mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v : V
inst✝ : Finite V
c : G.ConnectedComponent
h : G.IsCycles
hv : v ∈ c.supp
w : V
hw : w ∈ G.neighborSet v
u : V
p : G.Walk u u
hp : p.IsCycle ∧ s(v, w) ∈ p.edges
hvp : v ∈ p.support
⊢ p.toSubgraph.verts = c.supp

Tactic:
obtain ⟨c', hc'⟩ := p.toSubgraph_connected.exists_verts_eq_connectedComponentSupp (by
      intro v hv w hadj
      refine (Subgraph.adj_iff_of_neighborSet_equiv ?_ (Set.toFinite _)).mpr hadj
      have : (G.neighborSet v).Nonempty := by
        rw [Walk.mem_verts_toSubgraph] at hv
        refine (Set.nonempty_of_ncard_ne_zero ?_).mono (p.toSubgraph.neighborSet_subset v)
        rw [hp.1.ncard_neighborSet_toSubgraph_eq_two hv]
        lia
      refine @Classical.ofNonempty _ ?_
      rw [← Cardinal.eq, ← Set.cast_ncard (Set.toFinite _), ← Set.cast_ncard (Set.toFinite _),
        h this, hp.1.ncard_neighborSet_toSubgraph_eq_two (p.mem_verts_toSubgraph.mp hv)])

Goal state after:
V : Type u_1
G : SimpleGraph V
v : V
inst✝ : Finite V
c : G.ConnectedComponent
h : G.IsCycles
hv : v ∈ c.supp
w : V
hw : w ∈ G.neighborSet v
u : V
p : G.Walk u u
hp : p.IsCycle ∧ s(v, w) ∈ p.edges
hvp : v ∈ p.support
c' : G.ConnectedComponent
hc' : p.toSubgraph.verts = c'.supp
⊢ p.toSubgraph.verts = c.supp
Module: Mathlib.Combinatorics.SimpleGraph.FiveWheelLike

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
r k : ℕ
x v w₁ w₂ : α
s t : Finset α
inst✝¹ : DecidableEq α
hw : G.IsFiveWheelLike r k v w₁ w₂ s t
hcf : G.CliqueFree (r + 2)
inst✝ : DecidableRel G.Adj
hW : ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y
h2 : #({z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z}) ≤ 2
a b c d : α
haj : ¬G.Adj x a
hbj : ¬G.Adj x b
hab : a ≠ b
hat : a ∉ t
hbs : b ∉ s
W : Finset α := {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t)))
hcj : ¬G.Adj x a
hbc : b ≠ a
hca : True
hdj : ¬G.Adj x b
had : a ≠ b
hdb : True
ha : a = w₁ ∨ a ∈ s
hb : b = w₂ ∨ b ∈ t
hc : a = v ∨ a ∈ s
hd : b = v ∨ b ∈ t
has : a ∈ s
hbt : b ∈ t
hav : a ≠ v
hbv : b ≠ v
haw : a ≠ w₂
hbw : b ≠ w₁
hxv : v ≠ x
hxw₁ : w₁ ≠ x
hxw₂ : w₂ ≠ x
wa : ∀ ⦃w : α⦄, w ∈ W → w ≠ a → w ≠ b → G.Adj w x
h1s : insert w₁ s ⊆ W
h2t : insert w₂ t ⊆ W
⊢ #(insert x (((s ∩ t).erase b).erase a)) = k + 1

