file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/Antichain.lean
|
IsAntichain.isWeakAntichain
|
[] |
[
390,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
11
] |
Mathlib/Algebra/Order/WithZero.lean
|
le_div_iff₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nhc : c ≠ 0\n⊢ a ≤ b / c ↔ a * c ≤ b",
"tactic": "rw [div_eq_mul_inv, le_mul_inv_iff₀ hc]"
}
] |
[
249,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Order/Interval.lean
|
Interval.coe_inj
|
[] |
[
354,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
HomogeneousIdeal.homogeneousHull_toIdeal_eq_self
|
[] |
[
579,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIocDiv_eq_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp a b = toIocDiv hp 0 (b - a)",
"tactic": "rw [toIocDiv_sub_eq_toIocDiv_add, zero_add]"
}
] |
[
773,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
772,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.centroid_eq_of_inj_on_of_image_eq
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_4\nV : Type u_5\nP : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type u_3\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\np₂ : ι₂ → P\nhi₂ : ∀ (i : ι₂), i ∈ s₂ → ∀ (j : ι₂), j ∈ s₂ → p₂ i = p₂ j → i = j\nhe : p '' ↑s = p₂ '' ↑s₂\n⊢ centroid k s p = centroid k s₂ p₂",
"tactic": "classical rw [s.centroid_eq_centroid_image_of_inj_on k hi rfl,\n s₂.centroid_eq_centroid_image_of_inj_on k hi₂ he]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_4\nV : Type u_5\nP : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type u_3\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\np₂ : ι₂ → P\nhi₂ : ∀ (i : ι₂), i ∈ s₂ → ∀ (j : ι₂), j ∈ s₂ → p₂ i = p₂ j → i = j\nhe : p '' ↑s = p₂ '' ↑s₂\n⊢ centroid k s p = centroid k s₂ p₂",
"tactic": "rw [s.centroid_eq_centroid_image_of_inj_on k hi rfl,\ns₂.centroid_eq_centroid_image_of_inj_on k hi₂ he]"
}
] |
[
985,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
980,
1
] |
Mathlib/Analysis/InnerProductSpace/Dual.lean
|
InnerProductSpace.unique_continuousLinearMapOfBilin
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nB : E →L⋆[𝕜] E →L[𝕜] 𝕜\nv f : E\nis_lax_milgram : ∀ (w : E), inner f w = ↑(↑B v) w\n⊢ ∀ (v_1 : E), inner f v_1 = inner (↑B♯ v) v_1",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nB : E →L⋆[𝕜] E →L[𝕜] 𝕜\nv f : E\nis_lax_milgram : ∀ (w : E), inner f w = ↑(↑B v) w\n⊢ f = ↑B♯ v",
"tactic": "refine' ext_inner_right 𝕜 _"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nB : E →L⋆[𝕜] E →L[𝕜] 𝕜\nv f : E\nis_lax_milgram : ∀ (w : E), inner f w = ↑(↑B v) w\nw : E\n⊢ inner f w = inner (↑B♯ v) w",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nB : E →L⋆[𝕜] E →L[𝕜] 𝕜\nv f : E\nis_lax_milgram : ∀ (w : E), inner f w = ↑(↑B v) w\n⊢ ∀ (v_1 : E), inner f v_1 = inner (↑B♯ v) v_1",
"tactic": "intro w"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nB : E →L⋆[𝕜] E →L[𝕜] 𝕜\nv f : E\nis_lax_milgram : ∀ (w : E), inner f w = ↑(↑B v) w\nw : E\n⊢ inner f w = ↑(↑B v) w",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nB : E →L⋆[𝕜] E →L[𝕜] 𝕜\nv f : E\nis_lax_milgram : ∀ (w : E), inner f w = ↑(↑B v) w\nw : E\n⊢ inner f w = inner (↑B♯ v) w",
"tactic": "rw [continuousLinearMapOfBilin_apply]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nB : E →L⋆[𝕜] E →L[𝕜] 𝕜\nv f : E\nis_lax_milgram : ∀ (w : E), inner f w = ↑(↑B v) w\nw : E\n⊢ inner f w = ↑(↑B v) w",
"tactic": "exact is_lax_milgram w"
}
] |
[
188,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
SimpleGraph.cliqueSet_eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nG H : SimpleGraph α\nn : ℕ\na b c : α\ns : Finset α\n⊢ cliqueSet G n = ∅ ↔ CliqueFree G n",
"tactic": "simp_rw [CliqueFree, Set.eq_empty_iff_forall_not_mem, mem_cliqueSet_iff]"
}
] |
[
254,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.mem_span_singleton'
|
[] |
[
172,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformlyOnFilter.tendsto_at
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nh : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ (fun x_1 => (f x, F x_1 x)) ⁻¹' u ∈ p",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nh : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\n⊢ Tendsto (fun n => F n x) p (𝓝 (f x))",
"tactic": "refine' Uniform.tendsto_nhds_right.mpr fun u hu => mem_map.mpr _"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nh : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∀ (a : ι), (∀ᶠ (y : α) in p', (f y, F a y) ∈ u) → a ∈ (fun x_1 => (f x, F x_1 x)) ⁻¹' u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nh : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ (fun x_1 => (f x, F x_1 x)) ⁻¹' u ∈ p",
"tactic": "filter_upwards [(h u hu).curry]"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nh✝ : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\ni : ι\nh : ∀ᶠ (y : α) in p', (f y, F i y) ∈ u\n⊢ i ∈ (fun x_1 => (f x, F x_1 x)) ⁻¹' u",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nh : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ ∀ (a : ι), (∀ᶠ (y : α) in p', (f y, F a y) ∈ u) → a ∈ (fun x_1 => (f x, F x_1 x)) ⁻¹' u",
"tactic": "intro i h"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nh✝ : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\ni : ι\nh : ∀ᶠ (y : α) in p', (f y, F i y) ∈ u\n⊢ i ∈ (fun x_1 => (f x, F x_1 x)) ⁻¹' u",
"tactic": "simpa using h.filter_mono hx"
}
] |
[
173,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.inv_subset_closure
|
[
{
"state_after": "α : Type ?u.649\nG : Type u_1\nA : Type ?u.655\nS✝ : Type ?u.658\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ S : Set G\ns : G\nhs : s ∈ S⁻¹\n⊢ s⁻¹ ∈ closure S",
"state_before": "α : Type ?u.649\nG : Type u_1\nA : Type ?u.655\nS✝ : Type ?u.658\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ S : Set G\ns : G\nhs : s ∈ S⁻¹\n⊢ s ∈ ↑(closure S)",
"tactic": "rw [SetLike.mem_coe, ← Subgroup.inv_mem_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.649\nG : Type u_1\nA : Type ?u.655\nS✝ : Type ?u.658\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns✝ S : Set G\ns : G\nhs : s ∈ S⁻¹\n⊢ s⁻¹ ∈ closure S",
"tactic": "exact subset_closure (mem_inv.mp hs)"
}
] |
[
52,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Data/Nat/Cast/Defs.lean
|
Nat.cast_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : AddMonoidWithOne R\n⊢ ↑1 = 1",
"tactic": "rw [cast_succ, Nat.cast_zero, zero_add]"
}
] |
[
135,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformContinuous_iInf_dom
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nf : α → β\nu₁ : ι → UniformSpace α\nu₂ : UniformSpace β\ni : ι\nhf : UniformContinuous f\n⊢ Tendsto (fun x => (f x.fst, f x.snd)) (𝓤 α) (𝓤 β)",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nf : α → β\nu₁ : ι → UniformSpace α\nu₂ : UniformSpace β\ni : ι\nhf : UniformContinuous f\n⊢ UniformContinuous f",
"tactic": "delta UniformContinuous"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nf : α → β\nu₁ : ι → UniformSpace α\nu₂ : UniformSpace β\ni : ι\nhf : UniformContinuous f\n⊢ Tendsto (fun x => (f x.fst, f x.snd)) (⨅ (i : ι), 𝓤 α) (𝓤 β)",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nf : α → β\nu₁ : ι → UniformSpace α\nu₂ : UniformSpace β\ni : ι\nhf : UniformContinuous f\n⊢ Tendsto (fun x => (f x.fst, f x.snd)) (𝓤 α) (𝓤 β)",
"tactic": "rw [iInf_uniformity]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\nf : α → β\nu₁ : ι → UniformSpace α\nu₂ : UniformSpace β\ni : ι\nhf : UniformContinuous f\n⊢ Tendsto (fun x => (f x.fst, f x.snd)) (⨅ (i : ι), 𝓤 α) (𝓤 β)",
"tactic": "exact tendsto_iInf' i hf"
}
] |
[
1415,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1411,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.LeftInverse.cast_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort u_2\nγ : β → Sort v\nf : α → β\ng : β → α\nh : LeftInverse g f\nC : (a : α) → γ (f a)\na : α\n⊢ cast (_ : γ (f (g (f a))) = γ (f a)) (C (g (f a))) = C a",
"tactic": "rw [cast_eq_iff_heq, h]"
}
] |
[
1049,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1046,
1
] |
Mathlib/Order/Lattice.lean
|
Antitone.min
|
[] |
[
1185,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1182,
11
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.congr
|
[] |
[
486,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Topology/LocallyConstant/Algebra.lean
|
LocallyConstant.coe_mul
|
[] |
[
62,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/CategoryTheory/Bicategory/Free.lean
|
CategoryTheory.FreeBicategory.mk_right_unitor_inv
|
[] |
[
305,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Order/Basic.lean
|
ge_of_eq
|
[] |
[
336,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_lt_mul'
|
[] |
[
526,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.le_add_pred_of_pos
|
[
{
"state_after": "m n✝ k l n i : ℕ\nhi : i ≠ 0\n⊢ n ≤ i + n - 1",
"state_before": "m n✝ k l n i : ℕ\nhi : i ≠ 0\n⊢ n ≤ i + (n - 1)",
"tactic": "refine le_trans ?_ add_tsub_le_assoc"
},
{
"state_after": "no goals",
"state_before": "m n✝ k l n i : ℕ\nhi : i ≠ 0\n⊢ n ≤ i + n - 1",
"tactic": "simp [add_comm, Nat.add_sub_assoc, one_le_iff_ne_zero.2 hi]"
