file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Computability/NFA.lean
|
NFA.evalFrom_singleton
|
[] |
[
78,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Order/CompleteLattice.lean
|
le_iSup₂_of_le
|
[] |
[
808,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
806,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.ind
|
[] |
[
318,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Topology/Sober.lean
|
isGenericPoint_def
|
[] |
[
42,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Finset.partiallyWellOrderedOn
|
[] |
[
545,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
11
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_fun_eq_lift_mul
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.533809\ninst✝¹⁴ : Ring K\ninst✝¹³ : StrongRankCondition K\ninst✝¹² : AddCommGroup V\ninst✝¹¹ : Module K V\ninst✝¹⁰ : Module.Free K V\ninst✝⁹ : AddCommGroup V'\ninst✝⁸ : Module K V'\ninst✝⁷ : Module.Free K V'\ninst✝⁶ : AddCommGroup V₁\ninst✝⁵ : Module K V₁\ninst✝⁴ : Module.Free K V₁\ninst✝³ : (i : η) → AddCommGroup (φ i)\ninst✝² : (i : η) → Module K (φ i)\ninst✝¹ : ∀ (i : η), Module.Free K (φ i)\ninst✝ : Fintype η\n⊢ Module.rank K (η → V) = ↑(Fintype.card η) * lift (Module.rank K V)",
"tactic": "rw [rank_pi, Cardinal.sum_const, Cardinal.mk_fintype, Cardinal.lift_natCast]"
}
] |
[
1024,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1022,
1
] |
Mathlib/Logic/Equiv/Fintype.lean
|
Function.Embedding.toEquivRange_eq_ofInjective
|
[
{
"state_after": "case H.a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq β\ne : Equiv.Perm α\nf : α ↪ β\nx✝ : α\n⊢ ↑(↑(toEquivRange f) x✝) = ↑(↑(Equiv.ofInjective ↑f (_ : Injective ↑f)) x✝)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq β\ne : Equiv.Perm α\nf : α ↪ β\n⊢ toEquivRange f = Equiv.ofInjective ↑f (_ : Injective ↑f)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H.a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq β\ne : Equiv.Perm α\nf : α ↪ β\nx✝ : α\n⊢ ↑(↑(toEquivRange f) x✝) = ↑(↑(Equiv.ofInjective ↑f (_ : Injective ↑f)) x✝)",
"tactic": "simp"
}
] |
[
59,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.eq_of_associated_left
|
[] |
[
340,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isClopen_biUnion_finset
|
[] |
[
1597,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1595,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
eventually_nhds_norm_smul_sub_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.15386\nγ : Type ?u.15389\nι : Type ?u.15392\ninst✝⁵ : NormedField α\ninst✝⁴ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace α E\nF : Type ?u.15423\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace α F\nc : α\nx : E\nε : ℝ\nh : 0 < ε\n⊢ ‖c • (id x - x)‖ = 0",
"tactic": "simp"
}
] |
[
95,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Data/Int/Parity.lean
|
Int.not_even_bit1
|
[
{
"state_after": "no goals",
"state_before": "m n✝ n : ℤ\n⊢ ¬Even (bit1 n)",
"tactic": "simp [bit1, parity_simps]"
}
] |
[
118,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec₂.unpaired'
|
[] |
[
445,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.mk_finsupp_lift_of_infinite
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ (#α →₀ β) ≤ max (lift (#α)) (lift (#β))\n\ncase a\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ max (lift (#α)) (lift (#β)) ≤ (#α →₀ β)",
"state_before": "α : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ (#α →₀ β) = max (lift (#α)) (lift (#β))",
"tactic": "apply le_antisymm"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ (#α →₀ β) ≤ max (lift (#α)) (lift (#β))",
"tactic": "calc\n (#α →₀ β) ≤ (#Finset (α × β)) := mk_le_of_injective (Finsupp.graph_injective α β)\n _ = (#α × β) := mk_finset_of_infinite _\n _ = max (lift.{v} (#α)) (lift.{u} (#β)) :=\n by rw [mk_prod, mul_eq_max_of_aleph0_le_left] <;> simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ (#α × β) = max (lift (#α)) (lift (#β))",
"tactic": "rw [mk_prod, mul_eq_max_of_aleph0_le_left] <;> simp"
},
{
"state_after": "case a.h₁\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ lift (#α) ≤ lift (#α →₀ β)\n\ncase a.h₂\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ lift (#β) ≤ lift (#α →₀ β)",
"state_before": "case a\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ max (lift (#α)) (lift (#β)) ≤ (#α →₀ β)",
"tactic": "apply max_le <;> rw [← lift_id (#α →₀ β), ← lift_umax]"
},
{
"state_after": "case a.h₁.intro\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\nb : β\nhb : b ≠ 0\n⊢ lift (#α) ≤ lift (#α →₀ β)",
"state_before": "case a.h₁\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ lift (#α) ≤ lift (#α →₀ β)",
"tactic": "cases' exists_ne (0 : β) with b hb"
},
{
"state_after": "no goals",
"state_before": "case a.h₁.intro\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\nb : β\nhb : b ≠ 0\n⊢ lift (#α) ≤ lift (#α →₀ β)",
"tactic": "exact lift_mk_le.{u, max u v, v}.2 ⟨⟨_, Finsupp.single_left_injective hb⟩⟩"
},
{
"state_after": "case a.h₂\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\ninhabited_h : Inhabited α\n⊢ lift (#β) ≤ lift (#α →₀ β)",
"state_before": "case a.h₂\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\n⊢ lift (#β) ≤ lift (#α →₀ β)",
"tactic": "inhabit α"
},
{
"state_after": "no goals",
"state_before": "case a.h₂\nα : Type u\nβ : Type v\ninst✝² : Infinite α\ninst✝¹ : Zero β\ninst✝ : Nontrivial β\ninhabited_h : Inhabited α\n⊢ lift (#β) ≤ lift (#α →₀ β)",
"tactic": "exact lift_mk_le.{v, max u v, u}.2 ⟨⟨_, Finsupp.single_injective default⟩⟩"
}
] |
[
1046,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1032,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.disjoint_iff_ne
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.45481\nγ : Type ?u.45484\nf : α → β\ns t u : Finset α\na b : α\n⊢ _root_.Disjoint s t ↔ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b",
"tactic": "simp only [disjoint_left, imp_not_comm, forall_eq']"
}
] |
[
923,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/CategoryTheory/Types.lean
|
CategoryTheory.Iso.toEquiv_comp
|
[] |
[
372,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
IsOpen.exists_iUnion_isClosed
|
[
{
"state_after": "case intro.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\n⊢ ∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ (⋃ (n : ℕ), F n) = U ∧ Monotone F",
"state_before": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\n⊢ ∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ (⋃ (n : ℕ), F n) = U ∧ Monotone F",
"tactic": "obtain ⟨a, a_pos, a_lt_one⟩ : ∃ a : ℝ≥0∞, 0 < a ∧ a < 1 := exists_between zero_lt_one"
},
{
"state_after": "case intro.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\n⊢ ∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ (⋃ (n : ℕ), F n) = U ∧ Monotone F",
"state_before": "case intro.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\n⊢ ∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ (⋃ (n : ℕ), F n) = U ∧ Monotone F",
"tactic": "let F := fun n : ℕ => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)"
},
{
"state_after": "case intro.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ ∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ (⋃ (n : ℕ), F n) = U ∧ Monotone F",
"state_before": "case intro.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\n⊢ ∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ (⋃ (n : ℕ), F n) = U ∧ Monotone F",
"tactic": "have F_subset : ∀ n, F n ⊆ U := fun n x hx ↦ by\n by_contra h\n have : infEdist x (Uᶜ) ≠ 0 := ((ENNReal.pow_pos a_pos _).trans_le hx).ne'\n exact this (infEdist_zero_of_mem h)"
},
{
"state_after": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ (⋃ (n : ℕ), F n) = U\n\ncase intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ Monotone F",
"state_before": "case intro.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ ∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ (⋃ (n : ℕ), F n) = U ∧ Monotone F",
"tactic": "refine ⟨F, fun n => IsClosed.preimage continuous_infEdist isClosed_Ici, F_subset, ?_, ?_⟩"
},
{
"state_after": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ (⋃ (n : ℕ), F n) = U\n\ncase intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ Monotone F",
"state_before": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ (⋃ (n : ℕ), F n) = U\n\ncase intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ Monotone F",
"tactic": "show (⋃ n, F n) = U"
},
{
"state_after": "case intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ Monotone F",
"state_before": "case intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ Monotone F",
"tactic": "show Monotone F"
},
{
"state_after": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nn : ℕ\nx : α\nhx : x ∈ F n\nh : ¬x ∈ U\n⊢ False",
"state_before": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nn : ℕ\nx : α\nhx : x ∈ F n\n⊢ x ∈ U",
"tactic": "by_contra h"
},
{
"state_after": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nn : ℕ\nx : α\nhx : x ∈ F n\nh : ¬x ∈ U\nthis : infEdist x (Uᶜ) ≠ 0\n⊢ False",
"state_before": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nn : ℕ\nx : α\nhx : x ∈ F n\nh : ¬x ∈ U\n⊢ False",
"tactic": "have : infEdist x (Uᶜ) ≠ 0 := ((ENNReal.pow_pos a_pos _).trans_le hx).ne'"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nn : ℕ\nx : α\nhx : x ∈ F n\nh : ¬x ∈ U\nthis : infEdist x (Uᶜ) ≠ 0\n⊢ False",
"tactic": "exact this (infEdist_zero_of_mem h)"
},
{
"state_after": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\n⊢ x ∈ ⋃ (n : ℕ), F n",
"state_before": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ (⋃ (n : ℕ), F n) = U",
"tactic": "refine' Subset.antisymm (by simp only [iUnion_subset_iff, F_subset, forall_const]) fun x hx => _"
},
{
"state_after": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬x ∈ Uᶜ\n⊢ x ∈ ⋃ (n : ℕ), F n",
"state_before": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\n⊢ x ∈ ⋃ (n : ℕ), F n",
"tactic": "have : ¬x ∈ Uᶜ := by simpa using hx"
},
{
"state_after": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬infEdist x (Uᶜ) = 0\n⊢ x ∈ ⋃ (n : ℕ), F n",
"state_before": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬x ∈ Uᶜ\n⊢ x ∈ ⋃ (n : ℕ), F n",
"tactic": "rw [mem_iff_infEdist_zero_of_closed hU.isClosed_compl] at this"
},
{
"state_after": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\n⊢ x ∈ ⋃ (n : ℕ), F n",
"state_before": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬infEdist x (Uᶜ) = 0\n⊢ x ∈ ⋃ (n : ℕ), F n",
"tactic": "have B : 0 < infEdist x (Uᶜ) := by simpa [pos_iff_ne_zero] using this"
},
{
"state_after": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis✝ : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\nthis : Tendsto (fun n => a ^ n) atTop (𝓝 0)\n⊢ x ∈ ⋃ (n : ℕ), F n",
"state_before": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\n⊢ x ∈ ⋃ (n : ℕ), F n",
"tactic": "have : Filter.Tendsto (fun n => a ^ n) atTop (𝓝 0) :=\n ENNReal.tendsto_pow_atTop_nhds_0_of_lt_1 a_lt_one"
},
{
"state_after": "case intro.intro.refine_1.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis✝ : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\nthis : Tendsto (fun n => a ^ n) atTop (𝓝 0)\nn : ℕ\nhn : a ^ n < infEdist x (Uᶜ)\n⊢ x ∈ ⋃ (n : ℕ), F n",
"state_before": "case intro.intro.refine_1\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis✝ : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\nthis : Tendsto (fun n => a ^ n) atTop (𝓝 0)\n⊢ x ∈ ⋃ (n : ℕ), F n",
"tactic": "rcases((tendsto_order.1 this).2 _ B).exists with ⟨n, hn⟩"
},
{
"state_after": "case intro.intro.refine_1.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis✝ : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\nthis : Tendsto (fun n => a ^ n) atTop (𝓝 0)\nn : ℕ\nhn : a ^ n < infEdist x (Uᶜ)\n⊢ ∃ i, a ^ i ≤ infEdist x (Uᶜ)",