Tactic:
erase_eq_of_notMem <|
        notMem_mono inter_subset_left hbs,

Goal state after:
α : Type u_1
G : SimpleGraph α
r k : ℕ
x v w₁ w₂ : α
s t : Finset α
inst✝¹ : DecidableEq α
hw : G.IsFiveWheelLike r k v w₁ w₂ s t
hcf : G.CliqueFree (r + 2)
inst✝ : DecidableRel G.Adj
hW : ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y
h2 : #({z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z}) ≤ 2
a b c d : α
haj : ¬G.Adj x a
hbj : ¬G.Adj x b
hab : a ≠ b
hat : a ∉ t
hbs : b ∉ s
W : Finset α := {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t)))
hcj : ¬G.Adj x a
hbc : b ≠ a
hca : True
hdj : ¬G.Adj x b
had : a ≠ b
hdb : True
ha : a = w₁ ∨ a ∈ s
hb : b = w₂ ∨ b ∈ t
hc : a = v ∨ a ∈ s
hd : b = v ∨ b ∈ t
has : a ∈ s
hbt : b ∈ t
hav : a ≠ v
hbv : b ≠ v
haw : a ≠ w₂
hbw : b ≠ w₁
hxv : v ≠ x
hxw₁ : w₁ ≠ x
hxw₂ : w₂ ≠ x
wa : ∀ ⦃w : α⦄, w ∈ W → w ≠ a → w ≠ b → G.Adj w x
h1s : insert w₁ s ⊆ W
h2t : insert w₂ t ⊆ W
⊢ #(insert x ((s ∩ t).erase a)) = k + 1
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
ih :
  ∀ (b : ℕ),
    N' ≤ b →
      ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj],
        ↑G.minDegree ≥ (1 - 1 / ↑(r - 1) + ε') * ↑b → completeEquipartiteGraph (r - 1 + 1) t' ⊑ G
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
hrε_lt_1 : ↑r * ε < 1
ht_lt_t' : t < t'
hr_ne_1 : r ≠ 1
⊢ (1 - 1 / (↑r - 1) + 1 / (↑r - 1) * (↑r - (↑r - 1)) * (1 / ↑r) + ε) * ↑n = (1 - 1 / ↑(r - 1) + ε') * ↑n

Tactic:
sub_sub_cancel,

Goal state after:
ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
ih :
  ∀ (b : ℕ),
    N' ≤ b →
      ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj],
        ↑G.minDegree ≥ (1 - 1 / ↑(r - 1) + ε') * ↑b → completeEquipartiteGraph (r - 1 + 1) t' ⊑ G
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
hrε_lt_1 : ↑r * ε < 1
ht_lt_t' : t < t'
hr_ne_1 : r ≠ 1
⊢ (1 - 1 / (↑r - 1) + 1 / (↑r - 1) * 1 * (1 / ↑r) + ε) * ↑n = (1 - 1 / ↑(r - 1) + ε') * ↑n
Module: Mathlib.Combinatorics.SimpleGraph.FiveWheelLike

Goal state before:

Tactic:
_ ≤ ∑ w ∈ W, G.degree w + 2 * ∑ w ∈ s ∩ t, G.degree w := by
      simp_rw [add_assoc, add_comm k, ← add_assoc, ← Wc, add_assoc, ← two_mul, mul_add,
               ← hw.card_inter, card_eq_sum_ones, ← mul_assoc, mul_sum, mul_one, mul_comm 2]
      gcongr with i <;> exact minDegree_le_degree ..
    _ ≤ (#X * (#W - 3) + #Xᶜ * (#W - 1)) + 2 * (#X * #(s ∩ t) + #Xᶜ * (#(s ∩ t) - 1)) := by gcongr
    _ = #X * (#W - 3 + 2 * k) + #Xᶜ * ((#W - 1) + 2 * (k - 1)) := by grind
    _ ≤ _ := by
        by_cases hk : k = 0 -- so `s ∩ t = ∅` and hence `Xᶜ = ∅`
        · have Xu : X = univ := by
            rw [← hw.card_inter, card_eq_zero] at hk
            exact eq_univ_of_forall fun _ ↦ by simp [X, hk]
          subst k
          rw [add_zero] at Wc
          simp [Xu, Wc, mul_comm]
        have w3 : 3 ≤ #W := two_lt_card.2 ⟨_, mem_insert_self .., _, by simp [W], _, by simp [W],
          hw.isPathGraph3Compl.ne_fst, hw.isPathGraph3Compl.ne_snd, hw.isPathGraph3Compl.fst_ne_snd⟩
        have hap : #W - 1 + 2 * (k - 1) = #W - 3 + 2 * k := by lia
        rw [hap, ← add_mul, card_add_card_compl, mul_comm, two_mul, ← add_assoc]
        gcongr
        lia

