α
condβα : α → Prop
coeδγ : δ → γ
condδγ : γ → Prop
inst : CanLift α β coeβα condβα
inst_1 : CanLift γ δ coeδγ condδγ
x : α
y : γ
hx : condβα (x, y).fst
hy : condδγ (x, y).snd
⊢ ∃ (y_1 : β × δ), map coeβα coeδγ y_1 = (x, y) |
∀ (y : δ) (hx : condβα (coeβα x, coeδγ y).fst) (hy : condδγ (coeβα x, coeδγ y).snd),
∃ (y_1 : β × δ), map coeβα coeδγ y_1 = (coeβα x, coeδγ y) | rcases CanLift.prf (β := δ) y hy with ⟨y, rfl⟩ | [
"y",
"hx",
"hy"
] | rcases | goal | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
coeβα : β → α
condβα : α → Prop
coeδγ : δ → γ
condδγ : γ → Prop
inst : CanLift α β coeβα condβα
inst_1 : CanLift γ δ coeδγ condδγ
y : γ
x : β
hx : condβα (coeβα x, y).fst
hy : condδγ (coeβα x, y).snd
⊢ ∃ (y_1 : β × δ), map coeβα coeδγ y_1 = (coeβα x, y) |
∀ (this : DecidablePred p), ∃ (g : (i : ι) → α i), (fun (i : Subtype p) ↦ g (i.val (p := p))) = f | haveI : DecidablePred p := fun i ↦ Classical.propDecidable (p i) | [] | haveI | goal | ι : Sort u_1
α : ι → Sort u_2
ne : ∀ (i : ι), Nonempty.{u_2} (α i)
p : ι → Prop
f : (i : Subtype p) → α (i.val (p := p))
⊢ ∃ (g : (i : ι) → α i), (fun (i : Subtype p) ↦ g (i.val (p := p))) = f |
¬LT.lt (α := ℕ∞) ⊤ ⊤ | cases x | [
"x"
] | top | goal | x : ℕ∞
⊢ ¬⊤ < x |
∀ (a : ℕ), ¬LT.lt (α := ℕ∞) ⊤ (↑a : ℕ∞) | cases x | [
"x"
] | coe | goal | x : ℕ∞
top : ¬LT.lt (α := ℕ∞) ⊤ ⊤
⊢ ¬⊤ < x |
∀ (h : ¬r), r → p → r ∧ q | tauto | [
"h"
] | inl | goal | p q r : Prop
h : r → p → q
⊢ r ∧ p → r ∧ q |
∀ (h : p → q), r → p → r ∧ q | tauto | [
"h"
] | inr | goal | p q r : Prop
h : r → p → q
inl : ∀ (h : ¬r), r → p → r ∧ q
⊢ r ∧ p → r ∧ q |
∀ (h : p → q), r → p → r | tauto | [
"h"
] | inr₂ | goal | p q r : Prop
h : r → p → q
inl : ∀ (h : ¬r), r → p → r ∧ q
inr inr₁ : ∀ (h : p → q), r → p → r ∧ q
⊢ r ∧ p → r ∧ q |
∀ (h : p → q), r → p → q | tauto | [
"h"
] | inr₃ | goal | 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
⊢ r ∧ p → r ∧ q |
r → p → ∀ (h : ¬p), q | tauto | [
"h"
] | inr₄ | goal | 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
⊢ r ∧ p → r ∧ q |
r → p → ∀ (h : q), q | tauto | [
"h"
] | inr₅ | goal | 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
⊢ r ∧ p → r ∧ q |
∀ (h' : ∀ (a : α), β = β'), Eq (α := Sort (imax u v)) (α → β) (α → β') | cases h | [
"α'",
"h",
"h'"
] | refl | goal | α α' : Sort u
β β' : Sort v
h : α = α'
h' : ∀ (a : α'), β = β'
⊢ Eq (α := Sort (imax u v)) (α → β) (α' → β') |
Eq (α := Sort (imax u v)) ((x : α) → (fun (x : α) ↦ β') x) (α → β') | rw [funext h'] | [] | refl | goal | α : Sort u
β β' : Sort v
h' : ∀ (a : α), β = β'
⊢ Eq (α := Sort (imax u v)) ((x : α) → (fun (x : α) ↦ β) x) (α → β') |
∀ {p : x ≍ x → Prop} (h : ∀ (he : x = x), p sorry), p sorry | cases he | [