"state_before": "case intro.intro.refine_1.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis✝ : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\nthis : Tendsto (fun n => a ^ n) atTop (𝓝 0)\nn : ℕ\nhn : a ^ n < infEdist x (Uᶜ)\n⊢ x ∈ ⋃ (n : ℕ), F n",
"tactic": "simp only [mem_iUnion, mem_Ici, mem_preimage]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine_1.intro\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis✝ : ¬infEdist x (Uᶜ) = 0\nB : 0 < infEdist x (Uᶜ)\nthis : Tendsto (fun n => a ^ n) atTop (𝓝 0)\nn : ℕ\nhn : a ^ n < infEdist x (Uᶜ)\n⊢ ∃ i, a ^ i ≤ infEdist x (Uᶜ)",
"tactic": "exact ⟨n, hn.le⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ (⋃ (n : ℕ), F n) ⊆ U",
"tactic": "simp only [iUnion_subset_iff, F_subset, forall_const]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\n⊢ ¬x ∈ Uᶜ",
"tactic": "simpa using hx"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nx : α\nhx : x ∈ U\nthis : ¬infEdist x (Uᶜ) = 0\n⊢ 0 < infEdist x (Uᶜ)",
"tactic": "simpa [pos_iff_ne_zero] using this"
},
{
"state_after": "case intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nm n : ℕ\nhmn : m ≤ n\nx : α\nhx : x ∈ F m\n⊢ x ∈ F n",
"state_before": "case intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\n⊢ Monotone F",
"tactic": "intro m n hmn x hx"
},
{
"state_after": "case intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nm n : ℕ\nhmn : m ≤ n\nx : α\nhx : a ^ m ≤ infEdist x (Uᶜ)\n⊢ a ^ n ≤ infEdist x (Uᶜ)",
"state_before": "case intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nm n : ℕ\nhmn : m ≤ n\nx : α\nhx : x ∈ F m\n⊢ x ∈ F n",
"tactic": "simp only [mem_Ici, mem_preimage] at hx⊢"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine_2\nι : Sort ?u.21246\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx✝ y : α\ns t : Set α\nΦ : α → β\nU : Set α\nhU : IsOpen U\na : ℝ≥0∞\na_pos : 0 < a\na_lt_one : a < 1\nF : ℕ → Set α := fun n => (fun x => infEdist x (Uᶜ)) ⁻¹' Ici (a ^ n)\nF_subset : ∀ (n : ℕ), F n ⊆ U\nm n : ℕ\nhmn : m ≤ n\nx : α\nhx : a ^ m ≤ infEdist x (Uᶜ)\n⊢ a ^ n ≤ infEdist x (Uᶜ)",
"tactic": "apply le_trans (pow_le_pow_of_le_one' a_lt_one.le hmn) hx"
}
] |
[
229,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMin.comp_continuousOn
|
[] |
[
321,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
openSegment_subset_union
|
[
{
"state_after": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc : 𝕜\n⊢ openSegment 𝕜 x y ⊆\n insert (↑(lineMap x y) c) (openSegment 𝕜 x (↑(lineMap x y) c) ∪ openSegment 𝕜 (↑(lineMap x y) c) y)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z✝ x y z : E\nhz : z ∈ range ↑(lineMap x y)\n⊢ openSegment 𝕜 x y ⊆ insert z (openSegment 𝕜 x z ∪ openSegment 𝕜 z y)",
"tactic": "rcases hz with ⟨c, rfl⟩"
},
{
"state_after": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc : 𝕜\n⊢ MapsTo (fun a => ↑(lineMap x y) a) (Ioo 0 1)\n (insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1))",
"state_before": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc : 𝕜\n⊢ openSegment 𝕜 x y ⊆\n insert (↑(lineMap x y) c) (openSegment 𝕜 x (↑(lineMap x y) c) ∪ openSegment 𝕜 (↑(lineMap x y) c) y)",
"tactic": "simp only [openSegment_eq_image_lineMap, ← mapsTo']"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1)",
"state_before": "case intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc : 𝕜\n⊢ MapsTo (fun a => ↑(lineMap x y) a) (Ioo 0 1)\n (insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1))",
"tactic": "rintro a ⟨h₀, h₁⟩"
},
{
"state_after": "case intro.intro.inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1)\n\ncase intro.intro.inr.inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\na : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) a)\n ((fun a_1 => ↑(lineMap x (↑(lineMap x y) a)) a_1) '' Ioo 0 1 ∪\n (fun a_1 => ↑(lineMap (↑(lineMap x y) a) y) a_1) '' Ioo 0 1)\n\ncase intro.intro.inr.inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1)",
"state_before": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1)",
"tactic": "rcases lt_trichotomy a c with (hac | rfl | hca)"
},
{
"state_after": "case intro.intro.inl.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"state_before": "case intro.intro.inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1)",
"tactic": "right"
},
{
"state_after": "case intro.intro.inl.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1",
"state_before": "case intro.intro.inl.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"tactic": "left"
},
{
"state_after": "case intro.intro.inl.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\nhc : 0 < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1",
"state_before": "case intro.intro.inl.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1",
"tactic": "have hc : 0 < c := h₀.trans hac"
},
{
"state_after": "case intro.intro.inl.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\nhc : 0 < c\n⊢ (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) (a / c) = (fun a => ↑(lineMap x y) a) a",
"state_before": "case intro.intro.inl.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\nhc : 0 < c\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1",
"tactic": "refine' ⟨a / c, ⟨div_pos h₀ hc, (div_lt_one hc).2 hac⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhac : a < c\nhc : 0 < c\n⊢ (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) (a / c) = (fun a => ↑(lineMap x y) a) a",
"tactic": "simp only [← homothety_eq_lineMap, ← homothety_mul_apply, div_mul_cancel _ hc.ne']"
},
{
"state_after": "case intro.intro.inr.inl.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\na : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\n⊢ (fun a => ↑(lineMap x y) a) a = ↑(lineMap x y) a",
"state_before": "case intro.intro.inr.inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\na : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) a)\n ((fun a_1 => ↑(lineMap x (↑(lineMap x y) a)) a_1) '' Ioo 0 1 ∪\n (fun a_1 => ↑(lineMap (↑(lineMap x y) a) y) a_1) '' Ioo 0 1)",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inl.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\na : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\n⊢ (fun a => ↑(lineMap x y) a) a = ↑(lineMap x y) a",
"tactic": "rfl"
},
{
"state_after": "case intro.intro.inr.inr.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"state_before": "case intro.intro.inr.inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n insert (↑(lineMap x y) c)\n ((fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪\n (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1)",
"tactic": "right"
},
{
"state_after": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"state_before": "case intro.intro.inr.inr.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\n⊢ (fun a => ↑(lineMap x y) a) a ∈\n (fun a => ↑(lineMap x (↑(lineMap x y) c)) a) '' Ioo 0 1 ∪ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"tactic": "right"
},
{
"state_after": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\nhc : 0 < 1 - c\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"state_before": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"tactic": "have hc : 0 < 1 - c := sub_pos.2 (hca.trans h₁)"
},
{
"state_after": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\nhc : 0 < 1 - c\n⊢ ↑(lineMap y x) (1 - a) ∈ (fun a => ↑(lineMap y (↑(lineMap y x) (1 - c))) (1 - a)) '' Ioo 0 1",
"state_before": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\nhc : 0 < 1 - c\n⊢ (fun a => ↑(lineMap x y) a) a ∈ (fun a => ↑(lineMap (↑(lineMap x y) c) y) a) '' Ioo 0 1",
"tactic": "simp only [← lineMap_apply_one_sub y]"
},
{
"state_after": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\nhc : 0 < 1 - c\n⊢ (fun a => ↑(lineMap y (↑(lineMap y x) (1 - c))) (1 - a)) ((a - c) / (1 - c)) = ↑(lineMap y x) (1 - a)",
"state_before": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\nhc : 0 < 1 - c\n⊢ ↑(lineMap y x) (1 - a) ∈ (fun a => ↑(lineMap y (↑(lineMap y x) (1 - c))) (1 - a)) '' Ioo 0 1",
"tactic": "refine'\n ⟨(a - c) / (1 - c), ⟨div_pos (sub_pos.2 hca) hc, (div_lt_one hc).2 <| sub_lt_sub_right h₁ _⟩,\n _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr.inr.h.h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.220883\nG : Type ?u.220886\nι : Type ?u.220889\nπ : ι → Type ?u.220894\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx✝ y✝ z x y : E\nc a : 𝕜\nh₀ : 0 < a\nh₁ : a < 1\nhca : c < a\nhc : 0 < 1 - c\n⊢ (fun a => ↑(lineMap y (↑(lineMap y x) (1 - c))) (1 - a)) ((a - c) / (1 - c)) = ↑(lineMap y x) (1 - a)",
"tactic": "simp only [← homothety_eq_lineMap, ← homothety_mul_apply, sub_mul, one_mul,\n div_mul_cancel _ hc.ne', sub_sub_sub_cancel_right]"
}
] |
[
409,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
OpenSubgroup.mem_comap
|
[] |
[
293,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Cardinal.iSup_lt_of_isRegular
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.160773\nr : α → α → Prop\nι : Type u_1\nf : ι → Cardinal\nc : Cardinal\nhc : IsRegular c\nhι : (#ι) < c\n⊢ (#ι) < Ordinal.cof (ord c)",
"tactic": "rwa [hc.cof_eq]"
}
] |
[
1110,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1108,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.IsDiag.map
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nR : Type ?u.2825\nn : Type u_3\nm : Type ?u.2831\ninst✝¹ : Zero α\ninst✝ : Zero β\nA : Matrix n n α\nha : IsDiag A\nf : α → β\nhf : f 0 = 0\ni j : n\nh : i ≠ j\n⊢ Matrix.map A f i j = 0",
"state_before": "α : Type u_1\nβ : Type u_2\nR : Type ?u.2825\nn : Type u_3\nm : Type ?u.2831\ninst✝¹ : Zero α\ninst✝ : Zero β\nA : Matrix n n α\nha : IsDiag A\nf : α → β\nhf : f 0 = 0\n⊢ IsDiag (Matrix.map A f)",
"tactic": "intro i j h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nR : Type ?u.2825\nn : Type u_3\nm : Type ?u.2831\ninst✝¹ : Zero α\ninst✝ : Zero β\nA : Matrix n n α\nha : IsDiag A\nf : α → β\nhf : f 0 = 0\ni j : n\nh : i ≠ j\n⊢ Matrix.map A f i j = 0",
"tactic": "simp [ha h, hf]"
}
] |
[
82,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
nsmul_eq_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18224\nι : Type ?u.18227\nπ : ι → Type ?u.18232\ninst✝ : IdemSemiring α\na✝ b c : α\nn : ℕ\nx✝ : n + 2 ≠ 0\na : α\n⊢ (n + 2) • a = a",
"tactic": "rw [succ_nsmul, nsmul_eq_self n.succ_ne_zero, add_idem]"
}
] |
[
150,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometry.map_eq_iff
|
[] |
[
169,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.snd_surjective
|
[] |
[
90,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.zero_apply
|
[] |
[
320,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/NumberTheory/Multiplicity.lean
|
odd_sq_dvd_geom_sum₂_sub
|
[
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 : ∀ (i : ℕ), ↑p ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\n⊢ ↑p ^ 2 ∣ ∑ i in range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i) - ↑p * a ^ (p - 1)",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\n⊢ ↑p ^ 2 ∣ ∑ i in range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i) - ↑p * a ^ (p - 1)",
"tactic": "have h1 : ∀ (i : ℕ),\n (p : R) ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * i + a ^ i) := by\n intro i\n calc\n ↑p ^ 2 ∣ (↑p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]\n _ ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) := by\n simp only [sq_dvd_add_pow_sub_sub (↑p * b) a i, ← sub_sub]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ i in range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i)) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * a ^ (p - 1))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 : ∀ (i : ℕ), ↑p ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\n⊢ ↑p ^ 2 ∣ ∑ i in range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i) - ↑p * a ^ (p - 1)",
