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/Order/ConditionallyCompleteLattice/Basic.lean
|
ciSup_mem_Inter_Icc_of_antitone_Icc
|
[] |
[
904,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
899,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.cmp_eq
|
[
{
"state_after": "α : Type ?u.392175\nm n : PosNum\nthis : Ordering.casesOn (cmp m n) (↑m < ↑n) (m = n) (↑n < ↑m)\n⊢ cmp m n = Ordering.eq ↔ m = n",
"state_before": "α : Type ?u.392175\nm n : PosNum\n⊢ cmp m n = Ordering.eq ↔ m = n",
"tactic": "have := cmp_to_nat m n"
},
{
"state_after": "α : Type ?u.392175\nm n : PosNum\n⊢ Ordering.casesOn (cmp m n) (↑m < ↑n) (m = n) (↑n < ↑m) → (cmp m n = Ordering.eq ↔ m = n)",
"state_before": "α : Type ?u.392175\nm n : PosNum\nthis : Ordering.casesOn (cmp m n) (↑m < ↑n) (m = n) (↑n < ↑m)\n⊢ cmp m n = Ordering.eq ↔ m = n",
"tactic": "revert this"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.392175\nm n : PosNum\n⊢ Ordering.casesOn (cmp m n) (↑m < ↑n) (m = n) (↑n < ↑m) → (cmp m n = Ordering.eq ↔ m = n)",
"tactic": "cases cmp m n <;> intro this <;> simp at this ⊢ <;> try { exact this } <;>\nsimp [show m ≠ n from fun e => by rw [e] at this;exact lt_irrefl _ this]"
},
{
"state_after": "α : Type ?u.392175\nm n : PosNum\nthis : ↑n < ↑n\ne : m = n\n⊢ False",
"state_before": "α : Type ?u.392175\nm n : PosNum\nthis : ↑n < ↑m\ne : m = n\n⊢ False",
"tactic": "rw [e] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.392175\nm n : PosNum\nthis : ↑n < ↑n\ne : m = n\n⊢ False",
"tactic": "exact lt_irrefl _ this"
}
] |
[
697,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
693,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.def
|
[] |
[
167,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.sum_repr_mul_repr
|
[
{
"state_after": "ι : Type u_4\nι'✝ : Type ?u.761324\nR : Type u_2\nR₂ : Type ?u.761330\nK : Type ?u.761333\nM : Type u_3\nM' : Type ?u.761339\nM'' : Type ?u.761342\nV : Type u\nV' : Type ?u.761347\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nb : Basis ι R M\nb'✝ : Basis ι'✝ R M'\nι' : Type u_1\ninst✝ : Fintype ι'\nb' : Basis ι' R M\nx : M\ni : ι\n⊢ ∑ j : ι', ↑(↑b.repr (↑b' j)) i * ↑(↑b'.repr x) j = ↑(↑b.repr (∑ i : ι', ↑(↑b'.repr x) i • ↑b' i)) i",
"state_before": "ι : Type u_4\nι'✝ : Type ?u.761324\nR : Type u_2\nR₂ : Type ?u.761330\nK : Type ?u.761333\nM : Type u_3\nM' : Type ?u.761339\nM'' : Type ?u.761342\nV : Type u\nV' : Type ?u.761347\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nb : Basis ι R M\nb'✝ : Basis ι'✝ R M'\nι' : Type u_1\ninst✝ : Fintype ι'\nb' : Basis ι' R M\nx : M\ni : ι\n⊢ ∑ j : ι', ↑(↑b.repr (↑b' j)) i * ↑(↑b'.repr x) j = ↑(↑b.repr x) i",
"tactic": "conv_rhs => rw [← b'.sum_repr x]"
},
{
"state_after": "ι : Type u_4\nι'✝ : Type ?u.761324\nR : Type u_2\nR₂ : Type ?u.761330\nK : Type ?u.761333\nM : Type u_3\nM' : Type ?u.761339\nM'' : Type ?u.761342\nV : Type u\nV' : Type ?u.761347\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nb : Basis ι R M\nb'✝ : Basis ι'✝ R M'\nι' : Type u_1\ninst✝ : Fintype ι'\nb' : Basis ι' R M\nx : M\ni : ι\n⊢ ∑ x_1 : ι', ↑(↑b.repr (↑b' x_1)) i * ↑(↑b'.repr x) x_1 = ∑ x_1 : ι', ↑(↑(↑b'.repr x) x_1 • ↑b.repr (↑b' x_1)) i",
"state_before": "ι : Type u_4\nι'✝ : Type ?u.761324\nR : Type u_2\nR₂ : Type ?u.761330\nK : Type ?u.761333\nM : Type u_3\nM' : Type ?u.761339\nM'' : Type ?u.761342\nV : Type u\nV' : Type ?u.761347\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nb : Basis ι R M\nb'✝ : Basis ι'✝ R M'\nι' : Type u_1\ninst✝ : Fintype ι'\nb' : Basis ι' R M\nx : M\ni : ι\n⊢ ∑ j : ι', ↑(↑b.repr (↑b' j)) i * ↑(↑b'.repr x) j = ↑(↑b.repr (∑ i : ι', ↑(↑b'.repr x) i • ↑b' i)) i",
"tactic": "simp_rw [LinearEquiv.map_sum, LinearEquiv.map_smul, Finset.sum_apply']"
},
{
"state_after": "ι : Type u_4\nι'✝ : Type ?u.761324\nR : Type u_2\nR₂ : Type ?u.761330\nK : Type ?u.761333\nM : Type u_3\nM' : Type ?u.761339\nM'' : Type ?u.761342\nV : Type u\nV' : Type ?u.761347\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nb : Basis ι R M\nb'✝ : Basis ι'✝ R M'\nι' : Type u_1\ninst✝ : Fintype ι'\nb' : Basis ι' R M\nx : M\ni : ι\nj : ι'\nx✝ : j ∈ Finset.univ\n⊢ ↑(↑b.repr (↑b' j)) i * ↑(↑b'.repr x) j = ↑(↑(↑b'.repr x) j • ↑b.repr (↑b' j)) i",
"state_before": "ι : Type u_4\nι'✝ : Type ?u.761324\nR : Type u_2\nR₂ : Type ?u.761330\nK : Type ?u.761333\nM : Type u_3\nM' : Type ?u.761339\nM'' : Type ?u.761342\nV : Type u\nV' : Type ?u.761347\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nb : Basis ι R M\nb'✝ : Basis ι'✝ R M'\nι' : Type u_1\ninst✝ : Fintype ι'\nb' : Basis ι' R M\nx : M\ni : ι\n⊢ ∑ x_1 : ι', ↑(↑b.repr (↑b' x_1)) i * ↑(↑b'.repr x) x_1 = ∑ x_1 : ι', ↑(↑(↑b'.repr x) x_1 • ↑b.repr (↑b' x_1)) i",
"tactic": "refine' Finset.sum_congr rfl fun j _ => _"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_4\nι'✝ : Type ?u.761324\nR : Type u_2\nR₂ : Type ?u.761330\nK : Type ?u.761333\nM : Type u_3\nM' : Type ?u.761339\nM'' : Type ?u.761342\nV : Type u\nV' : Type ?u.761347\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\nb : Basis ι R M\nb'✝ : Basis ι'✝ R M'\nι' : Type u_1\ninst✝ : Fintype ι'\nb' : Basis ι' R M\nx : M\ni : ι\nj : ι'\nx✝ : j ∈ Finset.univ\n⊢ ↑(↑b.repr (↑b' j)) i * ↑(↑b'.repr x) j = ↑(↑(↑b'.repr x) j • ↑b.repr (↑b' j)) i",
"tactic": "rw [Finsupp.smul_apply, smul_eq_mul, mul_comm]"
}
] |
[
1067,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.natAbs_valMinAbs_add_le
|
[
{
"state_after": "case zero\nn a✝ : ℕ\na b : ZMod Nat.zero\n⊢ Int.natAbs (valMinAbs (a + b)) ≤ Int.natAbs (valMinAbs a + valMinAbs b)\n\ncase succ\nn✝ a✝ n : ℕ\na b : ZMod (Nat.succ n)\n⊢ Int.natAbs (valMinAbs (a + b)) ≤ Int.natAbs (valMinAbs a + valMinAbs b)",
"state_before": "n✝ a✝ n : ℕ\na b : ZMod n\n⊢ Int.natAbs (valMinAbs (a + b)) ≤ Int.natAbs (valMinAbs a + valMinAbs b)",
"tactic": "cases' n with n"
},
{
"state_after": "case succ.he\nn✝ a✝ n : ℕ\na b : ZMod (Nat.succ n)\n⊢ ↑(valMinAbs (a + b)) = ↑(valMinAbs a + valMinAbs b)\n\ncase succ.hl\nn✝ a✝ n : ℕ\na b : ZMod (Nat.succ n)\n⊢ Int.natAbs (valMinAbs (a + b)) ≤ Nat.succ n / 2",
"state_before": "case succ\nn✝ a✝ n : ℕ\na b : ZMod (Nat.succ n)\n⊢ Int.natAbs (valMinAbs (a + b)) ≤ Int.natAbs (valMinAbs a + valMinAbs b)",
"tactic": "apply natAbs_min_of_le_div_two n.succ"
},
{
"state_after": "no goals",
"state_before": "case zero\nn a✝ : ℕ\na b : ZMod Nat.zero\n⊢ Int.natAbs (valMinAbs (a + b)) ≤ Int.natAbs (valMinAbs a + valMinAbs b)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ.he\nn✝ a✝ n : ℕ\na b : ZMod (Nat.succ n)\n⊢ ↑(valMinAbs (a + b)) = ↑(valMinAbs a + valMinAbs b)",
"tactic": "simp_rw [Int.cast_add, coe_valMinAbs]"
},
{
"state_after": "no goals",
"state_before": "case succ.hl\nn✝ a✝ n : ℕ\na b : ZMod (Nat.succ n)\n⊢ Int.natAbs (valMinAbs (a + b)) ≤ Nat.succ n / 2",
"tactic": "apply natAbs_valMinAbs_le"
}
] |
[
1112,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1106,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_list_count_of_subset
|
[
{
"state_after": "ι : Type ?u.498342\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : List α\ns : Finset α\nhs : toFinset m ⊆ s\n⊢ ∏ m_1 in toFinset m, m_1 ^ count m_1 m = ∏ i in s, i ^ count i m",
"state_before": "ι : Type ?u.498342\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : List α\ns : Finset α\nhs : toFinset m ⊆ s\n⊢ prod m = ∏ i in s, i ^ count i m",
"tactic": "rw [prod_list_count]"
},
{
"state_after": "ι : Type ?u.498342\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : List α\ns : Finset α\nhs : toFinset m ⊆ s\nx : α\nx✝ : x ∈ s\nhx : ¬x ∈ toFinset m\n⊢ x ^ count x m = 1",
"state_before": "ι : Type ?u.498342\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : List α\ns : Finset α\nhs : toFinset m ⊆ s\n⊢ ∏ m_1 in toFinset m, m_1 ^ count m_1 m = ∏ i in s, i ^ count i m",
"tactic": "refine' prod_subset hs fun x _ hx => _"
},
{
"state_after": "ι : Type ?u.498342\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : List α\ns : Finset α\nhs : toFinset m ⊆ s\nx : α\nx✝ : x ∈ s\nhx : ¬x ∈ m\n⊢ x ^ count x m = 1",
"state_before": "ι : Type ?u.498342\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : List α\ns : Finset α\nhs : toFinset m ⊆ s\nx : α\nx✝ : x ∈ s\nhx : ¬x ∈ toFinset m\n⊢ x ^ count x m = 1",
"tactic": "rw [mem_toFinset] at hx"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.498342\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : List α\ns : Finset α\nhs : toFinset m ⊆ s\nx : α\nx✝ : x ∈ s\nhx : ¬x ∈ m\n⊢ x ^ count x m = 1",
"tactic": "rw [count_eq_zero_of_not_mem hx, pow_zero]"
}
] |
[
1306,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1301,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.snd_diag
|
[
{
"state_after": "case h\nn : ℕ\nα : TypeVec n\ni : Fin2 n\n⊢ (prod.snd ⊚ prod.diag) i = id i",
"state_before": "n : ℕ\nα : TypeVec n\n⊢ prod.snd ⊚ prod.diag = id",
"tactic": "funext i"
},
{
"state_after": "case h.fz\nn n✝ : ℕ\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) Fin2.fz = id Fin2.fz\n\ncase h.fs\nn n✝ : ℕ\na✝ : Fin2 n✝\na_ih✝ : ∀ {α : TypeVec n✝}, (prod.snd ⊚ prod.diag) a✝ = id a✝\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) (Fin2.fs a✝) = id (Fin2.fs a✝)",