"y",
"p",
"h",
"he"
] | refl | goal | α : Sort u_1
x y : α
p : x ≍ y → Prop
h : ∀ (he : x = y), p ⋯
he : x ≍ y
⊢ p he |
∀ {p : x = x → Prop} (h : ∀ (he : x ↔ x), p sorry), p sorry | cases he | [
"y",
"p",
"h",
"he"
] | refl | goal | x y : Prop
p : x = y → Prop
h : ∀ (he : x ↔ y), p ⋯
he : x = y
⊢ p he |
Nat → Ordering | cases p | [
"p"
] | var | goal | p q : IProp
⊢ Ordering |
AndKind → IProp → IProp → Ordering | cases p | [
"p"
] | and' | goal | p q : IProp
var : Nat → Ordering
true false : Ordering
⊢ Ordering |
Nat → Ordering | cases q | [
"q"
] | var | goal | q : IProp
a : Nat
⊢ Ordering |
AndKind → IProp → IProp → Ordering | cases q | [
"q"
] | var₃ | goal | q : IProp
a : Nat
var : Nat → Ordering
var₁ var₂ : Ordering
⊢ Ordering |
Nat → Ordering | cases q | [
"q"
] | true | goal | q : IProp
⊢ Ordering |
AndKind → IProp → IProp → Ordering | cases q | [
"q"
] | true₂ | goal | q : IProp
true : Nat → Ordering
cases true₁ : Ordering
⊢ Ordering |
Nat → Ordering | cases q | [
"q"
] | false | goal | q : IProp
⊢ Ordering |
AndKind → IProp → IProp → Ordering | cases q | [
"q"
] | false₂ | goal | q : IProp
false : Nat → Ordering
false₁ cases : Ordering
⊢ Ordering |
Nat → Ordering | cases q | [
"q"
] | and' | goal | q : IProp
a : AndKind
a_1 a_2 : IProp
⊢ Ordering |
AndKind → IProp → IProp → Ordering | cases q | [
"q"
] | and'₃ | goal | q : IProp
a : AndKind
a_1 a_2 : IProp
and' : Nat → Ordering
and'₁ and'₂ : Ordering
⊢ Ordering |
Nat → Ordering | cases q | [
"q"
] | or | goal | q a a_1 : IProp
⊢ Ordering |
AndKind → IProp → IProp → Ordering | cases q | [
"q"
] | or₃ | goal | q a a_1 : IProp
or : Nat → Ordering
or₁ or₂ : Ordering
⊢ Ordering |
Nat → Ordering | cases q | [
"q"
] | imp | goal | q a a_1 : IProp
⊢ Ordering |
AndKind → IProp → IProp → Ordering | cases q | [
"q"
] | imp₃ | goal | q a a_1 : IProp
imp : Nat → Ordering
imp₁ imp₂ : Ordering
⊢ Ordering |
NF.eval [] * zpow' («$r», «$x»).2 («$r», «$x»).1 = «$e» | rw [NF.eval_cons] | [] | rw | goal | «$v» : Level
«$M» : Type v
«$iM» : CommGroupWithZero «$M»
«$r» : ℤ
«$x» : «$M»
«$snd» : ℕ
«$e» : «$M»
«$pf» : zpow' «$x» «$r» = «$e»
⊢ ((«$r», «$x») ::ᵣ []).eval = «$e» |
¬p ∧ q ∨ p ∧ ¬q ↔ p ∧ ¬q ∨ ¬p ∧ q | rw [not_not, or_comm] | [] | rw | goal | p q : Prop
⊢ ¬p ∧ q ∨ ¬¬p ∧ ¬q ↔ p ∧ ¬q ∨ ¬p ∧ q |
p ∧ ¬q ∨ ¬p ∧ q ↔ p ∧ ¬q ∨ ¬p ∧ q | rw [not_not, or_comm] | [] | rw₁ | goal | p q : Prop
rw : ¬p ∧ q ∨ p ∧ ¬q ↔ p ∧ ¬q ∨ ¬p ∧ q
⊢ ¬p ∧ q ∨ ¬¬p ∧ ¬q ↔ p ∧ ¬q ∨ ¬p ∧ q |
∀ (a b : ℕ) (h : (Nat.unpair ((Nat.pairEquiv : (a : ℕ × ℕ) → ℕ) (a, b))).1 ≠ 0),
(Nat.unpair ((Nat.pairEquiv : (a : ℕ × ℕ) → ℕ) (a, b))).2 < (Nat.pairEquiv : (a : ℕ × ℕ) → ℕ) (a, b) | obtain ⟨⟨a, b⟩, rfl⟩ := Nat.pairEquiv.surjective n | [