"tactic": "simp_rw [← mem_span_singleton, ← Ideal.Quotient.eq] at *"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ i in range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i)) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * a ^ (p - 1))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ i in range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i)) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * a ^ (p - 1))",
"tactic": "let s : R := (p : R)^2"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\ni : ℕ\n⊢ ↑p ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\n⊢ ∀ (i : ℕ), ↑p ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\ni : ℕ\n⊢ ↑p ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)",
"tactic": "calc\n ↑p ^ 2 ∣ (↑p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]\n _ ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) := by\n simp only [sq_dvd_add_pow_sub_sub (↑p * b) a i, ← sub_sub]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\ni : ℕ\n⊢ ↑p ^ 2 ∣ (↑p * b) ^ 2",
"tactic": "simp only [mul_pow, dvd_mul_right]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\ni : ℕ\n⊢ (↑p * b) ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)",
"tactic": "simp only [sq_dvd_add_pow_sub_sub (↑p * b) a i, ← sub_sub]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ i in range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i)) =\n ∑ i in range p, ↑(Ideal.Quotient.mk (span {s})) ((a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) * a ^ (p - 1 - i))",
"tactic": "simp_rw [RingHom.map_geom_sum₂, ← map_pow, h1, ← _root_.map_mul]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∑ x in range p,\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (x - 1) * a ^ (p - 1 - x) * ↑p * b * ↑x + a ^ x * a ^ (p - 1 - x)) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x + (p - 1 - x)))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∑ i in range p, ↑(Ideal.Quotient.mk (span {s})) ((a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) * a ^ (p - 1 - i)) =\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x + (p - 1 - x)))",
"tactic": "ring"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x + (p - 1 - x))) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x + (p - 1 - x)))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∑ x in range p,\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (x - 1) * a ^ (p - 1 - x) * ↑p * b * ↑x + a ^ x * a ^ (p - 1 - x)) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x + (p - 1 - x)))",
"tactic": "simp only [← pow_add, map_add, Finset.sum_add_distrib, ← map_sum]"
},
{
"state_after": "case e_a.h.e_6.h.e_f\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ (fun x => a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x) = fun x => a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x + (p - 1 - x))) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x + (p - 1 - x)))",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_a.h.e_6.h.e_f\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ (fun x => a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x) = fun x => a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))",
"tactic": "simp [pow_add a, mul_assoc]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x + (p - 1 - x))) =\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (p - 1))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x + (p - 1 - x))) =\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (p - 1))",
"tactic": "rw [add_right_inj]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∀ (x : ℕ), x ∈ range p → a ^ (x + (p - 1 - x)) = a ^ (p - 1)\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x + (p - 1 - x))) =\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (p - 1))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x + (p - 1 - x))) =\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (p - 1))",
"tactic": "have : ∀ (x : ℕ), (hx : x ∈ range p) → a ^ (x + (p - 1 - x)) = a ^ (p - 1) := by\n intro x hx\n rw [← Nat.add_sub_assoc _ x, Nat.add_sub_cancel_left]\n exact Nat.le_pred_of_lt (Finset.mem_range.mp hx)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∀ (x : ℕ), x ∈ range p → a ^ (x + (p - 1 - x)) = a ^ (p - 1)\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x + (p - 1 - x))) =\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (p - 1))",
"tactic": "rw [Finset.sum_congr rfl this]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : x ∈ range p\n⊢ a ^ (x + (p - 1 - x)) = a ^ (p - 1)",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∀ (x : ℕ), x ∈ range p → a ^ (x + (p - 1 - x)) = a ^ (p - 1)",
"tactic": "intro x hx"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : x ∈ range p\n⊢ x ≤ p - 1",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : x ∈ range p\n⊢ a ^ (x + (p - 1 - x)) = a ^ (p - 1)",
"tactic": "rw [← Nat.add_sub_assoc _ x, Nat.add_sub_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : x ∈ range p\n⊢ x ≤ p - 1",
"tactic": "exact Nat.le_pred_of_lt (Finset.mem_range.mp hx)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (p - 1)) =\n ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1))",
"tactic": "simp only [add_right_inj, Finset.sum_const, Finset.card_range, nsmul_eq_mul]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * a ^ (p - 1)) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, ↑p * b * a ^ (p - 2) * ↑x) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * a ^ (p - 1))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {s})) (∑ x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1)) =\n ↑(Ideal.Quotient.mk (span {s})) (↑p * b * ∑ x in range p, a ^ (p - 2) * ↑x) +\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1))",
"tactic": "simp only [Finset.mul_sum, ← mul_assoc, ← pow_add]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∀ (x : ℕ), x ∈ range p → a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x = ↑p * b * a ^ (p - 2) * ↑x",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * a ^ (p - 1)) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (∑ x in range p, ↑p * b * a ^ (p - 2) * ↑x) +\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * a ^ (p - 1))",
"tactic": "rw [Finset.sum_congr rfl]"
},
{
"state_after": "case zero\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nhx : Nat.zero ∈ range p\n⊢ a ^ (Nat.zero - 1 + (p - 1 - Nat.zero)) * ↑p * b * ↑Nat.zero = ↑p * b * a ^ (p - 2) * ↑Nat.zero\n\ncase succ\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\n⊢ a ^ (Nat.succ x - 1 + (p - 1 - Nat.succ x)) * ↑p * b * ↑(Nat.succ x) = ↑p * b * a ^ (p - 2) * ↑(Nat.succ x)",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∀ (x : ℕ), x ∈ range p → a ^ (x - 1 + (p - 1 - x)) * ↑p * b * ↑x = ↑p * b * a ^ (p - 2) * ↑x",
"tactic": "rintro (⟨⟩ | ⟨x⟩) hx"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nhx : Nat.zero ∈ range p\n⊢ a ^ (Nat.zero - 1 + (p - 1 - Nat.zero)) * ↑p * b * ↑Nat.zero = ↑p * b * a ^ (p - 2) * ↑Nat.zero",
"tactic": "rw [Nat.cast_zero, MulZeroClass.mul_zero, MulZeroClass.mul_zero]"
},
{
"state_after": "case succ\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\nthis : Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2\n⊢ a ^ (Nat.succ x - 1 + (p - 1 - Nat.succ x)) * ↑p * b * ↑(Nat.succ x) = ↑p * b * a ^ (p - 2) * ↑(Nat.succ x)",
"state_before": "case succ\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\n⊢ a ^ (Nat.succ x - 1 + (p - 1 - Nat.succ x)) * ↑p * b * ↑(Nat.succ x) = ↑p * b * a ^ (p - 2) * ↑(Nat.succ x)",
"tactic": "have : x.succ - 1 + (p - 1 - x.succ) = p - 2 := by\n rw [← Nat.add_sub_assoc (Nat.le_pred_of_lt (Finset.mem_range.mp hx))]\n exact congr_arg Nat.pred (Nat.add_sub_cancel_left _ _)"
},
{
"state_after": "case succ\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\nthis : Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2\n⊢ a ^ (p - 2) * ↑p * b * ↑(Nat.succ x) = ↑p * b * a ^ (p - 2) * ↑(Nat.succ x)",
"state_before": "case succ\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\nthis : Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2\n⊢ a ^ (Nat.succ x - 1 + (p - 1 - Nat.succ x)) * ↑p * b * ↑(Nat.succ x) = ↑p * b * a ^ (p - 2) * ↑(Nat.succ x)",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\nthis : Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2\n⊢ a ^ (p - 2) * ↑p * b * ↑(Nat.succ x) = ↑p * b * a ^ (p - 2) * ↑(Nat.succ x)",
"tactic": "ring1"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\n⊢ Nat.succ x - 1 + (p - 1) - Nat.succ x = p - 2",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\n⊢ Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2",
"tactic": "rw [← Nat.add_sub_assoc (Nat.le_pred_of_lt (Finset.mem_range.mp hx))]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nx : ℕ\nhx : Nat.succ x ∈ range p\n⊢ Nat.succ x - 1 + (p - 1) - Nat.succ x = p - 2",
"tactic": "exact congr_arg Nat.pred (Nat.add_sub_cancel_left _ _)"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {s})) (↑p * b * ∑ x in range p, a ^ (p - 2) * ↑x) +\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1)) =\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1))",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ↑(Ideal.Quotient.mk (span {s})) (↑p * b * ∑ x in range p, a ^ (p - 2) * ↑x) +\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1)) =\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1))",
"tactic": "have : Finset.sum (range p) (fun (x : ℕ) ↦ (x : R)) =\n ((Finset.sum (range p) (fun (x : ℕ) ↦ (x : ℕ)))) := by simp only [Nat.cast_sum]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(∑ x in range p, x))) = 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {s})) (↑p * b * ∑ x in range p, a ^ (p - 2) * ↑x) +\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1)) =\n ↑(Ideal.Quotient.mk (span {s})) (↑p * a ^ (p - 1))",
"tactic": "simp only [add_left_eq_self, ← Finset.mul_sum, this]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(∑ x in range p, x))) = 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(∑ x in range p, x))) = 0",
"tactic": "norm_cast"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(p * (p - 1) / 2))) = 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(∑ x in range p, x))) = 0",
"tactic": "simp only [Finset.sum_range_id]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(p * (p - 1) / 2))) = 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(p * (p - 1) / 2))) = 0",
"tactic": "norm_cast"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) b *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (p - 2)) *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑((p - 1) / 2))) =\n 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (↑p * b * (a ^ (p - 2) * ↑(p * (p - 1) / 2))) = 0",
"tactic": "simp only [Nat.cast_mul, _root_.map_mul,\n Nat.mul_div_assoc p (even_iff_two_dvd.mp (Nat.Odd.sub_odd hp odd_one))]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p ^ 2 * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) b *\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (p - 2)) *\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑((p - 1) / 2) =\n 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) b *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (p - 2)) *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑((p - 1) / 2))) =\n 0",
"tactic": "ring"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p ^ 2 *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) b *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (p - 2)) * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑((p - 1) / 2))) =\n 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p ^ 2 * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) b *\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (p - 2)) *\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑((p - 1) / 2) =\n 0",