"state_before": "case h\nn : ℕ\nα : TypeVec n\ni : Fin2 n\n⊢ (prod.snd ⊚ prod.diag) i = id i",
"tactic": "induction i"
},
{
"state_after": "case h.fs\nn n✝ : ℕ\na✝ : Fin2 n✝\na_ih✝ : ∀ {α : TypeVec n✝}, (prod.snd ⊚ prod.diag) a✝ = id a✝\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) (Fin2.fs a✝) = id (Fin2.fs a✝)",
"state_before": "case h.fz\nn n✝ : ℕ\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) Fin2.fz = id Fin2.fz\n\ncase h.fs\nn n✝ : ℕ\na✝ : Fin2 n✝\na_ih✝ : ∀ {α : TypeVec n✝}, (prod.snd ⊚ prod.diag) a✝ = id a✝\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) (Fin2.fs a✝) = id (Fin2.fs a✝)",
"tactic": "case fz => rfl"
},
{
"state_after": "no goals",
"state_before": "case h.fs\nn n✝ : ℕ\na✝ : Fin2 n✝\na_ih✝ : ∀ {α : TypeVec n✝}, (prod.snd ⊚ prod.diag) a✝ = id a✝\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) (Fin2.fs a✝) = id (Fin2.fs a✝)",
"tactic": "case fs _ _ i_ih => apply i_ih"
},
{
"state_after": "no goals",
"state_before": "n n✝ : ℕ\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) Fin2.fz = id Fin2.fz",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "n n✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝}, (prod.snd ⊚ prod.diag) a✝ = id a✝\nα : TypeVec (succ n✝)\n⊢ (prod.snd ⊚ prod.diag) (Fin2.fs a✝) = id (Fin2.fs a✝)",
"tactic": "apply i_ih"
}
] |
[
593,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.IntegrableOn.add_measure
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1518598\nE : Type u_2\nF : Type ?u.1518604\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nhμ : IntegrableOn f s\nhν : IntegrableOn f s\n⊢ Integrable f",
"state_before": "α : Type u_1\nβ : Type ?u.1518598\nE : Type u_2\nF : Type ?u.1518604\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nhμ : IntegrableOn f s\nhν : IntegrableOn f s\n⊢ IntegrableOn f s",
"tactic": "delta IntegrableOn"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1518598\nE : Type u_2\nF : Type ?u.1518604\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nhμ : IntegrableOn f s\nhν : IntegrableOn f s\n⊢ Integrable f",
"state_before": "α : Type u_1\nβ : Type ?u.1518598\nE : Type u_2\nF : Type ?u.1518604\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nhμ : IntegrableOn f s\nhν : IntegrableOn f s\n⊢ Integrable f",
"tactic": "rw [Measure.restrict_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1518598\nE : Type u_2\nF : Type ?u.1518604\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nhμ : IntegrableOn f s\nhν : IntegrableOn f s\n⊢ Integrable f",
"tactic": "exact hμ.integrable.add_measure hν"
}
] |
[
226,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean
|
uniqueDiffOn_Icc_zero_one
|
[] |
[
453,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.coe_injective
|
[] |
[
569,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.sups_inter_subset_left
|
[] |
[
172,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.refl_prod_refl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.63296\nδ : Type ?u.63299\ne : LocalEquiv α β\ne' : LocalEquiv β γ\n⊢ prod (LocalEquiv.refl α) (LocalEquiv.refl β) = LocalEquiv.refl (α × β)",
"tactic": "ext ⟨x, y⟩ <;> simp"
}
] |
[
971,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
968,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.iff_def
|
[] |
[
340,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.mabs_sup_div_sup_le_mabs
|
[
{
"state_after": "case h\nα : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ abs ((a ⊔ c) / (b ⊔ c)) * ?b ≤ abs (a / b)\n\ncase hle\nα : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ 1 ≤ ?b\n\ncase b\nα : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ α",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ abs ((a ⊔ c) / (b ⊔ c)) ≤ abs (a / b)",
"tactic": "apply le_of_mul_le_of_one_le_left"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ abs ((a ⊔ c) / (b ⊔ c)) * ?b ≤ abs (a / b)",
"tactic": "rw [abs_div_sup_mul_abs_div_inf]"
},
{
"state_after": "no goals",
"state_before": "case hle\nα : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ 1 ≤ abs ((a ⊓ c) / (b ⊓ c))",
"tactic": "exact one_le_abs _"
}
] |
[
538,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/RingTheory/EuclideanDomain.lean
|
EuclideanDomain.dvd_or_coprime
|
[] |
[
103,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Analysis/Convex/Combination.lean
|
convexHull_eq_union_convexHull_finite_subsets
|
[
{
"state_after": "case refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Set E\n⊢ ↑(convexHull R).toOrderHom s ⊆ ⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t\n\ncase refine'_2\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Set E\n⊢ (⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t) ⊆ ↑(convexHull R).toOrderHom s",
"state_before": "R : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Set E\n⊢ ↑(convexHull R).toOrderHom s = ⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t",
"tactic": "refine' Subset.antisymm _ _"
},
{
"state_after": "case refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Set E\n⊢ {x | ∃ ι t w z x_1 x_2 x_3, centerMass t w z = x} ⊆ ⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t",
"state_before": "case refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Set E\n⊢ ↑(convexHull R).toOrderHom s ⊆ ⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t",
"tactic": "rw [_root_.convexHull_eq]"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ centerMass t w z ∈ ⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t",
"state_before": "case refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Set E\n⊢ {x | ∃ ι t w z x_1 x_2 x_3, centerMass t w z = x} ⊆ ⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t",
"tactic": "rintro x ⟨ι : Type u_1, t, w, z, hw₀, hw₁, hz, rfl⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ∃ i i_1, centerMass t w z ∈ ↑(convexHull R).toOrderHom ↑i",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ centerMass t w z ∈ ⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t",
"tactic": "simp only [mem_iUnion]"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ↑(Finset.image z t) ⊆ s\n\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ centerMass t w z ∈ ↑(convexHull R).toOrderHom ↑(Finset.image z t)",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ∃ i i_1, centerMass t w z ∈ ↑(convexHull R).toOrderHom ↑i",
"tactic": "refine' ⟨t.image z, _, _⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ↑t ⊆ z ⁻¹' s",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ↑(Finset.image z t) ⊆ s",
"tactic": "rw [coe_image, Set.image_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ↑t ⊆ z ⁻¹' s",
"tactic": "exact hz"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2.hws\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ 0 < ∑ i in t, w i\n\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2.hz\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ∀ (i : ι), i ∈ t → z i ∈ ↑(Finset.image z t)",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ centerMass t w z ∈ ↑(convexHull R).toOrderHom ↑(Finset.image z t)",
"tactic": "apply t.centerMass_mem_convexHull hw₀"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2.hws\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ 0 < ∑ i in t, w i",
"tactic": "simp only [hw₁, zero_lt_one]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2.hz\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι✝ : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz✝ : ι✝ → E\ns : Set E\nι : Type u_1\nt : Finset ι\nw : ι → R\nz : ι → E\nhw₀ : ∀ (i : ι), i ∈ t → 0 ≤ w i\nhw₁ : ∑ i in t, w i = 1\nhz : ∀ (i : ι), i ∈ t → z i ∈ s\n⊢ ∀ (i : ι), i ∈ t → z i ∈ ↑(Finset.image z t)",
"tactic": "exact fun i hi => Finset.mem_coe.2 (Finset.mem_image_of_mem _ hi)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_2\nE : Type u_1\nF : Type ?u.239379\nι : Type ?u.239382\nι' : Type ?u.239385\nα : Type ?u.239388\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ns : Set E\n⊢ (⋃ (t : Finset E) (_ : ↑t ⊆ s), ↑(convexHull R).toOrderHom ↑t) ⊆ ↑(convexHull R).toOrderHom s",
"tactic": "exact iUnion_subset fun i => iUnion_subset convexHull_mono"
}
] |
[
374,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.sum_integral_adjacent_intervals
|
[
{
"state_after": "ι : Type ?u.15613852\n𝕜 : Type ?u.15613855\nE : Type u_1\nF : Type ?u.15613861\nA : Type ?u.15613864\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nn : ℕ\nhint : ∀ (k : ℕ), k < n → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ (∑ k in Finset.Ico 0 n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a 0 ..a n, f x ∂μ",
"state_before": "ι : Type ?u.15613852\n𝕜 : Type ?u.15613855\nE : Type u_1\nF : Type ?u.15613861\nA : Type ?u.15613864\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nn : ℕ\nhint : ∀ (k : ℕ), k < n → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ (∑ k in Finset.range n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a 0 ..a n, f x ∂μ",
