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/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lt_sup
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.282487\nβ : Type ?u.282490\nγ : Type ?u.282493\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\na : Ordinal\n⊢ a < sup f ↔ ∃ i, a < f i",
"tactic": "simpa only [not_forall, not_le] using not_congr (@sup_le_iff.{_, v} _ f a)"
}
] |
[
1277,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1276,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.mem_iff_infEdist_zero_of_closed
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.17353\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nh : IsClosed s\n⊢ x ∈ s ↔ infEdist x s = 0",
"tactic": "rw [← mem_closure_iff_infEdist_zero, h.closure_eq]"
}
] |
[
163,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.support_swap_mul_eq
|
[
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ↑f x = x\n⊢ support (swap x (↑f x) * f) = support f \\ {x}\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\n⊢ support (swap x (↑f x) * f) = support f \\ {x}",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\n⊢ support (swap x (↑f x) * f) = support f \\ {x}",
"tactic": "by_cases hx : f x = x"
},
{
"state_after": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\n⊢ support (swap x (↑f x) * f) = support f \\ {x}",
"tactic": "ext z"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"state_before": "case neg.a\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"tactic": "by_cases hzx : z = x"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : z = ↑f x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : ¬z = ↑f x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"tactic": "by_cases hzf : z = f x"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : ¬z = ↑f x\nhzfx : ↑f z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : ¬z = ↑f x\nhzfx : ¬↑f z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : ¬z = ↑f x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"tactic": "by_cases hzfx : f z = x"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ↑f x = x\n⊢ support (swap x (↑f x) * f) = support f \\ {x}",
"tactic": "simp [hx, sdiff_singleton_eq_erase, not_mem_support.mpr hx, erase_eq_of_not_mem]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"tactic": "simp [hzx]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : z = ↑f x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"tactic": "simp [hzf, hx, h, swap_apply_of_ne_of_ne]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : ¬z = ↑f x\nhzfx : ↑f z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"tactic": "simp [Ne.symm hzx, hzx, Ne.symm hzf, hzfx]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ g f : Perm α\nx : α\nh : ↑f (↑f x) ≠ x\nhx : ¬↑f x = x\nz : α\nhzx : ¬z = x\nhzf : ¬z = ↑f x\nhzfx : ¬↑f z = x\n⊢ z ∈ support (swap x (↑f x) * f) ↔ z ∈ support f \\ {x}",
"tactic": "simp [Ne.symm hzx, hzx, Ne.symm hzf, hzfx, f.injective.ne hzx, swap_apply_of_ne_of_ne]"
}
] |
[
495,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.div_dvd_of_dvd
|
[
{
"state_after": "case pos\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : q ∣ p\nhq : q = 0\n⊢ p / q ∣ p\n\ncase neg\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : q ∣ p\nhq : ¬q = 0\n⊢ p / q ∣ p",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : q ∣ p\n⊢ p / q ∣ p",
"tactic": "by_cases hq : q = 0"
},
{
"state_after": "case neg\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : q ∣ p\nhq : ¬q = 0\n⊢ p = p / q * q",
"state_before": "case neg\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : q ∣ p\nhq : ¬q = 0\n⊢ p / q ∣ p",
"tactic": "use q"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : q ∣ p\nhq : ¬q = 0\n⊢ p = p / q * q",
"tactic": "rw [mul_comm, ← EuclideanDomain.mul_div_assoc _ hpq, mul_comm,\n EuclideanDomain.mul_div_cancel _ hq]"
},
{
"state_after": "case pos\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : p = 0\nhq : q = 0\n⊢ p / q ∣ p",
"state_before": "case pos\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : q ∣ p\nhq : q = 0\n⊢ p / q ∣ p",
"tactic": "rw [hq, zero_dvd_iff] at hpq"
},
{
"state_after": "case pos\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : p = 0\nhq : q = 0\n⊢ 0 / q ∣ 0",
"state_before": "case pos\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : p = 0\nhq : q = 0\n⊢ p / q ∣ p",
"tactic": "rw [hpq]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\ninst✝ : EuclideanDomain R\np q : R\nhpq : p = 0\nhq : q = 0\n⊢ 0 / q ∣ 0",
"tactic": "exact dvd_zero _"
}
] |
[
123,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sinh_ofReal_re
|
[] |
[
665,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_ite
|
[] |
[
643,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_eq_limsup_sum_nat
|
[] |
[
858,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
856,
11
] |
Mathlib/Data/Set/Lattice.lean
|
Set.mem_biInter
|
[] |
[
860,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
858,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.sdiff_singleton_eq_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.266452\nγ : Type ?u.266455\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\nha : ¬a ∈ s\n⊢ _root_.Disjoint {a} s",
"tactic": "simp [ha]"
}
] |
[
2287,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2286,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
Affine.Simplex.ext
|
[
{
"state_after": "case mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns2 : Simplex k P n\npoints✝ : Fin (n + 1) → P\nIndependent✝ : AffineIndependent k points✝\nh : ∀ (i : Fin (n + 1)), points { points := points✝, Independent := Independent✝ } i = points s2 i\n⊢ { points := points✝, Independent := Independent✝ } = s2",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns1 s2 : Simplex k P n\nh : ∀ (i : Fin (n + 1)), points s1 i = points s2 i\n⊢ s1 = s2",
"tactic": "cases s1"
},
{
"state_after": "case mk.mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\npoints✝¹ : Fin (n + 1) → P\nIndependent✝¹ : AffineIndependent k points✝¹\npoints✝ : Fin (n + 1) → P\nIndependent✝ : AffineIndependent k points✝\nh :\n ∀ (i : Fin (n + 1)),\n points { points := points✝¹, Independent := Independent✝¹ } i =\n points { points := points✝, Independent := Independent✝ } i\n⊢ { points := points✝¹, Independent := Independent✝¹ } = { points := points✝, Independent := Independent✝ }",
"state_before": "case mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns2 : Simplex k P n\npoints✝ : Fin (n + 1) → P\nIndependent✝ : AffineIndependent k points✝\nh : ∀ (i : Fin (n + 1)), points { points := points✝, Independent := Independent✝ } i = points s2 i\n⊢ { points := points✝, Independent := Independent✝ } = s2",
"tactic": "cases s2"
},
{
"state_after": "case mk.mk.e_points.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\npoints✝¹ : Fin (n + 1) → P\nIndependent✝¹ : AffineIndependent k points✝¹\npoints✝ : Fin (n + 1) → P\nIndependent✝ : AffineIndependent k points✝\nh :\n ∀ (i : Fin (n + 1)),\n points { points := points✝¹, Independent := Independent✝¹ } i =\n points { points := points✝, Independent := Independent✝ } i\ni : Fin (n + 1)\n⊢ points✝¹ i = points✝ i",
"state_before": "case mk.mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\npoints✝¹ : Fin (n + 1) → P\nIndependent✝¹ : AffineIndependent k points✝¹\npoints✝ : Fin (n + 1) → P\nIndependent✝ : AffineIndependent k points✝\nh :\n ∀ (i : Fin (n + 1)),\n points { points := points✝¹, Independent := Independent✝¹ } i =\n points { points := points✝, Independent := Independent✝ } i\n⊢ { points := points✝¹, Independent := Independent✝¹ } = { points := points✝, Independent := Independent✝ }",
"tactic": "congr with i"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.e_points.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\npoints✝¹ : Fin (n + 1) → P\nIndependent✝¹ : AffineIndependent k points✝¹\npoints✝ : Fin (n + 1) → P\nIndependent✝ : AffineIndependent k points✝\nh :\n ∀ (i : Fin (n + 1)),\n points { points := points✝¹, Independent := Independent✝¹ } i =\n points { points := points✝, Independent := Independent✝ } i\ni : Fin (n + 1)\n⊢ points✝¹ i = points✝ i",
"tactic": "exact h i"
}
] |
[
822,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Algebra/Associated.lean
|
associated_isUnit_mul_left_iff
|
[
{
"state_after": "α : Type ?u.166351\nβ✝ : Type ?u.166354\nγ : Type ?u.166357\nδ : Type ?u.166360\nβ : Type u_1\ninst✝ : CommMonoid β\nu a b : β\nhu : IsUnit u\n⊢ a * u ~ᵤ b ↔ a ~ᵤ b",
"state_before": "α : Type ?u.166351\nβ✝ : Type ?u.166354\nγ : Type ?u.166357\nδ : Type ?u.166360\nβ : Type u_1\ninst✝ : CommMonoid β\nu a b : β\nhu : IsUnit u\n⊢ u * a ~ᵤ b ↔ a ~ᵤ b",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.166351\nβ✝ : Type ?u.166354\nγ : Type ?u.166357\nδ : Type ?u.166360\nβ : Type u_1\ninst✝ : CommMonoid β\nu a b : β\nhu : IsUnit u\n⊢ a * u ~ᵤ b ↔ a ~ᵤ b",
"tactic": "exact associated_mul_isUnit_left_iff hu"
}
] |
[
482,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.toSet_inj
|
[] |
[
809,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
808,
1
] |
Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
|
ContinuousMap.attachBound_apply_coe
|
[] |
[
66,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
src/lean/Init/Core.lean
|
Iff.rfl
|
[] |
[
664,
13
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
663,
11
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.memℒp_neg_iff
|
[] |
[
1029,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1028,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
isOfFinOrder_one
|
[] |
[
124,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.hausdorffEdist_closure
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.54219\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t u : Set α\nΦ : α → β\n⊢ hausdorffEdist (closure s) (closure t) = hausdorffEdist s t",
"tactic": "simp"
}
] |
[
413,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/NumberTheory/Bernoulli.lean
|
sum_range_pow
|
[
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\n⊢ ∑ k in range n, ↑k ^ p = ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / (↑p + 1)",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\n⊢ ∑ k in range n, ↑k ^ p = ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / (↑p + 1)",