"n",
"h"
] | obtain | goal | n : ℕ
h : (Nat.unpair n).1 ≠ 0
⊢ (Nat.unpair n).2 < n |
∀ (h : a ≠ 0), b < Nat.pair a b | simp only [Nat.pairEquiv_apply, Function.uncurry_apply_pair, Nat.unpair_pair] at * | [
"h"
] | simp | goal | a b : ℕ
h : (Nat.unpair ((Nat.pairEquiv : (a : ℕ × ℕ) → ℕ) (a, b))).1 ≠ 0
⊢ (Nat.unpair ((Nat.pairEquiv : (a : ℕ × ℕ) → ℕ) (a, b))).2 < (Nat.pairEquiv : (a : ℕ × ℕ) → ℕ) (a, b) |
a ≤ a * a | have := Nat.le_mul_self a | [] | this | hypothesis | a b : ℕ
h : a ≠ 0
⊢ b < if a < b then b * b + a else a * a + a + b |
b ≤ b * b | have := Nat.le_mul_self b | [] | this₁ | hypothesis | a b : ℕ
h : a ≠ 0
this : a ≤ a * a
⊢ b < if a < b then b * b + a else a * a + a + b |
a < b → b < b * b + a | split | [] | isTrue | goal | a b : ℕ
h : a ≠ 0
this_1 : a ≤ a * a
this : b ≤ b * b
⊢ b < if a < b then b * b + a else a * a + a + b |
¬a < b → b < a * a + a + b | split | [] | isFalse | goal | a b : ℕ
h : a ≠ 0
this_1 : a ≤ a * a
this : b ≤ b * b
isTrue : a < b → b < b * b + a
⊢ b < if a < b then b * b + a else a * a + a + b |
(let p := Nat.unpair (Nat.pair 0 n);
if h : p.1 = 0 then S.nat p.2
else
have this : p.1 ≤ Nat.pair 0 n := sorry;
have this := sorry;
have this := sorry;
(S.decode (p.1 - 1)).cons (S.decode p.2)) =
S.nat n | unfold S.encode S.decode | [] | nat | goal | n : ℕ
⊢ S.decode (S.nat n).encode = S.nat n |
(let p := Nat.unpair (Nat.pair (a.encode + 1) b.encode);
if h : p.1 = 0 then S.nat p.2
else
have this : p.1 ≤ Nat.pair (a.encode + 1) b.encode := sorry;
have this := sorry;
have this := sorry;
(S.decode (p.1 - 1)).cons (S.decode p.2)) =
a.cons b | unfold S.encode S.decode | [] | cons | goal | a b : S
iha : S.decode a.encode = a
ihb : S.decode b.encode = b
⊢ S.decode (a.cons b).encode = a.cons b |
∀ (n : ℕ), S.decode (S.nat n).encode = S.nat n | induction s with
| nat n =>
unfold S.encode S.decode
simp
| cons a b iha ihb =>
unfold S.encode S.decode
simp [iha, ihb] | [
"s"
] | nat | goal | s : S
⊢ S.decode s.encode = s |
∀ (a b : S) (iha : S.decode a.encode = a) (ihb : S.decode b.encode = b), S.decode (a.cons b).encode = a.cons b | induction s with
| nat n =>
unfold S.encode S.decode
simp
| cons a b iha ihb =>
unfold S.encode S.decode
simp [iha, ihb] | [
"s"
] | cons | goal | s : S
nat : ∀ (n : ℕ), S.decode (S.nat n).encode = S.nat n
⊢ S.decode s.encode = s |
S.encode
(let p := Nat.unpair n;
if h : p.1 = 0 then S.nat p.2
else
have this := sorry;
have this := sorry;
have this := sorry;
(S.decode (p.1 - 1)).cons (S.decode p.2)) =
n | unfold S.decode | [] | ind | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
⊢ (S.decode n).encode = n |
(Nat.unpair n).1 = 0 → (S.nat (Nat.unpair n).2).encode = n | split | [] | ind | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
⊢ S.encode
(if h : (Nat.unpair n).1 = 0 then S.nat (Nat.unpair n).2
else (S.decode ((Nat.unpair n).1 - 1)).cons (S.decode (Nat.unpair n).2)) =
n |
¬(Nat.unpair n).1 = 0 → ((S.decode ((Nat.unpair n).1 - 1)).cons (S.decode (Nat.unpair n).2)).encode = n | split | [] | ind₁ | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
ind : (Nat.unpair n).1 = 0 → (S.nat (Nat.unpair n).2).encode = n
⊢ S.encode
(if h : (Nat.unpair n).1 = 0 then S.nat (Nat.unpair n).2
else (S.decode ((Nat.unpair n).1 - 1)).cons (S.decode (Nat.unpair n).2)) =
n |
Nat.pair (Nat.unpair n).1 (Nat.unpair n).2 = n | rw [← h, Nat.pair_unpair] | [] | rw | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : (Nat.unpair n).1 = 0
⊢ Nat.pair 0 (Nat.unpair n).2 = n |
n = n | rw [← h, Nat.pair_unpair] | [] | rw₁ | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : (Nat.unpair n).1 = 0
rw : Nat.pair (Nat.unpair n).1 (Nat.unpair n).2 = n
⊢ Nat.pair 0 (Nat.unpair n).2 = n |
1 ≤ (Nat.unpair n).1 | rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair] | [] | rw | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : ¬(Nat.unpair n).1 = 0
⊢ Nat.pair ((S.decode ((Nat.unpair n).1 - 1)).encode + 1) (S.decode (Nat.unpair n).2).encode = n |
(Nat.unpair n).2 < n | rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair] | [] | a | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : ¬(Nat.unpair n).1 = 0
rw : 1 ≤ (Nat.unpair n).1
⊢ Nat.pair ((S.decode ((Nat.unpair n).1 - 1)).encode + 1) (S.decode (Nat.unpair n).2).encode = n |
(Nat.unpair n).1 - 1 < n | rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair] | [] | a₁ | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : ¬(Nat.unpair n).1 = 0
rw : 1 ≤ (Nat.unpair n).1
a : (Nat.unpair n).2 < n
⊢ Nat.pair ((S.decode ((Nat.unpair n).1 - 1)).encode + 1) (S.decode (Nat.unpair n).2).encode = n |
(Nat.unpair n).1 ≠ 0 | rwa [Nat.one_le_iff_ne_zero] | [] | rwa | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : ¬(Nat.unpair n).1 = 0
⊢ 1 ≤ (Nat.unpair n).1 |
∀ (ih : ∀ m < 0, (S.decode m).encode = m) (h : ¬(Nat.unpair 0).1 = 0), (Nat.unpair 0).1 - 1 < 0 | obtain _ | n' := n | [
"n",
"ih",
"h"
] | a | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : ¬(Nat.unpair n).1 = 0
⊢ (Nat.unpair n).1 - 1 < n |
∀ (n' : ℕ) (ih : ∀ m < n' + 1, (S.decode m).encode = m) (h : ¬(Nat.unpair (n' + 1)).1 = 0),
(Nat.unpair (n' + 1)).1 - 1 < n' + 1 | obtain _ | n' := n | [
"n",
"ih",
"h"
] | a₁ | goal | n : ℕ
ih : ∀ m < n, (S.decode m).encode = m
h : ¬(Nat.unpair n).1 = 0
a : ∀ (ih : ∀ m < 0, (S.decode m).encode = m) (h : ¬(Nat.unpair 0).1 = 0), (Nat.unpair 0).1 - 1 < 0
⊢ (Nat.unpair n).1 - 1 < n |