"tactic": "rw [← Nat.Ico_zero_eq_range]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.15613852\n𝕜 : Type ?u.15613855\nE : Type u_1\nF : Type ?u.15613861\nA : Type ?u.15613864\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nn : ℕ\nhint : ∀ (k : ℕ), k < n → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ (∑ k in Finset.Ico 0 n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a 0 ..a n, f x ∂μ",
"tactic": "exact sum_integral_adjacent_intervals_Ico (zero_le n) fun k hk => hint k hk.2"
}
] |
[
928,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
924,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
SupHom.bot_apply
|
[] |
[
505,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
exp_neg_of_mem_ball
|
[] |
[
362,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Order/SymmDiff.lean
|
disjoint_symmDiff_inf
|
[
{
"state_after": "ι : Type ?u.59758\nα : Type u_1\nβ : Type ?u.59764\nπ : ι → Type ?u.59769\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ Disjoint ((a ⊔ b) \\ (a ⊓ b)) (a ⊓ b)",
"state_before": "ι : Type ?u.59758\nα : Type u_1\nβ : Type ?u.59764\nπ : ι → Type ?u.59769\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ Disjoint (a ∆ b) (a ⊓ b)",
"tactic": "rw [symmDiff_eq_sup_sdiff_inf]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.59758\nα : Type u_1\nβ : Type ?u.59764\nπ : ι → Type ?u.59769\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ Disjoint ((a ⊔ b) \\ (a ⊓ b)) (a ⊓ b)",
"tactic": "exact disjoint_sdiff_self_left"
}
] |
[
403,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.valMinAbs_natAbs_eq_min
|
[
{
"state_after": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\n⊢ Int.natAbs (if val a ≤ n / 2 then ↑(val a) else ↑(val a) - ↑n) = min (val a) (n - val a)",
"state_before": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\n⊢ Int.natAbs (valMinAbs a) = min (val a) (n - val a)",
"tactic": "rw [valMinAbs_def_pos]"
},
{
"state_after": "case inl\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ Int.natAbs ↑(val a) = min (val a) (n - val a)\n\ncase inr\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ Int.natAbs (↑(val a) - ↑n) = min (val a) (n - val a)",
"state_before": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\n⊢ Int.natAbs (if val a ≤ n / 2 then ↑(val a) else ↑(val a) - ↑n) = min (val a) (n - val a)",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ val a = min (val a) (n - val a)",
"state_before": "case inl\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ Int.natAbs ↑(val a) = min (val a) (n - val a)",
"tactic": "rw [Int.natAbs_ofNat]"
},
{
"state_after": "case inl\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ min (val a) (n - val a) = val a",
"state_before": "case inl\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ val a = min (val a) (n - val a)",
"tactic": "symm"
},
{
"state_after": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ n - (n - n / 2) ≤ n / 2",
"state_before": "case inl\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ min (val a) (n - val a) = val a",
"tactic": "apply\n min_eq_left (le_trans h (le_trans (Nat.half_le_of_sub_le_half _) (Nat.sub_le_sub_left n h)))"
},
{
"state_after": "no goals",
"state_before": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : val a ≤ n / 2\n⊢ n - (n - n / 2) ≤ n / 2",
"tactic": "rw [Nat.sub_sub_self (Nat.div_le_self _ _)]"
},
{
"state_after": "case inr\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ Int.natAbs ↑(n - val a) = min (val a) (n - val a)",
"state_before": "case inr\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ Int.natAbs (↑(val a) - ↑n) = min (val a) (n - val a)",
"tactic": "rw [← Int.natAbs_neg, neg_sub, ← Nat.cast_sub a.val_le]"
},
{
"state_after": "case inr\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ min (val a) (n - val a) = Int.natAbs ↑(n - val a)",
"state_before": "case inr\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ Int.natAbs ↑(n - val a) = min (val a) (n - val a)",
"tactic": "symm"
},
{
"state_after": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ n / 2 < n - (n - Nat.succ (n / 2))",
"state_before": "case inr\nn✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ min (val a) (n - val a) = Int.natAbs ↑(n - val a)",
"tactic": "apply\n min_eq_right\n (le_trans (le_trans (Nat.sub_le_sub_left n (lt_of_not_ge h)) (Nat.le_half_of_half_lt_sub _))\n (le_of_not_ge h))"
},
{
"state_after": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ n / 2 < Nat.succ (n / 2)",
"state_before": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ n / 2 < n - (n - Nat.succ (n / 2))",
"tactic": "rw [Nat.sub_sub_self (Nat.div_lt_self (lt_of_le_of_ne' (Nat.zero_le _) hpos.1) one_lt_two)]"
},
{
"state_after": "no goals",
"state_before": "n✝ a✝ n : ℕ\nhpos : NeZero n\na : ZMod n\nh : ¬val a ≤ n / 2\n⊢ n / 2 < Nat.succ (n / 2)",
"tactic": "apply Nat.lt_succ_self"
}
] |
[
1065,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1049,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.codisjoint_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4270\nγ : Type ?u.4273\ninst✝ : Fintype α\ns t : Finset α\n⊢ Codisjoint s t ↔ ∀ ⦃a : α⦄, ¬a ∈ s → a ∈ t",
"tactic": "classical simp [codisjoint_iff, eq_univ_iff_forall, or_iff_not_imp_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4270\nγ : Type ?u.4273\ninst✝ : Fintype α\ns t : Finset α\n⊢ Codisjoint s t ↔ ∀ ⦃a : α⦄, ¬a ∈ s → a ∈ t",
"tactic": "simp [codisjoint_iff, eq_univ_iff_forall, or_iff_not_imp_left]"
}
] |
[
146,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.continuous_eval_const
|
[] |
[
326,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.exists_image_iff
|
[] |
[
582,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.map_comp_proj
|
[] |
[
320,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
wbtw_self_iff
|
[
{
"state_after": "case refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : Wbtw R x y x\n⊢ y = x\n\ncase refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : y = x\n⊢ Wbtw R x y x",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\n⊢ Wbtw R x y x ↔ y = x",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : Wbtw R x y x\nw✝ : R\nleft✝ : w✝ ∈ Set.Icc 0 1\nh₂ : ↑(lineMap x x) w✝ = y\n⊢ y = x",
"state_before": "case refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : Wbtw R x y x\n⊢ y = x",
"tactic": "have ⟨_, _, h₂⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : Wbtw R x y x\nw✝ : R\nleft✝ : w✝ ∈ Set.Icc 0 1\nh₂ : ↑(lineMap x x) w✝ = y\n⊢ y = x",
"tactic": "rw [h₂.symm, lineMap_same_apply]"
},
{
"state_after": "case refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : y = x\n⊢ Wbtw R x x x",
"state_before": "case refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : y = x\n⊢ Wbtw R x y x",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type ?u.153895\nP : Type u_3\nP' : Type ?u.153901\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y : P\nh : y = x\n⊢ Wbtw R x x x",
"tactic": "exact wbtw_self_left R x x"
}
] |
[
307,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_lt_self_iff
|
[] |
[
482,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/LinearAlgebra/Orientation.lean
|
Orientation.map_refl
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type ?u.10058\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nι : Type u_3\n⊢ map ι (LinearEquiv.refl R M) = Equiv.refl (Orientation R M ι)",
"tactic": "rw [Orientation.map, AlternatingMap.domLCongr_refl, Module.Ray.map_refl]"
}
] |
[
83,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.associator_naturality
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nX₁ : ModuleCat R\nX₂ : ModuleCat R\nX₃ : ModuleCat R\nY₁ : ModuleCat R\nY₂ : ModuleCat R\nY₃ : ModuleCat R\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\nf₃ : X₃ ⟶ Y₃\n⊢ tensorHom (tensorHom f₁ f₂) f₃ ≫ (associator Y₁ Y₂ Y₃).hom =\n (associator X₁ X₂ X₃).hom ≫ tensorHom f₁ (tensorHom f₂ f₃)",
"tactic": "convert associator_naturality_aux f₁ f₂ f₃ using 1"
}
] |
[
127,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.head_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\ns : Seq α\n⊢ head (cons a s) = some a",
"tactic": "rw [head_eq_destruct, destruct_cons, Option.map_eq_map, Option.map_some']"
}
] |
[
262,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.toAddSubmonoid_eq
|
[] |
[
135,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.ind'
|
[] |
[
653,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
11
] |
Mathlib/RingTheory/Adjoin/Basic.lean
|
Algebra.adjoin_iUnion
|
[] |
[
75,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/FieldTheory/Minpoly/Field.lean
|
minpoly.eq_X_sub_C
|
[] |
[
212,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.coe_natCast
|
[] |
[
1290,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1290,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.zigzag_equivalence
|
[] |
[
274,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Data/List/Intervals.lean
|
List.Ico.chain'_succ
|
[
{