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.FiveWheelLike

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
r✝ : ℕ
inst✝¹ : Fintype α
inst✝ : DecidableRel G.Adj
r : ℕ
hf : G.CliqueFree (r + 2 + 1)
hd : (3 * (r + 2) - 4) * ‖α‖ / (3 * (r + 2) - 1) < G.minDegree
H : SimpleGraph α
hle : G ≤ H
hmcf : Maximal (fun H ↦ H.CliqueFree (r + 3)) H
hnotcol : ¬H.Colorable (r + 2)
hn : ¬H.IsCompleteMultipartite
k : ℕ
w✝⁴ w✝³ w✝² : α
w✝¹ w✝ : Finset α
hw : H.IsFiveWheelLike (r + 1) k w✝⁴ w✝³ w✝² w✝¹ w✝
hlt : k < r + 1
hm : ∀ (j : ℕ), k < j → H.FiveWheelLikeFree (r + 1) j
hD : H.minDegree ≤ (2 * (r + 1) + k) * ‖α‖ / (2 * (r + 1) + k + 3)
⊢ (?m.355 + 2).succ * ((2 * (r + 1) + k) * ‖α‖ / (2 * (r + 1) + k + 3) * (3 * r + 4).succ) ≤
    (?m.355 + 2).succ * ((3 * r + 2) * ‖α‖)

Tactic:
rw [← mul_assoc, mul_comm (2 * r + 2 + k + 3), mul_comm _ (_ * ‖α‖)]

Goal state after:
α : Type u_1
G : SimpleGraph α
r✝ : ℕ
inst✝¹ : Fintype α
inst✝ : DecidableRel G.Adj
r : ℕ
hf : G.CliqueFree (r + 2 + 1)
hd : (3 * (r + 2) - 4) * ‖α‖ / (3 * (r + 2) - 1) < G.minDegree
H : SimpleGraph α
hle : G ≤ H
hmcf : Maximal (fun H ↦ H.CliqueFree (r + 3)) H
hnotcol : ¬H.Colorable (r + 2)
hn : ¬H.IsCompleteMultipartite
k : ℕ
w✝⁴ w✝³ w✝² : α
w✝¹ w✝ : Finset α
hw : H.IsFiveWheelLike (r + 1) k w✝⁴ w✝³ w✝² w✝¹ w✝
hlt : k < r + 1
hm : ∀ (j : ℕ), k < j → H.FiveWheelLikeFree (r + 1) j
hD : H.minDegree ≤ (2 * (r + 1) + k) * ‖α‖ / (2 * (r + 1) + k + 3)
⊢ (2 * (r + 1) + k) * ‖α‖ / (2 * (r + 1) + k + 3) * (2 * r + 2 + k + 3) * (3 * r + 4).succ ≤
    (3 * r + 2) * ‖α‖ * (2 * r + 2 + k + 2).succ
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
K : G.deleteUniversalVerts.coe.ConnectedComponent
this : G.IsClique (Subtype.val '' K.supp \ M1.verts)
⊢ ∃ M, M.verts = Subtype.val '' K.supp \ M1.verts ∧ M.IsMatching

Tactic:
← this.even_iff_exists_isMatching (Set.toFinite _),

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
K : G.deleteUniversalVerts.coe.ConnectedComponent
this : G.IsClique (Subtype.val '' K.supp \ M1.verts)
⊢ Even (Subtype.val '' K.supp \ M1.verts).ncard
Module: Mathlib.Computability.Partrec

Goal state before:
pp: p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
⊢ ∀ (a : ℕ), Acc (lbp p) a

Tactic:
suffices ∀ m k, n ≤ k + m → Acc (lbp p) k by exact fun a => this _ _ (Nat.le_add_left _ _)

Goal state after:
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
⊢ ∀ (m k : ℕ), n ≤ k + m → Acc (lbp p) k
Module: Mathlib.Computability.Ackermann