}
] |
[
327,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.lineMap_apply_one_sub
|
[
{
"state_after": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.538087\nP2 : Type ?u.538090\nV3 : Type ?u.538093\nP3 : Type ?u.538096\nV4 : Type ?u.538099\nP4 : Type ?u.538102\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\nc : k\n⊢ ↑(lineMap p₁ p₀) (↑(lineMap 1 0) (1 - c)) = ↑(lineMap p₁ p₀) c",
"state_before": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.538087\nP2 : Type ?u.538090\nV3 : Type ?u.538093\nP3 : Type ?u.538096\nV4 : Type ?u.538099\nP4 : Type ?u.538102\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\nc : k\n⊢ ↑(lineMap p₀ p₁) (1 - c) = ↑(lineMap p₁ p₀) c",
"tactic": "rw [lineMap_symm p₀, comp_apply]"
},
{
"state_after": "case h.e_6.h\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.538087\nP2 : Type ?u.538090\nV3 : Type ?u.538093\nP3 : Type ?u.538096\nV4 : Type ?u.538099\nP4 : Type ?u.538102\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\nc : k\n⊢ ↑(lineMap 1 0) (1 - c) = c",
"state_before": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.538087\nP2 : Type ?u.538090\nV3 : Type ?u.538093\nP3 : Type ?u.538096\nV4 : Type ?u.538099\nP4 : Type ?u.538102\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\nc : k\n⊢ ↑(lineMap p₁ p₀) (↑(lineMap 1 0) (1 - c)) = ↑(lineMap p₁ p₀) c",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.e_6.h\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.538087\nP2 : Type ?u.538090\nV3 : Type ?u.538093\nP3 : Type ?u.538096\nV4 : Type ?u.538099\nP4 : Type ?u.538102\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\nc : k\n⊢ ↑(lineMap 1 0) (1 - c) = c",
"tactic": "simp [lineMap_apply]"
}
] |
[
631,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/RingTheory/Localization/Submodule.lean
|
IsLocalization.isNoetherianRing
|
[
{
"state_after": "R : Type u_1\ninst✝⁷ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Type ?u.117121\ninst✝⁴ : CommRing P\ng : R →+* P\nT : Submonoid P\nhy : M ≤ Submonoid.comap g T\nQ : Type ?u.117953\ninst✝³ : CommRing Q\ninst✝² : Algebra P Q\ninst✝¹ : IsLocalization T Q\ninst✝ : IsLocalization M S\nh : WellFounded fun x x_1 => x > x_1\n⊢ WellFounded fun x x_1 => x > x_1",
"state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Type ?u.117121\ninst✝⁴ : CommRing P\ng : R →+* P\nT : Submonoid P\nhy : M ≤ Submonoid.comap g T\nQ : Type ?u.117953\ninst✝³ : CommRing Q\ninst✝² : Algebra P Q\ninst✝¹ : IsLocalization T Q\ninst✝ : IsLocalization M S\nh : IsNoetherianRing R\n⊢ IsNoetherianRing S",
"tactic": "rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h⊢"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Type ?u.117121\ninst✝⁴ : CommRing P\ng : R →+* P\nT : Submonoid P\nhy : M ≤ Submonoid.comap g T\nQ : Type ?u.117953\ninst✝³ : CommRing Q\ninst✝² : Algebra P Q\ninst✝¹ : IsLocalization T Q\ninst✝ : IsLocalization M S\nh : WellFounded fun x x_1 => x > x_1\n⊢ WellFounded fun x x_1 => x > x_1",
"tactic": "exact OrderEmbedding.wellFounded (IsLocalization.orderEmbedding M S).dual h"
}
] |
[
103,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Data/Int/Sqrt.lean
|
Int.exists_mul_self
|
[
{
"state_after": "no goals",
"state_before": "x : ℤ\nx✝ : ∃ n, n * n = x\nn : ℤ\nhn : n * n = x\n⊢ sqrt x * sqrt x = x",
"tactic": "rw [← hn, sqrt_eq, ← Int.ofNat_mul, natAbs_mul_self]"
}
] |
[
36,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/CategoryTheory/Simple.lean
|
CategoryTheory.epi_of_nonzero_to_simple
|
[
{
"state_after": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\n⊢ Epi (factorThruImage f ≫ image.ι f)",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\n⊢ Epi f",
"tactic": "rw [← image.fac f]"
},
{
"state_after": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\nthis : IsIso (image.ι f)\n⊢ Epi (factorThruImage f ≫ image.ι f)",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\n⊢ Epi (factorThruImage f ≫ image.ι f)",
"tactic": "haveI : IsIso (image.ι f) := isIso_of_mono_of_nonzero fun h => w (eq_zero_of_image_eq_zero h)"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasEqualizers C\nX Y : C\ninst✝¹ : Simple Y\nf : X ⟶ Y\ninst✝ : HasImage f\nw : f ≠ 0\nthis : IsIso (image.ι f)\n⊢ Epi (factorThruImage f ≫ image.ι f)",
"tactic": "apply epi_comp"
}
] |
[
104,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.toNat_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "a : Int\nh : 0 ≤ a\n⊢ ↑(toNat a) = a",
"tactic": "rw [toNat_eq_max, Int.max_eq_left h]"
}
] |
[
1372,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1371,
9
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
SimpleGraph.isUniform_comm
|
[] |
[
79,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Algebra/Homology/QuasiIso.lean
|
quasiIso_of_comp_right
|
[] |
[
66,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_ennreal_ne_coe_ennreal_iff
|
[] |
[
516,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
nhds_basis_closed_balanced
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.126319\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\nU : Set E\ninst✝ : RegularSpace E\ns : Set E\nhs : s ∈ 𝓝 0 ∧ IsClosed s\n⊢ ∃ i', (i' ∈ 𝓝 0 ∧ IsClosed i' ∧ Balanced 𝕜 i') ∧ id i' ⊆ id s",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.126319\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\nU : Set E\ninst✝ : RegularSpace E\n⊢ HasBasis (𝓝 0) (fun s => s ∈ 𝓝 0 ∧ IsClosed s ∧ Balanced 𝕜 s) id",
"tactic": "refine'\n (closed_nhds_basis 0).to_hasBasis (fun s hs => _) fun s hs => ⟨s, ⟨hs.1, hs.2.1⟩, rfl.subset⟩"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.126319\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\nU : Set E\ninst✝ : RegularSpace E\ns : Set E\nhs : s ∈ 𝓝 0 ∧ IsClosed s\n⊢ IsClosed (balancedCore 𝕜 s) ∧ Balanced 𝕜 (balancedCore 𝕜 s)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.126319\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\nU : Set E\ninst✝ : RegularSpace E\ns : Set E\nhs : s ∈ 𝓝 0 ∧ IsClosed s\n⊢ ∃ i', (i' ∈ 𝓝 0 ∧ IsClosed i' ∧ Balanced 𝕜 i') ∧ id i' ⊆ id s",
"tactic": "refine' ⟨balancedCore 𝕜 s, ⟨balancedCore_mem_nhds_zero hs.1, _⟩, balancedCore_subset s⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.126319\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\nU : Set E\ninst✝ : RegularSpace E\ns : Set E\nhs : s ∈ 𝓝 0 ∧ IsClosed s\n⊢ IsClosed (balancedCore 𝕜 s) ∧ Balanced 𝕜 (balancedCore 𝕜 s)",
"tactic": "exact ⟨hs.2.balancedCore, balancedCore_balanced s⟩"
}
] |
[
279,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Deprecated/Subring.lean
|
RingHom.isSubring_preimage
|
[] |
[
61,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.inf_eq_right
|
[
{
"state_after": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\n⊢ a ⊓ b ≈ b\n\ncase inr\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nh : b ≈ a\n⊢ a ⊓ b ≈ b",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nh : b ≤ a\n⊢ a ⊓ b ≈ b",
"tactic": "obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h"
},
{
"state_after": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs (↑(a ⊓ b - b) j) < ε✝",
"state_before": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\n⊢ a ⊓ b ≈ b",
"tactic": "intro _ _"
},
{
"state_after": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ abs (↑(a ⊓ b - b) j) < ε✝",
"state_before": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs (↑(a ⊓ b - b) j) < ε✝",
"tactic": "refine' ⟨i, fun j hj => _⟩"
},
{
"state_after": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ abs (↑a j ⊓ ↑b j - ↑b j) < ε✝",
"state_before": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ abs (↑(a ⊓ b - b) j) < ε✝",
"tactic": "dsimp"
},
{
"state_after": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ abs (min (↑a j - ↑b j) (↑b j - ↑b j)) < ε✝",
"state_before": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ abs (↑a j ⊓ ↑b j - ↑b j) < ε✝",
"tactic": "erw [← min_sub_sub_right]"
},
{
"state_after": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ 0 ≤ ↑a j - ↑b j",
"state_before": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ abs (min (↑a j - ↑b j) (↑b j - ↑b j)) < ε✝",
"tactic": "rwa [sub_self, min_eq_right, abs_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro.intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nε : α\nε0 : 0 < ε\ni : ℕ\nh : ∀ (j : ℕ), j ≥ i → ε ≤ ↑(a - b) j\nε✝ : α\na✝ : ε✝ > 0\nj : ℕ\nhj : j ≥ i\n⊢ 0 ≤ ↑a j - ↑b j",
"tactic": "exact ε0.le.trans (h _ hj)"
},
{
"state_after": "case inr\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nh : b ≈ a\n⊢ b ⊓ b ≈ b",
"state_before": "case inr\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nh : b ≈ a\n⊢ a ⊓ b ≈ b",
"tactic": "refine' Setoid.trans (inf_equiv_inf (Setoid.symm h) (Setoid.refl _)) _"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝ : LinearOrderedField α\na b : CauSeq α abs\nh : b ≈ a\n⊢ b ⊓ b ≈ b",
"tactic": "rw [CauSeq.inf_idem]"
}
] |
[
940,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
931,
11
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.quot_hom_ext
|
[] |
[