"state_after": "case pos\nn m : ℕ\nh : n < m\n⊢ Chain' (fun a b => b = succ a) (Ico n m)\n\ncase neg\nn m : ℕ\nh : ¬n < m\n⊢ Chain' (fun a b => b = succ a) (Ico n m)",
"state_before": "n m : ℕ\n⊢ Chain' (fun a b => b = succ a) (Ico n m)",
"tactic": "by_cases n < m"
},
{
"state_after": "case pos\nn m : ℕ\nh : n < m\n⊢ Chain' (fun a b => b = succ a) (n :: Ico (n + 1) m)",
"state_before": "case pos\nn m : ℕ\nh : n < m\n⊢ Chain' (fun a b => b = succ a) (Ico n m)",
"tactic": "rw [eq_cons h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nn m : ℕ\nh : n < m\n⊢ Chain' (fun a b => b = succ a) (n :: Ico (n + 1) m)",
"tactic": "exact chain_succ_range' _ _ 1"
},
{
"state_after": "case neg\nn m : ℕ\nh : ¬n < m\n⊢ Chain' (fun a b => b = succ a) []",
"state_before": "case neg\nn m : ℕ\nh : ¬n < m\n⊢ Chain' (fun a b => b = succ a) (Ico n m)",
"tactic": "rw [eq_nil_of_le (le_of_not_gt h)]"
},
{
"state_after": "no goals",
"state_before": "case neg\nn m : ℕ\nh : ¬n < m\n⊢ Chain' (fun a b => b = succ a) []",
"tactic": "trivial"
}
] |
[
150,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
mul_eq_mul_iff_eq_and_eq_of_pos
|
[
{
"state_after": "case inl\nα : Type u_1\na c d : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhbd : c ≤ d\na0 : 0 < a\nd0 : 0 < d\nhac : a ≤ a\nh : a * c = a * d\n⊢ a = a ∧ c = d\n\ncase inr\nα : Type u_1\na b c d : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhac✝ : a ≤ b\nhbd : c ≤ d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\nhac : a < b\n⊢ a = b ∧ c = d",
"state_before": "α : Type u_1\na b c d : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhac : a ≤ b\nhbd : c ≤ d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\n⊢ a = b ∧ c = d",
"tactic": "rcases hac.eq_or_lt with (rfl | hac)"
},
{
"state_after": "case inr.inl\nα : Type u_1\na b c : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhac✝ : a ≤ b\na0 : 0 < a\nhac : a < b\nhbd : c ≤ c\nd0 : 0 < c\nh : a * c = b * c\n⊢ a = b ∧ c = c\n\ncase inr.inr\nα : Type u_1\na b c d : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhac✝ : a ≤ b\nhbd✝ : c ≤ d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\nhac : a < b\nhbd : c < d\n⊢ a = b ∧ c = d",
"state_before": "case inr\nα : Type u_1\na b c d : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhac✝ : a ≤ b\nhbd : c ≤ d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\nhac : a < b\n⊢ a = b ∧ c = d",
"tactic": "rcases eq_or_lt_of_le hbd with (rfl | hbd)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\na b c d : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhac✝ : a ≤ b\nhbd✝ : c ≤ d\na0 : 0 < a\nd0 : 0 < d\nh : a * c = b * d\nhac : a < b\nhbd : c < d\n⊢ a = b ∧ c = d",
"tactic": "exact ((mul_lt_mul_of_pos_of_pos hac hbd a0 d0).ne h).elim"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\na c d : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhbd : c ≤ d\na0 : 0 < a\nd0 : 0 < d\nhac : a ≤ a\nh : a * c = a * d\n⊢ a = a ∧ c = d",
"tactic": "exact ⟨rfl, (mul_left_cancel_iff_of_pos a0).mp h⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\na b c : α\ninst✝⁵ : MulZeroClass α\ninst✝⁴ : PartialOrder α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : PosMulMonoRev α\ninst✝ : MulPosMonoRev α\nhac✝ : a ≤ b\na0 : 0 < a\nhac : a < b\nhbd : c ≤ c\nd0 : 0 < c\nh : a * c = b * c\n⊢ a = b ∧ c = c",
"tactic": "exact ⟨(mul_right_cancel_iff_of_pos d0).mp h, rfl⟩"
}
] |
[
539,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.map_lt_add
|
[] |
[
696,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.continuousAt_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.100766\nι : Type ?u.100769\ninst✝¹ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ninst✝ : PseudoMetricSpace β\nf : α → β\na : α\n⊢ ContinuousAt f a ↔ ∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε",
"tactic": "rw [ContinuousAt, tendsto_nhds_nhds]"
}
] |
[
1054,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1052,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.log_eq_one_iff'
|
[
{
"state_after": "no goals",
"state_before": "b n : ℕ\n⊢ log b n = 1 ↔ b ≤ n ∧ n < b * b",
"tactic": "rw [log_eq_iff (Or.inl one_ne_zero), pow_add, pow_one]"
}
] |
[
165,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Order/Basic.lean
|
strongLT_of_strongLT_of_le
|
[] |
[
838,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
837,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.Nontrivial.ne_empty
|
[] |
[
2528,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2527,
11
] |
Mathlib/LinearAlgebra/Basis.lean
|
atom_iff_nonzero_span
|
[
{
"state_after": "case refine'_1\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nh : IsAtom W\n⊢ ∃ v x, W = span K {v}\n\ncase refine'_2\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nh : ∃ v x, W = span K {v}\n⊢ IsAtom W",
"state_before": "ι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\n⊢ IsAtom W ↔ ∃ v x, W = span K {v}",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\n⊢ ∃ v x, W = span K {v}",
"state_before": "case refine'_1\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nh : IsAtom W\n⊢ ∃ v x, W = span K {v}",
"tactic": "cases' h with hbot h"
},
{
"state_after": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\n⊢ ∃ v x, W = span K {v}",
"state_before": "case refine'_1.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\n⊢ ∃ v x, W = span K {v}",
"tactic": "rcases(Submodule.ne_bot_iff W).1 hbot with ⟨v, ⟨hW, hv⟩⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\n⊢ W = span K {v}",
"state_before": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\n⊢ ∃ v x, W = span K {v}",
"tactic": "refine' ⟨v, ⟨hv, _⟩⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\nheq : ¬W = span K {v}\n⊢ False",
"state_before": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\n⊢ W = span K {v}",
"tactic": "by_contra heq"
},
{
"state_after": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\nheq : ¬W = span K {v}\nh : span K {v} < W → span K {v} = ⊥\n⊢ False",
"state_before": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nh : ∀ (b : Submodule K V), b < W → b = ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\nheq : ¬W = span K {v}\n⊢ False",
"tactic": "specialize h (span K {v})"
},
{
"state_after": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\nheq : ¬W = span K {v}\nh : span K {v} ≤ W ∧ span K {v} ≠ W → v = 0\n⊢ False",
"state_before": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\nheq : ¬W = span K {v}\nh : span K {v} < W → span K {v} = ⊥\n⊢ False",
"tactic": "rw [span_singleton_eq_bot, lt_iff_le_and_ne] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nhbot : W ≠ ⊥\nv : V\nhW : v ∈ W\nhv : v ≠ 0\nheq : ¬W = span K {v}\nh : span K {v} ≤ W ∧ span K {v} ≠ W → v = 0\n⊢ False",
"tactic": "exact hv (h ⟨(span_singleton_le_iff_mem v W).2 hW, Ne.symm heq⟩)"
},
{
"state_after": "case refine'_2.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z v : V\nhv : v ≠ 0\n⊢ IsAtom (span K {v})",
"state_before": "case refine'_2\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nW : Submodule K V\nh : ∃ v x, W = span K {v}\n⊢ IsAtom W",
"tactic": "rcases h with ⟨v, ⟨hv, rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nι : Type ?u.1173697\nι' : Type ?u.1173700\nR : Type ?u.1173703\nR₂ : Type ?u.1173706\nK : Type u_1\nM : Type ?u.1173712\nM' : Type ?u.1173715\nM'' : Type ?u.1173718\nV : Type u\nV' : Type ?u.1173723\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv✝ : ι → V\ns t : Set V\nx y z v : V\nhv : v ≠ 0\n⊢ IsAtom (span K {v})",
"tactic": "exact nonzero_span_atom v hv"
}
] |
[
1550,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1539,
1
] |
Mathlib/Tactic/CancelDenoms.lean
|
CancelDenoms.sub_subst
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Ring α\nn e1 e2 t1 t2 : α\nh1 : n * e1 = t1\nh2 : n * e2 = t2\n⊢ n * (e1 - e2) = t1 - t2",
"tactic": "simp [left_distrib, *, sub_eq_add_neg]"
}
] |
[
58,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
rescale_to_shell
|
[] |
[
416,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval_assoc
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\n⊢ ↑(eval (↑(eval g) ∘ f)) p = eval₂ (RingHom.comp (eval g) C) (↑(eval g) ∘ f) p",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\n⊢ ↑(eval (↑(eval g) ∘ f)) p = ↑(eval g) (eval₂ C f p)",
"tactic": "rw [eval₂_comp_left (eval g)]"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\n⊢ ↑(eval₂Hom (RingHom.id R) (↑(eval₂Hom (RingHom.id R) g) ∘ f)) p =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id R) g) C) (↑(eval₂Hom (RingHom.id R) g) ∘ f) p",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\n⊢ ↑(eval (↑(eval g) ∘ f)) p = eval₂ (RingHom.comp (eval g) C) (↑(eval g) ∘ f) p",
"tactic": "unfold eval"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\n⊢ eval₂ (RingHom.id R) (eval₂ (RingHom.id R) g ∘ f) p =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id R) g) C) (eval₂ (RingHom.id R) g ∘ f) p",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\n⊢ ↑(eval₂Hom (RingHom.id R) (↑(eval₂Hom (RingHom.id R) g) ∘ f)) p =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id R) g) C) (↑(eval₂Hom (RingHom.id R) g) ∘ f) p",