"tactic": "have hne : ∀ m : ℕ, (m ! : ℚ) ≠ 0 := fun m => by exact_mod_cast factorial_ne_zero m"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhps :\n ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !\n⊢ ∑ x in range (p + 1), bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) / ↑(p + 1)! * ↑p ! =\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / (↑p + 1)",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhps :\n ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !\n⊢ ∑ k in range n, ↑k ^ p = ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / (↑p + 1)",
"tactic": "rw [hps, sum_mul]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhps :\n ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !\nx : ℕ\nx✝ : x ∈ range (p + 1)\n⊢ bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) / ↑(p + 1)! * ↑p ! =\n bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) / (↑p + 1)",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhps :\n ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !\n⊢ ∑ x in range (p + 1), bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) / ↑(p + 1)! * ↑p ! =\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / (↑p + 1)",
"tactic": "refine' sum_congr rfl fun x _ => _"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhps :\n ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !\nx : ℕ\nx✝ : x ∈ range (p + 1)\n⊢ bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) * ↑p ! * (↑p + 1) =\n bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) * ((↑p + 1) * ↑p !)",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhps :\n ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !\nx : ℕ\nx✝ : x ∈ range (p + 1)\n⊢ bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) / ↑(p + 1)! * ↑p ! =\n bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) / (↑p + 1)",
"tactic": "field_simp [mul_right_comm _ ↑p !, ← mul_assoc _ _ ↑p !, cast_add_one_ne_zero, hne]"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhps :\n ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !\nx : ℕ\nx✝ : x ∈ range (p + 1)\n⊢ bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) * ↑p ! * (↑p + 1) =\n bernoulli x * ↑(Nat.choose (p + 1) x) * ↑n ^ (p + 1 - x) * ((↑p + 1) * ↑p !)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p m : ℕ\n⊢ ↑m ! ≠ 0",
"tactic": "exact_mod_cast factorial_ne_zero m"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\n⊢ ↑(coeff ℚ q) ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n ↑(coeff ℚ q)\n (PowerSeries.mk fun p =>\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!)",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\n⊢ ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"tactic": "ext q : 1"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\n⊢ ↑(coeff ℚ q) ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n ↑(coeff ℚ q)\n (PowerSeries.mk fun p =>\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!)",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\n⊢ ↑(coeff ℚ q) ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n ↑(coeff ℚ q)\n (PowerSeries.mk fun p =>\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!)",
"tactic": "let f a b := bernoulli a / a ! * coeff ℚ (b + 1) (exp ℚ ^ n)"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\n⊢ ∑ x in range (succ q), bernoulli x / ↑x ! * ↑(coeff ℚ (q - x + 1)) (exp ℚ ^ n) =\n ∑ i in range (q + 1), bernoulli i * ↑(Nat.choose (q + 1) i) * ↑n ^ (q + 1 - i) / ↑(q + 1)!",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\n⊢ ↑(coeff ℚ q) ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n ↑(coeff ℚ q)\n (PowerSeries.mk fun p =>\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!)",
"tactic": "simp only [coeff_mul, coeff_mk, cast_mul, sum_antidiagonal_eq_sum_range_succ f]"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\n⊢ ∀ (x : ℕ),\n x ∈ range (succ q) →\n bernoulli x / ↑x ! * ↑(coeff ℚ (q - x + 1)) (exp ℚ ^ n) =\n bernoulli x * ↑(Nat.choose (q + 1) x) * ↑n ^ (q + 1 - x) / ↑(q + 1)!",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\n⊢ ∑ x in range (succ q), bernoulli x / ↑x ! * ↑(coeff ℚ (q - x + 1)) (exp ℚ ^ n) =\n ∑ i in range (q + 1), bernoulli i * ↑(Nat.choose (q + 1) i) * ↑n ^ (q + 1 - i) / ↑(q + 1)!",
"tactic": "apply sum_congr rfl"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range (succ q)\n⊢ bernoulli m / ↑m ! * ↑(coeff ℚ (q - m + 1)) (exp ℚ ^ n) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\n⊢ ∀ (x : ℕ),\n x ∈ range (succ q) →\n bernoulli x / ↑x ! * ↑(coeff ℚ (q - x + 1)) (exp ℚ ^ n) =\n bernoulli x * ↑(Nat.choose (q + 1) x) * ↑n ^ (q + 1 - x) / ↑(q + 1)!",
"tactic": "intros m h"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range (succ q)\n⊢ bernoulli m / ↑m ! * ↑(coeff ℚ (q - m + 1)) (exp ℚ ^ n) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range (succ q)\n⊢ bernoulli m / ↑m ! * ↑(coeff ℚ (q - m + 1)) (exp ℚ ^ n) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"tactic": "simp only [Finset.mem_range]"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range (succ q)\n⊢ bernoulli m / ↑m ! *\n ↑(coeff ℚ (q - m + 1))\n (↑{\n toOneHom :=\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries ℚ),\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n g) }\n (exp ℚ)) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range (succ q)\n⊢ bernoulli m / ↑m ! * ↑(coeff ℚ (q - m + 1)) (exp ℚ ^ n) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"tactic": "simp only [exp_pow_eq_rescale_exp, rescale, one_div, coeff_mk, RingHom.coe_mk, coeff_exp,\n RingHom.id_apply, cast_mul, algebraMap_rat_rat]"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m / ↑m ! *\n ↑(coeff ℚ (q - m + 1))\n (↑{\n toOneHom :=\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries ℚ),\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n g) }\n (exp ℚ)) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range (succ q)\n⊢ bernoulli m / ↑m ! *\n ↑(coeff ℚ (q - m + 1))\n (↑{\n toOneHom :=\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries ℚ),\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n g) }\n (exp ℚ)) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"tactic": "simp at h"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑(succ q)! / ↑m ! *\n ↑(coeff ℚ (q - m + 1))\n (↑{\n toOneHom :=\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries ℚ),\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n g) }\n (exp ℚ))",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m / ↑m ! *\n ↑(coeff ℚ (q - m + 1))\n (↑{\n toOneHom :=\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries ℚ),\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n g) }\n (exp ℚ)) =\n bernoulli m * ↑(Nat.choose (q + 1) m) * ↑n ^ (q + 1 - m) / ↑(q + 1)!",
"tactic": "rw [choose_eq_factorial_div_factorial h.le, eq_comm, div_eq_iff (hne q.succ), succ_eq_add_one,\n mul_assoc _ _ (q.succ ! : ℚ), mul_comm _ (q.succ ! : ℚ), ← mul_assoc, div_mul_eq_mul_div]"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑(succ q)! / ↑m ! * (↑n ^ (q - m + 1) * (↑(q - m + 1)!)⁻¹)",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑(succ q)! / ↑m ! *\n ↑(coeff ℚ (q - m + 1))\n (↑{\n toOneHom :=\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (f g : PowerSeries ℚ),\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n (f * g) =\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n f *\n OneHom.toFun\n { toFun := fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f,\n map_one' := (_ : (fun f => PowerSeries.mk fun n_1 => ↑n ^ n_1 * ↑(coeff ℚ n_1) f) 1 = 1) }\n g) }\n (exp ℚ))",
"tactic": "simp only [add_eq, add_zero, ge_iff_le, IsUnit.mul_iff, Nat.isUnit_iff, succ.injEq, cast_mul,\n cast_succ, MonoidHom.coe_mk, OneHom.coe_mk, coeff_exp, Algebra.id.map_eq_id, one_div,\n map_inv₀, map_natCast, coeff_mk, mul_inv_rev]"
},
{
"state_after": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m * (↑(q + 1)! / (↑m ! * ↑(q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑(succ q)! / (↑m ! * ↑(q + 1 - m)!) * ↑n ^ (q + 1 - m)\n\ncase h.n_dvd\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ m ! * (q + 1 - m)! ∣ (q + 1)!\n\ncase h.n_nonzero\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ ↑(m ! * (q + 1 - m)!) ≠ 0",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑(succ q)! / ↑m ! * (↑n ^ (q - m + 1) * (↑(q - m + 1)!)⁻¹)",
"tactic": "rw [mul_comm ((n : ℚ) ^ (q - m + 1)), ← mul_assoc _ _ ((n : ℚ) ^ (q - m + 1)), ← one_div,\n mul_one_div, div_div, tsub_add_eq_add_tsub (le_of_lt_succ h), cast_div, cast_mul]"
},
{
"state_after": "no goals",
"state_before": "case h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ bernoulli m * (↑(q + 1)! / (↑m ! * ↑(q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑(succ q)! / (↑m ! * ↑(q + 1 - m)!) * ↑n ^ (q + 1 - m)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "case h.n_dvd\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ m ! * (q + 1 - m)! ∣ (q + 1)!",
"tactic": "exact factorial_mul_factorial_dvd_factorial h.le"
},
{
"state_after": "no goals",
"state_before": "case h.n_nonzero\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * ↑(coeff ℚ (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m < succ q\n⊢ ↑(m ! * (q + 1 - m)!) ≠ 0",
"tactic": "simp [hne, factorial_ne_zero]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !",