315,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.IsCycleOn.pow_card_apply
|
[] |
[
894,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
892,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
fderiv_csinh
|
[] |
[
551,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Comp.lean
|
fderivWithin.comp₃
|
[
{
"state_after": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_4\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g✝ : E → F\nf' f₀' f₁' g'✝ e : E →L[𝕜] F\nx : E\ns t✝ : Set E\nL L₁ L₂ : Filter E\ng' : G → G'\ng : F → G\nt : Set F\nu : Set G\nhf : DifferentiableWithinAt 𝕜 f s x\nh2g : MapsTo g t u\nh2f : MapsTo f s t\nhxs : UniqueDiffWithinAt 𝕜 s x\nhg : DifferentiableWithinAt 𝕜 g t (f x)\nhg' : DifferentiableWithinAt 𝕜 g' u (g (f x))\n⊢ fderivWithin 𝕜 (g' ∘ g ∘ f) s x =\n ContinuousLinearMap.comp (fderivWithin 𝕜 g' u (g (f x)))\n (ContinuousLinearMap.comp (fderivWithin 𝕜 g t (f x)) (fderivWithin 𝕜 f s x))",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_4\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g✝ : E → F\nf' f₀' f₁' g'✝ e : E →L[𝕜] F\nx : E\ns t✝ : Set E\nL L₁ L₂ : Filter E\ng' : G → G'\ng : F → G\nt : Set F\nu : Set G\ny : F\ny' : G\nhg' : DifferentiableWithinAt 𝕜 g' u y'\nhg : DifferentiableWithinAt 𝕜 g t y\nhf : DifferentiableWithinAt 𝕜 f s x\nh2g : MapsTo g t u\nh2f : MapsTo f s t\nh3g : g y = y'\nh3f : f x = y\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderivWithin 𝕜 (g' ∘ g ∘ f) s x =\n ContinuousLinearMap.comp (fderivWithin 𝕜 g' u y')\n (ContinuousLinearMap.comp (fderivWithin 𝕜 g t y) (fderivWithin 𝕜 f s x))",
"tactic": "substs h3g h3f"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_4\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g✝ : E → F\nf' f₀' f₁' g'✝ e : E →L[𝕜] F\nx : E\ns t✝ : Set E\nL L₁ L₂ : Filter E\ng' : G → G'\ng : F → G\nt : Set F\nu : Set G\nhf : DifferentiableWithinAt 𝕜 f s x\nh2g : MapsTo g t u\nh2f : MapsTo f s t\nhxs : UniqueDiffWithinAt 𝕜 s x\nhg : DifferentiableWithinAt 𝕜 g t (f x)\nhg' : DifferentiableWithinAt 𝕜 g' u (g (f x))\n⊢ fderivWithin 𝕜 (g' ∘ g ∘ f) s x =\n ContinuousLinearMap.comp (fderivWithin 𝕜 g' u (g (f x)))\n (ContinuousLinearMap.comp (fderivWithin 𝕜 g t (f x)) (fderivWithin 𝕜 f s x))",
"tactic": "exact (hg'.hasFDerivWithinAt.comp x (hg.hasFDerivWithinAt.comp x hf.hasFDerivWithinAt h2f) <|\n h2g.comp h2f).fderivWithin hxs"
}
] |
[
167,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
Filter.EventuallyEq.fderiv
|
[] |
[
980,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
979,
11
] |
Std/Data/Int/DivMod.lean
|
Int.ofNat_dvd
|
[
{
"state_after": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\n⊢ m ∣ n",
"state_before": "m n : Nat\n⊢ ↑m ∣ ↑n ↔ m ∣ n",
"tactic": "refine ⟨fun ⟨a, ae⟩ => ?_, fun ⟨k, e⟩ => ⟨k, by rw [e, Int.ofNat_mul]⟩⟩"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\n⊢ m ∣ n",
"tactic": "match Int.le_total a 0 with\n| .inl h =>\n have := ae.symm ▸ Int.mul_nonpos_of_nonneg_of_nonpos (ofNat_zero_le _) h\n rw [Nat.le_antisymm (ofNat_le.1 this) (Nat.zero_le _)]\n apply Nat.dvd_zero\n| .inr h => match a, eq_ofNat_of_zero_le h with\n | _, ⟨k, rfl⟩ => exact ⟨k, Int.ofNat.inj ae⟩"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nx✝ : m ∣ n\nk : Nat\ne : n = m * k\n⊢ ↑n = ↑m * ↑k",
"tactic": "rw [e, Int.ofNat_mul]"
},
{
"state_after": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\nh : a ≤ 0\nthis : ↑n ≤ 0\n⊢ m ∣ n",
"state_before": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\nh : a ≤ 0\n⊢ m ∣ n",
"tactic": "have := ae.symm ▸ Int.mul_nonpos_of_nonneg_of_nonpos (ofNat_zero_le _) h"
},
{
"state_after": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\nh : a ≤ 0\nthis : ↑n ≤ 0\n⊢ m ∣ 0",
"state_before": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\nh : a ≤ 0\nthis : ↑n ≤ 0\n⊢ m ∣ n",
"tactic": "rw [Nat.le_antisymm (ofNat_le.1 this) (Nat.zero_le _)]"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\nh : a ≤ 0\nthis : ↑n ≤ 0\n⊢ m ∣ 0",
"tactic": "apply Nat.dvd_zero"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nae : ↑n = ↑m * a\nh : 0 ≤ a\n⊢ m ∣ n",
"tactic": "match a, eq_ofNat_of_zero_le h with\n| _, ⟨k, rfl⟩ => exact ⟨k, Int.ofNat.inj ae⟩"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nx✝ : ↑m ∣ ↑n\na : Int\nk : Nat\nae : ↑n = ↑m * ↑k\nh : 0 ≤ ↑k\n⊢ m ∣ n",
"tactic": "exact ⟨k, Int.ofNat.inj ae⟩"
}
] |
[
642,
49
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
634,
14
] |
Mathlib/Algebra/AddTorsor.lean
|
eq_of_vsub_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np1 p2 : P\nh : p1 -ᵥ p2 = 0\n⊢ p1 = p2",
"tactic": "rw [← vsub_vadd p1 p2, h, zero_vadd]"
}
] |
[
133,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean
|
newton_seq_norm_le
|
[] |
[
259,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
9
] |
Mathlib/Algebra/EuclideanDomain/Defs.lean
|
EuclideanDomain.gcdB_zero_left
|
[
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\ns : R\n⊢ (xgcd 0 s).snd = 1",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\ns : R\n⊢ gcdB 0 s = 1",
"tactic": "unfold gcdB"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\ns : R\n⊢ (xgcd 0 s).snd = 1",
"tactic": "rw [xgcd, xgcd_zero_left]"
}
] |
[
275,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.zero_nadd
|
[
{
"state_after": "no goals",
"state_before": "a b c : Ordinal\n⊢ 0 ♯ a = a",
"tactic": "rw [nadd_comm, nadd_zero]"
}
] |
[
294,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.units_coeff_zero_smul
|
[
{
"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]\nc : R[X]ˣ\np : R[X]\n⊢ coeff (↑c) 0 • p = ↑c * p",
"tactic": "rw [← Polynomial.C_mul', ← Polynomial.eq_C_of_degree_eq_zero (degree_coe_units c)]"
}
] |
[
845,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
844,
1
] |
Mathlib/Topology/Order.lean
|
nhds_sInf
|
[] |
[
666,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.toQuadraticForm_associated
|
[] |
[
814,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Algebra/GCDMonoid/Multiset.lean
|
Multiset.dvd_lcm
|
[] |
[
72,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Std/Data/Int/DivMod.lean
|
Int.emod_eq_of_lt
|
[] |
[
358,
87
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
355,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
norm_mk_lt
|
[] |
[
198,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.monotone_comap
|
[] |
[
312,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.nat_iff
|
[] |
[
237,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.neg_one_le_cos
|
[] |
[
1279,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1278,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_subtype_of_mem
|
[
{
"state_after": "ι : Type ?u.380694\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → β\np : α → Prop\ninst✝ : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\n⊢ ∀ (x : α), x ∈ s → p x",
"state_before": "ι : Type ?u.380694\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → β\np : α → Prop\ninst✝ : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\n⊢ ∏ x in Finset.subtype p s, f ↑x = ∏ x in s, f x",
"tactic": "rw [prod_subtype_eq_prod_filter, filter_true_of_mem]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.380694\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → β\np : α → Prop\ninst✝ : DecidablePred p\nh : ∀ (x : α), x ∈ s → p x\n⊢ ∀ (x : α), x ∈ s → p x",
"tactic": "simpa using h"
}
] |
[
881,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
878,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.isLocalization_iff_of_algEquiv
|
[] |
[
769,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.sum_mul_sum
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns s₁✝ s₂✝ : Finset α\na : α\nb : β\nf g : α → β\ninst✝ : NonUnitalNonAssocSemiring β\nι₁ : Type u_1\nι₂ : Type u_2\ns₁ : Finset ι₁\ns₂ : Finset ι₂\nf₁ : ι₁ → β\nf₂ : ι₂ → β\n⊢ ∀ (x : ι₁), x ∈ s₁ → f₁ x * ∑ x₂ in s₂, f₂ x₂ = ∑ y in s₂, f₁ (x, y).fst * f₂ (x, y).snd",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns s₁✝ s₂✝ : Finset α\na : α\nb : β\nf g : α → β\ninst✝ : NonUnitalNonAssocSemiring β\nι₁ : Type u_1\nι₂ : Type u_2\ns₁ : Finset ι₁\ns₂ : Finset ι₂\nf₁ : ι₁ → β\nf₂ : ι₂ → β\n⊢ (∑ x₁ in s₁, f₁ x₁) * ∑ x₂ in s₂, f₂ x₂ = ∑ p in s₁ ×ˢ s₂, f₁ p.fst * f₂ p.snd",
"tactic": "rw [sum_product, sum_mul, sum_congr rfl]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns s₁✝ s₂✝ : Finset α\na : α\nb : β\nf g : α → β\ninst✝ : NonUnitalNonAssocSemiring β\nι₁ : Type u_1\nι₂ : Type u_2\ns₁ : Finset ι₁\ns₂ : Finset ι₂\nf₁ : ι₁ → β\nf₂ : ι₂ → β\nx✝ : ι₁\na✝ : x✝ ∈ s₁\n⊢ f₁ x✝ * ∑ x₂ in s₂, f₂ x₂ = ∑ y in s₂, f₁ (x✝, y).fst * f₂ (x✝, y).snd",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns s₁✝ s₂✝ : Finset α\na : α\nb : β\nf g : α → β\ninst✝ : NonUnitalNonAssocSemiring β\nι₁ : Type u_1\nι₂ : Type u_2\ns₁ : Finset ι₁\ns₂ : Finset ι₂\nf₁ : ι₁ → β\nf₂ : ι₂ → β\n⊢ ∀ (x : ι₁), x ∈ s₁ → f₁ x * ∑ x₂ in s₂, f₂ x₂ = ∑ y in s₂, f₁ (x, y).fst * f₂ (x, y).snd",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns s₁✝ s₂✝ : Finset α\na : α\nb : β\nf g : α → β\ninst✝ : NonUnitalNonAssocSemiring β\nι₁ : Type u_1\nι₂ : Type u_2\ns₁ : Finset ι₁\ns₂ : Finset ι₂\nf₁ : ι₁ → β\nf₂ : ι₂ → β\nx✝ : ι₁\na✝ : x✝ ∈ s₁\n⊢ f₁ x✝ * ∑ x₂ in s₂, f₂ x₂ = ∑ y in s₂, f₁ (x✝, y).fst * f₂ (x✝, y).snd",