"tactic": "simp only [coe_eval₂Hom]"
},
{
"state_after": "case e_f.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\na : R\n⊢ ↑(RingHom.id R) a = ↑(RingHom.comp (eval₂Hom (RingHom.id R) g) C) a",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\n⊢ eval₂ (RingHom.id R) (eval₂ (RingHom.id R) g ∘ f) p =\n eval₂ (RingHom.comp (eval₂Hom (RingHom.id R) g) C) (eval₂ (RingHom.id R) g ∘ f) p",
"tactic": "congr with a"
},
{
"state_after": "no goals",
"state_before": "case e_f.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_2\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : σ → R\nτ : Type u_1\nf : σ → MvPolynomial τ R\ng : τ → R\np : MvPolynomial σ R\na : R\n⊢ ↑(RingHom.id R) a = ↑(RingHom.comp (eval₂Hom (RingHom.id R) g) C) a",
"tactic": "simp"
}
] |
[
1171,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1167,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.lintegral_bind
|
[] |
[
188,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Logic/Basic.lean
|
pi_congr
|
[] |
[
621,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
620,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.norm_eq_ciSup
|
[
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\n⊢ (if hp : ⊤ = 0 then\n Eq.rec (motive := fun x x_1 => { x_2 // x_2 ∈ lp E x } → ℝ)\n (fun f => ↑(Finset.card (Set.Finite.toFinset (_ : Set.Finite {i | ↑f i ≠ 0})))) (_ : 0 = ⊤) f\n else if ⊤ = ⊤ then ⨆ (i : α), ‖↑f i‖ else (∑' (i : α), ‖↑f i‖ ^ 0) ^ (1 / 0)) =\n ⨆ (i : α), ‖↑f i‖",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\n⊢ ‖f‖ = ⨆ (i : α), ‖↑f i‖",
"tactic": "dsimp [norm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : { x // x ∈ lp E ⊤ }\n⊢ (if hp : ⊤ = 0 then\n Eq.rec (motive := fun x x_1 => { x_2 // x_2 ∈ lp E x } → ℝ)\n (fun f => ↑(Finset.card (Set.Finite.toFinset (_ : Set.Finite {i | ↑f i ≠ 0})))) (_ : 0 = ⊤) f\n else if ⊤ = ⊤ then ⨆ (i : α), ‖↑f i‖ else (∑' (i : α), ‖↑f i‖ ^ 0) ^ (1 / 0)) =\n ⨆ (i : α), ‖↑f i‖",
"tactic": "rw [dif_neg ENNReal.top_ne_zero, if_pos rfl]"
}
] |
[
401,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.le_of_top_imp_top_of_toNNReal_le
|
[
{
"state_after": "α : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nh : a = ⊤ → b = ⊤\nh_nnreal : a ≠ ⊤ → b ≠ ⊤ → ENNReal.toNNReal a ≤ ENNReal.toNNReal b\nhlt : b < a\n⊢ False",
"state_before": "α : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nh : a = ⊤ → b = ⊤\nh_nnreal : a ≠ ⊤ → b ≠ ⊤ → ENNReal.toNNReal a ≤ ENNReal.toNNReal b\n⊢ a ≤ b",
"tactic": "by_contra' hlt"
},
{
"state_after": "case intro\nα : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nb : ℝ≥0\nh : a = ⊤ → ↑b = ⊤\nh_nnreal : a ≠ ⊤ → ↑b ≠ ⊤ → ENNReal.toNNReal a ≤ ENNReal.toNNReal ↑b\nhlt : ↑b < a\n⊢ False",
"state_before": "α : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nh : a = ⊤ → b = ⊤\nh_nnreal : a ≠ ⊤ → b ≠ ⊤ → ENNReal.toNNReal a ≤ ENNReal.toNNReal b\nhlt : b < a\n⊢ False",
"tactic": "lift b to ℝ≥0 using hlt.ne_top"
},
{
"state_after": "case intro.intro\nα : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q b a : ℝ≥0\nh : ↑a = ⊤ → ↑b = ⊤\nh_nnreal : ↑a ≠ ⊤ → ↑b ≠ ⊤ → ENNReal.toNNReal ↑a ≤ ENNReal.toNNReal ↑b\nhlt : ↑b < ↑a\n⊢ False",
"state_before": "case intro\nα : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nb : ℝ≥0\nh : a = ⊤ → ↑b = ⊤\nh_nnreal : a ≠ ⊤ → ↑b ≠ ⊤ → ENNReal.toNNReal a ≤ ENNReal.toNNReal ↑b\nhlt : ↑b < a\n⊢ False",
"tactic": "lift a to ℝ≥0 using mt h coe_ne_top"
},
{
"state_after": "case intro.intro\nα : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q b a : ℝ≥0\nh : ↑a = ⊤ → ↑b = ⊤\nh_nnreal : ↑a ≠ ⊤ → ↑b ≠ ⊤ → ENNReal.toNNReal ↑a ≤ ENNReal.toNNReal ↑b\nhlt : ↑b < ↑a\n⊢ ↑a ≤ ↑b",
"state_before": "case intro.intro\nα : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q b a : ℝ≥0\nh : ↑a = ⊤ → ↑b = ⊤\nh_nnreal : ↑a ≠ ⊤ → ↑b ≠ ⊤ → ENNReal.toNNReal ↑a ≤ ENNReal.toNNReal ↑b\nhlt : ↑b < ↑a\n⊢ False",
"tactic": "refine hlt.not_le ?_"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type ?u.142320\nβ : Type ?u.142323\na✝ b✝ c d : ℝ≥0∞\nr p q b a : ℝ≥0\nh : ↑a = ⊤ → ↑b = ⊤\nh_nnreal : ↑a ≠ ⊤ → ↑b ≠ ⊤ → ENNReal.toNNReal ↑a ≤ ENNReal.toNNReal ↑b\nhlt : ↑b < ↑a\n⊢ ↑a ≤ ↑b",
"tactic": "simpa using h_nnreal"
}
] |
[
926,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
920,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.div_div_cancel
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.66666\nG : Type ?u.66669\nα : Type u_1\nM : Type ?u.66675\nN : Type ?u.66678\ninst✝ : DivisionCommMonoid α\na b c d : α\nh : IsUnit a\n⊢ a / (a / b) = b",
"tactic": "rw [div_div_eq_mul_div, h.mul_div_cancel_left]"
}
] |
[
505,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
11
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.dual
|
[] |
[
82,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Logic/Nontrivial.lean
|
nontrivial_of_lt
|
[] |
[
67,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/List/Basic.lean
|
List.nthLe_set_eq
|
[] |
[
1572,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1571,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.lintegral_fn_integral_sub
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.2423125\nβ : Type u_2\nβ' : Type ?u.2423131\nγ : Type ?u.2423134\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2423399\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → ℝ≥0∞\nhf : Integrable f\nhg : Integrable g\n⊢ (fun x => F (∫ (y : β), f (x, y) - g (x, y) ∂ν)) =ᶠ[ae μ] fun x =>\n F ((∫ (y : β), f (x, y) ∂ν) - ∫ (y : β), g (x, y) ∂ν)",
"state_before": "α : Type u_1\nα' : Type ?u.2423125\nβ : Type u_2\nβ' : Type ?u.2423131\nγ : Type ?u.2423134\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2423399\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → ℝ≥0∞\nhf : Integrable f\nhg : Integrable g\n⊢ (∫⁻ (x : α), F (∫ (y : β), f (x, y) - g (x, y) ∂ν) ∂μ) =\n ∫⁻ (x : α), F ((∫ (y : β), f (x, y) ∂ν) - ∫ (y : β), g (x, y) ∂ν) ∂μ",
"tactic": "refine' lintegral_congr_ae _"
},
{
"state_after": "case h\nα : Type u_1\nα' : Type ?u.2423125\nβ : Type u_2\nβ' : Type ?u.2423131\nγ : Type ?u.2423134\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2423399\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → ℝ≥0∞\nhf : Integrable f\nhg : Integrable g\na✝ : α\nh2f : Integrable fun y => f (a✝, y)\nh2g : Integrable fun y => g (a✝, y)\n⊢ F (∫ (y : β), f (a✝, y) - g (a✝, y) ∂ν) = F ((∫ (y : β), f (a✝, y) ∂ν) - ∫ (y : β), g (a✝, y) ∂ν)",
"state_before": "α : Type u_1\nα' : Type ?u.2423125\nβ : Type u_2\nβ' : Type ?u.2423131\nγ : Type ?u.2423134\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2423399\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → ℝ≥0∞\nhf : Integrable f\nhg : Integrable g\n⊢ (fun x => F (∫ (y : β), f (x, y) - g (x, y) ∂ν)) =ᶠ[ae μ] fun x =>\n F ((∫ (y : β), f (x, y) ∂ν) - ∫ (y : β), g (x, y) ∂ν)",
"tactic": "filter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nα' : Type ?u.2423125\nβ : Type u_2\nβ' : Type ?u.2423131\nγ : Type ?u.2423134\nE : Type u_3\ninst✝¹² : MeasurableSpace α\ninst✝¹¹ : MeasurableSpace α'\ninst✝¹⁰ : MeasurableSpace β\ninst✝⁹ : MeasurableSpace β'\ninst✝⁸ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SigmaFinite ν\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : SigmaFinite μ\nE' : Type ?u.2423399\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nf g : α × β → E\nF : E → ℝ≥0∞\nhf : Integrable f\nhg : Integrable g\na✝ : α\nh2f : Integrable fun y => f (a✝, y)\nh2g : Integrable fun y => g (a✝, y)\n⊢ F (∫ (y : β), f (a✝, y) - g (a✝, y) ∂ν) = F ((∫ (y : β), f (a✝, y) ∂ν) - ∫ (y : β), g (a✝, y) ∂ν)",
"tactic": "simp [integral_sub h2f h2g]"
}
] |
[
386,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl
|
[] |
[
1947,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1945,
1
] |
Mathlib/Algebra/Lie/Matrix.lean
|
Matrix.lieConj_symm_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nP A : Matrix n n R\nh : Invertible P\n⊢ ↑(LieEquiv.symm (lieConj P h)) A = P⁻¹ ⬝ A ⬝ P",
"tactic": "simp [LinearEquiv.symm_conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,\n LinearMap.toMatrix'_toLin']"
}
] |
[
83,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.surj
|
[] |
[
129,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Data/Set/Function.lean
|
StrictMonoOn.comp_strictAntiOn
|
[] |
[
1561,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1558,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.extend_empty
|
[] |
[
1378,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1377,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
exists_ratio_hasDerivAt_eq_ratio_slope'
|
[
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"tactic": "let h x := (lgb - lga) * f x - (lfb - lfa) * g x"
},
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"tactic": "have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by\n have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) :=\n (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)\n convert this using 2\n ring"
},