"tactic": "rw [mul_assoc, mul_assoc]"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p ^ 2 = 0",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p ^ 2 *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) b *\n (↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (p - 2)) * ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑((p - 1) / 2))) =\n 0",
"tactic": "refine' mul_eq_zero_of_left _ _"
},
{
"state_after": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ (fun x => x ^ 2) ↑p ∈ span {↑p ^ 2}",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ↑p ^ 2 = 0",
"tactic": "refine' Ideal.Quotient.eq_zero_iff_mem.mpr _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x in range p, ↑x = ↑(∑ x in range p, x)\n⊢ (fun x => x ^ 2) ↑p ∈ span {↑p ^ 2}",
"tactic": "simp [mem_span_singleton]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x y : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n ↑(Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∑ x in range p, ↑x = ↑(∑ x in range p, x)",
"tactic": "simp only [Nat.cast_sum]"
}
] |
[
152,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Topology/Constructions.lean
|
isOpenMap_sigma_map
|
[] |
[
1555,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1552,
1
] |
Mathlib/RingTheory/Finiteness.lean
|
Submodule.fg_iff_add_subgroup_fg
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.2420\nM : Type ?u.2423\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nG : Type u_1\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : FG P\nS : Finset G\nhS : span ℤ ↑S = P\n⊢ AddSubgroup.closure ↑S = toAddSubgroup P",
"tactic": "simpa [← span_int_eq_addSubgroup_closure] using hS"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.2420\nM : Type ?u.2423\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nG : Type u_1\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : AddSubgroup.FG (toAddSubgroup P)\nS : Finset G\nhS : AddSubgroup.closure ↑S = toAddSubgroup P\n⊢ span ℤ ↑S = P",
"tactic": "simpa [← span_int_eq_addSubgroup_closure] using hS"
}
] |
[
68,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.monotoneOn_iff_monotone
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\n⊢ MonotoneOn f s ↔ Monotone fun a => f ↑a",
"tactic": "simp [Monotone, MonotoneOn]"
}
] |
[
2653,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2651,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
SMul.comp.isScalarTower
|
[
{
"state_after": "M : Type ?u.9269\nN : Type ?u.9272\nG : Type ?u.9275\nA : Type ?u.9278\nB : Type ?u.9281\nα : Type ?u.9284\nβ : Type ?u.9287\nγ : Type ?u.9290\nδ : Type ?u.9293\ninst✝³ : SMul M α\ninst✝² : SMul M β\ninst✝¹ : SMul α β\ninst✝ : IsScalarTower M α β\ng : N → M\nthis : SMul N α\n⊢ Sort ?u.9343",
"state_before": "M : Type ?u.9269\nN : Type ?u.9272\nG : Type ?u.9275\nA : Type ?u.9278\nB : Type ?u.9281\nα : Type ?u.9284\nβ : Type ?u.9287\nγ : Type ?u.9290\nδ : Type ?u.9293\ninst✝³ : SMul M α\ninst✝² : SMul M β\ninst✝¹ : SMul α β\ninst✝ : IsScalarTower M α β\ng : N → M\n⊢ Sort ?u.9343",
"tactic": "haveI := comp α g"
},
{
"state_after": "M : Type ?u.9269\nN : Type ?u.9272\nG : Type ?u.9275\nA : Type ?u.9278\nB : Type ?u.9281\nα : Type ?u.9284\nβ : Type ?u.9287\nγ : Type ?u.9290\nδ : Type ?u.9293\ninst✝³ : SMul M α\ninst✝² : SMul M β\ninst✝¹ : SMul α β\ninst✝ : IsScalarTower M α β\ng : N → M\nthis✝ : SMul N α\nthis : SMul N β\n⊢ Sort ?u.9343",
"state_before": "M : Type ?u.9269\nN : Type ?u.9272\nG : Type ?u.9275\nA : Type ?u.9278\nB : Type ?u.9281\nα : Type ?u.9284\nβ : Type ?u.9287\nγ : Type ?u.9290\nδ : Type ?u.9293\ninst✝³ : SMul M α\ninst✝² : SMul M β\ninst✝¹ : SMul α β\ninst✝ : IsScalarTower M α β\ng : N → M\nthis : SMul N α\n⊢ Sort ?u.9343",
"tactic": "haveI := comp β g"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.9269\nN : Type ?u.9272\nG : Type ?u.9275\nA : Type ?u.9278\nB : Type ?u.9281\nα : Type ?u.9284\nβ : Type ?u.9287\nγ : Type ?u.9290\nδ : Type ?u.9293\ninst✝³ : SMul M α\ninst✝² : SMul M β\ninst✝¹ : SMul α β\ninst✝ : IsScalarTower M α β\ng : N → M\nthis✝ : SMul N α\nthis : SMul N β\n⊢ Sort ?u.9343",
"tactic": "exact IsScalarTower N α β"
}
] |
[
370,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin
|
[] |
[
548,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/CategoryTheory/Subobject/Basic.lean
|
CategoryTheory.Subobject.pullback_map_self
|
[
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ ∀ (g : Subobject X), (pullback f).obj ((map f).obj g) = g",
"state_before": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nf : X ⟶ Y\ninst✝ : Mono f\ng : Subobject X\n⊢ (pullback f).obj ((map f).obj g) = g",
"tactic": "revert g"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ ∀ (g : Subobject X), (pullback f).obj ((map f).obj g) = g",
"tactic": "exact Quotient.ind (fun g' => Quotient.sound ⟨(MonoOver.pullbackMapSelf f).app _⟩)"
}
] |
[
645,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
|
Finsupp.toFreeAbelianGroup_comp_toFinsupp
|
[
{
"state_after": "case H\nX : Type u_1\nx✝ : X\n⊢ ↑(AddMonoidHom.comp toFreeAbelianGroup toFinsupp) (of x✝) = ↑(AddMonoidHom.id (FreeAbelianGroup X)) (of x✝)",
"state_before": "X : Type u_1\n⊢ AddMonoidHom.comp toFreeAbelianGroup toFinsupp = AddMonoidHom.id (FreeAbelianGroup X)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nX : Type u_1\nx✝ : X\n⊢ ↑(AddMonoidHom.comp toFreeAbelianGroup toFinsupp) (of x✝) = ↑(AddMonoidHom.id (FreeAbelianGroup X)) (of x✝)",
"tactic": "rw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of,\n liftAddHom_apply_single, AddMonoidHom.flip_apply, smulAddHom_apply, one_smul,\n AddMonoidHom.id_apply]"
}
] |
[
75,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.NonemptyCompacts.carrier_eq_coe
|
[] |
[
253,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.inv_mk
|
[] |
[
610,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
609,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
BumpCovering.exists_isSubordinate_of_locallyFinite
|
[] |
[
328,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.map_move
|
[
{
"state_after": "case mk\nΓ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nd : Dir\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ map f (move d { head := head✝, left := left✝, right := right✝ }) =\n move d (map f { head := head✝, left := left✝, right := right✝ })",
"state_before": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nT : Tape Γ\nd : Dir\n⊢ map f (move d T) = move d (map f T)",
"tactic": "cases T"
},
{
"state_after": "no goals",
"state_before": "case mk\nΓ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nd : Dir\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ map f (move d { head := head✝, left := left✝, right := right✝ }) =\n move d (map f { head := head✝, left := left✝, right := right✝ })",
"tactic": "cases d <;> simp only [Tape.move, Tape.map, ListBlank.head_map, eq_self_iff_true,\n ListBlank.map_cons, and_self_iff, ListBlank.tail_map]"
}
] |
[
718,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
714,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_sub_lt_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\n⊢ abs (a - b) < c ↔ a - b < c ∧ b - a < c",
"tactic": "rw [@abs_lt α, neg_lt_sub_iff_lt_add', sub_lt_iff_lt_add', and_comm, sub_lt_iff_lt_add']"
}
] |
[
289,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean
|
ZMod.charpoly_pow_card
|
[
{
"state_after": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nM : Matrix n n (ZMod p)\nh : charpoly (M ^ Fintype.card (ZMod p)) = charpoly M\n⊢ charpoly (M ^ p) = charpoly M",
"state_before": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nM : Matrix n n (ZMod p)\n⊢ charpoly (M ^ p) = charpoly M",
"tactic": "have h := FiniteField.Matrix.charpoly_pow_card M"
},
{
"state_after": "no goals",
"state_before": "n : Type u_1\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nM : Matrix n n (ZMod p)\nh : charpoly (M ^ Fintype.card (ZMod p)) = charpoly M\n⊢ charpoly (M ^ p) = charpoly M",
"tactic": "rwa [ZMod.card] at h"
}
] |
[
53,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_inter_null_of_null_right
|
[] |
[
344,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Data/List/Lemmas.lean
|
List.foldl_range_eq_of_range_eq
|
[] |
[
82,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.prod_le_prod_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.175540\nι : Sort ?u.175543\nκ : ι → Sort ?u.175548\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns s₁ s₂ : UpperSet α\nt t₁ t₂ : UpperSet β\nx : α × β\n⊢ ↑s₂ ⊆ ↑s₁ ∧ ↑t₂ ⊆ ↑t₁ ∨ ↑s₂ = ∅ ∨ ↑t₂ = ∅ ↔ s₁ ≤ s₂ ∧ t₁ ≤ t₂ ∨ s₂ = ⊤ ∨ t₂ = ⊤",
"tactic": "simp"
}
] |
[
1625,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1624,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.exists_get?_of_mem
|
[
{
"state_after": "case h1\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh : a ∈ s\n⊢ ∀ (b : α) (s' : WSeq α), (a = b ∨ ∃ n, some a ∈ get? s' n) → ∃ n, some a ∈ get? (cons b s') n\n\ncase h2\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh : a ∈ s\n⊢ ∀ (s : WSeq α), (∃ n, some a ∈ get? s n) → ∃ n, some a ∈ get? (think s) n",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh : a ∈ s\n⊢ ∃ n, some a ∈ get? s n",
"tactic": "apply mem_rec_on h"
},
{
"state_after": "case h1\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a' ∨ ∃ n, some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (cons a' s') n",
"state_before": "case h1\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh : a ∈ s\n⊢ ∀ (b : α) (s' : WSeq α), (a = b ∨ ∃ n, some a ∈ get? s' n) → ∃ n, some a ∈ get? (cons b s') n",
"tactic": "intro a' s' h"
},
{
"state_after": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ ∃ n, some a ∈ get? (cons a' s') n\n\ncase h1.inr\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : ∃ n, some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (cons a' s') n",
"state_before": "case h1\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a' ∨ ∃ n, some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (cons a' s') n",
"tactic": "cases' h with h h"
},
{
"state_after": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ some a ∈ get? (cons a' s') 0",
"state_before": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ ∃ n, some a ∈ get? (cons a' s') n",
"tactic": "exists 0"
},
{
"state_after": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ some a ∈ Computation.pure (some a')",
"state_before": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ some a ∈ get? (cons a' s') 0",
"tactic": "simp only [get?, drop, head_cons]"
},
{
"state_after": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ some a' ∈ Computation.pure (some a')",
"state_before": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ some a ∈ Computation.pure (some a')",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case h1.inl\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : a = a'\n⊢ some a' ∈ Computation.pure (some a')",
"tactic": "apply ret_mem"
},
{
"state_after": "case h1.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (cons a' s') n",
"state_before": "case h1.inr\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nh : ∃ n, some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (cons a' s') n",
"tactic": "cases' h with n h"
},
{
"state_after": "case h1.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ some a ∈ get? (cons a' s') (n + 1)",
"state_before": "case h1.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (cons a' s') n",
"tactic": "exists n + 1"
},
{
"state_after": "no goals",
"state_before": "case h1.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\na' : α\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ some a ∈ get? (cons a' s') (n + 1)",
"tactic": "simpa [get?]"