"tactic": "rw [mul_sum]"
}
] |
[
67,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/List/Basic.lean
|
List.drop_left'
|
[
{
"state_after": "ι : Type ?u.202919\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ l₁ l₂ : List α\nn : ℕ\nh : length l₁ = n\n⊢ drop (length l₁) (l₁ ++ l₂) = l₂",
"state_before": "ι : Type ?u.202919\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ l₁ l₂ : List α\nn : ℕ\nh : length l₁ = n\n⊢ drop n (l₁ ++ l₂) = l₂",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.202919\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ l₁ l₂ : List α\nn : ℕ\nh : length l₁ = n\n⊢ drop (length l₁) (l₁ ++ l₂) = l₂",
"tactic": "apply drop_left"
}
] |
[
2152,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2151,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
isSubmonoid_iUnion_of_directed
|
[] |
[
121,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.coe_mul
|
[
{
"state_after": "no goals",
"state_before": "q : ℕ+\nn m : ℤ\n⊢ ↑(n * m) = ↑n * ↑m",
"tactic": "ext <;> simp"
}
] |
[
334,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.lift.mk
|
[] |
[
729,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
PerfectClosure.eq_iff'
|
[
{
"state_after": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\n⊢ mk K p x = mk K p y → ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd\n\ncase mpr\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\n⊢ (∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd) → mk K p x = mk K p y",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\n⊢ mk K p x = mk K p y ↔ ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd",
"tactic": "constructor"
},
{
"state_after": "case mpr\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\nH : ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd\n⊢ mk K p x = mk K p y",
"state_before": "case mpr\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\n⊢ (∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd) → mk K p x = mk K p y",
"tactic": "intro H"
},
{
"state_after": "case mpr.mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\ny : ℕ × K\nm : ℕ\nx : K\nH : ∃ z, (↑(frobenius K p)^[y.fst + z]) (m, x).snd = (↑(frobenius K p)^[(m, x).fst + z]) y.snd\n⊢ mk K p (m, x) = mk K p y",
"state_before": "case mpr\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\nH : ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd\n⊢ mk K p x = mk K p y",
"tactic": "cases' x with m x"
},
{
"state_after": "case mpr.mk.mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nH : ∃ z, (↑(frobenius K p)^[(n, y).fst + z]) (m, x).snd = (↑(frobenius K p)^[(m, x).fst + z]) (n, y).snd\n⊢ mk K p (m, x) = mk K p (n, y)",
"state_before": "case mpr.mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\ny : ℕ × K\nm : ℕ\nx : K\nH : ∃ z, (↑(frobenius K p)^[y.fst + z]) (m, x).snd = (↑(frobenius K p)^[(m, x).fst + z]) y.snd\n⊢ mk K p (m, x) = mk K p y",
"tactic": "cases' y with n y"
},
{
"state_after": "case mpr.mk.mk.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nz : ℕ\nH : (↑(frobenius K p)^[(n, y).fst + z]) (m, x).snd = (↑(frobenius K p)^[(m, x).fst + z]) (n, y).snd\n⊢ mk K p (m, x) = mk K p (n, y)",
"state_before": "case mpr.mk.mk\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nH : ∃ z, (↑(frobenius K p)^[(n, y).fst + z]) (m, x).snd = (↑(frobenius K p)^[(m, x).fst + z]) (n, y).snd\n⊢ mk K p (m, x) = mk K p (n, y)",
"tactic": "cases' H with z H"
},
{
"state_after": "case mpr.mk.mk.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nz : ℕ\nH : (↑(frobenius K p)^[n + z]) x = (↑(frobenius K p)^[m + z]) y\n⊢ mk K p (m, x) = mk K p (n, y)",
"state_before": "case mpr.mk.mk.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nz : ℕ\nH : (↑(frobenius K p)^[(n, y).fst + z]) (m, x).snd = (↑(frobenius K p)^[(m, x).fst + z]) (n, y).snd\n⊢ mk K p (m, x) = mk K p (n, y)",
"tactic": "dsimp only at H"
},
{
"state_after": "case mpr.mk.mk.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nz : ℕ\nH : (↑(frobenius K p)^[n + z]) x = (↑(frobenius K p)^[m + z]) y\n⊢ mk K p (n + z + m, (↑(frobenius K p)^[m + z]) y) = mk K p (m + z + n, (↑(frobenius K p)^[m + z]) y)",
"state_before": "case mpr.mk.mk.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nz : ℕ\nH : (↑(frobenius K p)^[n + z]) x = (↑(frobenius K p)^[m + z]) y\n⊢ mk K p (m, x) = mk K p (n, y)",
"tactic": "rw [R.sound K p (n + z) m x _ rfl, R.sound K p (m + z) n y _ rfl, H]"
},
{
"state_after": "no goals",
"state_before": "case mpr.mk.mk.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nm : ℕ\nx : K\nn : ℕ\ny : K\nz : ℕ\nH : (↑(frobenius K p)^[n + z]) x = (↑(frobenius K p)^[m + z]) y\n⊢ mk K p (n + z + m, (↑(frobenius K p)^[m + z]) y) = mk K p (m + z + n, (↑(frobenius K p)^[m + z]) y)",
"tactic": "rw [add_assoc, add_comm, add_comm z]"
},
{
"state_after": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\nH : mk K p x = mk K p y\n⊢ ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd",
"state_before": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\n⊢ mk K p x = mk K p y → ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd",
"tactic": "intro H"
},
{
"state_after": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\nH : EqvGen (R K p) x y\n⊢ ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd",
"state_before": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\nH : mk K p x = mk K p y\n⊢ ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd",
"tactic": "replace H := Quot.exact _ H"
},
{
"state_after": "case mp.rel\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : R K p x✝ y✝\n⊢ ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n\ncase mp.refl\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ : ℕ × K\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) x✝.snd\n\ncase mp.symm\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : EqvGen (R K p) x✝ y✝\na_ih✝ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) x✝.snd\n\ncase mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"state_before": "case mp\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y : ℕ × K\nH : EqvGen (R K p) x y\n⊢ ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd",
"tactic": "induction H"
},
{
"state_after": "case mp.refl\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ : ℕ × K\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) x✝.snd\n\ncase mp.symm\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : EqvGen (R K p) x✝ y✝\na_ih✝ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) x✝.snd\n\ncase mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"state_before": "case mp.rel\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : R K p x✝ y✝\n⊢ ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n\ncase mp.refl\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ : ℕ × K\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) x✝.snd\n\ncase mp.symm\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : EqvGen (R K p) x✝ y✝\na_ih✝ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) x✝.snd\n\ncase mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"tactic": "case rel x y H => cases' H with n x; exact ⟨0, rfl⟩"
},
{
"state_after": "case mp.symm\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : EqvGen (R K p) x✝ y✝\na_ih✝ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) x✝.snd\n\ncase mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"state_before": "case mp.refl\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ : ℕ × K\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) x✝.snd\n\ncase mp.symm\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : EqvGen (R K p) x✝ y✝\na_ih✝ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) x✝.snd\n\ncase mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"tactic": "case refl H => exact ⟨0, rfl⟩"
},
{
"state_after": "case mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"state_before": "case mp.symm\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ : ℕ × K\na✝ : EqvGen (R K p) x✝ y✝\na_ih✝ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[x✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) x✝.snd\n\ncase mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"tactic": "case symm x y H ih => cases' ih with w ih; exact ⟨w, ih.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.trans\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y x✝ y✝ z✝ : ℕ × K\na✝¹ : EqvGen (R K p) x✝ y✝\na✝ : EqvGen (R K p) y✝ z✝\na_ih✝¹ : ∃ z, (↑(frobenius K p)^[y✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) y✝.snd\na_ih✝ : ∃ z, (↑(frobenius K p)^[z✝.fst + z]) y✝.snd = (↑(frobenius K p)^[y✝.fst + z]) z✝.snd\n⊢ ∃ z, (↑(frobenius K p)^[z✝.fst + z]) x✝.snd = (↑(frobenius K p)^[x✝.fst + z]) z✝.snd",
"tactic": "case trans x y z H1 H2 ih1 ih2 =>\n cases' ih1 with z1 ih1\n cases' ih2 with z2 ih2\n exists z2 + (y.1 + z1)\n rw [← add_assoc, iterate_add_apply, ih1]\n rw [← iterate_add_apply, add_comm, iterate_add_apply, ih2]\n rw [← iterate_add_apply]\n simp only [add_comm, add_left_comm]"
},
{
"state_after": "case intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y : ℕ × K\nn : ℕ\nx : K\n⊢ ∃ z,\n (↑(frobenius K p)^[(n + 1, ↑(frobenius K p) x).fst + z]) (n, x).snd =\n (↑(frobenius K p)^[(n, x).fst + z]) (n + 1, ↑(frobenius K p) x).snd",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y : ℕ × K\nH : R K p x y\n⊢ ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd",
"tactic": "cases' H with n x"
},
{
"state_after": "no goals",
"state_before": "case intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y : ℕ × K\nn : ℕ\nx : K\n⊢ ∃ z,\n (↑(frobenius K p)^[(n + 1, ↑(frobenius K p) x).fst + z]) (n, x).snd =\n (↑(frobenius K p)^[(n, x).fst + z]) (n + 1, ↑(frobenius K p) x).snd",
"tactic": "exact ⟨0, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx y H : ℕ × K\n⊢ ∃ z, (↑(frobenius K p)^[H.fst + z]) H.snd = (↑(frobenius K p)^[H.fst + z]) H.snd",
"tactic": "exact ⟨0, rfl⟩"
},
{
"state_after": "case intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y : ℕ × K\nH : EqvGen (R K p) x y\nw : ℕ\nih : (↑(frobenius K p)^[y.fst + w]) x.snd = (↑(frobenius K p)^[x.fst + w]) y.snd\n⊢ ∃ z, (↑(frobenius K p)^[x.fst + z]) y.snd = (↑(frobenius K p)^[y.fst + z]) x.snd",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y : ℕ × K\nH : EqvGen (R K p) x y\nih : ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd\n⊢ ∃ z, (↑(frobenius K p)^[x.fst + z]) y.snd = (↑(frobenius K p)^[y.fst + z]) x.snd",
"tactic": "cases' ih with w ih"
},
{
"state_after": "no goals",
"state_before": "case intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y : ℕ × K\nH : EqvGen (R K p) x y\nw : ℕ\nih : (↑(frobenius K p)^[y.fst + w]) x.snd = (↑(frobenius K p)^[x.fst + w]) y.snd\n⊢ ∃ z, (↑(frobenius K p)^[x.fst + z]) y.snd = (↑(frobenius K p)^[y.fst + z]) x.snd",
"tactic": "exact ⟨w, ih.symm⟩"
},
{
"state_after": "case intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nih2 : ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) y.snd = (↑(frobenius K p)^[y.fst + z_1]) z.snd\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\n⊢ ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) x.snd = (↑(frobenius K p)^[x.fst + z_1]) z.snd",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nih1 : ∃ z, (↑(frobenius K p)^[y.fst + z]) x.snd = (↑(frobenius K p)^[x.fst + z]) y.snd\nih2 : ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) y.snd = (↑(frobenius K p)^[y.fst + z_1]) z.snd\n⊢ ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) x.snd = (↑(frobenius K p)^[x.fst + z_1]) z.snd",
"tactic": "cases' ih1 with z1 ih1"
},
{
"state_after": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) x.snd = (↑(frobenius K p)^[x.fst + z_1]) z.snd",
"state_before": "case intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nih2 : ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) y.snd = (↑(frobenius K p)^[y.fst + z_1]) z.snd\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\n⊢ ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) x.snd = (↑(frobenius K p)^[x.fst + z_1]) z.snd",
"tactic": "cases' ih2 with z2 ih2"
},
{
"state_after": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[z.fst + (z2 + (y.fst + z1))]) x.snd = (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"state_before": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ ∃ z_1, (↑(frobenius K p)^[z.fst + z_1]) x.snd = (↑(frobenius K p)^[x.fst + z_1]) z.snd",
"tactic": "exists z2 + (y.1 + z1)"
},
{
"state_after": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[z.fst + z2]) ((↑(frobenius K p)^[x.fst + z1]) y.snd) =\n (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"state_before": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[z.fst + (z2 + (y.fst + z1))]) x.snd = (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"tactic": "rw [← add_assoc, iterate_add_apply, ih1]"
},
{
"state_after": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[x.fst + z1]) ((↑(frobenius K p)^[y.fst + z2]) z.snd) =\n (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"state_before": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[z.fst + z2]) ((↑(frobenius K p)^[x.fst + z1]) y.snd) =\n (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"tactic": "rw [← iterate_add_apply, add_comm, iterate_add_apply, ih2]"
},
{
"state_after": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[x.fst + z1 + (y.fst + z2)]) z.snd = (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"state_before": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[x.fst + z1]) ((↑(frobenius K p)^[y.fst + z2]) z.snd) =\n (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"tactic": "rw [← iterate_add_apply]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx✝ y✝ x y z : ℕ × K\nH1 : EqvGen (R K p) x y\nH2 : EqvGen (R K p) y z\nz1 : ℕ\nih1 : (↑(frobenius K p)^[y.fst + z1]) x.snd = (↑(frobenius K p)^[x.fst + z1]) y.snd\nz2 : ℕ\nih2 : (↑(frobenius K p)^[z.fst + z2]) y.snd = (↑(frobenius K p)^[y.fst + z2]) z.snd\n⊢ (↑(frobenius K p)^[x.fst + z1 + (y.fst + z2)]) z.snd = (↑(frobenius K p)^[x.fst + (z2 + (y.fst + z1))]) z.snd",
"tactic": "simp only [add_comm, add_left_comm]"
}
] |
[
416,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
Finsupp.sum_inner
|
[
{
"state_after": "case h.e'_3\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.1418390\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\nl : ι →₀ 𝕜\nv : ι → E\nx : E\n⊢ (sum l fun i a => ↑(starRingEnd 𝕜) a • inner (v i) x) = ∑ i in l.support, inner (↑l i • v i) x",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.1418390\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\nl : ι →₀ 𝕜\nv : ι → E\nx : E\n⊢ inner (sum l fun i a => a • v i) x = sum l fun i a => ↑(starRingEnd 𝕜) a • inner (v i) x",
"tactic": "convert _root_.sum_inner (𝕜 := 𝕜) l.support (fun a => l a • v a) x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\n𝕜 : Type u_2\nE : Type u_3\nF : Type ?u.1418390\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_1\nl : ι →₀ 𝕜\nv : ι → E\nx : E\n⊢ (sum l fun i a => ↑(starRingEnd 𝕜) a • inner (v i) x) = ∑ i in l.support, inner (↑l i • v i) x",
"tactic": "simp only [inner_smul_left, Finsupp.sum, smul_eq_mul]"
}
] |
[
537,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/CategoryTheory/Functor/FullyFaithful.lean
|
CategoryTheory.natIsoOfCompFullyFaithful_inv
|
[
{
"state_after": "case w.h\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nE : Type u_2\ninst✝² : Category E\nF G : C ⥤ D\nH : D ⥤ E\ninst✝¹ : Full H\ninst✝ : Faithful H\ni : F ⋙ H ≅ G ⋙ H\nx✝ : C\n⊢ (natIsoOfCompFullyFaithful H i).inv.app x✝ = (natTransOfCompFullyFaithful H i.inv).app x✝",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nE : Type u_2\ninst✝² : Category E\nF G : C ⥤ D\nH : D ⥤ E\ninst✝¹ : Full H\ninst✝ : Faithful H\ni : F ⋙ H ≅ G ⋙ H\n⊢ (natIsoOfCompFullyFaithful H i).inv = natTransOfCompFullyFaithful H i.inv",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nE : Type u_2\ninst✝² : Category E\nF G : C ⥤ D\nH : D ⥤ E\ninst✝¹ : Full H\ninst✝ : Faithful H\ni : F ⋙ H ≅ G ⋙ H\nx✝ : C\n⊢ (natIsoOfCompFullyFaithful H i).inv.app x✝ = (natTransOfCompFullyFaithful H i.inv).app x✝",
"tactic": "simp [← preimage_comp]"
}
] |
[
235,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Reachable.rfl
|
[] |
[
1872,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1872,
11
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
ciSup_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.58776\nγ : Type ?u.58779\nι : Sort u_1\ninst✝ : ConditionallyCompleteLattice α\ns t : Set α\na✝ b : α\nhι : Nonempty ι\na : α\n⊢ (⨆ (x : ι), a) = a",
"tactic": "rw [iSup, range_const, csSup_singleton]"
}
] |
[
832,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
831,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.induction_on_max_value
|
[
{
"state_after": "case a\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\n⊢ p s",
"state_before": "F : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\ns : Finset ι\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\n⊢ p s",
"tactic": "induction' s using Finset.strongInductionOn with s ihs"
},
{
"state_after": "case a.inl\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : image f s = ∅\n⊢ p s\n\ncase a.inr\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\n⊢ p s",
"state_before": "case a\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\n⊢ p s",
"tactic": "rcases(s.image f).eq_empty_or_nonempty with (hne | hne)"
},
{
"state_after": "case a.inl\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : s = ∅\n⊢ p s",
"state_before": "case a.inl\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : image f s = ∅\n⊢ p s",
"tactic": "simp only [image_eq_empty] at hne"
},
{
"state_after": "no goals",
"state_before": "case a.inl\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : s = ∅\n⊢ p s",
"tactic": "simp only [hne, h0]"
},
{
"state_after": "case a.inr\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\nH : max' (image f s) hne ∈ image f s\n⊢ p s",
"state_before": "case a.inr\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\n⊢ p s",
"tactic": "have H : (s.image f).max' hne ∈ s.image f := max'_mem (s.image f) hne"
},
{
"state_after": "case a.inr\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\nH : ∃ a, a ∈ s ∧ f a = max' (image f s) hne\n⊢ p s",
"state_before": "case a.inr\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\nH : max' (image f s) hne ∈ image f s\n⊢ p s",
"tactic": "simp only [mem_image, exists_prop] at H"
},
{
"state_after": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\n⊢ p s",
"state_before": "case a.inr\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\nH : ∃ a, a ∈ s ∧ f a = max' (image f s) hne\n⊢ p s",
"tactic": "rcases H with ⟨a, has, hfa⟩"
},
{
"state_after": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\n⊢ p (insert a (erase s a))",
"state_before": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\n⊢ p s",
"tactic": "rw [← insert_erase has]"
},
{
"state_after": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\nx : ι\nhx : x ∈ erase s a\n⊢ f x ≤ f a",
"state_before": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\n⊢ p (insert a (erase s a))",
"tactic": "refine' step _ _ (not_mem_erase a s) (fun x hx => _) (ihs _ <| erase_ssubset has)"
},
{
"state_after": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\nx : ι\nhx : x ∈ erase s a\n⊢ f x ≤ max' (image f s) hne",
"state_before": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\nx : ι\nhx : x ∈ erase s a\n⊢ f x ≤ f a",
"tactic": "rw [hfa]"
},
{
"state_after": "no goals",
"state_before": "case a.inr.intro.intro\nF : Type ?u.399951\nα : Type u_2\nβ : Type ?u.399957\nγ : Type ?u.399960\nι : Type u_1\nκ : Type ?u.399966\ninst✝² : LinearOrder α\ninst✝¹ : LinearOrder β\ninst✝ : DecidableEq ι\nf : ι → α\np : Finset ι → Prop\nh0 : p ∅\nstep : ∀ (a : ι) (s : Finset ι), ¬a ∈ s → (∀ (x : ι), x ∈ s → f x ≤ f a) → p s → p (insert a s)\ns : Finset ι\nihs : ∀ (t : Finset ι), t ⊂ s → p t\nhne : Finset.Nonempty (image f s)\na : ι\nhas : a ∈ s\nhfa : f a = max' (image f s) hne\nx : ι\nhx : x ∈ erase s a\n⊢ f x ≤ max' (image f s) hne",
"tactic": "exact le_max' _ _ (mem_image_of_mem _ <| mem_of_mem_erase hx)"
}
] |
[
1673,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1661,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
|
Matrix.det_eq_sign_charpoly_coeff
|
[
{
"state_after": "R : Type u\ninst✝³ : CommRing R\nn G : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nα β : Type v\ninst✝ : DecidableEq α\nM✝ M : Matrix n n R\n⊢ det M = det (-1 • eval (↑(scalar n) 0) (X - ↑C M))",
"state_before": "R : Type u\ninst✝³ : CommRing R\nn G : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nα β : Type v\ninst✝ : DecidableEq α\nM✝ M : Matrix n n R\n⊢ det M = (-1) ^ Fintype.card n * coeff (charpoly M) 0",
"tactic": "rw [coeff_zero_eq_eval_zero, charpoly, eval_det, matPolyEquiv_charmatrix, ← det_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝³ : CommRing R\nn G : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nα β : Type v\ninst✝ : DecidableEq α\nM✝ M : Matrix n n R\n⊢ det M = det (-1 • eval (↑(scalar n) 0) (X - ↑C M))",
"tactic": "simp"
}
] |
[
198,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.take_inter
|
[
{
"state_after": "α✝ : Type uu\nβ : Type vv\nl₁ l₂ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nxs ys : List α\nn : ℕ\nh : xs ~ ys\nh' : Nodup ys\n⊢ take n xs ~ List.filter (fun x => decide (x ∈ take n xs)) ys",
"state_before": "α✝ : Type uu\nβ : Type vv\nl₁ l₂ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nxs ys : List α\nn : ℕ\nh : xs ~ ys\nh' : Nodup ys\n⊢ take n xs ~ List.inter ys (take n xs)",
"tactic": "simp only [List.inter]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type uu\nβ : Type vv\nl₁ l₂ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nxs ys : List α\nn : ℕ\nh : xs ~ ys\nh' : Nodup ys\n⊢ take n xs ~ List.filter (fun x => decide (x ∈ take n xs)) ys",
"tactic": "exact Perm.trans (show xs.take n ~ xs.filter (. ∈ xs.take n) by\n conv_lhs => rw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')])\n (Perm.filter _ h)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type uu\nβ : Type vv\nl₁ l₂ : List α✝\nα : Type u_1\ninst✝ : DecidableEq α\nxs ys : List α\nn : ℕ\nh : xs ~ ys\nh' : Nodup ys\n⊢ take n xs ~ List.filter (fun x => decide (x ∈ take n xs)) xs",
"tactic": "conv_lhs => rw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')]"
}
] |
[
1163,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1158,
1
] |
Mathlib/MeasureTheory/Group/MeasurableEquiv.lean
|
MeasurableEquiv.coe_smul₀
|
[] |
[
79,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.coeff_X_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\n⊢ coeff (X * p) (n + 1) = coeff p n",
"tactic": "rw [(commute_X p).eq, coeff_mul_X]"
}
] |
[
270,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix_transpose_apply'
|
[] |
[
599,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
597,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.IsImage.symm_eqOn_of_inter_eq_of_eqOn
|
[] |
[
582,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
eventually_norm_pow_le
|
[] |
[
387,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/GroupTheory/Subgroup/Saturated.lean
|
Subgroup.saturated_iff_zpow
|
[
{
"state_after": "case mp\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ Saturated H → ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H\n\ncase mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H) → Saturated H",
"state_before": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ Saturated H ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H",
"tactic": "constructor"
},
{
"state_after": "case mp\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn : ℤ\ng : G\nhgn : g ^ n ∈ H\n⊢ n = 0 ∨ g ∈ H",
"state_before": "case mp\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ Saturated H → ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H",
"tactic": "intros hH n g hgn"
},
{
"state_after": "case mp.ofNat\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.ofNat n ∈ H\n⊢ Int.ofNat n = 0 ∨ g ∈ H\n\ncase mp.negSucc\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.negSucc n ∈ H\n⊢ Int.negSucc n = 0 ∨ g ∈ H",
"state_before": "case mp\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn : ℤ\ng : G\nhgn : g ^ n ∈ H\n⊢ n = 0 ∨ g ∈ H",
"tactic": "induction' n with n n"
},
{
"state_after": "case mp.ofNat\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ n ∈ H\n⊢ n = 0 ∨ g ∈ H",
"state_before": "case mp.ofNat\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.ofNat n ∈ H\n⊢ Int.ofNat n = 0 ∨ g ∈ H",
"tactic": "simp only [Int.coe_nat_eq_zero, Int.ofNat_eq_coe, zpow_ofNat] at hgn⊢"
},
{
"state_after": "no goals",
"state_before": "case mp.ofNat\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ n ∈ H\n⊢ n = 0 ∨ g ∈ H",
"tactic": "exact hH hgn"
},
{
"state_after": "case mp.negSucc\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.negSucc n ∈ H\n⊢ g ^ (n + 1) ∈ H",
"state_before": "case mp.negSucc\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.negSucc n ∈ H\n⊢ Int.negSucc n = 0 ∨ g ∈ H",
"tactic": "suffices g ^ (n + 1) ∈ H by\n refine' (hH this).imp _ id\n simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case mp.negSucc\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.negSucc n ∈ H\n⊢ g ^ (n + 1) ∈ H",
"tactic": "simpa only [inv_mem_iff, zpow_negSucc] using hgn"
},
{
"state_after": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.negSucc n ∈ H\nthis : g ^ (n + 1) ∈ H\n⊢ n + 1 = 0 → Int.negSucc n = 0",
"state_before": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.negSucc n ∈ H\nthis : g ^ (n + 1) ∈ H\n⊢ Int.negSucc n = 0 ∨ g ∈ H",
"tactic": "refine' (hH this).imp _ id"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : Saturated H\nn✝ : ℤ\ng : G\nhgn✝ : g ^ n✝ ∈ H\nn : ℕ\nhgn : g ^ Int.negSucc n ∈ H\nthis : g ^ (n + 1) ∈ H\n⊢ n + 1 = 0 → Int.negSucc n = 0",
"tactic": "simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]"
},
{
"state_after": "case mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H\nn : ℕ\ng : G\nhgn : g ^ n ∈ H\n⊢ n = 0 ∨ g ∈ H",
"state_before": "case mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H) → Saturated H",
"tactic": "intro h n g hgn"
},
{
"state_after": "case mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn : ℕ\ng : G\nhgn : g ^ n ∈ H\nh : g ^ ↑n ∈ H → ↑n = 0 ∨ g ∈ H\n⊢ n = 0 ∨ g ∈ H",
"state_before": "case mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H\nn : ℕ\ng : G\nhgn : g ^ n ∈ H\n⊢ n = 0 ∨ g ∈ H",
"tactic": "specialize h n g"
},
{
"state_after": "case mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn : ℕ\ng : G\nhgn : g ^ n ∈ H\nh : g ^ n ∈ H → n = 0 ∨ g ∈ H\n⊢ n = 0 ∨ g ∈ H",
"state_before": "case mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn : ℕ\ng : G\nhgn : g ^ n ∈ H\nh : g ^ ↑n ∈ H → ↑n = 0 ∨ g ∈ H\n⊢ n = 0 ∨ g ∈ H",
"tactic": "simp only [Int.coe_nat_eq_zero, zpow_ofNat] at h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn : ℕ\ng : G\nhgn : g ^ n ∈ H\nh : g ^ n ∈ H → n = 0 ∨ g ∈ H\n⊢ n = 0 ∨ g ∈ H",
"tactic": "apply h hgn"
}
] |
[
58,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
IsCoprime.of_mul_add_right_right
|
[
{
"state_after": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + z * x)\n⊢ IsCoprime x y",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (z * x + y)\n⊢ IsCoprime x y",
"tactic": "rw [add_comm] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + z * x)\n⊢ IsCoprime x y",
"tactic": "exact h.of_add_mul_right_right"
}
] |
[
221,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.piecewise_insert
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ (fun i => if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j)",
"state_before": "α : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ piecewise (insert j s) f g = update (piecewise s f g) j (f j)",
"tactic": "simp [piecewise]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\n⊢ (if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j) i",
"state_before": "α : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ (fun i => if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j)",
"tactic": "ext i"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\nh : i = j\n⊢ (if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j) i\n\ncase neg\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\nh : ¬i = j\n⊢ (if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j) i",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\n⊢ (if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j) i",
"tactic": "by_cases h : i = j"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\nh : i = j\n⊢ (if j = j ∨ j ∈ s then f j else g j) = update (fun i => if i ∈ s then f i else g i) j (f j) j",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\nh : i = j\n⊢ (if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j) i",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\nh : i = j\n⊢ (if j = j ∨ j ∈ s then f j else g j) = update (fun i => if i ∈ s then f i else g i) j (f j) j",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.73354\nγ : Type ?u.73357\nι : Sort ?u.73360\nπ : α → Type ?u.73365\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\ni : α\nh : ¬i = j\n⊢ (if i = j ∨ i ∈ s then f i else g i) = update (fun i => if i ∈ s then f i else g i) j (f j) i",
"tactic": "by_cases h' : i ∈ s <;> simp [h, h']"
}
] |
[
1380,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1373,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
DifferentiableAt.smul
|
[] |
[
205,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Bounded.weak
|
[] |
[
927,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
926,
1
] |
Mathlib/Data/MvPolynomial/Comap.lean
|
MvPolynomial.comap_comp
|
[
{
"state_after": "case h\nσ : Type u_2\nτ : Type u_3\nυ : Type u_4\nR : Type u_1\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial τ R\ng : MvPolynomial τ R →ₐ[R] MvPolynomial υ R\nx : υ → R\n⊢ comap (AlgHom.comp g f) x = (comap f ∘ comap g) x",
"state_before": "σ : Type u_2\nτ : Type u_3\nυ : Type u_4\nR : Type u_1\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial τ R\ng : MvPolynomial τ R →ₐ[R] MvPolynomial υ R\n⊢ comap (AlgHom.comp g f) = comap f ∘ comap g",
"tactic": "funext x"
},
{
"state_after": "no goals",
"state_before": "case h\nσ : Type u_2\nτ : Type u_3\nυ : Type u_4\nR : Type u_1\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial τ R\ng : MvPolynomial τ R →ₐ[R] MvPolynomial υ R\nx : υ → R\n⊢ comap (AlgHom.comp g f) x = (comap f ∘ comap g) x",
"tactic": "exact comap_comp_apply _ _ _"
}
] |
[
83,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Topology/Covering.lean
|
IsCoveringMap.isLocallyHomeomorph
|
[] |
[
169,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
11
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mex_le_lsub
|
[] |
[
2035,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2034,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.insertNth_add
|
[] |
[
777,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.Neg.aux_of
|
[] |
[
1193,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1192,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.Tendsto.atBot_mul_atTop
|
[
{
"state_after": "ι : Type ?u.197158\nι' : Type ?u.197161\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.197170\ninst✝ : StrictOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atBot\nhg : Tendsto g l atTop\nthis : Tendsto (fun x => -f x * g x) l atTop\n⊢ Tendsto (fun x => f x * g x) l atBot",
"state_before": "ι : Type ?u.197158\nι' : Type ?u.197161\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.197170\ninst✝ : StrictOrderedRing α\nl : Filter β\nf g : β → α\nhf : Tendsto f l atBot\nhg : Tendsto g l atTop\n⊢ Tendsto (fun x => f x * g x) l atBot",
"tactic": "have : Tendsto (fun x => -f x * g x) l atTop :=\n (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop hg"
}
] |
[
916,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
912,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
MeasureTheory.Measure.add_haar_closedBall'
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2003002\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\nx : E\nr : ℝ\nhr : 0 ≤ r\n⊢ ↑↑μ (closedBall x r) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ (closedBall 0 1)",
"tactic": "rw [← add_haar_closedBall_mul μ x hr zero_le_one, mul_one]"
}
] |
[
460,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.toFractionRing_injective
|
[
{
"state_after": "K : Type u\ninst✝ : CommRing K\nx : FractionRing K[X]\n⊢ { toFractionRing := x } = { toFractionRing := { toFractionRing := x }.toFractionRing }",
"state_before": "K : Type u\ninst✝ : CommRing K\nx y : FractionRing K[X]\nxy : { toFractionRing := x }.toFractionRing = { toFractionRing := y }.toFractionRing\n⊢ { toFractionRing := x } = { toFractionRing := y }",
"tactic": "subst xy"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : CommRing K\nx : FractionRing K[X]\n⊢ { toFractionRing := x } = { toFractionRing := { toFractionRing := x }.toFractionRing }",
"tactic": "rfl"
}
] |
[
136,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean
|
iSup_eq_iSup_subseq_of_monotone
|
[] |
[
322,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
TopologicalGroup.tendstoUniformlyOn_iff
|
[] |
[
628,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
623,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.SimpleFunc.FinMeasSupp.integrable
|
[] |
[
382,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.coe_eq_univ
|
[] |
[
203,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.HD_below_aux2
|
[] |
[
313,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.induction_on_monomial
|
[
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ M (↑(monomial s) a)",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\n⊢ ∀ (s : σ →₀ ℕ) (a : R), M (↑(monomial s) a)",
"tactic": "intro s a"
},
{
"state_after": "case h0\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ M (↑(monomial 0) a)\n\ncase ha\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ ∀ (a_1 : σ) (b : ℕ) (f : σ →₀ ℕ),\n ¬a_1 ∈ f.support → b ≠ 0 → M (↑(monomial f) a) → M (↑(monomial (Finsupp.single a_1 b + f)) a)",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ M (↑(monomial s) a)",
"tactic": "apply @Finsupp.induction σ ℕ _ _ s"
},
{
"state_after": "case h0\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ M (↑(monomial 0) a)",
"state_before": "case h0\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ M (↑(monomial 0) a)",
"tactic": "show M (monomial 0 a)"
},
{
"state_after": "no goals",
"state_before": "case h0\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ M (↑(monomial 0) a)",
"tactic": "exact h_C a"
},
{
"state_after": "case ha\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e ≠ 0\nih : M (↑(monomial p) a)\n⊢ M (↑(monomial (Finsupp.single n e + p)) a)",
"state_before": "case ha\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ ∀ (a_1 : σ) (b : ℕ) (f : σ →₀ ℕ),\n ¬a_1 ∈ f.support → b ≠ 0 → M (↑(monomial f) a) → M (↑(monomial (Finsupp.single a_1 b + f)) a)",
"tactic": "intro n e p _hpn _he ih"
},
{
"state_after": "case ha\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e ≠ 0\nih : M (↑(monomial p) a)\nthis : ∀ (e : ℕ), M (↑(monomial p) a * X n ^ e)\n⊢ M (↑(monomial (Finsupp.single n e + p)) a)",
"state_before": "case ha\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e ≠ 0\nih : M (↑(monomial p) a)\n⊢ M (↑(monomial (Finsupp.single n e + p)) a)",
"tactic": "have : ∀ e : ℕ, M (monomial p a * X n ^ e) := by\n intro e\n induction e with\n | zero => simp [ih]\n | succ e e_ih => simp [ih, pow_succ', (mul_assoc _ _ _).symm, h_X, e_ih]"
},
{
"state_after": "no goals",
"state_before": "case ha\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e ≠ 0\nih : M (↑(monomial p) a)\nthis : ∀ (e : ℕ), M (↑(monomial p) a * X n ^ e)\n⊢ M (↑(monomial (Finsupp.single n e + p)) a)",
"tactic": "simp [add_comm, monomial_add_single, this]"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝¹ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne✝ : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e✝ ≠ 0\nih : M (↑(monomial p) a)\ne : ℕ\n⊢ M (↑(monomial p) a * X n ^ e)",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e ≠ 0\nih : M (↑(monomial p) a)\n⊢ ∀ (e : ℕ), M (↑(monomial p) a * X n ^ e)",
"tactic": "intro e"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝¹ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne✝ : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e✝ ≠ 0\nih : M (↑(monomial p) a)\ne : ℕ\n⊢ M (↑(monomial p) a * X n ^ e)",
"tactic": "induction e with\n| zero => simp [ih]\n| succ e e_ih => simp [ih, pow_succ', (mul_assoc _ _ _).symm, h_X, e_ih]"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e ≠ 0\nih : M (↑(monomial p) a)\n⊢ M (↑(monomial p) a * X n ^ Nat.zero)",
"tactic": "simp [ih]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne✝¹ : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nM : MvPolynomial σ R → Prop\nh_C : ∀ (a : R), M (↑C a)\nh_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)\ns : σ →₀ ℕ\na : R\nn : σ\ne✝ : ℕ\np : σ →₀ ℕ\n_hpn : ¬n ∈ p.support\n_he : e✝ ≠ 0\nih : M (↑(monomial p) a)\ne : ℕ\ne_ih : M (↑(monomial p) a * X n ^ e)\n⊢ M (↑(monomial p) a * X n ^ Nat.succ e)",
"tactic": "simp [ih, pow_succ', (mul_assoc _ _ _).symm, h_X, e_ih]"
}
] |
[
406,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.neg_im
|
[] |
[
919,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
919,
16
] |
Mathlib/CategoryTheory/Localization/Predicate.lean
|
CategoryTheory.Functor.IsLocalization.of_equivalence_target
|
[
{
"state_after": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\n⊢ IsLocalization L' W",
"state_before": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\n⊢ IsLocalization L' W",
"tactic": "have h : W.IsInvertedBy L' := by\n rw [← MorphismProperty.IsInvertedBy.iff_of_iso W e]\n exact MorphismProperty.IsInvertedBy.of_comp W L (Localization.inverts L W) eq.functor"
},
{
"state_after": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\nF₁ : MorphismProperty.Localization W ⥤ D := Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)\n⊢ IsLocalization L' W",
"state_before": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\n⊢ IsLocalization L' W",
"tactic": "let F₁ := Localization.Construction.lift L (Localization.inverts L W)"
},
{
"state_after": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\nF₁ : MorphismProperty.Localization W ⥤ D := Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)\nF₂ : MorphismProperty.Localization W ⥤ E := Construction.lift L' h\n⊢ IsLocalization L' W",
"state_before": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\nF₁ : MorphismProperty.Localization W ⥤ D := Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)\n⊢ IsLocalization L' W",
"tactic": "let F₂ := Localization.Construction.lift L' h"
},
{
"state_after": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\nF₁ : MorphismProperty.Localization W ⥤ D := Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)\nF₂ : MorphismProperty.Localization W ⥤ E := Construction.lift L' h\ne' : F₁ ⋙ eq.functor ≅ F₂ := liftNatIso (MorphismProperty.Q W) W (L ⋙ eq.functor) L' (F₁ ⋙ eq.functor) F₂ e\n⊢ IsLocalization L' W",
"state_before": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\nF₁ : MorphismProperty.Localization W ⥤ D := Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)\nF₂ : MorphismProperty.Localization W ⥤ E := Construction.lift L' h\n⊢ IsLocalization L' W",
"tactic": "let e' : F₁ ⋙ eq.functor ≅ F₂ := liftNatIso W.Q W (L ⋙ eq.functor) L' _ _ e"
},
{
"state_after": "no goals",
"state_before": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\nh : MorphismProperty.IsInvertedBy W L'\nF₁ : MorphismProperty.Localization W ⥤ D := Construction.lift L (_ : MorphismProperty.IsInvertedBy W L)\nF₂ : MorphismProperty.Localization W ⥤ E := Construction.lift L' h\ne' : F₁ ⋙ eq.functor ≅ F₂ := liftNatIso (MorphismProperty.Q W) W (L ⋙ eq.functor) L' (F₁ ⋙ eq.functor) F₂ e\n⊢ IsLocalization L' W",
"tactic": "exact\n { inverts := h\n nonempty_isEquivalence := Nonempty.intro (IsEquivalence.ofIso e' inferInstance) }"
},
{
"state_after": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\n⊢ MorphismProperty.IsInvertedBy W (L ⋙ eq.functor)",
"state_before": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\n⊢ MorphismProperty.IsInvertedBy W L'",
"tactic": "rw [← MorphismProperty.IsInvertedBy.iff_of_iso W e]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_4\nD : Type u_6\ninst✝⁴ : Category C\ninst✝³ : Category D\nL : C ⥤ D\nW : MorphismProperty C\nE✝ : Type ?u.77743\ninst✝² : Category E✝\nE : Type u_1\ninst✝¹ : Category E\nL' : C ⥤ E\neq : D ≌ E\ninst✝ : IsLocalization L W\ne : L ⋙ eq.functor ≅ L'\n⊢ MorphismProperty.IsInvertedBy W (L ⋙ eq.functor)",
"tactic": "exact MorphismProperty.IsInvertedBy.of_comp W L (Localization.inverts L W) eq.functor"
}
] |
[
435,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Topology/Separation.lean
|
exists_compact_superset_iff
|
[] |
[
1309,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1306,
1
] |
Mathlib/Data/Setoid/Basic.lean
|
Quotient.subsingleton_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\n⊢ (∀ (x y : Quotient s), x = y) ↔ ∀ {x y : α}, ⊤ → Setoid.Rel s x y",
"state_before": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\n⊢ Subsingleton (Quotient s) ↔ s = ⊤",
"tactic": "simp only [_root_.subsingleton_iff, eq_top_iff, Setoid.le_def, Setoid.top_def, Pi.top_apply,\n forall_const]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\na : α\n⊢ (∀ (y : Quotient s), Quotient.mk s a = y) ↔ ∀ {y : α}, ⊤ → Setoid.Rel s a y",
"state_before": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\n⊢ (∀ (x y : Quotient s), x = y) ↔ ∀ {x y : α}, ⊤ → Setoid.Rel s x y",
"tactic": "refine' (surjective_quotient_mk _).forall.trans (forall_congr' fun a => _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\na b : α\n⊢ Quotient.mk s a = Quotient.mk s b ↔ ⊤ → Setoid.Rel s a b",
"state_before": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\na : α\n⊢ (∀ (y : Quotient s), Quotient.mk s a = y) ↔ ∀ {y : α}, ⊤ → Setoid.Rel s a y",
"tactic": "refine' (surjective_quotient_mk _).forall.trans (forall_congr' fun b => _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\na b : α\n⊢ Setoid.r a b ↔ Setoid.Rel s a b",
"state_before": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\na b : α\n⊢ Quotient.mk s a = Quotient.mk s b ↔ ⊤ → Setoid.Rel s a b",
"tactic": "simp_rw [←Quotient.mk''_eq_mk, Prop.top_eq_true, true_implies, Quotient.eq'']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.22286\ns : Setoid α\na b : α\n⊢ Setoid.r a b ↔ Setoid.Rel s a b",
"tactic": "rfl"
}
] |
[
468,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/RingTheory/WittVector/IsPoly.lean
|
WittVector.IsPoly.comp
|
[
{
"state_after": "case mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\n⊢ IsPoly p fun R _Rcr => g ∘ f",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nhf : IsPoly p f\n⊢ IsPoly p fun R _Rcr => g ∘ f",
"tactic": "obtain ⟨φ, hf⟩ := hf"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ IsPoly p fun R _Rcr => g ∘ f",
"state_before": "case mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\n⊢ IsPoly p fun R _Rcr => g ∘ f",
"tactic": "obtain ⟨ψ, hg⟩ := hg"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R),\n ((g ∘ f) x).coeff = fun n => ↑(aeval x.coeff) ((fun n => ↑(bind₁ φ) (ψ n)) n)",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ IsPoly p fun R _Rcr => g ∘ f",
"tactic": "use fun n => bind₁ φ (ψ n)"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝ : 𝕎 R✝\n⊢ ((g ∘ f) x✝).coeff = fun n => ↑(aeval x✝.coeff) ((fun n => ↑(bind₁ φ) (ψ n)) n)",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝¹ : CommRing R\ninst✝ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R),\n ((g ∘ f) x).coeff = fun n => ↑(aeval x.coeff) ((fun n => ↑(bind₁ φ) (ψ n)) n)",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.528965\nidx : Type ?u.528968\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ng f : ⦃R : Type u_1⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u_1⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u_1\ninst✝ : CommRing R✝\nx✝ : 𝕎 R✝\n⊢ ((g ∘ f) x✝).coeff = fun n => ↑(aeval x✝.coeff) ((fun n => ↑(bind₁ φ) (ψ n)) n)",
"tactic": "simp only [aeval_bind₁, Function.comp, hg, hf]"
}
] |
[
279,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Order/Hom/Basic.lean
|
WithBot.toDualTopEquiv_symm_bot
|
[] |
[
1264,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1263,
1
] |
Mathlib/Algebra/Order/Monoid/Defs.lean
|
Mul.to_covariantClass_right
|
[] |
[
83,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.mem_ae_of_mem_ae_map
|
[] |
[
2704,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2702,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
IsLowerSet.top_mem
|
[] |
[
290,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Std/Data/List/Lemmas.lean
|
List.suffix_or_suffix_of_suffix
|
[] |
[
1678,
21
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1676,
1
] |
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