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"tactic": "have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by\n have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) :=\n (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)\n convert this using 2\n ring"
},
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"tactic": "let h' x := (lgb - lga) * f' x - (lfb - lfa) * g' x"
},
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"tactic": "have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := by\n intro x hx\n exact ((hff' x hx).const_mul _).sub ((hgg' x hx).const_mul _)"
},
{
"state_after": "case intro.intro\nE : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : h' c = 0\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"tactic": "rcases exists_hasDerivAt_eq_zero' hab hha hhb hhh' with ⟨c, cmem, hc⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nE : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : h' c = 0\n⊢ ∃ c, c ∈ Ioo a b ∧ (lgb - lga) * f' c = (lfb - lfa) * g' c",
"tactic": "exact ⟨c, cmem, sub_eq_zero.1 hc⟩"
},
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nthis : Tendsto h (𝓝[Ioi a] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))\n⊢ Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\n⊢ Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))",
"tactic": "have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) :=\n (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)"
},
{
"state_after": "case h.e'_5.h.e'_3\nE : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nthis : Tendsto h (𝓝[Ioi a] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))\n⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfa - (lfb - lfa) * lga",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nthis : Tendsto h (𝓝[Ioi a] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))\n⊢ Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))",
"tactic": "convert this using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e'_3\nE : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nthis : Tendsto h (𝓝[Ioi a] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga))\n⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfa - (lfb - lfa) * lga",
"tactic": "ring"
},
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nthis : Tendsto h (𝓝[Iio b] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))\n⊢ Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\n⊢ Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))",
"tactic": "have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) :=\n (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)"
},
{
"state_after": "case h.e'_5.h.e'_3\nE : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nthis : Tendsto h (𝓝[Iio b] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))\n⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfb - (lfb - lfa) * lgb",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nthis : Tendsto h (𝓝[Iio b] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))\n⊢ Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))",
"tactic": "convert this using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e'_3\nE : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nthis : Tendsto h (𝓝[Iio b] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb))\n⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfb - (lfb - lfa) * lgb",
"tactic": "ring"
},
{
"state_after": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nx : ℝ\nhx : x ∈ Ioo a b\n⊢ HasDerivAt h (h' x) x",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\n⊢ ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.311271\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.311367\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg'✝ : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nlfa lga lfb lgb : ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhfa : Tendsto f (𝓝[Ioi a] a) (𝓝 lfa)\nhga : Tendsto g (𝓝[Ioi a] a) (𝓝 lga)\nhfb : Tendsto f (𝓝[Iio b] b) (𝓝 lfb)\nhgb : Tendsto g (𝓝[Iio b] b) (𝓝 lgb)\nh : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (𝓝[Ioi a] a) (𝓝 (lgb * lfa - lfb * lga))\nhhb : Tendsto h (𝓝[Iio b] b) (𝓝 (lgb * lfa - lfb * lga))\nh' : ℝ → ℝ := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nx : ℝ\nhx : x ∈ Ioo a b\n⊢ HasDerivAt h (h' x) x",
"tactic": "exact ((hff' x hx).const_mul _).sub ((hgg' x hx).const_mul _)"
}
] |
[
749,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.Ioc_pred_left
|
[] |
[
741,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Std/Data/Int/DivMod.lean
|
Int.div_sign
|
[
{
"state_after": "no goals",
"state_before": "x✝ : Int\nn✝ : Nat\n⊢ div x✝ (sign ↑(succ n✝)) = x✝ * sign ↑(succ n✝)",
"tactic": "simp [sign, Int.mul_one]"
},
{
"state_after": "no goals",
"state_before": "x✝ : Int\n⊢ div x✝ (sign 0) = x✝ * sign 0",
"tactic": "simp [sign, Int.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "x✝ : Int\na✝ : Nat\n⊢ div x✝ (sign -[a✝+1]) = x✝ * sign -[a✝+1]",
"tactic": "simp [sign, Int.mul_neg, Int.mul_one]"
}
] |
[
827,
58
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
824,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.mul_mem
|
[] |
[
177,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
11
] |
Mathlib/Data/Real/Basic.lean
|
Real.ofCauchy_zero
|
[] |
[
116,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.Nonempty.to_type
|
[] |
[
523,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.coeFn_abs
|
[
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.1579337\nγ : Type ?u.1579340\nδ : Type ?u.1579343\ninst✝⁸ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁷ : TopologicalSpace β✝\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : TopologicalSpace δ\nβ : Type u_1\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Lattice β\ninst✝² : TopologicalLattice β\ninst✝¹ : AddGroup β\ninst✝ : TopologicalAddGroup β\nf : α →ₘ[μ] β\n⊢ ↑(f ⊔ -f) =ᵐ[μ] fun x => ↑f x ⊔ -↑f x",
"state_before": "α : Type u_2\nβ✝ : Type ?u.1579337\nγ : Type ?u.1579340\nδ : Type ?u.1579343\ninst✝⁸ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁷ : TopologicalSpace β✝\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : TopologicalSpace δ\nβ : Type u_1\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Lattice β\ninst✝² : TopologicalLattice β\ninst✝¹ : AddGroup β\ninst✝ : TopologicalAddGroup β\nf : α →ₘ[μ] β\n⊢ ↑(abs f) =ᵐ[μ] fun x => abs (↑f x)",
"tactic": "simp_rw [abs_eq_sup_neg]"
},
{
"state_after": "case h\nα : Type u_2\nβ✝ : Type ?u.1579337\nγ : Type ?u.1579340\nδ : Type ?u.1579343\ninst✝⁸ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁷ : TopologicalSpace β✝\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : TopologicalSpace δ\nβ : Type u_1\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Lattice β\ninst✝² : TopologicalLattice β\ninst✝¹ : AddGroup β\ninst✝ : TopologicalAddGroup β\nf : α →ₘ[μ] β\nx : α\nhx_sup : ↑(f ⊔ -f) x = ↑f x ⊔ ↑(-f) x\nhx_neg : ↑(-f) x = (-↑f) x\n⊢ ↑(f ⊔ -f) x = ↑f x ⊔ -↑f x",
"state_before": "α : Type u_2\nβ✝ : Type ?u.1579337\nγ : Type ?u.1579340\nδ : Type ?u.1579343\ninst✝⁸ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁷ : TopologicalSpace β✝\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : TopologicalSpace δ\nβ : Type u_1\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Lattice β\ninst✝² : TopologicalLattice β\ninst✝¹ : AddGroup β\ninst✝ : TopologicalAddGroup β\nf : α →ₘ[μ] β\n⊢ ↑(f ⊔ -f) =ᵐ[μ] fun x => ↑f x ⊔ -↑f x",
"tactic": "filter_upwards [AEEqFun.coeFn_sup f (-f), AEEqFun.coeFn_neg f] with x hx_sup hx_neg"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ✝ : Type ?u.1579337\nγ : Type ?u.1579340\nδ : Type ?u.1579343\ninst✝⁸ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁷ : TopologicalSpace β✝\ninst✝⁶ : TopologicalSpace γ\ninst✝⁵ : TopologicalSpace δ\nβ : Type u_1\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Lattice β\ninst✝² : TopologicalLattice β\ninst✝¹ : AddGroup β\ninst✝ : TopologicalAddGroup β\nf : α →ₘ[μ] β\nx : α\nhx_sup : ↑(f ⊔ -f) x = ↑f x ⊔ ↑(-f) x\nhx_neg : ↑(-f) x = (-↑f) x\n⊢ ↑(f ⊔ -f) x = ↑f x ⊔ -↑f x",
"tactic": "rw [hx_sup, hx_neg, Pi.neg_apply]"
}
] |
[
881,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
877,
1
] |
Mathlib/Data/List/Basic.lean
|
List.getLast_concat'
|
[] |
[
721,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
720,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.mem_range
|
[] |
[
651,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
650,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.finset_card_lt_aleph0
|
[] |
[
2063,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2062,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.HasFiniteIntegral.add_measure
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.718349\nδ : Type ?u.718352\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤\nhν : (∫⁻ (a : α), ↑‖f a‖₊ ∂ν) < ⊤\n⊢ ((∫⁻ (a : α), ↑‖f a‖₊ ∂μ) + ∫⁻ (a : α), ↑‖f a‖₊ ∂ν) < ⊤",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.718349\nδ : Type ?u.718352\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : HasFiniteIntegral f\nhν : HasFiniteIntegral f\n⊢ HasFiniteIntegral f",