"tactic": "suffices\n (mk fun p => ∑ k in range n, (k : ℚ) ^ p * algebraMap ℚ ℚ p !⁻¹) =\n mk fun p =>\n ∑ i in range (p + 1), bernoulli i * (p + 1).choose i * (n : ℚ) ^ (p + 1 - i) / (p + 1)! by\n rw [← div_eq_iff (hne p), div_eq_mul_inv, sum_mul]\n rw [PowerSeries.ext_iff] at this\n simpa using this p"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"tactic": "have hexp : exp ℚ - 1 ≠ 0 := by\n simp only [exp, PowerSeries.ext_iff, Ne, not_forall]\n use 1\n simp"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"tactic": "have h_r : exp ℚ ^ n - 1 = X * mk fun p => coeff ℚ (p + 1) (exp ℚ ^ n) := by\n have h_const : C ℚ (constantCoeff ℚ (exp ℚ ^ n)) = 1 := by simp\n rw [← h_const, sub_const_eq_X_mul_shift]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) * (exp ℚ - 1) =\n (exp ℚ - 1) *\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"tactic": "rw [← mul_right_inj' hexp, mul_comm]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ x in range n, ↑(x ^ p) * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) * (exp ℚ - 1) =\n (exp ℚ - 1) *\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), bernoulli x * ↑(Nat.choose (p + 1) x) * ↑(n ^ (p + 1 - x)) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) * (exp ℚ - 1) =\n (exp ℚ - 1) *\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"tactic": "simp only [← cast_pow]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ ((PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ((fun x => ℚ) (exp ℚ ^ n))) (bernoulli n_1 / ↑n_1 !)) *\n PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)) *\n (exp ((fun x => ℚ) (exp ℚ ^ n)) - 1) =\n (exp ℚ - 1) *\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), bernoulli x * ↑(Nat.choose (p + 1) x) * ↑(n ^ (p + 1 - x)) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ x in range n, ↑(x ^ p) * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) * (exp ℚ - 1) =\n (exp ℚ - 1) *\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), bernoulli x * ↑(Nat.choose (p + 1) x) * ↑(n ^ (p + 1 - x)) / ↑(p + 1)!",
"tactic": "rw [←exp_pow_sum, geom_sum_mul, h_r, ← bernoulliPowerSeries_mul_exp_sub_one,\nbernoulliPowerSeries, mul_right_comm]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ ((PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ℚ) (bernoulli n_1 / ↑n_1 !)) *\n PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), ↑(n ^ (p + 1 - x)) * (bernoulli x * ↑(Nat.choose (p + 1) x)) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ ((PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ((fun x => ℚ) (exp ℚ ^ n))) (bernoulli n_1 / ↑n_1 !)) *\n PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)) *\n (exp ((fun x => ℚ) (exp ℚ ^ n)) - 1) =\n (exp ℚ - 1) *\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), bernoulli x * ↑(Nat.choose (p + 1) x) * ↑(n ^ (p + 1 - x)) / ↑(p + 1)!",
"tactic": "simp only [mul_comm, mul_eq_mul_left_iff, hexp, or_false]"
},
{
"state_after": "case refine'_1\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ ((PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ℚ) (bernoulli n_1 / ↑n_1 !)) =\n PowerSeries.mk fun p => bernoulli p / ↑p !) ∨\n (PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)) = 0\n\ncase refine'_2\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) =\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), ↑(n ^ (p + 1 - x)) * (bernoulli x * ↑(Nat.choose (p + 1) x)) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ ((PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ℚ) (bernoulli n_1 / ↑n_1 !)) *\n PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), ↑(n ^ (p + 1 - x)) * (bernoulli x * ↑(Nat.choose (p + 1) x)) / ↑(p + 1)!",
"tactic": "refine' Eq.trans (mul_eq_mul_right_iff.mpr _) (Eq.trans h_cauchy _)"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ∑ x in range n, ↑x ^ p * (↑p !)⁻¹ =\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ∑ k in range n, ↑k ^ p =\n (∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) * ↑p !",
"tactic": "rw [← div_eq_iff (hne p), div_eq_mul_inv, sum_mul]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis✝ :\n (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n ∀ (n_1 : ℕ),\n ↑(coeff ℚ n_1) (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n ↑(coeff ℚ n_1)\n (PowerSeries.mk fun p =>\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!)\n⊢ ∑ x in range n, ↑x ^ p * (↑p !)⁻¹ =\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ∑ x in range n, ↑x ^ p * (↑p !)⁻¹ =\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"tactic": "rw [PowerSeries.ext_iff] at this"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis✝ :\n (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n ∀ (n_1 : ℕ),\n ↑(coeff ℚ n_1) (PowerSeries.mk fun p => ∑ k in range n, ↑k ^ p * ↑(algebraMap ℚ ℚ) (↑p !)⁻¹) =\n ↑(coeff ℚ n_1)\n (PowerSeries.mk fun p =>\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!)\n⊢ ∑ x in range n, ↑x ^ p * (↑p !)⁻¹ =\n ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!",
"tactic": "simpa using this p"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ∃ x, ¬↑(coeff ℚ x) ((PowerSeries.mk fun n => ↑(algebraMap ℚ ℚ) (1 / ↑n !)) - 1) = ↑(coeff ℚ x) 0",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ exp ℚ - 1 ≠ 0",
"tactic": "simp only [exp, PowerSeries.ext_iff, Ne, not_forall]"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ¬↑(coeff ℚ 1) ((PowerSeries.mk fun n => ↑(algebraMap ℚ ℚ) (1 / ↑n !)) - 1) = ↑(coeff ℚ 1) 0",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ∃ x, ¬↑(coeff ℚ x) ((PowerSeries.mk fun n => ↑(algebraMap ℚ ℚ) (1 / ↑n !)) - 1) = ↑(coeff ℚ x) 0",
"tactic": "use 1"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\n⊢ ¬↑(coeff ℚ 1) ((PowerSeries.mk fun n => ↑(algebraMap ℚ ℚ) (1 / ↑n !)) - 1) = ↑(coeff ℚ 1) 0",
"tactic": "simp"
},
{
"state_after": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_const : ↑(C ℚ) (↑(constantCoeff ℚ) (exp ℚ ^ n)) = 1\n⊢ exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\n⊢ exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)",
"tactic": "have h_const : C ℚ (constantCoeff ℚ (exp ℚ ^ n)) = 1 := by simp"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_const : ↑(C ℚ) (↑(constantCoeff ℚ) (exp ℚ ^ n)) = 1\n⊢ exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)",
"tactic": "rw [← h_const, sub_const_eq_X_mul_shift]"
},
{
"state_after": "no goals",
"state_before": "A : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\n⊢ ↑(C ℚ) (↑(constantCoeff ℚ) (exp ℚ ^ n)) = 1",
"tactic": "simp"
},
{
"state_after": "case refine'_1.h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ℚ) (bernoulli n_1 / ↑n_1 !)) = PowerSeries.mk fun p => bernoulli p / ↑p !",
"state_before": "case refine'_1\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ ((PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ℚ) (bernoulli n_1 / ↑n_1 !)) =\n PowerSeries.mk fun p => bernoulli p / ↑p !) ∨\n (PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)) = 0",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.h\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun n_1 => ↑(algebraMap ℚ ℚ) (bernoulli n_1 / ↑n_1 !)) = PowerSeries.mk fun p => bernoulli p / ↑p !",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nA : Type ?u.791691\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p => bernoulli p / ↑p !) * PowerSeries.mk fun q => ↑(coeff ℚ (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nhexp : exp ℚ - 1 ≠ 0\nh_r : exp ℚ ^ n - 1 = X * PowerSeries.mk fun p => ↑(coeff ℚ (p + 1)) (exp ℚ ^ n)\n⊢ (PowerSeries.mk fun p => ∑ i in range (p + 1), bernoulli i * ↑(Nat.choose (p + 1) i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!) =\n PowerSeries.mk fun p =>\n ∑ x in range (p + 1), ↑(n ^ (p + 1 - x)) * (bernoulli x * ↑(Nat.choose (p + 1) x)) / ↑(p + 1)!",
"tactic": "simp only [mul_comm, factorial, cast_succ, cast_pow]"
}
] |
[
380,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/CategoryTheory/Localization/Predicate.lean
|
CategoryTheory.Functor.IsLocalization.for_id
|
[] |
[
155,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioo_insert_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.94924\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b c : α\ninst✝ : DecidableEq α\nh : a < b\n⊢ insert b (Ioo a b) = Ioc a b",
"tactic": "rw [← coe_inj, coe_insert, coe_Ioo, coe_Ioc, Set.insert_eq, Set.union_comm, Set.Ioo_union_right h]"
}
] |
[
581,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.add_mul_subset
|
[] |
[
820,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.ext_iff_blocks
|
[
{
"state_after": "no goals",
"state_before": "l : Type u_4\nm : Type u_3\nn : Type u_2\no : Type u_1\np : Type ?u.5734\nq : Type ?u.5737\nm' : o → Type ?u.5742\nn' : o → Type ?u.5747\np' : o → Type ?u.5752\nR : Type ?u.5755\nS : Type ?u.5758\nα : Type u_5\nβ : Type ?u.5764\nA B : Matrix (n ⊕ o) (l ⊕ m) α\nx✝ :\n toBlocks₁₁ A = toBlocks₁₁ B ∧ toBlocks₁₂ A = toBlocks₁₂ B ∧ toBlocks₂₁ A = toBlocks₂₁ B ∧ toBlocks₂₂ A = toBlocks₂₂ B\nh₁₁ : toBlocks₁₁ A = toBlocks₁₁ B\nh₁₂ : toBlocks₁₂ A = toBlocks₁₂ B\nh₂₁ : toBlocks₂₁ A = toBlocks₂₁ B\nh₂₂ : toBlocks₂₂ A = toBlocks₂₂ B\n⊢ A = B",
"tactic": "rw [← fromBlocks_toBlocks A, ← fromBlocks_toBlocks B, h₁₁, h₁₂, h₂₁, h₂₂]"
}
] |
[
137,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/GroupTheory/GroupAction/Sum.lean
|
Sum.smul_inl
|
[] |
[
48,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.nodup_replicate
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nl l₁ l₂ : List α\nr : α → α → Prop\na✝ b a : α\n⊢ Nodup (replicate 0 a) ↔ 0 ≤ 1",
"tactic": "simp [Nat.zero_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nl l₁ l₂ : List α\nr : α → α → Prop\na✝ b a : α\n⊢ Nodup (replicate 1 a) ↔ 1 ≤ 1",
"tactic": "simp"
}
] |
[
194,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
TopologicalGroup.exists_antitone_basis_nhds_one
|
[
{
"state_after": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"tactic": "rcases(𝓝 (1 : G)).exists_antitone_basis with ⟨u, hu, u_anti⟩"
},
{
"state_after": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), True → ∃ ia, (True ∧ True) ∧ ∀ (x : G × G), x ∈ u ia.fst ×ˢ u ia.snd → x.fst * x.snd ∈ u ib\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"state_before": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"tactic": "have :=\n ((hu.prod_nhds hu).tendsto_iff hu).mp\n (by simpa only [mul_one] using continuous_mul.tendsto ((1, 1) : G × G))"
},
{
"state_after": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"state_before": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), True → ∃ ia, (True ∧ True) ∧ ∀ (x : G × G), x ∈ u ia.fst ×ˢ u ia.snd → x.fst * x.snd ∈ u ib\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"tactic": "simp only [and_self_iff, mem_prod, and_imp, Prod.forall, exists_true_left, Prod.exists,\n forall_true_left] at this"
},
{
"state_after": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nevent_mul : ∀ (n : ℕ), ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"state_before": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"tactic": "have event_mul : ∀ n : ℕ, ∀ᶠ m in atTop, u m * u m ⊆ u n := by\n intro n\n rcases this n with ⟨j, k, -, h⟩\n refine' atTop_basis.eventually_iff.mpr ⟨max j k, True.intro, fun m hm => _⟩\n rintro - ⟨a, b, ha, hb, rfl⟩\n exact h a b (u_anti ((le_max_left _ _).trans hm) ha) (u_anti ((le_max_right _ _).trans hm) hb)"
},
{
"state_after": "case intro.mk.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nevent_mul : ∀ (n : ℕ), ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n\nφ : ℕ → ℕ\nhφ : ∀ ⦃m n : ℕ⦄, m < n → u (φ n) * u (φ n) ⊆ u (φ m)\nφ_anti_basis : HasAntitoneBasis (𝓝 1) (u ∘ φ)\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"state_before": "case intro.mk\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nevent_mul : ∀ (n : ℕ), ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"tactic": "obtain ⟨φ, -, hφ, φ_anti_basis⟩ := HasAntitoneBasis.subbasis_with_rel ⟨hu, u_anti⟩ event_mul"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nevent_mul : ∀ (n : ℕ), ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n\nφ : ℕ → ℕ\nhφ : ∀ ⦃m n : ℕ⦄, m < n → u (φ n) * u (φ n) ⊆ u (φ m)\nφ_anti_basis : HasAntitoneBasis (𝓝 1) (u ∘ φ)\n⊢ ∃ u, HasAntitoneBasis (𝓝 1) u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n",
"tactic": "exact ⟨u ∘ φ, φ_anti_basis, fun n => hφ n.lt_succ_self⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\n⊢ Tendsto ?m.464910 (𝓝 (1, 1)) (𝓝 1)",
"tactic": "simpa only [mul_one] using continuous_mul.tendsto ((1, 1) : G × G)"
},
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn : ℕ\n⊢ ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\n⊢ ∀ (n : ℕ), ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n",
"tactic": "intro n"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn j k : ℕ\nh : ∀ (a b : G), a ∈ u j → b ∈ u k → a * b ∈ u n\n⊢ ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn : ℕ\n⊢ ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n",
"tactic": "rcases this n with ⟨j, k, -, h⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn j k : ℕ\nh : ∀ (a b : G), a ∈ u j → b ∈ u k → a * b ∈ u n\nm : ℕ\nhm : m ∈ Ici (max j k)\n⊢ u m * u m ⊆ u n",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn j k : ℕ\nh : ∀ (a b : G), a ∈ u j → b ∈ u k → a * b ∈ u n\n⊢ ∀ᶠ (m : ℕ) in atTop, u m * u m ⊆ u n",
"tactic": "refine' atTop_basis.eventually_iff.mpr ⟨max j k, True.intro, fun m hm => _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn j k : ℕ\nh : ∀ (a b : G), a ∈ u j → b ∈ u k → a * b ∈ u n\nm : ℕ\nhm : m ∈ Ici (max j k)\na b : G\nha : a ∈ u m\nhb : b ∈ u m\n⊢ (fun x x_1 => x * x_1) a b ∈ u n",
"state_before": "case intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn j k : ℕ\nh : ∀ (a b : G), a ∈ u j → b ∈ u k → a * b ∈ u n\nm : ℕ\nhm : m ∈ Ici (max j k)\n⊢ u m * u m ⊆ u n",
"tactic": "rintro - ⟨a, b, ha, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\nN : Subgroup G\nn✝ : Subgroup.Normal N\ninst✝ : FirstCountableTopology G\nu : ℕ → Set G\nhu : HasBasis (𝓝 1) (fun x => True) u\nu_anti : Antitone u\nthis : ∀ (ib : ℕ), ∃ a b, True ∧ ∀ (a_1 b_1 : G), a_1 ∈ u a → b_1 ∈ u b → a_1 * b_1 ∈ u ib\nn j k : ℕ\nh : ∀ (a b : G), a ∈ u j → b ∈ u k → a * b ∈ u n\nm : ℕ\nhm : m ∈ Ici (max j k)\na b : G\nha : a ∈ u m\nhb : b ∈ u m\n⊢ (fun x x_1 => x * x_1) a b ∈ u n",
"tactic": "exact h a b (u_anti ((le_max_left _ _).trans hm) ha) (u_anti ((le_max_right _ _).trans hm) hb)"
}
] |
[
1036,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1021,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_min_of_nonneg
|
[] |
[
973,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
972,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.coe_min
|
[] |
[
485,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.prod_eq'
|
[
{
"state_after": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ (x : α), x ∈ l₁ → ∀ (y : α), y ∈ l₁ → ∀ (z : α), z * x * y = z * y * x",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ prod l₁ = prod l₂",
"tactic": "refine h.foldl_eq' ?_ _"
},
{
"state_after": "case H\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Symmetric fun x y => ∀ (z : α), z * x * y = z * y * x\n\ncase H₁\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ (x : α), x ∈ l₁ → ∀ (z : α), z * x * x = z * x * x\n\ncase H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Pairwise (fun x y => ∀ (z : α), z * x * y = z * y * x) l₁",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ (x : α), x ∈ l₁ → ∀ (y : α), y ∈ l₁ → ∀ (z : α), z * x * y = z * y * x",
"tactic": "apply Pairwise.forall_of_forall"
},
{
"state_after": "case H₁\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ (x : α), x ∈ l₁ → ∀ (z : α), z * x * x = z * x * x\n\ncase H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Pairwise (fun x y => ∀ (z : α), z * x * y = z * y * x) l₁",
"state_before": "case H\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Symmetric fun x y => ∀ (z : α), z * x * y = z * y * x\n\ncase H₁\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ (x : α), x ∈ l₁ → ∀ (z : α), z * x * x = z * x * x\n\ncase H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Pairwise (fun x y => ∀ (z : α), z * x * y = z * y * x) l₁",
"tactic": ". intro x y h z\n exact (h z).symm"
},
{
"state_after": "case H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Pairwise (fun x y => ∀ (z : α), z * x * y = z * y * x) l₁",
"state_before": "case H₁\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ (x : α), x ∈ l₁ → ∀ (z : α), z * x * x = z * x * x\n\ncase H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Pairwise (fun x y => ∀ (z : α), z * x * y = z * y * x) l₁",
"tactic": ". intros; rfl"
},
{
"state_after": "no goals",
"state_before": "case H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Pairwise (fun x y => ∀ (z : α), z * x * y = z * y * x) l₁",
"tactic": ". apply hc.imp\n intro a b h z\n rw [mul_assoc z, mul_assoc z, h]"
},
{
"state_after": "case H\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh✝ : l₁ ~ l₂\nhc : Pairwise Commute l₁\nx y : α\nh : ∀ (z : α), z * x * y = z * y * x\nz : α\n⊢ z * y * x = z * x * y",
"state_before": "case H\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Symmetric fun x y => ∀ (z : α), z * x * y = z * y * x",
"tactic": "intro x y h z"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh✝ : l₁ ~ l₂\nhc : Pairwise Commute l₁\nx y : α\nh : ∀ (z : α), z * x * y = z * y * x\nz : α\n⊢ z * y * x = z * x * y",
"tactic": "exact (h z).symm"
},
{
"state_after": "case H₁\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\nx✝ : α\na✝ : x✝ ∈ l₁\nz✝ : α\n⊢ z✝ * x✝ * x✝ = z✝ * x✝ * x✝",
"state_before": "case H₁\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ (x : α), x ∈ l₁ → ∀ (z : α), z * x * x = z * x * x",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case H₁\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\nx✝ : α\na✝ : x✝ ∈ l₁\nz✝ : α\n⊢ z✝ * x✝ * x✝ = z✝ * x✝ * x✝",
"tactic": "rfl"
},
{
"state_after": "case H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ {a b : α}, Commute a b → ∀ (z : α), z * a * b = z * b * a",
"state_before": "case H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ Pairwise (fun x y => ∀ (z : α), z * x * y = z * y * x) l₁",
"tactic": "apply hc.imp"
},
{
"state_after": "case H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh✝ : l₁ ~ l₂\nhc : Pairwise Commute l₁\na b : α\nh : Commute a b\nz : α\n⊢ z * a * b = z * b * a",
"state_before": "case H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh : l₁ ~ l₂\nhc : Pairwise Commute l₁\n⊢ ∀ {a b : α}, Commute a b → ∀ (z : α), z * a * b = z * b * a",
"tactic": "intro a b h z"
},
{
"state_after": "no goals",
"state_before": "case H₂\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nM : Monoid α\nl₁ l₂ : List α\nh✝ : l₁ ~ l₂\nhc : Pairwise Commute l₁\na b : α\nh : Commute a b\nz : α\n⊢ z * a * b = z * b * a",
"tactic": "rw [mul_assoc z, mul_assoc z, h]"
}
] |
[
575,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_ofMul
|
[] |
[
2208,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2207,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_subset_image_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82868\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt✝ : Finset β\na : α\nb c : β\nt : Finset α\nhf : Injective f\n⊢ ↑(image f s) ⊆ ↑(image f t) ↔ ↑s ⊆ ↑t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82868\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt✝ : Finset β\na : α\nb c : β\nt : Finset α\nhf : Injective f\n⊢ image f s ⊆ image f t ↔ s ⊆ t",
"tactic": "simp_rw [← coe_subset]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82868\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt✝ : Finset β\na : α\nb c : β\nt : Finset α\nhf : Injective f\n⊢ f '' ↑s ⊆ f '' ↑t ↔ ↑s ⊆ ↑t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82868\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt✝ : Finset β\na : α\nb c : β\nt : Finset α\nhf : Injective f\n⊢ ↑(image f s) ⊆ ↑(image f t) ↔ ↑s ⊆ ↑t",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.82868\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt✝ : Finset β\na : α\nb c : β\nt : Finset α\nhf : Injective f\n⊢ f '' ↑s ⊆ f '' ↑t ↔ ↑s ⊆ ↑t",
"tactic": "exact Set.image_subset_image_iff hf"
}
] |
[
457,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_opow
|
[
{
"state_after": "case h.e'_2\nb : Ordinal\nhb : 1 < b\nx : Ordinal\n⊢ log b (b ^ x) = log b (b ^ x * 1 + 0)",
"state_before": "b : Ordinal\nhb : 1 < b\nx : Ordinal\n⊢ log b (b ^ x) = x",
"tactic": "convert log_opow_mul_add hb zero_ne_one.symm hb (opow_pos x (zero_lt_one.trans hb))\n using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nb : Ordinal\nhb : 1 < b\nx : Ordinal\n⊢ log b (b ^ x) = log b (b ^ x * 1 + 0)",
"tactic": "rw [add_zero, mul_one]"
}
] |
[
428,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.insertNth_apply_same
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\ni : Fin (n + 1)\nx : α i\np : (j : Fin n) → α (↑(succAbove i) j)\n⊢ insertNth i x p i = x",
"tactic": "simp [insertNth, succAboveCases]"
}
] |
[
665,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.isPartitionTop
|
[] |
[
730,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
729,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.map_eq_none
|
[] |
[
134,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
134,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.coe_vaddConst_symm
|
[] |
[
711,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
710,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.pureMonoidHom_apply
|
[] |
[
654,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
AddSubgroup.toIntSubmodule_symm
|
[] |
[
400,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Algebra/Module/BigOperators.lean
|
Multiset.sum_smul
|
[] |
[
31,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/Data/List/Basic.lean
|
List.lookmap_some
|
[] |
[
3308,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3306,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
SubsemiringClass.coe_subtype
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.countable_coe_iff
|
[] |
[
40,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Algebra/CharZero/Lemmas.lean
|
bit0_eq_zero
|
[] |
[
81,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Topology/Order/Basic.lean
|
tendsto_nhds_top_mono
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderTop β\ninst✝ : OrderTopology β\nl : Filter α\nf g : α → β\nhg : f ≤ᶠ[l] g\nhf : ∀ (i : β), i < ⊤ → ∀ᶠ (a : α) in l, f a ∈ Ioi i\n⊢ ∀ (i : β), i < ⊤ → ∀ᶠ (a : α) in l, g a ∈ Ioi i",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderTop β\ninst✝ : OrderTopology β\nl : Filter α\nf g : α → β\nhf : Tendsto f l (𝓝 ⊤)\nhg : f ≤ᶠ[l] g\n⊢ Tendsto g l (𝓝 ⊤)",
"tactic": "simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderTop β\ninst✝ : OrderTopology β\nl : Filter α\nf g : α → β\nhg : f ≤ᶠ[l] g\nhf : ∀ (i : β), i < ⊤ → ∀ᶠ (a : α) in l, f a ∈ Ioi i\nx : β\nhx : x < ⊤\n⊢ ∀ᶠ (a : α) in l, g a ∈ Ioi x",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderTop β\ninst✝ : OrderTopology β\nl : Filter α\nf g : α → β\nhg : f ≤ᶠ[l] g\nhf : ∀ (i : β), i < ⊤ → ∀ᶠ (a : α) in l, f a ∈ Ioi i\n⊢ ∀ (i : β), i < ⊤ → ∀ᶠ (a : α) in l, g a ∈ Ioi i",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderTop β\ninst✝ : OrderTopology β\nl : Filter α\nf g : α → β\nhg : f ≤ᶠ[l] g\nhf : ∀ (i : β), i < ⊤ → ∀ᶠ (a : α) in l, f a ∈ Ioi i\nx : β\nhx : x < ⊤\n⊢ ∀ᶠ (a : α) in l, g a ∈ Ioi x",
"tactic": "filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le"
}
] |
[
1165,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1161,
1
] |
Mathlib/MeasureTheory/Measure/OpenPos.lean
|
IsOpen.eq_empty_of_measure_zero
|
[] |
[
84,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.smul_toNNReal
|
[
{
"state_after": "α : Type ?u.819259\nβ : Type ?u.819262\na✝ b✝ c d : ℝ≥0∞\nr p q a : ℝ≥0\nb : ℝ≥0∞\n⊢ ENNReal.toNNReal (↑a * b) = a * ENNReal.toNNReal b",
"state_before": "α : Type ?u.819259\nβ : Type ?u.819262\na✝ b✝ c d : ℝ≥0∞\nr p q a : ℝ≥0\nb : ℝ≥0∞\n⊢ ENNReal.toNNReal (a • b) = a * ENNReal.toNNReal b",
"tactic": "change ((a : ℝ≥0∞) * b).toNNReal = a * b.toNNReal"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.819259\nβ : Type ?u.819262\na✝ b✝ c d : ℝ≥0∞\nr p q a : ℝ≥0\nb : ℝ≥0∞\n⊢ ENNReal.toNNReal (↑a * b) = a * ENNReal.toNNReal b",
"tactic": "simp only [ENNReal.toNNReal_mul, ENNReal.toNNReal_coe]"
}
] |
[
2209,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2207,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.measure_union_add_inter₀
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.17157\nα : Type u_1\nβ : Type ?u.17163\nγ : Type ?u.17166\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t s : Set α\nht : NullMeasurableSet t\n⊢ ↑↑μ (s ∪ t) + ↑↑μ (s ∩ t) = ↑↑μ s + ↑↑μ t",
"tactic": "rw [← measure_inter_add_diff₀ (s ∪ t) ht, union_inter_cancel_right, union_diff_right, ←\n measure_inter_add_diff₀ s ht, add_comm, ← add_assoc, add_right_comm]"
}
] |
[
327,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_mul_X
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nk : ℕ\n⊢ Commute (↑f (coeff X k)) x",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\n⊢ eval₂ f x (p * X) = eval₂ f x p * x",
"tactic": "refine' _root_.trans (eval₂_mul_noncomm _ _ fun k => _) (by rw [eval₂_X])"
},
{
"state_after": "case inl\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\n⊢ Commute (↑f (coeff X 1)) x\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nk : ℕ\nhk : ¬k = 1\n⊢ Commute (↑f (coeff X k)) x",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nk : ℕ\n⊢ Commute (↑f (coeff X k)) x",
"tactic": "rcases em (k = 1) with (rfl | hk)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\n⊢ eval₂ f x p * eval₂ f x X = eval₂ f x p * x",
"tactic": "rw [eval₂_X]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\n⊢ Commute (↑f (coeff X 1)) x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nk : ℕ\nhk : ¬k = 1\n⊢ Commute (↑f (coeff X k)) x",
"tactic": "simp [coeff_X_of_ne_one hk]"
}
] |
[
192,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Order/Atoms.lean
|
IsSimpleOrder.eq_bot_of_lt
|
[] |
[
509,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.memℒp_one_iff_integrable
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.827340\nδ : Type ?u.827343\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\n⊢ Memℒp f 1 ↔ Integrable f",
"tactic": "simp_rw [Integrable, HasFiniteIntegral, Memℒp, snorm_one_eq_lintegral_nnnorm]"
}
] |
[
447,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/MeasureTheory/Integral/Average.lean
|
MeasureTheory.set_average_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.195874\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ : Measure α\ns✝ : Set E\nf : α → E\ns : Set α\n⊢ (⨍ (x : α) in s, f x ∂μ) = (ENNReal.toReal (↑↑μ s))⁻¹ • ∫ (x : α) in s, f x ∂μ",
"tactic": "rw [average_eq, restrict_apply_univ]"
}
] |
[
116,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Algebra/Group/Commute.lean
|
mul_inv_cancel_comm_assoc
|
[] |
[
418,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.support_subset_singleton'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.136560\nγ : Type ?u.136563\nι : Type ?u.136566\nM : Type u_2\nM' : Type ?u.136572\nN : Type ?u.136575\nP : Type ?u.136578\nG : Type ?u.136581\nH : Type ?u.136584\nR : Type ?u.136587\nS : Type ?u.136590\ninst✝ : Zero M\na✝ a' : α\nb✝ : M\nf : α →₀ M\na : α\nx✝ : ∃ b, f = single a b\nb : M\nhb : f = single a b\n⊢ f.support ⊆ {a}",
"tactic": "rw [hb, support_subset_singleton, single_eq_same]"
}
] |
[
499,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
src/lean/Init/SimpLemmas.lean
|
or_true
|
[] |
[
88,
81
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
88,
9
] |
Mathlib/LinearAlgebra/DirectSum/Finsupp.lean
|
finsuppTensorFinsupp_apply
|
[
{
"state_after": "case h0\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (0 ⊗ₜ[R] g)) (i, k) = ↑0 i ⊗ₜ[R] ↑g k\n\ncase hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\n⊢ ∀ (f g_1 : ι →₀ M),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (f ⊗ₜ[R] g)) (i, k) = ↑f i ⊗ₜ[R] ↑g k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (g_1 ⊗ₜ[R] g)) (i, k) = ↑g_1 i ⊗ₜ[R] ↑g k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) ((f + g_1) ⊗ₜ[R] g)) (i, k) = ↑(f + g_1) i ⊗ₜ[R] ↑g k\n\ncase hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\n⊢ ∀ (a : ι) (b : M),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single a b ⊗ₜ[R] g)) (i, k) = ↑(Finsupp.single a b) i ⊗ₜ[R] ↑g k",
"state_before": "R✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (f ⊗ₜ[R] g)) (i, k) = ↑f i ⊗ₜ[R] ↑g k",
"tactic": "apply Finsupp.induction_linear f"
},
{
"state_after": "no goals",
"state_before": "case h0\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (0 ⊗ₜ[R] g)) (i, k) = ↑0 i ⊗ₜ[R] ↑g k",
"tactic": "simp"
},
{
"state_after": "case hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\nf₁ f₂ : ι →₀ M\nhf₁ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (f₁ ⊗ₜ[R] g)) (i, k) = ↑f₁ i ⊗ₜ[R] ↑g k\nhf₂ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (f₂ ⊗ₜ[R] g)) (i, k) = ↑f₂ i ⊗ₜ[R] ↑g k\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) ((f₁ + f₂) ⊗ₜ[R] g)) (i, k) = ↑(f₁ + f₂) i ⊗ₜ[R] ↑g k",
"state_before": "case hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\n⊢ ∀ (f g_1 : ι →₀ M),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (f ⊗ₜ[R] g)) (i, k) = ↑f i ⊗ₜ[R] ↑g k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (g_1 ⊗ₜ[R] g)) (i, k) = ↑g_1 i ⊗ₜ[R] ↑g k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) ((f + g_1) ⊗ₜ[R] g)) (i, k) = ↑(f + g_1) i ⊗ₜ[R] ↑g k",
"tactic": "intro f₁ f₂ hf₁ hf₂"
},
{
"state_after": "no goals",
"state_before": "case hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\nf₁ f₂ : ι →₀ M\nhf₁ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (f₁ ⊗ₜ[R] g)) (i, k) = ↑f₁ i ⊗ₜ[R] ↑g k\nhf₂ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (f₂ ⊗ₜ[R] g)) (i, k) = ↑f₂ i ⊗ₜ[R] ↑g k\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) ((f₁ + f₂) ⊗ₜ[R] g)) (i, k) = ↑(f₁ + f₂) i ⊗ₜ[R] ↑g k",
"tactic": "simp [add_tmul, hf₁, hf₂]"
},
{
"state_after": "case hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g k",
"state_before": "case hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\n⊢ ∀ (a : ι) (b : M),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single a b ⊗ₜ[R] g)) (i, k) = ↑(Finsupp.single a b) i ⊗ₜ[R] ↑g k",
"tactic": "intro i' m"
},
{
"state_after": "case hsingle.h0\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] 0)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑0 k\n\ncase hsingle.hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ∀ (f g : κ →₀ N),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] f)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑f k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] (f + g))) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(f + g) k\n\ncase hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ∀ (a : κ) (b : N),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] Finsupp.single a b)) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(Finsupp.single a b) k",
"state_before": "case hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g k",
"tactic": "apply Finsupp.induction_linear g"
},
{
"state_after": "no goals",
"state_before": "case hsingle.h0\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] 0)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑0 k",
"tactic": "simp"
},
{
"state_after": "case hsingle.hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\ng₁ g₂ : κ →₀ N\nhg₁ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g₁)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g₁ k\nhg₂ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g₂)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g₂ k\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] (g₁ + g₂))) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(g₁ + g₂) k",
"state_before": "case hsingle.hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ∀ (f g : κ →₀ N),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] f)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑f k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g k →\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] (f + g))) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(f + g) k",
"tactic": "intro g₁ g₂ hg₁ hg₂"
},
{
"state_after": "no goals",
"state_before": "case hsingle.hadd\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\ng₁ g₂ : κ →₀ N\nhg₁ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g₁)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g₁ k\nhg₂ : ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] g₂)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑g₂ k\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] (g₁ + g₂))) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(g₁ + g₂) k",
"tactic": "simp [tmul_add, hg₁, hg₂]"
},
{
"state_after": "case hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] Finsupp.single k' n)) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(Finsupp.single k' n) k",
"state_before": "case hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\n⊢ ∀ (a : κ) (b : N),\n ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] Finsupp.single a b)) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(Finsupp.single a b) k",
"tactic": "intro k' n"
},
{
"state_after": "case hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\n⊢ ↑(Finsupp.single (i', k') (m ⊗ₜ[R] n)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(Finsupp.single k' n) k",
"state_before": "case hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\n⊢ ↑(↑(finsuppTensorFinsupp R M N ι κ) (Finsupp.single i' m ⊗ₜ[R] Finsupp.single k' n)) (i, k) =\n ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(Finsupp.single k' n) k",
"tactic": "simp only [finsuppTensorFinsupp_single]"
},
{
"state_after": "case hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"state_before": "case hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\n⊢ ↑(Finsupp.single (i', k') (m ⊗ₜ[R] n)) (i, k) = ↑(Finsupp.single i' m) i ⊗ₜ[R] ↑(Finsupp.single k' n) k",
"tactic": "simp only [Finsupp.single_apply]"
},
{
"state_after": "case pos\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : (i', k') = (i, k)\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0\n\ncase neg\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : ¬(i', k') = (i, k)\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"state_before": "case hsingle.hsingle\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"tactic": "by_cases h1 : (i', k') = (i, k)"
},
{
"state_after": "case pos\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : i' = i ∧ k' = k\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"state_before": "case pos\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : (i', k') = (i, k)\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"tactic": "simp only [Prod.mk.inj_iff] at h1"
},
{
"state_after": "no goals",
"state_before": "case pos\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : i' = i ∧ k' = k\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"tactic": "simp [h1]"
},
{
"state_after": "case neg\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : ¬(i', k') = (i, k)\n⊢ 0 = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"state_before": "case neg\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : ¬(i', k') = (i, k)\n⊢ (if (i', k') = (i, k) then m ⊗ₜ[R] n else 0) = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"tactic": "simp only [h1, if_false]"
},
{
"state_after": "case neg\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : ¬i' = i ∨ ¬k' = k\n⊢ 0 = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"state_before": "case neg\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : ¬(i', k') = (i, k)\n⊢ 0 = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"tactic": "simp only [Prod.mk.inj_iff, not_and_or] at h1"
},
{
"state_after": "no goals",
"state_before": "case neg\nR✝ : Type u\nM✝ : Type v\nN✝ : Type w\ninst✝⁹ : Ring R✝\ninst✝⁸ : AddCommGroup M✝\ninst✝⁷ : Module R✝ M✝\ninst✝⁶ : AddCommGroup N✝\ninst✝⁵ : Module R✝ N✝\nR : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\nκ : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : ι →₀ M\ng : κ →₀ N\ni : ι\nk : κ\ni' : ι\nm : M\nk' : κ\nn : N\nh1 : ¬i' = i ∨ ¬k' = k\n⊢ 0 = (if i' = i then m else 0) ⊗ₜ[R] if k' = k then n else 0",
"tactic": "cases' h1 with h1 h1 <;> simp [h1]"
}
] |
[
78,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
RingHom.coe_range
|
[] |
[
654,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.pow_lt_ascFactorial'
|
[
{
"state_after": "n k : ℕ\n⊢ (n + 1) * (n + 1) ^ (k + 1) < (n + (k + 1) + 1) * ascFactorial n (k + 1)",
"state_before": "n k : ℕ\n⊢ (n + 1) ^ (k + 2) < ascFactorial n (k + 2)",
"tactic": "rw [pow_succ, ascFactorial, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ (n + 1) * (n + 1) ^ (k + 1) < (n + (k + 1) + 1) * ascFactorial n (k + 1)",
"tactic": "exact\n Nat.mul_lt_mul (Nat.add_lt_add_right (Nat.lt_add_of_pos_right succ_pos') 1)\n (pow_succ_le_ascFactorial n _) (pow_pos succ_pos' _)"
}
] |
[
296,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Topology/Support.lean
|
range_eq_image_mulTSupport_or
|
[] |
[
86,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Order/BoundedOrder.lean
|
not_bot_lt_iff
|
[] |
[
386,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean
|
differentiableAt_iff_restrictScalars
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableAt 𝕜 f x\n⊢ DifferentiableWithinAt 𝕜' f univ x ↔ ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f univ x",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableAt 𝕜 f x\n⊢ DifferentiableAt 𝕜' f x ↔ ∃ g', restrictScalars 𝕜 g' = fderiv 𝕜 f x",
"tactic": "rw [← differentiableWithinAt_univ, ← fderivWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableAt 𝕜 f x\n⊢ DifferentiableWithinAt 𝕜' f univ x ↔ ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f univ x",
"tactic": "exact\n differentiableWithinAt_iff_restrictScalars 𝕜 hf.differentiableWithinAt uniqueDiffWithinAt_univ"
}
] |
[
123,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integrableOn_Ioi_deriv_of_nonpos'
|
[
{
"state_after": "E : Type ?u.164089\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ (x : ℝ), x ∈ Ici a → HasDerivAt g (g' x) x\ng'neg : ∀ (x : ℝ), x ∈ Ioi a → g' x ≤ 0\nhg : Tendsto g atTop (𝓝 l)\nx : ℝ\nhx : x ∈ Ici a\n⊢ ContinuousWithinAt g (Ici a) x",
"state_before": "E : Type ?u.164089\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ (x : ℝ), x ∈ Ici a → HasDerivAt g (g' x) x\ng'neg : ∀ (x : ℝ), x ∈ Ioi a → g' x ≤ 0\nhg : Tendsto g atTop (𝓝 l)\n⊢ IntegrableOn g' (Ioi a)",
"tactic": "refine integrableOn_Ioi_deriv_of_nonpos (fun x hx ↦ ?_) (fun x hx ↦ hderiv x hx.out.le) g'neg hg"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.164089\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ (x : ℝ), x ∈ Ici a → HasDerivAt g (g' x) x\ng'neg : ∀ (x : ℝ), x ∈ Ioi a → g' x ≤ 0\nhg : Tendsto g atTop (𝓝 l)\nx : ℝ\nhx : x ∈ Ici a\n⊢ ContinuousWithinAt g (Ici a) x",
"tactic": "exact (hderiv x hx).continuousAt.continuousWithinAt"
}
] |
[
773,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
|
SimpleGraph.ComponentCompl.hom_refl
|
[
{
"state_after": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\n⊢ ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Lᶜ)) C = C",
"state_before": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\n⊢ hom (_ : L ⊆ L) C = C",
"tactic": "change C.map _ = C"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\n⊢ ConnectedComponent.map (InduceHom Hom.id (_ : Lᶜ ⊆ Lᶜ)) C = C",
"tactic": "erw [induceHom_id G (Lᶜ), ConnectedComponent.map_id]"
}
] |
[
212,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
AddSubgroup.smul_mem_pointwise_smul_iff₀
|
[] |
[
531,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.sym_inv
|
[] |
[
219,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/NumberTheory/Multiplicity.lean
|
multiplicity.pow_sub_pow_of_prime
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn✝ : ℕ\ninst✝² : CommRing R\na b x✝ y✝ : R\np✝ : ℕ\ninst✝¹ : IsDomain R\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np : R\nhp : Prime p\nx y : R\nhxy : p ∣ x - y\nhx : ¬p ∣ x\nn : ℕ\nhn : ¬p ∣ ↑n\n⊢ multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y)",
"tactic": "rw [← geom_sum₂_mul, multiplicity.mul hp, multiplicity_eq_zero.2 (not_dvd_geom_sum₂ hp hxy hx hn),\n zero_add]"
}
] |
[
164,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Topology/Order.lean
|
continuous_iInf_rng
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.42166\nf : α → β\nι : Sort u_1\nt₁ : TopologicalSpace α\nt₂ : ι → TopologicalSpace β\n⊢ Continuous f ↔ ∀ (i : ι), Continuous f",
"tactic": "simp only [continuous_iff_coinduced_le, le_iInf_iff]"
}
] |
[
805,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image_subset_image₂_right
|
[] |
[
99,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.map_nil
|
[] |
[
695,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_le_sub_left_of_le_add
|
[
{
"state_after": "a b c : Int\nh✝ : c ≤ a + b\nh : -a ≤ -c + b\n⊢ -a ≤ b - c",
"state_before": "a b c : Int\nh : c ≤ a + b\n⊢ -a ≤ b - c",
"tactic": "have h := Int.le_neg_add_of_add_le (Int.sub_left_le_of_le_add h)"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh✝ : c ≤ a + b\nh : -a ≤ -c + b\n⊢ -a ≤ b - c",
"tactic": "rwa [Int.add_comm] at h"
}
] |
[
1019,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1017,
11
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_sUnion_le
|
[
{
"state_after": "α✝ β α : Type u\nA : Set (Set α)\n⊢ (#↑(⋃ (i : ↑A), ↑i)) ≤ (#↑A) * ⨆ (s : ↑A), #↑↑s",
"state_before": "α✝ β α : Type u\nA : Set (Set α)\n⊢ (#↑(⋃₀ A)) ≤ (#↑A) * ⨆ (s : ↑A), #↑↑s",
"tactic": "rw [sUnion_eq_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α✝ β α : Type u\nA : Set (Set α)\n⊢ (#↑(⋃ (i : ↑A), ↑i)) ≤ (#↑A) * ⨆ (s : ↑A), #↑↑s",
"tactic": "apply mk_iUnion_le"
}
] |
[
2053,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2051,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
strictMonoOn_univ
|
[] |
[
404,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
9
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.Prepartition.IsPartition.biUnionTagged
|
[] |
[
170,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.preimage_metric_closedBall
|
[
{
"state_after": "case h\nR : Type ?u.1012446\nR' : Type ?u.1012449\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1012455\n𝕜₃ : Type ?u.1012458\n𝕝 : Type ?u.1012461\nE : Type u_1\nE₂ : Type ?u.1012467\nE₃ : Type ?u.1012470\nF : Type ?u.1012473\nG : Type ?u.1012476\nι : Type ?u.1012479\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜 E\nr : ℝ\nx : E\n⊢ x ∈ ↑p ⁻¹' Metric.closedBall 0 r ↔ x ∈ {x | ↑p x ≤ r}",
"state_before": "R : Type ?u.1012446\nR' : Type ?u.1012449\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1012455\n𝕜₃ : Type ?u.1012458\n𝕝 : Type ?u.1012461\nE : Type u_1\nE₂ : Type ?u.1012467\nE₃ : Type ?u.1012470\nF : Type ?u.1012473\nG : Type ?u.1012476\nι : Type ?u.1012479\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜 E\nr : ℝ\n⊢ ↑p ⁻¹' Metric.closedBall 0 r = {x | ↑p x ≤ r}",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type ?u.1012446\nR' : Type ?u.1012449\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1012455\n𝕜₃ : Type ?u.1012458\n𝕝 : Type ?u.1012461\nE : Type u_1\nE₂ : Type ?u.1012467\nE₃ : Type ?u.1012470\nF : Type ?u.1012473\nG : Type ?u.1012476\nι : Type ?u.1012479\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜 E\nr : ℝ\nx : E\n⊢ x ∈ ↑p ⁻¹' Metric.closedBall 0 r ↔ x ∈ {x | ↑p x ≤ r}",
"tactic": "simp only [mem_setOf, mem_preimage, mem_closedBall_zero_iff,\n Real.norm_of_nonneg (map_nonneg p _)]"
}
] |
[
815,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
812,
1
] |
Mathlib/Topology/Algebra/Order/Compact.lean
|
Continuous.exists_forall_ge_of_hasCompactMulSupport
|
[] |
[
362,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Data/List/Sections.lean
|
List.mem_sections
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.11\nL : List (List α)\nf : List α\nh : f ∈ sections L\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f L\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.11\nL : List (List α)\nf : List α\nh : Forall₂ (fun x x_1 => x ∈ x_1) f L\n⊢ f ∈ sections L",
"state_before": "α : Type u_1\nβ : Type ?u.11\nL : List (List α)\nf : List α\n⊢ f ∈ sections L ↔ Forall₂ (fun x x_1 => x ∈ x_1) f L",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1.nil\nα : Type u_1\nβ : Type ?u.11\nf : List α\nh : f ∈ sections []\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f []\n\ncase refine'_1.cons\nα : Type u_1\nβ : Type ?u.11\nhead✝ : List α\ntail✝ : List (List α)\ntail_ih✝ : ∀ {f : List α}, f ∈ sections tail✝ → Forall₂ (fun x x_1 => x ∈ x_1) f tail✝\nf : List α\nh : f ∈ sections (head✝ :: tail✝)\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f (head✝ :: tail✝)",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.11\nL : List (List α)\nf : List α\nh : f ∈ sections L\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f L",
"tactic": "induction L generalizing f"
},
{
"state_after": "case refine'_1.cons\nα : Type u_1\nβ : Type ?u.11\nhead✝ : List α\ntail✝ : List (List α)\ntail_ih✝ : ∀ {f : List α}, f ∈ sections tail✝ → Forall₂ (fun x x_1 => x ∈ x_1) f tail✝\nf : List α\nh : ∃ a, a ∈ sections tail✝ ∧ ∃ a_1, a_1 ∈ head✝ ∧ a_1 :: a = f\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f (head✝ :: tail✝)",
"state_before": "case refine'_1.cons\nα : Type u_1\nβ : Type ?u.11\nhead✝ : List α\ntail✝ : List (List α)\ntail_ih✝ : ∀ {f : List α}, f ∈ sections tail✝ → Forall₂ (fun x x_1 => x ∈ x_1) f tail✝\nf : List α\nh : f ∈ sections (head✝ :: tail✝)\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f (head✝ :: tail✝)",
"tactic": "simp only [sections, bind_eq_bind, mem_bind, mem_map] at h"
},
{
"state_after": "case refine'_1.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.11\nhead✝ : List α\ntail✝ : List (List α)\ntail_ih✝ : ∀ {f : List α}, f ∈ sections tail✝ → Forall₂ (fun x x_1 => x ∈ x_1) f tail✝\nw✝¹ : List α\nleft✝¹ : w✝¹ ∈ sections tail✝\nw✝ : α\nleft✝ : w✝ ∈ head✝\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) (w✝ :: w✝¹) (head✝ :: tail✝)",
"state_before": "case refine'_1.cons\nα : Type u_1\nβ : Type ?u.11\nhead✝ : List α\ntail✝ : List (List α)\ntail_ih✝ : ∀ {f : List α}, f ∈ sections tail✝ → Forall₂ (fun x x_1 => x ∈ x_1) f tail✝\nf : List α\nh : ∃ a, a ∈ sections tail✝ ∧ ∃ a_1, a_1 ∈ head✝ ∧ a_1 :: a = f\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f (head✝ :: tail✝)",
"tactic": "rcases h with ⟨_, _, _, _, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.11\nhead✝ : List α\ntail✝ : List (List α)\ntail_ih✝ : ∀ {f : List α}, f ∈ sections tail✝ → Forall₂ (fun x x_1 => x ∈ x_1) f tail✝\nw✝¹ : List α\nleft✝¹ : w✝¹ ∈ sections tail✝\nw✝ : α\nleft✝ : w✝ ∈ head✝\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) (w✝ :: w✝¹) (head✝ :: tail✝)",
"tactic": "simp only [*, forall₂_cons, true_and_iff]"
},
{
"state_after": "case refine'_1.nil.refl\nα : Type u_1\nβ : Type ?u.11\nh : [] ∈ sections []\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) [] []",
"state_before": "case refine'_1.nil\nα : Type u_1\nβ : Type ?u.11\nf : List α\nh : f ∈ sections []\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) f []",
"tactic": "cases mem_singleton.1 h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.nil.refl\nα : Type u_1\nβ : Type ?u.11\nh : [] ∈ sections []\n⊢ Forall₂ (fun x x_1 => x ∈ x_1) [] []",
"tactic": "exact Forall₂.nil"
},
{
"state_after": "case refine'_2.nil\nα : Type u_1\nβ : Type ?u.11\nL : List (List α)\nf : List α\n⊢ [] ∈ sections []\n\ncase refine'_2.cons\nα : Type u_1\nβ : Type ?u.11\nL✝ : List (List α)\nf✝ : List α\na : α\nl f : List α\nL : List (List α)\nal : a ∈ l\nfL : Forall₂ (fun x x_1 => x ∈ x_1) f L\nfs : f ∈ sections L\n⊢ a :: f ∈ sections (l :: L)",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.11\nL : List (List α)\nf : List α\nh : Forall₂ (fun x x_1 => x ∈ x_1) f L\n⊢ f ∈ sections L",
"tactic": "induction' h with a l f L al fL fs"
},
{
"state_after": "case refine'_2.cons\nα : Type u_1\nβ : Type ?u.11\nL✝ : List (List α)\nf✝ : List α\na : α\nl f : List α\nL : List (List α)\nal : a ∈ l\nfL : Forall₂ (fun x x_1 => x ∈ x_1) f L\nfs : f ∈ sections L\n⊢ ∃ a_1, a_1 ∈ sections L ∧ ∃ a_2, a_2 ∈ l ∧ a_2 :: a_1 = a :: f",
"state_before": "case refine'_2.cons\nα : Type u_1\nβ : Type ?u.11\nL✝ : List (List α)\nf✝ : List α\na : α\nl f : List α\nL : List (List α)\nal : a ∈ l\nfL : Forall₂ (fun x x_1 => x ∈ x_1) f L\nfs : f ∈ sections L\n⊢ a :: f ∈ sections (l :: L)",
"tactic": "simp only [sections, bind_eq_bind, mem_bind, mem_map]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.cons\nα : Type u_1\nβ : Type ?u.11\nL✝ : List (List α)\nf✝ : List α\na : α\nl f : List α\nL : List (List α)\nal : a ∈ l\nfL : Forall₂ (fun x x_1 => x ∈ x_1) f L\nfs : f ∈ sections L\n⊢ ∃ a_1, a_1 ∈ sections L ∧ ∃ a_2, a_2 ∈ l ∧ a_2 :: a_1 = a :: f",
"tactic": "exact ⟨f, fs, a, al, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.nil\nα : Type u_1\nβ : Type ?u.11\nL : List (List α)\nf : List α\n⊢ [] ∈ sections []",
"tactic": "simp only [sections, mem_singleton]"
}
] |
[
37,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
26,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.mem_roots'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\n⊢ a ∈ roots p ↔ p ≠ 0 ∧ IsRoot p a",
"tactic": "rw [← count_pos, count_roots p, rootMultiplicity_pos']"
}
] |
[
577,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
576,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.vec3_dotProduct
|
[] |
[
507,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_disjoint
|
[] |
[
2893,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2892,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Icc.coe_ne_zero
|
[] |
[
86,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
Filter.Tendsto.const_smul
|
[] |
[
84,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.eval_eq_eval_mv_eval'
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ↑(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (↑(finSuccEquiv R n) f))",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\n⊢ ↑(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (↑(finSuccEquiv R n) f))",
"tactic": "let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n { Polynomial.mapRingHom (eval s) with\n commutes' := fun r => by\n convert Polynomial.map_C (eval s)\n exact (eval_C _).symm }"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ↑(aeval (Fin.cons y s)) f = ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) f",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ↑(eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (eval s) (↑(finSuccEquiv R n) f))",
"tactic": "show\n aeval (Fin.cons y s : Fin (n + 1) → R) f =\n (Polynomial.aeval y).comp (φ.comp (finSuccEquiv R n).toAlgHom) f"
},
{
"state_after": "case e_a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ aeval (Fin.cons y s) = AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ↑(aeval (Fin.cons y s)) f = ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) f",
"tactic": "congr 2"
},
{
"state_after": "case e_a.hf\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ∀ (i : Fin (n + 1)),\n ↑(aeval (Fin.cons y s)) (X i) = ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) (X i)",
"state_before": "case e_a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ aeval (Fin.cons y s) = AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))",
"tactic": "apply MvPolynomial.algHom_ext"