Goal state before:
pp: case succ
n : ℕ
IH : ack 2 n = 2 * n + 3
⊢ ack 2 (n + 1) = 2 * (n + 1) + 3

Tactic:
simpa [mul_succ]

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ f x = 0

Tactic:
refine hmain ⌊-logb 2 (x / x₀)⌋₊ x le_rfl ⟨?lb, ?ub⟩

Goal state after:
case lb
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ 2 ^ (-↑⌊-logb 2 (x / x₀)⌋₊ - 1) * x₀ ≤ x

case ub
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ x ≤ 2 ^ (-↑⌊-logb 2 (x / x₀)⌋₊) * x₀
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ x ≤ 2 ^ (-↑⌊-logb 2 (x / x₀)⌋₊) * x₀

Tactic:
← div_le_iff₀ x₀_pos

Goal state after:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : max x 1 ≤ x₀
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ x / x₀ ≤ 2 ^ (-↑⌊-logb 2 (x / x₀)⌋₊)
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
h₁ : ∑ x, 0 = 0
⊢ Tendsto (fun p ↦ ∑ i, a i * b i ^ p) atTop (𝓝 (∑ x, 0))

Tactic:
refine tendsto_finsetSum (univ : Finset α) (fun i _ => ?_)

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
h₁ : ∑ x, 0 = 0
i : α
x✝ : i ∈ univ
⊢ Tendsto (fun p ↦ a i * b i ^ p) atTop (𝓝 0)
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ 0 < b (max_bi b)

Tactic:
have := R.b_pos (max_bi b)

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
this : 0 < b (max_bi b)
⊢ 0 < b (max_bi b)
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: ⊢ GrowsPolynomially fun x ↦ x

Tactic:
by
  intro b hb
  refine ⟨b, hb.1, ?_⟩
  refine ⟨1, by norm_num, ?_⟩
  filter_upwards with x u hu
  simp only [one_mul, Set.mem_Icc]
  exact ⟨hu.1, hu.2⟩

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ c₂ * g ↑n
hc₁_pos : 0 < c₁
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ c₂ * g ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 ≤ p a b + 1
⊢ ↑n ^ p a b * ∑ u ∈ Ico (r i n) n, c₂ * g ↑n / ↑u ^ (p a b + 1) ≤
    ↑n ^ p a b * ∑ _u ∈ Ico (r i n) n, c₂ * g ↑n / ↑(r i n) ^ (p a b + 1)

Tactic:
gcongr with u hu; rw [Finset.mem_Ico] at hu; exact hu.1

Goal state after:
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |f x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x|) x) (c₂ * (fun x ↦ |f x|) x)
hg :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |g x|) u ∈ Set.Icc (c₁ * (fun x ↦ |g x|) x) (c₂ * (fun x ↦ |g x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x ↦ |f x| * |g x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x| * |g x|) x) (c₂ * (fun x ↦ |f x| * |g x|) x)

Tactic:
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf

Goal state after:
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hg :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |g x|) u ∈ Set.Icc (c₁ * (fun x ↦ |g x|) x) (c₂ * (fun x ↦ |g x|) x)
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |f x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x|) x) (c₂ * (fun x ↦ |f x|) x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x ↦ |f x| * |g x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x| * |g x|) x) (c₂ * (fun x ↦ |f x| * |g x|) x)
Module: Mathlib.Computability.AkraBazzi.GrowsPolynomially

Goal state before:
pp: f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hf : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hg : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₁ * g x) (c₂ * g x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x ↦ f x + g x) u ∈ Set.Icc (c₁ * (fun x ↦ f x + g x) x) (c₂ * (fun x ↦ f x + g x) x)

Tactic:
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf

Goal state after:
f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
hg✝ : GrowsPolynomially g
hf' : 0 ≤ᶠ[atTop] f
hg' : 0 ≤ᶠ[atTop] g
b : ℝ
hb : b ∈ Set.Ioo 0 1
hg : ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, g u ∈ Set.Icc (c₁ * g x) (c₂ * g x)
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun x ↦ f x + g x) u ∈ Set.Icc (c₁ * (fun x ↦ f x + g x) x) (c₂ * (fun x ↦ f x + g x) x)
Module: Mathlib.Computability.PartrecCode