"tactic": "simp only [HasFiniteIntegral, lintegral_add_measure] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.718349\nδ : Type ?u.718352\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhμ : (∫⁻ (a : α), ↑‖f a‖₊ ∂μ) < ⊤\nhν : (∫⁻ (a : α), ↑‖f a‖₊ ∂ν) < ⊤\n⊢ ((∫⁻ (a : α), ↑‖f a‖₊ ∂μ) + ∫⁻ (a : α), ↑‖f a‖₊ ∂ν) < ⊤",
"tactic": "exact add_lt_top.2 ⟨hμ, hν⟩"
}
] |
[
198,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/PNat/Prime.lean
|
Nat.Primes.coe_pnat_inj
|
[] |
[
44,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Std/Logic.lean
|
Decidable.not_iff_not
|
[
{
"state_after": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (¬a → ¬b) ∧ (¬b → ¬a) ↔ (b → a) ∧ (a → b)",
"state_before": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (¬a ↔ ¬b) ↔ (a ↔ b)",
"tactic": "rw [@iff_def (¬a), @iff_def' a]"
},
{
"state_after": "no goals",
"state_before": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ (¬a → ¬b) ∧ (¬b → ¬a) ↔ (b → a) ∧ (a → b)",
"tactic": "exact and_congr not_imp_not not_imp_not"
}
] |
[
577,
75
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
576,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_neg_uIcc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\n⊢ -[[a, b]] = [[-a, -b]]",
"tactic": "simp only [← Icc_min_max, preimage_neg_Icc, min_neg_neg, max_neg_neg]"
}
] |
[
441,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Order/BoundedOrder.lean
|
max_bot_left
|
[] |
[
847,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
846,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.idZeroEquivIsoZero_apply_inv
|
[] |
[
431,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Data/Set/Image.lean
|
Function.Surjective.preimage_injective
|
[] |
[
1283,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1282,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.none_eq_bot
|
[] |
[
79,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_update_of_not_mem
|
[
{
"state_after": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\n⊢ ∀ (x : α), x ∈ s → Function.update f i b x = f x",
"state_before": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\n⊢ ∏ x in s, Function.update f i b x = ∏ x in s, f x",
"tactic": "apply prod_congr rfl"
},
{
"state_after": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\nj : α\nhj : j ∈ s\n⊢ Function.update f i b j = f j",
"state_before": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\n⊢ ∀ (x : α), x ∈ s → Function.update f i b x = f x",
"tactic": "intros j hj"
},
{
"state_after": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\nj : α\nhj : j ∈ s\nthis : j ≠ i\n⊢ Function.update f i b j = f j",
"state_before": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\nj : α\nhj : j ∈ s\n⊢ Function.update f i b j = f j",
"tactic": "have : j ≠ i := by\n rintro rfl\n exact h hj"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\nj : α\nhj : j ∈ s\nthis : j ≠ i\n⊢ Function.update f i b j = f j",
"tactic": "simp [this]"
},
{
"state_after": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\nf : α → β\nb : β\nj : α\nhj : j ∈ s\nh : ¬j ∈ s\n⊢ False",
"state_before": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\ni : α\nh : ¬i ∈ s\nf : α → β\nb : β\nj : α\nhj : j ∈ s\n⊢ j ≠ i",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.790472\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns : Finset α\nf : α → β\nb : β\nj : α\nhj : j ∈ s\nh : ¬j ∈ s\n⊢ False",
"tactic": "exact h hj"
}
] |
[
1615,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1608,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.inter_div_union_subset_union
|
[] |
[
686,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
685,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_add_aux
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (∫⁻ (a : α), f a + g a ∂μ) = ∫⁻ (a : α), (⨆ (n : ℕ), ↑(eapprox f n) a) + ⨆ (n : ℕ), ↑(eapprox g n) a ∂μ",
"tactic": "simp only [iSup_eapprox_apply, hf, hg]"
},
{
"state_after": "case e_f\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (fun a => (⨆ (n : ℕ), ↑(eapprox f n) a) + ⨆ (n : ℕ), ↑(eapprox g n) a) = fun a =>\n ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a",
"state_before": "α : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (∫⁻ (a : α), (⨆ (n : ℕ), ↑(eapprox f n) a) + ⨆ (n : ℕ), ↑(eapprox g n) a ∂μ) =\n ∫⁻ (a : α), ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ",
"tactic": "congr"
},
{
"state_after": "case e_f.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ ((⨆ (n : ℕ), ↑(eapprox f n) a) + ⨆ (n : ℕ), ↑(eapprox g n) a) = ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a",
"state_before": "case e_f\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (fun a => (⨆ (n : ℕ), ↑(eapprox f n) a) + ⨆ (n : ℕ), ↑(eapprox g n) a) = fun a =>\n ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a",
"tactic": "funext a"
},
{
"state_after": "case e_f.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ (⨆ (a_1 : ℕ), ↑(eapprox f a_1) a + ↑(eapprox g a_1) a) = ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a\n\ncase e_f.h.hf\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ Monotone fun n => ↑(eapprox f n) a\n\ncase e_f.h.hg\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ Monotone fun n => ↑(eapprox g n) a",
"state_before": "case e_f.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ ((⨆ (n : ℕ), ↑(eapprox f n) a) + ⨆ (n : ℕ), ↑(eapprox g n) a) = ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a",
"tactic": "rw [ENNReal.iSup_add_iSup_of_monotone]"
},
{
"state_after": "no goals",
"state_before": "case e_f.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ (⨆ (a_1 : ℕ), ↑(eapprox f a_1) a + ↑(eapprox g a_1) a) = ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a",
"tactic": "rfl"
},
{
"state_after": "case e_f.h.hf\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\ni j : ℕ\nh : i ≤ j\n⊢ (fun n => ↑(eapprox f n) a) i ≤ (fun n => ↑(eapprox f n) a) j",
"state_before": "case e_f.h.hf\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ Monotone fun n => ↑(eapprox f n) a",
"tactic": "intro i j h"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.hf\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\ni j : ℕ\nh : i ≤ j\n⊢ (fun n => ↑(eapprox f n) a) i ≤ (fun n => ↑(eapprox f n) a) j",
"tactic": "exact monotone_eapprox _ h a"
},
{
"state_after": "case e_f.h.hg\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\ni j : ℕ\nh : i ≤ j\n⊢ (fun n => ↑(eapprox g n) a) i ≤ (fun n => ↑(eapprox g n) a) j",
"state_before": "case e_f.h.hg\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\n⊢ Monotone fun n => ↑(eapprox g n) a",
"tactic": "intro i j h"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.hg\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\ni j : ℕ\nh : i ≤ j\n⊢ (fun n => ↑(eapprox g n) a) i ≤ (fun n => ↑(eapprox g n) a) j",
"tactic": "exact monotone_eapprox _ h a"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (⨆ (n : ℕ), ∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) =\n ⨆ (n : ℕ), SimpleFunc.lintegral (eapprox f n) μ + SimpleFunc.lintegral (eapprox g n) μ\n\ncase hf\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ ∀ (n : ℕ), Measurable fun a => (↑(eapprox f n) + ↑(eapprox g n)) a\n\ncase h_mono\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ Monotone fun n a => (↑(eapprox f n) + ↑(eapprox g n)) a",
"state_before": "α : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (∫⁻ (a : α), ⨆ (n : ℕ), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) =\n ⨆ (n : ℕ), SimpleFunc.lintegral (eapprox f n) μ + SimpleFunc.lintegral (eapprox g n) μ",
"tactic": "rw [lintegral_iSup]"
},
{
"state_after": "case e_s\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (fun n => ∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) = fun n =>\n SimpleFunc.lintegral (eapprox f n) μ + SimpleFunc.lintegral (eapprox g n) μ",
"state_before": "α : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (⨆ (n : ℕ), ∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) =\n ⨆ (n : ℕ), SimpleFunc.lintegral (eapprox f n) μ + SimpleFunc.lintegral (eapprox g n) μ",
"tactic": "congr"
},
{
"state_after": "case e_s.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nn : ℕ\n⊢ (∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) =\n SimpleFunc.lintegral (eapprox f n) μ + SimpleFunc.lintegral (eapprox g n) μ",
"state_before": "case e_s\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ (fun n => ∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) = fun n =>\n SimpleFunc.lintegral (eapprox f n) μ + SimpleFunc.lintegral (eapprox g n) μ",
"tactic": "funext n"
},
{
"state_after": "case e_s.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nn : ℕ\n⊢ (∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) = ∫⁻ (a : α), ↑(eapprox f n + eapprox g n) a ∂μ",
"state_before": "case e_s.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nn : ℕ\n⊢ (∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) =\n SimpleFunc.lintegral (eapprox f n) μ + SimpleFunc.lintegral (eapprox g n) μ",
"tactic": "rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral]"
},
{
"state_after": "no goals",
"state_before": "case e_s.h\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nn : ℕ\n⊢ (∫⁻ (a : α), (↑(eapprox f n) + ↑(eapprox g n)) a ∂μ) = ∫⁻ (a : α), ↑(eapprox f n + eapprox g n) a ∂μ",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ ∀ (n : ℕ), Measurable fun a => (↑(eapprox f n) + ↑(eapprox g n)) a",
"tactic": "measurability"
},
{
"state_after": "case h_mono\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ni j : ℕ\nh : i ≤ j\na : α\n⊢ (fun n a => (↑(eapprox f n) + ↑(eapprox g n)) a) i a ≤ (fun n a => (↑(eapprox f n) + ↑(eapprox g n)) a) j a",
"state_before": "case h_mono\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ Monotone fun n a => (↑(eapprox f n) + ↑(eapprox g n)) a",
"tactic": "intro i j h a"
},
{
"state_after": "no goals",
"state_before": "case h_mono\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ni j : ℕ\nh : i ≤ j\na : α\n⊢ (fun n a => (↑(eapprox f n) + ↑(eapprox g n)) a) i a ≤ (fun n a => (↑(eapprox f n) + ↑(eapprox g n)) a) j a",
"tactic": "exact add_le_add (monotone_eapprox _ h _) (monotone_eapprox _ h _)"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ni j : ℕ\nh : i ≤ j\n⊢ (fun n => SimpleFunc.lintegral (eapprox g n) μ) i ≤ (fun n => SimpleFunc.lintegral (eapprox g n) μ) j",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ Monotone fun n => SimpleFunc.lintegral (eapprox g n) μ",
"tactic": "intro i j h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\ni j : ℕ\nh : i ≤ j\n⊢ (fun n => SimpleFunc.lintegral (eapprox g n) μ) i ≤ (fun n => SimpleFunc.lintegral (eapprox g n) μ) j",
"tactic": "exact SimpleFunc.lintegral_mono (monotone_eapprox _ h) (le_refl μ)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.853934\nγ : Type ?u.853937\nδ : Type ?u.853940\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ ((⨆ (n : ℕ), SimpleFunc.lintegral (eapprox f n) μ) + ⨆ (n : ℕ), SimpleFunc.lintegral (eapprox g n) μ) =\n (∫⁻ (a : α), f a ∂μ) + ∫⁻ (a : α), g a ∂μ",
"tactic": "rw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral hg]"
}
] |
[
563,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
|
mul_mul_div
|
[] |
[
128,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.upperClosure
|
[] |
[
1317,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1316,
11
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.induction_on'
|
[] |
[
1235,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1229,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.zpow_sub
|
[
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nha : IsUnit (det A)\nz1 z2 : ℤ\n⊢ A ^ (z1 - z2) = A ^ z1 / A ^ z2",
"tactic": "rw [sub_eq_add_neg, zpow_add ha, zpow_neg ha, div_eq_mul_inv]"
}
] |
[
307,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisInsertion.l_sSup_u_image
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.39544\na a₁ a₂ : α\nb b₁ b₂ : β\nl : α → β\nu : β → α\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\ngi : GaloisInsertion l u\ns : Set β\n⊢ l (sSup (u '' s)) = sSup s",
"tactic": "rw [sSup_image, gi.l_biSup_u, sSup_eq_iSup]"
}
] |
[
553,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/Data/TwoPointing.lean
|
TwoPointing.bool_snd
|
[] |
[
153,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.invRev_injective
|
[] |
[
584,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/RingTheory/Ideal/IdempotentFG.lean
|
Ideal.isIdempotentElem_iff_eq_bot_or_top
|
[
{
"state_after": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nh : FG I\n⊢ IsIdempotentElem I → I = ⊥ ∨ I = ⊤\n\ncase mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nh : FG I\n⊢ I = ⊥ ∨ I = ⊤ → IsIdempotentElem I",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nh : FG I\n⊢ IsIdempotentElem I ↔ I = ⊥ ∨ I = ⊤",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nh : FG I\nH : IsIdempotentElem I\n⊢ I = ⊥ ∨ I = ⊤",
"state_before": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nh : FG I\n⊢ IsIdempotentElem I → I = ⊥ ∨ I = ⊤",
"tactic": "intro H"
},
{
"state_after": "case mp.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n⊢ Submodule.span R {e} = ⊥ ∨ Submodule.span R {e} = ⊤",
"state_before": "case mp\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nh : FG I\nH : IsIdempotentElem I\n⊢ I = ⊥ ∨ I = ⊤",
"tactic": "obtain ⟨e, he, rfl⟩ := (I.isIdempotentElem_iff_of_fg h).mp H"
},
{
"state_after": "case mp.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n⊢ e = 0 ∨ span {e} = ⊤",
"state_before": "case mp.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n⊢ Submodule.span R {e} = ⊥ ∨ Submodule.span R {e} = ⊤",
"tactic": "simp only [Ideal.submodule_span_eq, Ideal.span_singleton_eq_bot]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n⊢ e = 1 → span {e} = ⊤",
"state_before": "case mp.intro.intro\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n⊢ e = 0 ∨ span {e} = ⊤",
"tactic": "apply Or.imp id _ (IsIdempotentElem.iff_eq_zero_or_one.mp he)"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhe : IsIdempotentElem 1\nh : FG (Submodule.span R {1})\nH : IsIdempotentElem (Submodule.span R {1})\n⊢ span {1} = ⊤",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n⊢ e = 1 → span {e} = ⊤",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhe : IsIdempotentElem 1\nh : FG (Submodule.span R {1})\nH : IsIdempotentElem (Submodule.span R {1})\n⊢ span {1} = ⊤",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nI : Ideal R\nh : FG I\n⊢ I = ⊥ ∨ I = ⊤ → IsIdempotentElem I",
"tactic": "rintro (rfl | rfl) <;> simp [IsIdempotentElem]"
}
] |
[
49,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.max_eq_zero
|
[] |
[
1014,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1013,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.mem_ball
|
[] |
[
531,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
9
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.isCompact_image
|
[] |
[
262,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
Module.Ray.ind
|
[] |
[
263,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorBundleCore.localTriv_coordChange_eq
|
[
{
"state_after": "case a\nR : Type u_2\nB : Type u_1\nF : Type u_3\nE : B → Type ?u.414625\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ baseSet Z i ∩ baseSet Z j\nv : F\n⊢ { fst := (b, v).fst, snd := ↑(coordChange Z i (indexAt Z (b, v).fst) (b, v).fst) (b, v).snd }.fst ∈\n baseSet Z i ∩\n baseSet Z\n (indexAt Z\n { fst := (b, v).fst, snd := ↑(coordChange Z i (indexAt Z (b, v).fst) (b, v).fst) (b, v).snd }.fst) ∩\n baseSet Z j\n\ncase hb\nR : Type u_2\nB : Type u_1\nF : Type u_3\nE : B → Type ?u.414625\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ baseSet Z i ∩ baseSet Z j\nv : F\n⊢ b ∈ (localTriv Z i).baseSet ∩ (localTriv Z j).baseSet",
"state_before": "R : Type u_2\nB : Type u_1\nF : Type u_3\nE : B → Type ?u.414625\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ baseSet Z i ∩ baseSet Z j\nv : F\n⊢ ↑(Trivialization.coordChangeL R (localTriv Z i) (localTriv Z j) b) v = ↑(coordChange Z i j b) v",
"tactic": "rw [Trivialization.coordChangeL_apply', localTriv_symm_fst, localTriv_apply, coordChange_comp]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_2\nB : Type u_1\nF : Type u_3\nE : B → Type ?u.414625\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ baseSet Z i ∩ baseSet Z j\nv : F\n⊢ { fst := (b, v).fst, snd := ↑(coordChange Z i (indexAt Z (b, v).fst) (b, v).fst) (b, v).snd }.fst ∈\n baseSet Z i ∩\n baseSet Z\n (indexAt Z\n { fst := (b, v).fst, snd := ↑(coordChange Z i (indexAt Z (b, v).fst) (b, v).fst) (b, v).snd }.fst) ∩\n baseSet Z j\n\ncase hb\nR : Type u_2\nB : Type u_1\nF : Type u_3\nE : B → Type ?u.414625\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ baseSet Z i ∩ baseSet Z j\nv : F\n⊢ b ∈ (localTriv Z i).baseSet ∩ (localTriv Z j).baseSet",
"tactic": "exacts [⟨⟨hb.1, Z.mem_baseSet_at b⟩, hb.2⟩, hb]"
}
] |
[
732,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
729,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.sup_factors_of_factors_right
|
[] |
[
535,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Topology/Order/Basic.lean
|
pi_Ioi_mem_nhds
|
[] |
[
1496,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1495,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.top_prod_top
|
[] |
[
680,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
IsAbsoluteValue.abv_sum
|
[] |
[
749,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/Topology/Separation.lean
|
IsCompact.isClosed
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\nγ : Type ?u.129636\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space α\ns : Set α\nhs : IsCompact s\nx : α\nhx : x ∈ sᶜ\nu v : Set α\nleft✝ : IsOpen u\nvo : IsOpen v\nsu : s ⊆ u\nxv : {x} ⊆ v\nuv : Disjoint u v\n⊢ x ∈ v",
"tactic": "simpa using xv"
}
] |
[
1266,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1262,
1
] |
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