},
{
"state_after": "case e_a.hf\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ↑(aeval (Fin.cons y s)) (X 0) = ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) (X 0) ∧\n ∀ (i : Fin n),\n ↑(aeval (Fin.cons y s)) (X (Fin.succ i)) =\n ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) (X (Fin.succ i))",
"state_before": "case e_a.hf\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ∀ (i : Fin (n + 1)),\n ↑(aeval (Fin.cons y s)) (X i) = ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) (X i)",
"tactic": "rw [Fin.forall_fin_succ]"
},
{
"state_after": "no goals",
"state_before": "case e_a.hf\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nφ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=\n let src := mapRingHom (eval s);\n {\n toRingHom :=\n { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },\n commutes' :=\n (_ :\n ∀ (r : R),\n OneHom.toFun\n (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r) }\n⊢ ↑(aeval (Fin.cons y s)) (X 0) = ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) (X 0) ∧\n ∀ (i : Fin n),\n ↑(aeval (Fin.cons y s)) (X (Fin.succ i)) =\n ↑(AlgHom.comp (Polynomial.aeval y) (AlgHom.comp φ ↑(finSuccEquiv R n))) (X (Fin.succ i))",
"tactic": "simp only [aeval_X, Fin.cons_zero, AlgEquiv.toAlgHom_eq_coe, AlgHom.coe_comp,\n Polynomial.coe_aeval_eq_eval, Polynomial.map_C, AlgHom.coe_mk, RingHom.toFun_eq_coe,\n Polynomial.coe_mapRingHom, comp_apply, finSuccEquiv_apply, eval₂Hom_X',\n Fin.cases_zero, Polynomial.map_X, Polynomial.eval_X, Fin.cons_succ,\n Fin.cases_succ, eval_X, Polynomial.eval_C,\n RingHom.coe_mk, MonoidHom.coe_coe, AlgHom.coe_coe, implies_true, and_self,\n RingHom.toMonoidHom_eq_coe]"
},
{
"state_after": "case h.e'_3.h.e'_6\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nsrc✝ : (MvPolynomial (Fin n) R)[X] →+* R[X] := mapRingHom (eval s)\nr : R\n⊢ r = ↑(eval s) (↑(algebraMap R (MvPolynomial (Fin n) R)) r)",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nsrc✝ : (MvPolynomial (Fin n) R)[X] →+* R[X] := mapRingHom (eval s)\nr : R\n⊢ OneHom.toFun\n (↑↑{ toMonoidHom := ↑src✝, map_zero' := (_ : OneHom.toFun (↑↑src✝) 0 = 0),\n map_add' :=\n (_ :\n ∀ (x y : (MvPolynomial (Fin n) R)[X]),\n OneHom.toFun (↑↑src✝) (x + y) = OneHom.toFun (↑↑src✝) x + OneHom.toFun (↑↑src✝) y) })\n (↑(algebraMap R (MvPolynomial (Fin n) R)[X]) r) =\n ↑(algebraMap R R[X]) r",
"tactic": "convert Polynomial.map_C (eval s)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1054353\na a' a₁ a₂ : R\ne : ℕ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ns : Fin n → R\ny : R\nf : MvPolynomial (Fin (n + 1)) R\nsrc✝ : (MvPolynomial (Fin n) R)[X] →+* R[X] := mapRingHom (eval s)\nr : R\n⊢ r = ↑(eval s) (↑(algebraMap R (MvPolynomial (Fin n) R)) r)",
"tactic": "exact (eval_C _).symm"
}
] |
[
413,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.comap_bot
|
[] |
[
2806,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2805,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Eventually.mono
|
[] |
[
1132,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1130,
1
] |
Mathlib/ModelTheory/Types.lean
|
FirstOrder.Language.Theory.exists_modelType_is_realized_in
|
[
{
"state_after": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\n⊢ ∃ M, p ∈ realizedTypes T (↑M) α",
"state_before": "L : Language\nT : Theory L\nα : Type w\nM : Type w'\ninst✝² : Structure L M\ninst✝¹ : Nonempty M\ninst✝ : M ⊨ T\np : CompleteType T α\n⊢ ∃ M, p ∈ realizedTypes T (↑M) α",
"tactic": "obtain ⟨M⟩ := p.isMaximal.1"
},
{
"state_after": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\n⊢ (typeOf T fun a => ↑(Language.con L a)) = p",
"state_before": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\n⊢ ∃ M, p ∈ realizedTypes T (↑M) α",
"tactic": "refine' ⟨(M.subtheoryModel p.subset).reduct (L.lhomWithConstants α), fun a => (L.con a : M), _⟩"
},
{
"state_after": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\nφ : Sentence (L[[α]])\n⊢ (φ ∈ typeOf T fun a => ↑(Language.con L a)) ↔ φ ∈ p",
"state_before": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\n⊢ (typeOf T fun a => ↑(Language.con L a)) = p",
"tactic": "refine' SetLike.ext fun φ => _"
},
{
"state_after": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\nφ : Sentence (L[[α]])\n⊢ (Formula.Realize (↑Formula.equivSentence.symm φ) fun a => ↑(Language.con L a)) ↔ φ ∈ p",
"state_before": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\nφ : Sentence (L[[α]])\n⊢ (φ ∈ typeOf T fun a => ↑(Language.con L a)) ↔ φ ∈ p",
"tactic": "simp only [CompleteType.mem_typeOf]"
},
{
"state_after": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\nφ : Sentence (L[[α]])\n⊢ ↑(ModelType.reduct (lhomWithConstants L α)\n (ModelType.subtheoryModel M (_ : LHom.onTheory (lhomWithConstants L α) T ⊆ ↑p))) ⊨\n φ ↔\n ↑M ⊨ φ",
"state_before": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\nφ : Sentence (L[[α]])\n⊢ (Formula.Realize (↑Formula.equivSentence.symm φ) fun a => ↑(Language.con L a)) ↔ φ ∈ p",
"tactic": "refine'\n (@Formula.realize_equivSentence_symm_con _\n ((M.subtheoryModel p.subset).reduct (L.lhomWithConstants α)) _ _ M.struc _ φ).trans\n (_root_.trans (_root_.trans _ (p.isMaximal.isComplete.realize_sentence_iff φ M))\n (p.isMaximal.mem_iff_models φ).symm)"
},
{
"state_after": "no goals",
"state_before": "case intro\nL : Language\nT : Theory L\nα : Type w\nM✝ : Type w'\ninst✝² : Structure L M✝\ninst✝¹ : Nonempty M✝\ninst✝ : M✝ ⊨ T\np : CompleteType T α\nM : ModelType ↑p\nφ : Sentence (L[[α]])\n⊢ ↑(ModelType.reduct (lhomWithConstants L α)\n (ModelType.subtheoryModel M (_ : LHom.onTheory (lhomWithConstants L α) T ⊆ ↑p))) ⊨\n φ ↔\n ↑M ⊨ φ",
"tactic": "rfl"
}
] |
[
227,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.mem_disjUnion
|
[] |
[
364,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finsupp.prod_hom_add_index
|
[] |
[
415,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.coe_natAdd
|
[] |
[
1398,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1397,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.prev_reverse_eq_next'
|
[] |
[
856,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
853,
8
] |
Mathlib/Data/List/Basic.lean
|
List.mem_pmap
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.328936\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\nl : List α\nH : ∀ (a : α), a ∈ l → p a\nb : β\n⊢ b ∈ pmap f l H ↔ ∃ a h, f a (_ : p a) = b",
"tactic": "simp only [pmap_eq_map_attach, mem_map, mem_attach, true_and_iff, Subtype.exists, eq_comm]"
}
] |
[
3143,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3141,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
subset_interior_smul
|
[] |
[
1249,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1248,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Uniform.lean
|
Pmf.ofMultiset_apply_of_not_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.183679\nγ : Type ?u.183682\ns : Multiset α\nhs : s ≠ 0\na : α\nha : ¬a ∈ s\n⊢ ↑(ofMultiset s hs) a = 0",
"tactic": "simpa only [ofMultiset_apply, ENNReal.div_eq_zero_iff, Nat.cast_eq_zero, Multiset.count_eq_zero,\n ENNReal.nat_ne_top, or_false_iff] using ha"
}
] |
[
195,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.map_smul_of_tower
|
[] |
[
423,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Order/Heyting/Regular.lean
|
Heyting.IsRegular.himp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : HeytingAlgebra α\na b : α\nha : IsRegular a\nhb : IsRegular b\n⊢ IsRegular (a ⇨ b)",
"tactic": "rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]"
}
] |
[
74,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
balanced_iUnion₂
|
[] |
[
196,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.map_add_right_Ico
|
[
{
"state_after": "ι : Type ?u.218659\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addRightEmbedding c) '' Set.Ico a b = Set.Ico (a + c) (b + c)",
"state_before": "ι : Type ?u.218659\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (addRightEmbedding c) (Ico a b) = Ico (a + c) (b + c)",
"tactic": "rw [← coe_inj, coe_map, coe_Ico, coe_Ico]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.218659\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addRightEmbedding c) '' Set.Ico a b = Set.Ico (a + c) (b + c)",
"tactic": "exact Set.image_add_const_Ico _ _ _"
}
] |
[
1065,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_eq_of_eq
|
[] |
[
377,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
BumpCovering.le_one
|
[] |
[
252,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/FieldTheory/Separable.lean
|
Polynomial.Separable.isCoprime
|
[
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (↑derivative (f * g))\n⊢ IsCoprime f g",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\n⊢ IsCoprime f g",
"tactic": "have := h.of_mul_left_left"
},
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (↑derivative f * g + f * ↑derivative g)\n⊢ IsCoprime f g",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (↑derivative (f * g))\n⊢ IsCoprime f g",
"tactic": "rw [derivative_mul] at this"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\nf g : R[X]\nh : Separable (f * g)\nthis : IsCoprime f (↑derivative f * g + f * ↑derivative g)\n⊢ IsCoprime f g",
"tactic": "exact IsCoprime.of_mul_right_right (IsCoprime.of_add_mul_left_right this)"
}
] |
[
112,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.Lifts.exists_lift_of_splits
|
[] |
[
1143,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1141,
1
] |
Mathlib/Algebra/Hom/Equiv/Units/GroupWithZero.lean
|
mulLeft_bijective₀
|
[] |
[
37,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.sub_sub_sub
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c d : X ⟶ Y\n⊢ prod.lift (prod.lift a b ≫ σ) (prod.lift c d ≫ σ) ≫ σ = a - b - (c - d)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c d : X ⟶ Y\n⊢ a - c - (b - d) = a - b - (c - d)",
"tactic": "rw [sub_def, ← lift_sub_lift, sub_def, Category.assoc, σ_comp, prod.lift_map_assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c d : X ⟶ Y\n⊢ prod.lift (prod.lift a b ≫ σ) (prod.lift c d ≫ σ) ≫ σ = a - b - (c - d)",
"tactic": "rfl"
}
] |
[
375,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
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