(Nat.unpair (n' + 1)).1 < n' + 1 | have := Nat.unpair_lt (by lia : 1 ≤ n' + 1) | [] | this | hypothesis | n' : ℕ
ih : ∀ m < n' + 1, (S.decode m).encode = m
h : ¬(Nat.unpair (n' + 1)).1 = 0
⊢ (Nat.unpair (n' + 1)).1 - 1 < n' + 1 |
∀ (n : ℕ) (ih : ∀ m < n, (S.decode m).encode = m), (S.decode n).encode = n | induction n using Nat.strongRecOn with | _ n ih =>
unfold S.decode
dsimp only
split
next h =>
unfold S.encode
rw [← h, Nat.pair_unpair]
next h =>
unfold S.encode
rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair]
· rwa [Nat.one_le_iff_ne_zero]
· exact nat_unpair_lt_... | [
"n"
] | ind | goal | n : ℕ
⊢ (S.decode n).encode = n |
r₁ • x + (l₁.eval + (r₂ • x + l₂.eval)) = r₁ • x + (r₂ • x + (l₁.eval + l₂.eval)) | simp only [← h, eval_cons, add_smul, add_assoc] | [] | simp | goal | 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
⊢ ((r₁, x) ::ᵣ l₁).eval + ((r₂, x) ::ᵣ l₂).eval = ((r₁ + r₂, x) ::ᵣ l).eval |
l₁.eval + (r₂ • x + l₂.eval) = r₂ • x + (l₁.eval + l₂.eval) | congr! 1 | [] | h₁ | goal | 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
⊢ r₁ • x + (l₁.eval + (r₂ • x + l₂.eval)) = r₁ • x + (r₂ • x + (l₁.eval + l₂.eval)) |
l₁.eval + r₂ • x + l₂.eval = r₂ • x + l₁.eval + l₂.eval | simp only [← add_assoc] | [] | h₁ | goal | 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
⊢ l₁.eval + (r₂ • x + l₂.eval) = r₂ • x + (l₁.eval + l₂.eval) |
l₁.eval + r₂ • x = r₂ • x + l₁.eval | congr! 1 | [] | h₁ | goal | 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
⊢ l₁.eval + r₂ • x + l₂.eval = r₂ • x + l₁.eval + l₂.eval |
r₂ • x + l₁.eval = r₂ • x + l₁.eval | rw [add_comm] | [] | h₁ | goal | 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
⊢ l₁.eval + r₂ • x = r₂ • x + l₁.eval |
a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) = a₂.1 • a₂.2 + (a₁.1 • a₁.2 + l₁.eval + l₂.eval) | simp only [eval_cons, ← h] | [] | simp | goal | 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
⊢ (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₂ ::ᵣ l).eval |
a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) = a₁.1 • a₁.2 + l₁.eval + l₂.eval + a₂.1 • a₂.2 | nth_rw 4 [add_comm] | [] | nth_rw | goal | 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
⊢ a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) = a₂.1 • a₂.2 + (a₁.1 • a₁.2 + l₁.eval + l₂.eval) |
a₁.1 • a₁.2 + (l₁.eval + (a₂.1 • a₂.2 + l₂.eval)) = a₁.1 • a₁.2 + (l₁.eval + (l₂.eval + a₂.1 • a₂.2)) | simp only [add_assoc] | [] | simp | goal | 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
⊢ a₁.1 • a₁.2 + l₁.eval + (a₂.1 • a₂.2 + l₂.eval) = a₁.1 • a₁.2 + l₁.eval + l₂.eval + a₂.1 • a₂.2 |
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