},
{
"state_after": "case h2\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nh : ∃ n, some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (think s') n",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh : a ∈ s\n⊢ ∀ (s : WSeq α), (∃ n, some a ∈ get? s n) → ∃ n, some a ∈ get? (think s) n",
"tactic": "intro s' h"
},
{
"state_after": "case h2.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (think s') n",
"state_before": "case h2\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nh : ∃ n, some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (think s') n",
"tactic": "cases' h with n h"
},
{
"state_after": "case h2.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ some a ∈ get? (think s') n",
"state_before": "case h2.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ ∃ n, some a ∈ get? (think s') n",
"tactic": "exists n"
},
{
"state_after": "case h2.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ some a ∈ Computation.think (head (drop s' n))",
"state_before": "case h2.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ some a ∈ get? (think s') n",
"tactic": "simp [get?]"
},
{
"state_after": "no goals",
"state_before": "case h2.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\na : α\nh✝ : a ∈ s\ns' : WSeq α\nn : ℕ\nh : some a ∈ get? s' n\n⊢ some a ∈ Computation.think (head (drop s' n))",
"tactic": "apply think_mem h"
}
] |
[
1021,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1005,
1
] |
Mathlib/Analysis/MeanInequalitiesPow.lean
|
NNReal.rpow_add_le_mul_rpow_add_rpow
|
[
{
"state_after": "case inl\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\nhp : 1 ≤ 1\n⊢ (z₁ + z₂) ^ 1 ≤ 2 ^ (1 - 1) * (z₁ ^ 1 + z₂ ^ 1)\n\ncase inr\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)",
"state_before": "ι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\n⊢ (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)",
"tactic": "rcases eq_or_lt_of_le hp with (rfl | h'p)"
},
{
"state_after": "case h.e'_3\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p = (1 / 2 * (2 * z₁) + 1 / 2 * (2 * z₂)) ^ p\n\ncase h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"state_before": "case inr\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)",
"tactic": "convert rpow_arith_mean_le_arith_mean2_rpow (1 / 2) (1 / 2) (2 * z₁) (2 * z₂) (add_halves 1) hp\n using 1"
},
{
"state_after": "case inl\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\nhp : 1 ≤ 1\n⊢ z₁ + z₂ ≤ z₁ + z₂",
"state_before": "case inl\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\nhp : 1 ≤ 1\n⊢ (z₁ + z₂) ^ 1 ≤ 2 ^ (1 - 1) * (z₁ ^ 1 + z₂ ^ 1)",
"tactic": "simp only [rpow_one, sub_self, rpow_zero, one_mul]"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\nhp : 1 ≤ 1\n⊢ z₁ + z₂ ≤ z₁ + z₂",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p = (1 / 2 * (2 * z₁) + 1 / 2 * (2 * z₂)) ^ p",
"tactic": "simp only [one_div, inv_mul_cancel_left₀, Ne.def, mul_eq_zero, two_ne_zero, one_ne_zero,\n not_false_iff]"
},
{
"state_after": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nA : p - 1 ≠ 0\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"state_before": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"tactic": "have A : p - 1 ≠ 0 := ne_of_gt (sub_pos.2 h'p)"
},
{
"state_after": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nA : p - 1 ≠ 0\n⊢ 2⁻¹ * 2 ^ p * (z₁ ^ p + z₂ ^ p) = 2⁻¹ * (2 ^ p * z₁ ^ p) + 2⁻¹ * (2 ^ p * z₂ ^ p)",
"state_before": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nA : p - 1 ≠ 0\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"tactic": "simp only [mul_rpow, rpow_sub' _ A, _root_.div_eq_inv_mul, rpow_one, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nA : p - 1 ≠ 0\n⊢ 2⁻¹ * 2 ^ p * (z₁ ^ p + z₂ ^ p) = 2⁻¹ * (2 ^ p * z₁ ^ p) + 2⁻¹ * (2 ^ p * z₂ ^ p)",
"tactic": "ring"
}
] |
[
164,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Std/Data/String/Lemmas.lean
|
String.valid_next
|
[
{
"state_after": "no goals",
"state_before": "s : String\np : Pos\nh : Pos.Valid s p\nh₂ : p < endPos s\n⊢ Pos.Valid s (next s p)",
"tactic": "match s, p, h with\n| ⟨_⟩, ⟨_⟩, ⟨cs, [], rfl, rfl⟩ => simp at h₂\n| ⟨_⟩, ⟨_⟩, ⟨cs, c::cs', rfl, rfl⟩ =>\n rw [utf8ByteSize.go_eq, next_of_valid]\n simpa using Pos.Valid.mk (cs ++ [c]) cs'"
},
{
"state_after": "no goals",
"state_before": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nh₂ : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ [] }\n⊢ Pos.Valid { data := cs ++ [] } (next { data := cs ++ [] } { byteIdx := utf8ByteSize.go cs })",
"tactic": "simp at h₂"
},
{
"state_after": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nc : Char\ncs' : List Char\nh₂ : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' }\n⊢ Pos.Valid { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c }",
"state_before": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nc : Char\ncs' : List Char\nh₂ : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' }\n⊢ Pos.Valid { data := cs ++ c :: cs' } (next { data := cs ++ c :: cs' } { byteIdx := utf8ByteSize.go cs })",
"tactic": "rw [utf8ByteSize.go_eq, next_of_valid]"
},
{
"state_after": "no goals",
"state_before": "s : String\np : Pos\nh : Pos.Valid s p\ncs : List Char\nc : Char\ncs' : List Char\nh₂ : { byteIdx := utf8ByteSize.go cs } < endPos { data := cs ++ c :: cs' }\n⊢ Pos.Valid { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c }",
"tactic": "simpa using Pos.Valid.mk (cs ++ [c]) cs'"
}
] |
[
270,
45
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
265,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Fintype.total_apply
|
[] |
[
1059,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/Order/Filter/Pi.lean
|
Filter.compl_mem_coprodᵢ
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns✝ : (i : ι) → Set (α i)\ns : Set ((i : ι) → α i)\n⊢ sᶜ ∈ Filter.coprodᵢ f ↔ ∀ (i : ι), (eval i '' s)ᶜ ∈ f i",
"tactic": "simp only [Filter.coprodᵢ, mem_iSup, compl_mem_comap]"
}
] |
[
210,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pushoutComparison_map_desc
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW✝ X Y Z : C\nG : C ⥤ D\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasPushout f g\ninst✝ : HasPushout (G.map f) (G.map g)\nW : C\nh : Y ⟶ W\nk : Z ⟶ W\nw : f ≫ h = g ≫ k\n⊢ G.map f ≫ G.map h = G.map g ≫ G.map k",
"tactic": "simp only [← G.map_comp, w]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW✝ X Y Z : C\nG : C ⥤ D\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝¹ : HasPushout f g\ninst✝ : HasPushout (G.map f) (G.map g)\nW : C\nh : Y ⟶ W\nk : Z ⟶ W\nw : f ≫ h = g ≫ k\n⊢ pushoutComparison G f g ≫ G.map (pushout.desc h k w) =\n pushout.desc (G.map h) (G.map k) (_ : G.map f ≫ G.map h = G.map g ≫ G.map k)",
"tactic": "ext <;> simp [← G.map_comp]"
}
] |
[
1495,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1491,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.swap_apply_of_ne_of_ne
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\ninst✝ : DecidableEq α\na b x : α\n⊢ x ≠ a → x ≠ b → ↑(swap a b) x = x",
"tactic": "simp (config := { contextual := true }) [swap_apply_def]"
}
] |
[
1573,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1572,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_le_of_injective
|
[] |
[
273,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.ProbabilityMeasure.continuous_testAgainstNN_eval
|
[] |
[
256,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Algebra/Lie/Classical.lean
|
LieAlgebra.SpecialLinear.sl_bracket
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
image_norm_le_of_norm_deriv_right_le_deriv_boundary
|
[] |
[
335,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
Function.monotone_eval
|
[] |
[
342,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.mem_toSubsemigroup
|
[] |
[
237,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
inf_sdiff_right_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.39578\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x \\ z ⊓ y = (x ⊓ y) \\ z",
"tactic": "rw [@inf_comm _ _ x, inf_comm, inf_sdiff_assoc]"
}
] |
[
450,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.negOfNat_eq
|
[] |
[
45,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
45,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.sum_smul_index
|
[] |
[
1631,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1629,
1
] |
Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean
|
GeneralizedContinuedFraction.continuants_recurrence
|
[
{
"state_after": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn : ℕ\ninst✝ : DivisionRing K\ngp ppred pred : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_conts_eq : continuantsAux g (n + 1) = ppred\nsucc_nth_conts_eq : continuantsAux g (n + 1 + 1) = pred\n⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn : ℕ\ninst✝ : DivisionRing K\ngp ppred pred : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_conts_eq : continuants g n = ppred\nsucc_nth_conts_eq : continuants g (n + 1) = pred\n⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }",
"tactic": "rw [nth_cont_eq_succ_nth_cont_aux] at nth_conts_eq succ_nth_conts_eq"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn : ℕ\ninst✝ : DivisionRing K\ngp ppred pred : Pair K\nsucc_nth_s_eq : Stream'.Seq.get? g.s (n + 1) = some gp\nnth_conts_eq : continuantsAux g (n + 1) = ppred\nsucc_nth_conts_eq : continuantsAux g (n + 1 + 1) = pred\n⊢ continuants g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }",
"tactic": "exact continuants_recurrenceAux succ_nth_s_eq nth_conts_eq succ_nth_conts_eq"
}
] |
[
49,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.prod_eq_one
|
[
{
"state_after": "case nil\nι : Type ?u.143653\nα : Type ?u.143656\nM : Type u_1\nN : Type ?u.143662\nP : Type ?u.143665\nM₀ : Type ?u.143668\nG : Type ?u.143671\nR : Type ?u.143674\ninst✝ : Monoid M\nl : List M\nhl✝ : ∀ (x : M), x ∈ l → x = 1\nhl : ∀ (x : M), x ∈ [] → x = 1\n⊢ prod [] = 1\n\ncase cons\nι : Type ?u.143653\nα : Type ?u.143656\nM : Type u_1\nN : Type ?u.143662\nP : Type ?u.143665\nM₀ : Type ?u.143668\nG : Type ?u.143671\nR : Type ?u.143674\ninst✝ : Monoid M\nl✝ : List M\nhl✝ : ∀ (x : M), x ∈ l✝ → x = 1\ni : M\nl : List M\nhil : (∀ (x : M), x ∈ l → x = 1) → prod l = 1\nhl : ∀ (x : M), x ∈ i :: l → x = 1\n⊢ prod (i :: l) = 1",
"state_before": "ι : Type ?u.143653\nα : Type ?u.143656\nM : Type u_1\nN : Type ?u.143662\nP : Type ?u.143665\nM₀ : Type ?u.143668\nG : Type ?u.143671\nR : Type ?u.143674\ninst✝ : Monoid M\nl : List M\nhl : ∀ (x : M), x ∈ l → x = 1\n⊢ prod l = 1",
"tactic": "induction' l with i l hil"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.143653\nα : Type ?u.143656\nM : Type u_1\nN : Type ?u.143662\nP : Type ?u.143665\nM₀ : Type ?u.143668\nG : Type ?u.143671\nR : Type ?u.143674\ninst✝ : Monoid M\nl✝ : List M\nhl✝ : ∀ (x : M), x ∈ l✝ → x = 1\ni : M\nl : List M\nhil : (∀ (x : M), x ∈ l → x = 1) → prod l = 1\nhl : ∀ (x : M), x ∈ i :: l → x = 1\n⊢ prod (i :: l) = 1",