Goal state before:
pp: c : Code
n x : ℕ
⊢ (c.curry n).eval x = c.eval (Nat.pair n x)

Tactic:
simp! [Seq.seq, curry]

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:

Tactic:
sumTransform (p a b) g (r i n) n
      _ = n ^ (p a b) * (∑ u ∈ Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1)) := rfl
      _ ≥ n ^ (p a b) * (∑ u ∈ Finset.Ico (r i n) n, c₂ * g n / u ^ ((p a b) + 1)) := by
        gcongr with u hu
        rw [Finset.mem_Ico] at hu
        have hu' : u ∈ Set.Icc (r i n) n := ⟨hu.1, by lia⟩
        refine hn₂ u ?_
        rw [Set.mem_Icc]
        refine ⟨?_, by norm_cast; lia⟩
        calc c₁ * n ≤ r i n := by exact hn₁ i
                  _ ≤ u := by exact_mod_cast hu'.1
      _ ≥ n ^ (p a b) * (∑ _u ∈ Finset.Ico (r i n) n, c₂ * g n / n ^ ((p a b) + 1)) := by
        gcongr with u hu
        · rw [Finset.mem_Ico] at hu
          have := calc 0 < r i n := hrpos_i
                      _ ≤ u := hu.1
          positivity
        · rw [Finset.mem_Ico] at hu
          exact le_of_lt hu.2
      _ ≥ n ^ p a b * #(Ico (r i n) n) • (c₂ * g n / n ^ (p a b + 1)) := by
        gcongr; exact Finset.card_nsmul_le_sum _ _ _ (fun x _ => by rfl)
      _ = n ^ p a b * #(Ico (r i n) n) * (c₂ * g n / n ^ (p a b + 1)) := by
        rw [nsmul_eq_mul, mul_assoc]
      _ = n ^ (p a b) * (n - r i n) * (c₂ * g n / n ^ ((p a b) + 1)) := by
        congr; rw [Nat.card_Ico, Nat.cast_sub (le_of_lt <| hr_lt_n i)]
      _ ≥ n ^ (p a b) * (n - c₃ * n) * (c₂ * g n / n ^ ((p a b) + 1)) := by
        gcongr; exact hn₃ i
      _ = n ^ (p a b) * n * (1 - c₃) * (c₂ * g n / n ^ ((p a b) + 1)) := by ring
      _ = c₂ * (1 - c₃) * g n * (n ^ ((p a b) + 1) / n ^ ((p a b) + 1)) := by
        rw [← Real.rpow_add_one (by positivity) (p a b)]; ring
      _ = c₂ * (1 - c₃) * g n := by rw [div_self (by positivity), mul_one]
      _ ≥ min (c₂ * (1 - c₃)) ((1 - c₃) * c₂ / c₁ ^ ((p a b) + 1)) * g n := by
        gcongr; exact min_le_left _ _

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ ↑n ^ p a b * ↑(#(Ico (r i n) n)) * (c₂ * g ↑n / ↑(r i n) ^ (p a b + 1)) ≥
    ↑n ^ p a b * ↑(#(Ico (r i n) n)) * (c₂ * g ↑n / (c₁ * ↑n) ^ (p a b + 1))

Tactic:
gcongr n ^ p a b * #(Ico (r i n) n) * (c₂ * g n / ?_)

Goal state after:
case hdb
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ ↑(r i n) ^ (p a b + 1) ≤ (c₁ * ↑n) ^ (p a b + 1)
Module: Mathlib.Data.Nat.Size

Goal state before:
pp: case bit
b✝ : Bool
n✝ : ℕ
h : n✝ = 0 → b✝ = true
ih : n✝.bits.length = n✝.size
⊢ (bit b✝ n✝).bits.length = (bit b✝ n✝).size