"tactic": "rw [List.prod_cons, hil fun x hx => hl _ (mem_cons_of_mem i hx), hl _ (mem_cons_self i l),\n one_mul]"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.143653\nα : Type ?u.143656\nM : Type u_1\nN : Type ?u.143662\nP : Type ?u.143665\nM₀ : Type ?u.143668\nG : Type ?u.143671\nR : Type ?u.143674\ninst✝ : Monoid M\nl : List M\nhl✝ : ∀ (x : M), x ∈ l → x = 1\nhl : ∀ (x : M), x ∈ [] → x = 1\n⊢ prod [] = 1",
"tactic": "rfl"
}
] |
[
532,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
528,
1
] |
Mathlib/Data/Sum/Order.lean
|
Sum.inr_strictMono
|
[] |
[
197,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
IsUnit.ne_zero
|
[] |
[
55,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
intermediate_value_Ico
|
[] |
[
554,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean
|
CategoryTheory.hasFiniteProducts_of_has_binary_and_terminal
|
[
{
"state_after": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nK : Discrete (Fin n) ⥤ C\n⊢ HasLimit K",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\n⊢ HasFiniteProducts C",
"tactic": "refine' ⟨fun n => ⟨fun K => _⟩⟩"
},
{
"state_after": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nK : Discrete (Fin n) ⥤ C\nthis : HasProduct fun n_1 => K.obj { as := n_1 } := CategoryTheory.hasProduct_fin n fun n_1 => K.obj { as := n_1 }\n⊢ HasLimit K",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nK : Discrete (Fin n) ⥤ C\n⊢ HasLimit K",
"tactic": "letI := hasProduct_fin n fun n => K.obj ⟨n⟩"
},
{
"state_after": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nK : Discrete (Fin n) ⥤ C\nthis : HasProduct fun n_1 => K.obj { as := n_1 } := CategoryTheory.hasProduct_fin n fun n_1 => K.obj { as := n_1 }\nthat : (Discrete.functor fun n_1 => K.obj { as := n_1 }) ≅ K :=\n Discrete.natIso fun x =>\n match x with\n | { as := i } => Iso.refl ((Discrete.functor fun n_1 => K.obj { as := n_1 }).obj { as := i })\n⊢ HasLimit K",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nK : Discrete (Fin n) ⥤ C\nthis : HasProduct fun n_1 => K.obj { as := n_1 } := CategoryTheory.hasProduct_fin n fun n_1 => K.obj { as := n_1 }\n⊢ HasLimit K",
"tactic": "let that : (Discrete.functor fun n => K.obj ⟨n⟩) ≅ K := Discrete.natIso fun ⟨i⟩ => Iso.refl _"
},
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nK : Discrete (Fin n) ⥤ C\nthis : HasProduct fun n_1 => K.obj { as := n_1 } := CategoryTheory.hasProduct_fin n fun n_1 => K.obj { as := n_1 }\nthat : (Discrete.functor fun n_1 => K.obj { as := n_1 }) ≅ K :=\n Discrete.natIso fun x =>\n match x with\n | { as := i } => Iso.refl ((Discrete.functor fun n_1 => K.obj { as := n_1 }).obj { as := i })\n⊢ HasLimit K",
"tactic": "apply @hasLimitOfIso _ _ _ _ _ _ this that"
}
] |
[
115,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castAdd_cast
|
[] |
[
1166,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1164,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.coe_aeval_eq_eval
|
[] |
[
299,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.withDensity_add_left
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1755617\nγ : Type ?u.1755620\nδ : Type ?u.1755623\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\ng : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(withDensity μ (f + g)) s = ↑↑(withDensity μ f + withDensity μ g) s",
"state_before": "α : Type u_1\nβ : Type ?u.1755617\nγ : Type ?u.1755620\nδ : Type ?u.1755623\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\ng : α → ℝ≥0∞\n⊢ withDensity μ (f + g) = withDensity μ f + withDensity μ g",
"tactic": "refine' Measure.ext fun s hs => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1755617\nγ : Type ?u.1755620\nδ : Type ?u.1755623\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\ng : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (a : α) in s, (f + g) a ∂μ) = ∫⁻ (a : α) in s, f a + g a ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.1755617\nγ : Type ?u.1755620\nδ : Type ?u.1755623\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\ng : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(withDensity μ (f + g)) s = ↑↑(withDensity μ f + withDensity μ g) s",
"tactic": "rw [withDensity_apply _ hs, Measure.add_apply, withDensity_apply _ hs, withDensity_apply _ hs,\n ← lintegral_add_left hf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1755617\nγ : Type ?u.1755620\nδ : Type ?u.1755623\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\ng : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (a : α) in s, (f + g) a ∂μ) = ∫⁻ (a : α) in s, f a + g a ∂μ",
"tactic": "rfl"
}
] |
[
1566,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1561,
1
] |
Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean
|
CategoryTheory.FreeMonoidalCategory.tensorFunc_obj_map
|
[
{
"state_after": "case mk\nC : Type u\nZ : F C\nn' : (Discrete ∘ NormalMonoidalObject) C\nas✝ : NormalMonoidalObject C\nf : { as := as✝ } ⟶ n'\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ⊗ 𝟙 Z",
"state_before": "C : Type u\nZ : F C\nn n' : (Discrete ∘ NormalMonoidalObject) C\nf : n ⟶ n'\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ⊗ 𝟙 Z",
"tactic": "cases n"
},
{
"state_after": "case mk.mk\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nf : { as := as✝¹ } ⟶ { as := as✝ }\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ⊗ 𝟙 Z",
"state_before": "case mk\nC : Type u\nZ : F C\nn' : (Discrete ∘ NormalMonoidalObject) C\nas✝ : NormalMonoidalObject C\nf : { as := as✝ } ⟶ n'\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ⊗ 𝟙 Z",
"tactic": "cases n'"
},
{
"state_after": "case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : { as := as✝¹ }.as = { as := as✝ }.as\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ⊗ 𝟙 Z",
"state_before": "case mk.mk\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nf : { as := as✝¹ } ⟶ { as := as✝ }\n⊢ ((tensorFunc C).obj Z).map f = inclusion.map f ⊗ 𝟙 Z",
"tactic": "rcases f with ⟨⟨h⟩⟩"
},
{
"state_after": "case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : as✝¹ = as✝\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ⊗ 𝟙 Z",
"state_before": "case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : { as := as✝¹ }.as = { as := as✝ }.as\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ⊗ 𝟙 Z",
"tactic": "dsimp at h"
},
{
"state_after": "case mk.mk.up.up\nC : Type u\nZ : F C\nas✝ : NormalMonoidalObject C\n⊢ ((tensorFunc C).obj Z).map { down := { down := (_ : as✝ = as✝) } } =\n inclusion.map { down := { down := (_ : as✝ = as✝) } } ⊗ 𝟙 Z",
"state_before": "case mk.mk.up.up\nC : Type u\nZ : F C\nas✝¹ as✝ : NormalMonoidalObject C\nh : as✝¹ = as✝\n⊢ ((tensorFunc C).obj Z).map { down := { down := h } } = inclusion.map { down := { down := h } } ⊗ 𝟙 Z",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.up.up\nC : Type u\nZ : F C\nas✝ : NormalMonoidalObject C\n⊢ ((tensorFunc C).obj Z).map { down := { down := (_ : as✝ = as✝) } } =\n inclusion.map { down := { down := (_ : as✝ = as✝) } } ⊗ 𝟙 Z",
"tactic": "simp"
}
] |
[
183,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.uIcc_of_not_ge
|
[] |
[
1000,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
999,
1
] |
Mathlib/Data/Rat/Lemmas.lean
|
Rat.num_den_mk
|
[
{
"state_after": "case inl\nq : ℚ\nd : ℤ\nhd : d ≠ 0\nqdf : q = 0 /. d\n⊢ ∃ c, 0 = c * q.num ∧ d = c * ↑q.den\n\ncase inr\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\n⊢ ∃ c, n = c * q.num ∧ d = c * ↑q.den",
"state_before": "q : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\n⊢ ∃ c, n = c * q.num ∧ d = c * ↑q.den",
"tactic": "obtain rfl | hn := eq_or_ne n 0"
},
{
"state_after": "case inr\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ ∃ c, n = c * q.num ∧ d = c * ↑q.den",
"state_before": "case inr\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\n⊢ ∃ c, n = c * q.num ∧ d = c * ↑q.den",
"tactic": "have hqdn : q.num ∣ n := by\n rw [qdf]\n exact Rat.num_dvd _ hd"
},
{
"state_after": "case inr.refine'_1\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ n = n / q.num * q.num\n\ncase inr.refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ d = n / q.num * ↑q.den",
"state_before": "case inr\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ ∃ c, n = c * q.num ∧ d = c * ↑q.den",
"tactic": "refine' ⟨n / q.num, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nq : ℚ\nd : ℤ\nhd : d ≠ 0\nqdf : q = 0 /. d\n⊢ ∃ c, 0 = c * q.num ∧ d = c * ↑q.den",
"tactic": "simp [qdf]"
},
{
"state_after": "case refine'_1\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\n⊢ ↑q.den ≠ 0\n\ncase refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\n⊢ q.num /. ↑q.den = n /. d",
"state_before": "q : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\n⊢ q.num * d = n * ↑q.den",
"tactic": "refine' (divInt_eq_iff _ hd).mp _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\n⊢ ↑q.den ≠ 0",
"tactic": "exact Int.coe_nat_ne_zero.mpr (Rat.den_nz _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\n⊢ q.num /. ↑q.den = n /. d",
"tactic": "rwa [num_den]"
},
{
"state_after": "q : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\n⊢ (n /. d).num ∣ n",
"state_before": "q : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\n⊢ q.num ∣ n",
"tactic": "rw [qdf]"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\n⊢ (n /. d).num ∣ n",
"tactic": "exact Rat.num_dvd _ hd"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_1\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ n = n / q.num * q.num",
"tactic": "rw [Int.ediv_mul_cancel hqdn]"
},
{
"state_after": "case inr.refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ q.num ≠ 0",
"state_before": "case inr.refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ d = n / q.num * ↑q.den",
"tactic": "refine' Int.eq_mul_div_of_mul_eq_mul_of_dvd_left _ hqdn this"
},
{
"state_after": "case inr.refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ (n /. d).num ≠ 0",
"state_before": "case inr.refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ q.num ≠ 0",
"tactic": "rw [qdf]"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_2\nq : ℚ\nn d : ℤ\nhd : d ≠ 0\nqdf : q = n /. d\nhn : n ≠ 0\nthis : q.num * d = n * ↑q.den\nhqdn : q.num ∣ n\n⊢ (n /. d).num ≠ 0",
"tactic": "exact Rat.num_ne_zero_of_ne_zero ((divInt_ne_zero hd).mpr hn)"
}
] |
[
57,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.mk_eq_zero
|
[] |
[
965,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
964,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.first_countable
|
[
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\nthis : Filter.IsCountablyGenerated (𝓝 0)\n⊢ FirstCountableTopology E",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\n⊢ FirstCountableTopology E",
"tactic": "have : (𝓝 (0 : E)).IsCountablyGenerated := by\n rw [p.withSeminorms_iff_nhds_eq_iInf.mp hp]\n exact Filter.iInf.isCountablyGenerated _"
},
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\nthis✝ : Filter.IsCountablyGenerated (𝓝 0)\nthis : Filter.IsCountablyGenerated (uniformity E)\n⊢ FirstCountableTopology E",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\nthis : Filter.IsCountablyGenerated (𝓝 0)\n⊢ FirstCountableTopology E",
"tactic": "haveI : (uniformity E).IsCountablyGenerated := UniformAddGroup.uniformity_countably_generated"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\nthis✝ : Filter.IsCountablyGenerated (𝓝 0)\nthis : Filter.IsCountablyGenerated (uniformity E)\n⊢ FirstCountableTopology E",
"tactic": "exact UniformSpace.firstCountableTopology E"
},
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\n⊢ Filter.IsCountablyGenerated (⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0))",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\n⊢ Filter.IsCountablyGenerated (𝓝 0)",
"tactic": "rw [p.withSeminorms_iff_nhds_eq_iInf.mp hp]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.848749\n𝕝 : Type ?u.848752\n𝕝₂ : Type ?u.848755\nE : Type u_2\nF : Type ?u.848761\nG : Type ?u.848764\nι : Type u_3\nι' : Type ?u.848770\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝¹ : UniformSpace E\ninst✝ : UniformAddGroup E\nhp : WithSeminorms p\n⊢ Filter.IsCountablyGenerated (⨅ (i : ι), Filter.comap (↑(p i)) (𝓝 0))",
"tactic": "exact Filter.iInf.isCountablyGenerated _"
}
] |
[
770,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Std/Data/String/Lemmas.lean
|
String.all_iff
|
[
{
"state_after": "no goals",
"state_before": "s : String\np : Char → Bool\n⊢ all s p = true ↔ ∀ (c : Char), c ∈ s.data → p c = true",
"tactic": "simp [all_eq]"
}
] |
[
725,
94
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
725,
1
] |
Mathlib/Topology/Bornology/Basic.lean
|
Bornology.isCobounded_biInter
|
[] |
[
264,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Data/List/OfFn.lean
|
List.ofFn_inj
|
[] |
[
291,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_inf_eq
|
[] |
[
1238,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1237,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.comp_liftAddHom
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nδ : Type u_1\ninst✝² : (i : ι) → AddZeroClass (β i)\ninst✝¹ : AddCommMonoid γ\ninst✝ : AddCommMonoid δ\ng : γ →+ δ\nf : (i : ι) → β i →+ γ\na : ι\n⊢ ↑(AddEquiv.symm liftAddHom) (AddMonoidHom.comp g (↑liftAddHom f)) a = AddMonoidHom.comp g (f a)",
"tactic": "rw [liftAddHom_symm_apply, AddMonoidHom.comp_assoc, liftAddHom_comp_single]"
}
] |
[
2050,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2045,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.linear_mk'
|
[] |
[
190,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
upperSemicontinuousAt_iInf
|
[] |
[
1026,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean
|
Complex.measurable_arg
|
[] |
[
124,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.isOpen_Iic_principal
|
[] |
[
55,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.div_lt_top
|
[] |
[
1415,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1414,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean
|
CategoryTheory.Limits.has_limits_of_finite_and_cofiltered
|
[] |
[
102,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Algebra/GCDMonoid/Finset.lean
|
Finset.lcm_congr
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.7148\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ : Finset β\nf✝ f g : β → α\nhfg : ∀ (a : β), a ∈ s₁ → f a = g a\n⊢ lcm s₁ f = lcm s₁ g",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.7148\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ s₂ : Finset β\nf✝ f g : β → α\nhs : s₁ = s₂\nhfg : ∀ (a : β), a ∈ s₂ → f a = g a\n⊢ lcm s₁ f = lcm s₂ g",
"tactic": "subst hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.7148\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns s₁ : Finset β\nf✝ f g : β → α\nhfg : ∀ (a : β), a ∈ s₁ → f a = g a\n⊢ lcm s₁ f = lcm s₁ g",
"tactic": "exact Finset.fold_congr hfg"
}
] |
[
106,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.add_assoc
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - -b + c = a + (b + c)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a + b + c = a + (b + c)",
"tactic": "conv_lhs =>\n congr; rw [add_def]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - -b + c = a + (b + c)",
"tactic": "rw [sub_add, ← add_neg, neg_sub', neg_neg]"
}
] |
[
425,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Algebra/Module/Torsion.lean
|
Module.isTorsionBySet_span_singleton_iff
|
[] |
[
348,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Std/Data/List/Lemmas.lean
|
List.diff_append
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ l₃ : List α\n⊢ List.diff l₁ (l₂ ++ l₃) = List.diff (List.diff l₁ l₂) l₃",
"tactic": "simp only [diff_eq_foldl, foldl_append]"
}
] |
[
1523,
42
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1522,
9
] |
Mathlib/Order/Heyting/Hom.lean
|
map_symmDiff
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_3\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.40368\nδ : Type ?u.40371\ninst✝² : CoheytingAlgebra α\ninst✝¹ : CoheytingAlgebra β\ninst✝ : CoheytingHomClass F α β\nf : F\na b : α\n⊢ ↑f (a ∆ b) = ↑f a ∆ ↑f b",
"tactic": "simp_rw [symmDiff, map_sup, map_sdiff]"
}
] |
[
212,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.inl_smul
|
[] |
[
336,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mem_iSup_of_directed
|
[
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ (x ∈ ⨆ (i : ι), S i) → ∃ i, x ∈ S i",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ (x ∈ ⨆ (i : ι), S i) ↔ ∃ i, x ∈ S i",
"tactic": "refine' ⟨_, fun ⟨i, hi⟩ => (SetLike.le_def.1 <| le_iSup S i) hi⟩"
},
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : Subsemiring R :=\n Subsemiring.mk' (⋃ (i : ι), ↑(S i)) (⨆ (i : ι), (S i).toSubmonoid)\n (_ : ↑(⨆ (i : ι), (S i).toSubmonoid) = ⋃ (i : ι), ↑(S i).toSubmonoid) (⨆ (i : ι), toAddSubmonoid (S i))\n (_ : ↑(⨆ (i : ι), toAddSubmonoid (S i)) = ⋃ (i : ι), ↑(toAddSubmonoid (S i)))\n⊢ (x ∈ ⨆ (i : ι), S i) → ∃ i, x ∈ S i",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\n⊢ (x ∈ ⨆ (i : ι), S i) → ∃ i, x ∈ S i",
"tactic": "let U : Subsemiring R :=\n Subsemiring.mk' (⋃ i, (S i : Set R)) (⨆ i, (S i).toSubmonoid)\n (Submonoid.coe_iSup_of_directed <| hS.mono_comp _ fun _ _ => id) (⨆ i, (S i).toAddSubmonoid)\n (AddSubmonoid.coe_iSup_of_directed <| hS.mono_comp _ fun _ _ => id)"
},
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : Subsemiring R :=\n Subsemiring.mk' (⋃ (i : ι), ↑(S i)) (⨆ (i : ι), (S i).toSubmonoid)\n (_ : ↑(⨆ (i : ι), (S i).toSubmonoid) = ⋃ (i : ι), ↑(S i).toSubmonoid) (⨆ (i : ι), toAddSubmonoid (S i))\n (_ : ↑(⨆ (i : ι), toAddSubmonoid (S i)) = ⋃ (i : ι), ↑(toAddSubmonoid (S i)))\n⊢ (⨆ (i : ι), S i) ≤ U",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : Subsemiring R :=\n Subsemiring.mk' (⋃ (i : ι), ↑(S i)) (⨆ (i : ι), (S i).toSubmonoid)\n (_ : ↑(⨆ (i : ι), (S i).toSubmonoid) = ⋃ (i : ι), ↑(S i).toSubmonoid) (⨆ (i : ι), toAddSubmonoid (S i))\n (_ : ↑(⨆ (i : ι), toAddSubmonoid (S i)) = ⋃ (i : ι), ↑(toAddSubmonoid (S i)))\n⊢ (x ∈ ⨆ (i : ι), S i) → ∃ i, x ∈ S i",
"tactic": "suffices h : (⨆ i, S i) ≤ U by simpa using @h x"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : Subsemiring R :=\n Subsemiring.mk' (⋃ (i : ι), ↑(S i)) (⨆ (i : ι), (S i).toSubmonoid)\n (_ : ↑(⨆ (i : ι), (S i).toSubmonoid) = ⋃ (i : ι), ↑(S i).toSubmonoid) (⨆ (i : ι), toAddSubmonoid (S i))\n (_ : ↑(⨆ (i : ι), toAddSubmonoid (S i)) = ⋃ (i : ι), ↑(toAddSubmonoid (S i)))\n⊢ (⨆ (i : ι), S i) ≤ U",
"tactic": "exact iSup_le fun i x hx => Set.mem_iUnion.2 ⟨i, hx⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : NonAssocSemiring R\nM : Submonoid R\ninst✝¹ : NonAssocSemiring S✝\ninst✝ : NonAssocSemiring T\nι : Sort u_1\nhι : Nonempty ι\nS : ι → Subsemiring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : Subsemiring R :=\n Subsemiring.mk' (⋃ (i : ι), ↑(S i)) (⨆ (i : ι), (S i).toSubmonoid)\n (_ : ↑(⨆ (i : ι), (S i).toSubmonoid) = ⋃ (i : ι), ↑(S i).toSubmonoid) (⨆ (i : ι), toAddSubmonoid (S i))\n (_ : ↑(⨆ (i : ι), toAddSubmonoid (S i)) = ⋃ (i : ι), ↑(toAddSubmonoid (S i)))\nh : (⨆ (i : ι), S i) ≤ U\n⊢ (x ∈ ⨆ (i : ι), S i) → ∃ i, x ∈ S i",
"tactic": "simpa using @h x"
}
] |
[
1098,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1089,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Commute.units_zpow_left
|
[] |
[
1146,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1144,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.withConstants_funMap_sum_inl
|
[] |
[
512,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.count_eq_of_nodup
|
[
{
"state_after": "case inl\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : DecidableEq α\na : α\nl : List α\nd : Nodup l\nh : a ∈ l\n⊢ count a l = 1\n\ncase inr\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : DecidableEq α\na : α\nl : List α\nd : Nodup l\nh : ¬a ∈ l\n⊢ count a l = 0",
"state_before": "α : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : DecidableEq α\na : α\nl : List α\nd : Nodup l\n⊢ count a l = if a ∈ l then 1 else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : DecidableEq α\na : α\nl : List α\nd : Nodup l\nh : a ∈ l\n⊢ count a l = 1",
"tactic": "exact count_eq_one_of_mem d h"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b : α\ninst✝ : DecidableEq α\na : α\nl : List α\nd : Nodup l\nh : ¬a ∈ l\n⊢ count a l = 0",
"tactic": "exact count_eq_zero_of_not_mem h"
}
] |
[
207,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Data/Real/Pi/Bounds.lean
|
Real.pi_lt_sqrtTwoAddSeries
|
[
{
"state_after": "n : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2) =\n 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n",
"state_before": "n : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ π < 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n",
"tactic": "apply lt_of_lt_of_le this (le_of_eq _)"
},
{
"state_after": "n : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) + 1 / (2 ^ n) ^ 3 / 4 * 2 ^ (n + 2) =\n 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n",
"state_before": "n : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2) =\n 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n",
"tactic": "rw [add_mul]"
},
{
"state_after": "case e_a\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) = 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n)\n\ncase e_a\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 1 / (2 ^ n) ^ 3 / 4 * 2 ^ (n + 2) = 1 / 4 ^ n",
"state_before": "n : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) + 1 / (2 ^ n) ^ 3 / 4 * 2 ^ (n + 2) =\n 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n",
"tactic": "congr 1"
},
{
"state_after": "case e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ^ n ≠ 0\n\ncase e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 4 ≠ 0",
"state_before": "case e_a\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 1 / (2 ^ n) ^ 3 / 4 * 2 ^ (n + 2) = 1 / 4 ^ n",
"tactic": "rw [pow_succ, ← pow_mul, mul_comm n 2, pow_mul, show (2 : ℝ) ^ 2 = 4 by norm_num, pow_succ,\n pow_succ, ← mul_assoc (2 : ℝ), show (2 : ℝ) * 2 = 4 by norm_num, ← mul_assoc, div_mul_cancel,\n mul_comm ((2 : ℝ) ^ n), ← div_div, div_mul_cancel]"
},
{
"state_after": "case e_a.h.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ≠ 0\n\ncase e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 4 ≠ 0",
"state_before": "case e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ^ n ≠ 0\n\ncase e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 4 ≠ 0",
"tactic": "apply pow_ne_zero"
},
{
"state_after": "case e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 4 ≠ 0",
"state_before": "case e_a.h.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ≠ 0\n\ncase e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 4 ≠ 0",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 4 ≠ 0",
"tactic": "norm_num"
},
{
"state_after": "n : ℕ\n⊢ π / 2 ^ (n + 2) < sin (π / 2 ^ (n + 2)) + 1 / (2 ^ n) ^ 3 / 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "n : ℕ\n⊢ π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)",
"tactic": "rw [← div_lt_iff, ← sin_pi_over_two_pow_succ]"
},
{
"state_after": "case refine'_1\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2)\n\ncase refine'_2\nn : ℕ\n⊢ π / 2 ^ (n + 2) ≤ 1\n\ncase refine'_3\nn : ℕ\n⊢ sin (π / 2 ^ (n + 2)) + (π / 2 ^ (n + 2)) ^ 3 / 4 ≤ sin (π / 2 ^ (n + 2)) + 1 / (2 ^ n) ^ 3 / 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "n : ℕ\n⊢ π / 2 ^ (n + 2) < sin (π / 2 ^ (n + 2)) + 1 / (2 ^ n) ^ 3 / 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "refine' lt_of_lt_of_le (lt_add_of_sub_right_lt (sin_gt_sub_cube _ _)) _"
},
{
"state_after": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2)) ^ 3 / 4 ≤ 1 / (2 ^ n) ^ 3 / 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3\nn : ℕ\n⊢ sin (π / 2 ^ (n + 2)) + (π / 2 ^ (n + 2)) ^ 3 / 4 ≤ sin (π / 2 ^ (n + 2)) + 1 / (2 ^ n) ^ 3 / 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "apply add_le_add_left"
},
{
"state_after": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2)) ^ 3 ≤ 1 / (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2)) ^ 3 / 4 ≤ 1 / (2 ^ n) ^ 3 / 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "rw [div_le_div_right]"
},
{
"state_after": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2) * 2 ^ n) ^ 3 ≤ 1\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2)) ^ 3 ≤ 1 / (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "rw [le_div_iff, ← mul_pow]"
},
{
"state_after": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2) * 2 ^ n) ^ 3 ≤ 1 ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2) * 2 ^ n) ^ 3 ≤ 1\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "refine' le_trans _ (le_of_eq (one_pow 3))"
},
{
"state_after": "case refine'_3.bc.ha\nn : ℕ\n⊢ 0 ≤ π / 2 ^ (n + 2) * 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ π / 2 ^ (n + 2) * 2 ^ n ≤ 1\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc\nn : ℕ\n⊢ (π / 2 ^ (n + 2) * 2 ^ n) ^ 3 ≤ 1 ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "apply pow_le_pow_of_le_left"
},
{
"state_after": "case refine'_3.bc.hab\nn : ℕ\n⊢ π / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc.hab\nn : ℕ\n⊢ π / 2 ^ (n + 2) * 2 ^ n ≤ 1\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "rw [← le_div_iff]"
},
{
"state_after": "case refine'_3.bc.hab.refine'_1\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc.hab\nn : ℕ\n⊢ π / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "refine' le_trans ((div_le_div_right _).mpr pi_le_four) _"
},
{
"state_after": "case refine'_3.bc.hab.refine'_1.H\nn : ℕ\n⊢ 0 < 2\n\ncase refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc.hab.refine'_1\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "apply pow_pos"
},
{
"state_after": "case refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc.hab.refine'_1.H\nn : ℕ\n⊢ 0 < 2\n\ncase refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "norm_num"
},
{
"state_after": "case refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / (2 * 2) / 2 ^ n ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / 2 ^ (n + 2) ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "rw [pow_succ, pow_succ, ← mul_assoc, ← div_div]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.bc.hab.refine'_2\nn : ℕ\n⊢ 4 / (2 * 2) / 2 ^ n ≤ 1 / 2 ^ n\n\ncase refine'_3.bc.hab\nn : ℕ\n⊢ 0 < 2 ^ n\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < (2 ^ n) ^ 3\n\ncase refine'_3.bc\nn : ℕ\n⊢ 0 < 4\n\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "all_goals (repeat' apply pow_pos); norm_num"
},
{
"state_after": "case refine'_1\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_1\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2)",
"tactic": "apply div_pos pi_pos"
},
{
"state_after": "case refine'_1.H\nn : ℕ\n⊢ 0 < 2",
"state_before": "case refine'_1\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "apply pow_pos"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.H\nn : ℕ\n⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "case refine'_2\nn : ℕ\n⊢ π ≤ 2 ^ (n + 2) * 1\n\ncase refine'_2\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"state_before": "case refine'_2\nn : ℕ\n⊢ π / 2 ^ (n + 2) ≤ 1",
"tactic": "rw [div_le_iff']"
},
{
"state_after": "case refine'_2\nn : ℕ\n⊢ 4 ≤ 2 ^ (n + 2) * 1",
"state_before": "case refine'_2\nn : ℕ\n⊢ π ≤ 2 ^ (n + 2) * 1",
"tactic": "refine' le_trans pi_le_four _"
},
{
"state_after": "case refine'_2\nn : ℕ\n⊢ 2 ^ 2 ≤ 2 ^ (n + 2)",
"state_before": "case refine'_2\nn : ℕ\n⊢ 4 ≤ 2 ^ (n + 2) * 1",
"tactic": "simp only [show (4 : ℝ) = (2 : ℝ) ^ 2 by norm_num, mul_one]"
},
{
"state_after": "case refine'_2.ha\nn : ℕ\n⊢ 1 ≤ 2\n\ncase refine'_2.h\nn : ℕ\n⊢ 2 ≤ n + 2",
"state_before": "case refine'_2\nn : ℕ\n⊢ 2 ^ 2 ≤ 2 ^ (n + 2)",
"tactic": "apply pow_le_pow"
},
{
"state_after": "case refine'_2.h\nn : ℕ\n⊢ 2 ≤ n + 2",
"state_before": "case refine'_2.ha\nn : ℕ\n⊢ 1 ≤ 2\n\ncase refine'_2.h\nn : ℕ\n⊢ 2 ≤ n + 2",
"tactic": "norm_num"
},
{
"state_after": "case refine'_2.h.h\nn : ℕ\n⊢ 0 ≤ n",
"state_before": "case refine'_2.h\nn : ℕ\n⊢ 2 ≤ n + 2",
"tactic": "apply le_add_of_nonneg_left"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.h.h\nn : ℕ\n⊢ 0 ≤ n",
"tactic": "apply Nat.zero_le"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 4 = 2 ^ 2",
"tactic": "norm_num"
},
{
"state_after": "case refine'_2.H\nn : ℕ\n⊢ 0 < 2",
"state_before": "case refine'_2\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "apply pow_pos"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.H\nn : ℕ\n⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "case refine'_3.bc.ha.a\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2) * 2 ^ n",
"state_before": "case refine'_3.bc.ha\nn : ℕ\n⊢ 0 ≤ π / 2 ^ (n + 2) * 2 ^ n",
"tactic": "apply le_of_lt"
},
{
"state_after": "case refine'_3.bc.ha.a.ha\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2)\n\ncase refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"state_before": "case refine'_3.bc.ha.a\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2) * 2 ^ n",
"tactic": "apply mul_pos"
},
{
"state_after": "case refine'_3.bc.ha.a.ha\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"state_before": "case refine'_3.bc.ha.a.ha\nn : ℕ\n⊢ 0 < π / 2 ^ (n + 2)\n\ncase refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"tactic": "apply div_pos pi_pos"
},
{
"state_after": "case refine'_3.bc.ha.a.ha.H\nn : ℕ\n⊢ 0 < 2\n\ncase refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"state_before": "case refine'_3.bc.ha.a.ha\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)\n\ncase refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"tactic": "apply pow_pos"
},
{
"state_after": "case refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"state_before": "case refine'_3.bc.ha.a.ha.H\nn : ℕ\n⊢ 0 < 2\n\ncase refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"tactic": "norm_num"
},
{
"state_after": "case refine'_3.bc.ha.a.hb.H\nn : ℕ\n⊢ 0 < 2",
"state_before": "case refine'_3.bc.ha.a.hb\nn : ℕ\n⊢ 0 < 2 ^ n",
"tactic": "apply pow_pos"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.bc.ha.a.hb.H\nn : ℕ\n⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "case H\nn : ℕ\n⊢ 0 < 2",
"state_before": "n : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "(repeat' apply pow_pos)"
},
{
"state_after": "no goals",
"state_before": "case H\nn : ℕ\n⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "case H\nn : ℕ\n⊢ 0 < 2",
"state_before": "n : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"tactic": "repeat' apply pow_pos"
},
{
"state_after": "case H\nn : ℕ\n⊢ 0 < 2",
"state_before": "case H\nn : ℕ\n⊢ 0 < 2",
"tactic": "apply pow_pos"
},
{
"state_after": "case e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ≠ 0",
"state_before": "case e_a\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) = 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n)",
"tactic": "rw [pow_succ _ (n + 1), ← mul_assoc, div_mul_cancel, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case e_a.h\nn : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ≠ 0",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ^ 2 = 4",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nthis : π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 * 2 = 4",
"tactic": "norm_num"
}
] |
[
69,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.comp_add_le
|
[] |
[
353,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.restrict_univ
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nι : Type ?u.1444502\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\ns t : Set β\na : α\n⊢ ↑(kernel.restrict κ (_ : MeasurableSet Set.univ)) a = ↑κ a",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type ?u.1444502\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\ns t : Set β\n⊢ kernel.restrict κ (_ : MeasurableSet Set.univ) = κ",
"tactic": "ext1 a"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nι : Type ?u.1444502\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\ns t : Set β\na : α\n⊢ ↑(kernel.restrict κ (_ : MeasurableSet Set.univ)) a = ↑κ a",
"tactic": "rw [kernel.restrict_apply, Measure.restrict_univ]"
}
] |
[
506,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean
|
SimplexCategory.δ_comp_δ_self'
|
[
{
"state_after": "n : ℕ\ni : Fin (n + 2)\n⊢ δ i ≫ δ (↑Fin.castSucc i) = δ i ≫ δ (Fin.succ i)",
"state_before": "n : ℕ\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : j = ↑Fin.castSucc i\n⊢ δ i ≫ δ j = δ i ≫ δ (Fin.succ i)",
"tactic": "subst H"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\ni : Fin (n + 2)\n⊢ δ i ≫ δ (↑Fin.castSucc i) = δ i ≫ δ (Fin.succ i)",
"tactic": "rw [δ_comp_δ_self]"
}
] |
[
255,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.coe_mk
|
[] |
[
169,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.spanCompIso_app_left
|
[] |
[
344,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_ite_empty_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t✝ t₁ t₂ u : Set α\np : Prop\ninst✝ : Decidable p\nt : Set α\nx : α\n⊢ (∃ h, x ∈ t) ↔ ¬p ∧ x ∈ t",
"tactic": "simp"
}
] |
[
2233,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2231,
1
] |
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