Tactic:
| bit _ _ h ih =>

Goal state after:
Module: Mathlib.Data.Nat.Bitwise

Goal state before:
pp: n : ℕ
ih :
  List.foldl (fun x1 x2 ↦ x1 ^^^ x2) 0 (List.range (n + 1)) =
    match Fin.ofNat 4 n with
    | 0 => n
    | 1 => 1
    | 2 => n + 1
    | 3 => 0
h : Fin.ofNat 4 n = 1
⊢ (match 1 with
      | 0 => n
      | 1 => 1
      | 2 => n + 1
      | 3 => 0) ^^^
      n + 1 =
    match 1 + Fin.ofNat 4 1 with
    | 0 => n + 1
    | 1 => 1
    | 2 => n + 1 + 1
    | 3 => 0

Tactic:
rw [Nat.xor_comm]

Goal state after:
n : ℕ
ih :
  List.foldl (fun x1 x2 ↦ x1 ^^^ x2) 0 (List.range (n + 1)) =
    match Fin.ofNat 4 n with
    | 0 => n
    | 1 => 1
    | 2 => n + 1
    | 3 => 0
h : Fin.ofNat 4 n = 1
⊢ (n + 1 ^^^
      match 1 with
      | 0 => n
      | 1 => 1
      | 2 => n + 1
      | 3 => 0) =
    match 1 + Fin.ofNat 4 1 with
    | 0 => n + 1
    | 1 => 1
    | 2 => n + 1 + 1
    | 3 => 0
Module: Mathlib.Computability.NFA

Goal state before:
pp: α : Type u
σ : Type v
M : NFA α σ
S : Set σ
x : List α
⊢ M.evalFrom S x = ⋃ s ∈ S, M.evalFrom {s} x

Tactic:
simp [← evalFrom_iUnion₂]

Goal state after:
Module: Mathlib.Computability.EpsilonNFA

Goal state before:
pp: α : Type u
σ : Type v
M : εNFA α σ
S : Set σ
x : List α
a : α
⊢ M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a

Tactic:
by
  rw [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]

Goal state after:
Module: Mathlib.Data.Num.Lemmas

Goal state before:
pp: n : PosNum
⊢ ↑n.bit0.size = n.bit0.natSize

Tactic:
by rw [size, succ_to_nat, natSize, size_eq_natSize n]

Goal state after:
Module: Mathlib.Data.Num.Lemmas

Goal state before:
pp: n : PosNum
⊢ ↑n.bit1.size = n.bit1.natSize

Tactic:
by rw [size, succ_to_nat, natSize, size_eq_natSize n]

Goal state after:
Module: Mathlib.Data.Num.Lemmas

Goal state before:
pp: m n : PosNum
this : Ordering.casesOn (m.cmp n) (↑m < ↑n) (m = n) (↑n < ↑m)
⊢ m.cmp n = Ordering.eq ↔ m = n

Tactic:
norm_cast at this

Goal state after:
m n : PosNum
this : Ordering.casesOn (m.cmp n) (m < n) (m = n) (n < m)
⊢ m.cmp n = Ordering.eq ↔ m = n
Module: Mathlib.Data.Num.Lemmas

Goal state before:
pp: α : Type u_1
inst✝² : Semiring α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
m n : PosNum
⊢ ¬↑n < ↑m ↔ m ≤ n

Tactic:
exact not_congr cast_lt

Goal state after:
Module: Mathlib.Data.Num.Lemmas

Goal state before:
pp: m n : Num
this : Ordering.casesOn (m.cmp n) (↑m < ↑n) (m = n) (↑n < ↑m)
⊢ m.cmp n = Ordering.eq ↔ m = n

Tactic:
norm_cast at this

Goal state after:
m n : Num
this : Ordering.casesOn (m.cmp n) (m < n) (m = n) (n < m)
⊢ m.cmp n = Ordering.eq ↔ m = n
Module: Mathlib.Data.Num.Lemmas

Goal state before:
pp: α : Type u_1
inst✝² : Semiring α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
m n : Num
⊢ ¬↑n < ↑m ↔ m ≤ n

Tactic:
exact not_congr cast_lt

Goal state after: