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/Data/Real/ENNReal.lean
|
ENNReal.le_inv_iff_le_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.269244\nβ : Type ?u.269247\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ a ≤ b⁻¹ ↔ b ≤ a⁻¹",
"tactic": "simpa only [inv_inv] using @ENNReal.inv_le_inv a⁻¹ b"
}
] |
[
1500,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1499,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.get_bind
|
[] |
[
823,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
821,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lt_congr_right
|
[] |
[
862,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
861,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
Measurable.measurable_of_countable_ne
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (g ⁻¹' t)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\n⊢ Measurable g",
"tactic": "intro t ht"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (g ⁻¹' t)",
"tactic": "have : g ⁻¹' t = g ⁻¹' t ∩ { x | f x = g x }ᶜ ∪ g ⁻¹' t ∩ { x | f x = g x } := by\n simp [← inter_union_distrib_left]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x})",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t)",
"tactic": "rw [this]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x})",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x})",
"tactic": "refine (h.mono (inter_subset_right _ _)).measurableSet.union ?_"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\nthis : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x})",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x})",
"tactic": "have : g ⁻¹' t ∩ { x : α | f x = g x } = f ⁻¹' t ∩ { x : α | f x = g x } := by\n ext x\n simp (config := { contextual := true })"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\nthis : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (f ⁻¹' t ∩ {x | f x = g x})",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\nthis : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (g ⁻¹' t ∩ {x | f x = g x})",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\nthis : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x}\n⊢ MeasurableSet (f ⁻¹' t ∩ {x | f x = g x})",
"tactic": "exact (hf ht).inter h.measurableSet.of_compl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\n⊢ g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}",
"tactic": "simp [← inter_union_distrib_left]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\nx : α\n⊢ x ∈ g ⁻¹' t ∩ {x | f x = g x} ↔ x ∈ f ⁻¹' t ∩ {x | f x = g x}",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\n⊢ g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x}",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.44496\nδ : Type ?u.44499\nδ' : Type ?u.44502\nι : Sort uι\ns t✝ u : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nhf : Measurable f\nh : Set.Countable {x | f x ≠ g x}\nt : Set β\nht : MeasurableSet t\nthis : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}\nx : α\n⊢ x ∈ g ⁻¹' t ∩ {x | f x = g x} ↔ x ∈ f ⁻¹' t ∩ {x | f x = g x}",
"tactic": "simp (config := { contextual := true })"
}
] |
[
365,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_map
|
[] |
[
253,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.powersetLen_coe
|
[] |
[
242,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.exists_mem_subset_iff
|
[] |
[
212,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.permCongr_symm_apply
|
[] |
[
446,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
Filter.coprodᵢ_cocompact
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\n⊢ cocompact ((d : δ) → κ d) ≤ Filter.coprodᵢ fun d => cocompact (κ d)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\n⊢ (Filter.coprodᵢ fun d => cocompact (κ d)) = cocompact ((d : δ) → κ d)",
"tactic": "refine' le_antisymm (iSup_le fun i => Filter.comap_cocompact_le (continuous_apply i)) _"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns✝ t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\ns : Set ((i : δ) → κ i)\nH : sᶜ ∈ Filter.coprodᵢ fun d => cocompact (κ d)\n⊢ sᶜ ∈ cocompact ((d : δ) → κ d)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\n⊢ cocompact ((d : δ) → κ d) ≤ Filter.coprodᵢ fun d => cocompact (κ d)",
"tactic": "refine' compl_surjective.forall.2 fun s H => _"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns✝ t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\ns : Set ((i : δ) → κ i)\nH : ∀ (i : δ), ∃ t, IsCompact t ∧ s ⊆ Function.eval i ⁻¹' t\n⊢ ∃ t, IsCompact t ∧ s ⊆ t",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns✝ t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\ns : Set ((i : δ) → κ i)\nH : sᶜ ∈ Filter.coprodᵢ fun d => cocompact (κ d)\n⊢ sᶜ ∈ cocompact ((d : δ) → κ d)",
"tactic": "simp only [compl_mem_coprodᵢ, Filter.mem_cocompact, compl_subset_compl, image_subset_iff] at H⊢"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns✝ t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\ns : Set ((i : δ) → κ i)\nK : (i : δ) → Set (κ i)\nhKc : ∀ (i : δ), IsCompact (K i)\nhtK : ∀ (i : δ), s ⊆ Function.eval i ⁻¹' K i\n⊢ ∃ t, IsCompact t ∧ s ⊆ t",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns✝ t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\ns : Set ((i : δ) → κ i)\nH : ∀ (i : δ), ∃ t, IsCompact t ∧ s ⊆ Function.eval i ⁻¹' t\n⊢ ∃ t, IsCompact t ∧ s ⊆ t",
"tactic": "choose K hKc htK using H"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.140164\nπ : ι → Type ?u.140169\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ns✝ t : Set α\ninst✝¹ : (i : ι) → TopologicalSpace (π i)\nδ : Type u_1\nκ : δ → Type u_2\ninst✝ : (d : δ) → TopologicalSpace (κ d)\ns : Set ((i : δ) → κ i)\nK : (i : δ) → Set (κ i)\nhKc : ∀ (i : δ), IsCompact (K i)\nhtK : ∀ (i : δ), s ⊆ Function.eval i ⁻¹' K i\n⊢ ∃ t, IsCompact t ∧ s ⊆ t",
"tactic": "exact ⟨Set.pi univ K, isCompact_univ_pi hKc, fun f hf i _ => htK i hf⟩"
}
] |
[
1057,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1051,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.BoundedFormula.realize_restrictFreeVar
|
[
{
"state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nh : ↑(freeVarFinset falsum) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar falsum (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize falsum v xs\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nh : ↑(freeVarFinset (equal t₁✝ t₂✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (equal t₁✝ t₂✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (equal t₁✝ t₂✝) v xs\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nh : ↑(freeVarFinset (rel R✝ ts✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (rel R✝ ts✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (rel R✝ ts✝) v xs\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 :\n ∀ (h : ↑(freeVarFinset f₁✝) ⊆ s) {xs : Fin n✝ → M},\n Realize (restrictFreeVar f₁✝ (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize f₁✝ v xs\nih2 :\n ∀ (h : ↑(freeVarFinset f₂✝) ⊆ s) {xs : Fin n✝ → M},\n Realize (restrictFreeVar f₂✝ (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize f₂✝ v xs\nh : ↑(freeVarFinset (f₁✝ ⟹ f₂✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (f₁✝ ⟹ f₂✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (f₁✝ ⟹ f₂✝) v xs\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 :\n ∀ (h : ↑(freeVarFinset f✝) ⊆ s) {xs : Fin (n✝ + 1) → M},\n Realize (restrictFreeVar f✝ (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize f✝ v xs\nh : ↑(freeVarFinset (∀'f✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (∀'f✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (∀'f✝) v xs",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs : Fin n → M\n⊢ Realize (restrictFreeVar φ (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize φ v xs",
"tactic": "induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3"
},
{
"state_after": "no goals",
"state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nh : ↑(freeVarFinset falsum) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar falsum (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize falsum v xs",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nh : ↑(freeVarFinset (equal t₁✝ t₂✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (equal t₁✝ t₂✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (equal t₁✝ t₂✝) v xs",
"tactic": "simp [restrictFreeVar, Realize]"
},
{
"state_after": "no goals",
"state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nh : ↑(freeVarFinset (rel R✝ ts✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (rel R✝ ts✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (rel R✝ ts✝) v xs",
"tactic": "simp [restrictFreeVar, Realize]"
},
{
"state_after": "no goals",
"state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 :\n ∀ (h : ↑(freeVarFinset f₁✝) ⊆ s) {xs : Fin n✝ → M},\n Realize (restrictFreeVar f₁✝ (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize f₁✝ v xs\nih2 :\n ∀ (h : ↑(freeVarFinset f₂✝) ⊆ s) {xs : Fin n✝ → M},\n Realize (restrictFreeVar f₂✝ (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize f₂✝ v xs\nh : ↑(freeVarFinset (f₁✝ ⟹ f₂✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (f₁✝ ⟹ f₂✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (f₁✝ ⟹ f₂✝) v xs",
"tactic": "simp [restrictFreeVar, Realize, ih1, ih2]"
},
{
"state_after": "no goals",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.135581\nP : Type ?u.135584\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝¹ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝¹ : Fin l → M\ninst✝ : DecidableEq α\nn : ℕ\nφ : BoundedFormula L α n\ns : Set α\nh✝ : ↑(freeVarFinset φ) ⊆ s\nv : α → M\nxs✝ : Fin n → M\nn✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 :\n ∀ (h : ↑(freeVarFinset f✝) ⊆ s) {xs : Fin (n✝ + 1) → M},\n Realize (restrictFreeVar f✝ (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize f✝ v xs\nh : ↑(freeVarFinset (∀'f✝)) ⊆ s\nxs : Fin n✝ → M\n⊢ Realize (restrictFreeVar (∀'f✝) (Set.inclusion h)) (v ∘ Subtype.val) xs ↔ Realize (∀'f✝) v xs",
"tactic": "simp [restrictFreeVar, Realize, ih3]"
}
] |
[
474,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_div_le_one_div_of_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.82335\nα : Type u_1\nβ : Type ?u.82341\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\nh : a ≤ b\n⊢ 1 / b ≤ 1 / a",
"tactic": "simpa using inv_le_inv_of_le ha h"
}
] |
[
454,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.lift_unique
|
[
{
"state_after": "case a\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\n⊢ ↑(lift f_compat) r = ↑g r",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\n⊢ lift f_compat = g",
"tactic": "ext1 r"
},
{
"state_after": "case a.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\n⊢ ∀ (ε : ℝ), ε > 0 → dist (↑(lift f_compat) r) (↑g r) ≤ ε",
"state_before": "case a\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\n⊢ ↑(lift f_compat) r = ↑g r",
"tactic": "apply eq_of_forall_dist_le"
},
{
"state_after": "case a.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\nε : ℝ\nhε : ε > 0\n⊢ dist (↑(lift f_compat) r) (↑g r) ≤ ε",
"state_before": "case a.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\n⊢ ∀ (ε : ℝ), ε > 0 → dist (↑(lift f_compat) r) (↑g r) ≤ ε",
"tactic": "intro ε hε"
},
{
"state_after": "case a.h.intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\nε : ℝ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ dist (↑(lift f_compat) r) (↑g r) ≤ ε",
"state_before": "case a.h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\nε : ℝ\nhε : ε > 0\n⊢ dist (↑(lift f_compat) r) (↑g r) ≤ ε",
"tactic": "obtain ⟨n, hn⟩ := exists_pow_neg_lt p hε"
},
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\nε : ℝ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ dist (↑(lift f_compat) r) (↑g r) ≤ ↑p ^ (-↑n)",
"state_before": "case a.h.intro\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\nε : ℝ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ dist (↑(lift f_compat) r) (↑g r) ≤ ε",
"tactic": "apply le_trans _ (le_of_lt hn)"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R →+* ℤ_[p]\nhg : ∀ (n : ℕ), RingHom.comp (toZModPow n) g = f n\nr : R\nε : ℝ\nhε : ε > 0\nn : ℕ\nhn : ↑p ^ (-↑n) < ε\n⊢ dist (↑(lift f_compat) r) (↑g r) ≤ ↑p ^ (-↑n)",
"tactic": "rw [dist_eq_norm, norm_le_pow_iff_mem_span_pow, ← ker_toZModPow, RingHom.mem_ker,\n RingHom.map_sub, ← RingHom.comp_apply, ← RingHom.comp_apply, lift_spec, hg, sub_self]"
}
] |
[
659,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
EulerSine.integral_cos_pow_eq
|
[
{
"state_after": "z : ℂ\nn✝ n : ℕ\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π, sin x ^ n",
"state_before": "z : ℂ\nn✝ n : ℕ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = 1 / 2 * ∫ (x : ℝ) in 0 ..π, sin x ^ n",
"tactic": "rw [mul_comm (1 / 2 : ℝ), ← div_eq_iff (one_div_ne_zero (two_ne_zero' ℝ)), ← div_mul, div_one,\n mul_two]"
},
{
"state_after": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π, sin x ^ n",
"state_before": "z : ℂ\nn✝ n : ℕ\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π, sin x ^ n",
"tactic": "have L : IntervalIntegrable _ volume 0 (π / 2) := (continuous_sin.pow n).intervalIntegrable _ _"
},
{
"state_after": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π, sin x ^ n",
"state_before": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π, sin x ^ n",
"tactic": "have R : IntervalIntegrable _ volume (π / 2) π := (continuous_sin.pow n).intervalIntegrable _ _"
},
{
"state_after": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) =\n (∫ (x : ℝ) in 0 ..π / 2, sin x ^ n) + ∫ (x : ℝ) in π / 2 ..π, sin x ^ n",
"state_before": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π, sin x ^ n",
"tactic": "rw [← integral_add_adjacent_intervals L R]"
},
{
"state_after": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n\n\ncase refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in π / 2 ..π, sin x ^ n",
"state_before": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ ((∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) + ∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) =\n (∫ (x : ℝ) in 0 ..π / 2, sin x ^ n) + ∫ (x : ℝ) in π / 2 ..π, sin x ^ n",
"tactic": "refine congr_arg₂ _ ?_ ?_"
},
{
"state_after": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in π / 2 - π / 2 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n",
"state_before": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n",
"tactic": "nth_rw 1 [(by ring : 0 = π / 2 - π / 2)]"
},
{
"state_after": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in π / 2 - π / 2 ..π / 2 - 0, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n",
"state_before": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in π / 2 - π / 2 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n",
"tactic": "nth_rw 3 [(by ring : π / 2 = π / 2 - 0)]"
},
{
"state_after": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos (π / 2 - x) ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n",
"state_before": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in π / 2 - π / 2 ..π / 2 - 0, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n",
"tactic": "rw [← integral_comp_sub_left]"
},
{
"state_after": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ cos (π / 2 - x) ^ n = sin x ^ n",
"state_before": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos (π / 2 - x) ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin x ^ n",
"tactic": "refine' integral_congr fun x _ => _"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ cos (π / 2 - x) ^ n = sin x ^ n",
"tactic": "rw [cos_pi_div_two_sub]"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ 0 = π / 2 - π / 2",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ π / 2 = π / 2 - 0",
"tactic": "ring"
},
{
"state_after": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in π / 2 ..π / 2 + π / 2, sin x ^ n",
"state_before": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in π / 2 ..π, sin x ^ n",
"tactic": "nth_rw 3 [(by ring : π = π / 2 + π / 2)]"
},
{
"state_after": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 + π / 2 ..π / 2 + π / 2, sin x ^ n",
"state_before": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in π / 2 ..π / 2 + π / 2, sin x ^ n",
"tactic": "nth_rw 2 [(by ring : π / 2 = 0 + π / 2)]"
},
{
"state_after": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin (x + π / 2) ^ n",
"state_before": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 + π / 2 ..π / 2 + π / 2, sin x ^ n",
"tactic": "rw [← integral_comp_add_right]"
},
{
"state_after": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ cos x ^ n = sin (x + π / 2) ^ n",
"state_before": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ (∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) = ∫ (x : ℝ) in 0 ..π / 2, sin (x + π / 2) ^ n",
"tactic": "refine' integral_congr fun x _ => _"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nz : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ cos x ^ n = sin (x + π / 2) ^ n",
"tactic": "rw [sin_add_pi_div_two]"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ π = π / 2 + π / 2",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nn✝ n : ℕ\nL : IntervalIntegrable (fun b => sin b ^ n) volume 0 (π / 2)\nR : IntervalIntegrable (fun b => sin b ^ n) volume (π / 2) π\n⊢ π / 2 = 0 + π / 2",
"tactic": "ring"
}
] |
[
206,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_two_mul_add_two
|
[
{
"state_after": "n : ℕ\n⊢ fib n * (2 * fib (n + 1) - fib n) + (fib (n + 1) ^ 2 + fib n ^ 2) = fib (n + 1) * (2 * fib n + fib (n + 1))",
"state_before": "n : ℕ\n⊢ fib (2 * n + 2) = fib (n + 1) * (2 * fib n + fib (n + 1))",
"tactic": "rw [fib_add_two, fib_two_mul, fib_two_mul_add_one]"
},
{
"state_after": "n : ℕ\nthis : fib n ≤ 2 * fib (n + 1)\n⊢ fib n * (2 * fib (n + 1) - fib n) + (fib (n + 1) ^ 2 + fib n ^ 2) = fib (n + 1) * (2 * fib n + fib (n + 1))",
"state_before": "n : ℕ\n⊢ fib n * (2 * fib (n + 1) - fib n) + (fib (n + 1) ^ 2 + fib n ^ 2) = fib (n + 1) * (2 * fib n + fib (n + 1))",
"tactic": "have : fib n ≤ 2 * fib (n + 1) :=\n le_trans (fib_le_fib_succ) (mul_comm 2 _ ▸ le_mul_of_pos_right two_pos)"
},
{
"state_after": "n : ℕ\nthis : fib n ≤ 2 * fib (n + 1)\n⊢ ↑(fib n) * (2 * ↑(fib (n + 1)) - ↑(fib n)) + (↑(fib (n + 1)) ^ 2 + ↑(fib n) ^ 2) =\n ↑(fib (n + 1)) * (2 * ↑(fib n) + ↑(fib (n + 1)))",
"state_before": "n : ℕ\nthis : fib n ≤ 2 * fib (n + 1)\n⊢ fib n * (2 * fib (n + 1) - fib n) + (fib (n + 1) ^ 2 + fib n ^ 2) = fib (n + 1) * (2 * fib n + fib (n + 1))",
"tactic": "zify [this]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nthis : fib n ≤ 2 * fib (n + 1)\n⊢ ↑(fib n) * (2 * ↑(fib (n + 1)) - ↑(fib n)) + (↑(fib (n + 1)) ^ 2 + ↑(fib n) ^ 2) =\n ↑(fib (n + 1)) * (2 * ↑(fib n) + ↑(fib (n + 1)))",
"tactic": "ring"
}
] |
[
180,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Order/Filter/Cofinite.lean
|
Filter.Tendsto.exists_forall_le
|
[] |
[
178,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Algebra/Category/Mon/FilteredColimits.lean
|
MonCat.FilteredColimits.colimitMulAux_eq_of_rel_left
|
[
{
"state_after": "case mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx' y : (j : J) × ↑(F.obj j)\nj₁ : J\nx : ↑(F.obj j₁)\nhxx' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := x } x'\n⊢ colimitMulAux F { fst := j₁, snd := x } y = colimitMulAux F x' y",
"state_before": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx x' y : (j : J) × ↑(F.obj j)\nhxx' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) x x'\n⊢ colimitMulAux F x y = colimitMulAux F x' y",
"tactic": "cases' x with j₁ x"
},
{
"state_after": "case mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx' : (j : J) × ↑(F.obj j)\nj₁ : J\nx : ↑(F.obj j₁)\nhxx' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := x } x'\nj₂ : J\ny : ↑(F.obj j₂)\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } = colimitMulAux F x' { fst := j₂, snd := y }",
"state_before": "case mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx' y : (j : J) × ↑(F.obj j)\nj₁ : J\nx : ↑(F.obj j₁)\nhxx' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := x } x'\n⊢ colimitMulAux F { fst := j₁, snd := x } y = colimitMulAux F x' y",
"tactic": "cases' y with j₂ y"
},
{
"state_after": "case mk.mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nhxx' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := x } { fst := j₃, snd := x' }\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"state_before": "case mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx' : (j : J) × ↑(F.obj j)\nj₁ : J\nx : ↑(F.obj j₁)\nhxx' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := x } x'\nj₂ : J\ny : ↑(F.obj j₂)\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } = colimitMulAux F x' { fst := j₂, snd := y }",
"tactic": "cases' x' with j₃ x'"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : (F ⋙ forget MonCat).map f { fst := j₁, snd := x }.snd = (F ⋙ forget MonCat).map g { fst := j₃, snd := x' }.snd\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"state_before": "case mk.mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nhxx' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := x } { fst := j₃, snd := x' }\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"tactic": "obtain ⟨l, f, g, hfg⟩ := hxx'"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"state_before": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : (F ⋙ forget MonCat).map f { fst := j₁, snd := x }.snd = (F ⋙ forget MonCat).map g { fst := j₃, snd := x' }.snd\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"tactic": "simp at hfg"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"state_before": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"tactic": "obtain ⟨s, α, β, γ, h₁, h₂, h₃⟩ :=\n IsFiltered.tulip (IsFiltered.leftToMax j₁ j₂) (IsFiltered.rightToMax j₁ j₂)\n (IsFiltered.rightToMax j₃ j₂) (IsFiltered.leftToMax j₃ j₂) f g"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ∃ k f g,\n ↑(F.map f)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map g)\n { fst := IsFiltered.max { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₃, snd := x' }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ colimitMulAux F { fst := j₁, snd := x } { fst := j₂, snd := y } =\n colimitMulAux F { fst := j₃, snd := x' } { fst := j₂, snd := y }",
"tactic": "apply M.mk_eq"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map α)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map γ)\n { fst := IsFiltered.max { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₃, snd := x' }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ∃ k f g,\n ↑(F.map f)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map g)\n { fst := IsFiltered.max { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₃, snd := x' }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd",
"tactic": "use s, α, γ"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₃ j₂)) x' * ↑(F.map (IsFiltered.rightToMax j₃ j₂)) y)",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map α)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map γ)\n { fst := IsFiltered.max { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₃, snd := x' }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₃, snd := x' }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd",
"tactic": "dsimp"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x) * ↑(F.map α) (↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₃ j₂)) x') * ↑(F.map γ) (↑(F.map (IsFiltered.rightToMax j₃ j₂)) y)",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₃ j₂)) x' * ↑(F.map (IsFiltered.rightToMax j₃ j₂)) y)",
"tactic": "simp_rw [MonoidHom.map_mul]"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₁ j₂) ≫ F.map α) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂) ≫ F.map α) y =\n ↑(F.map (IsFiltered.leftToMax j₃ j₂) ≫ F.map γ) x' * ↑(F.map (IsFiltered.rightToMax j₃ j₂) ≫ F.map γ) y",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x) * ↑(F.map α) (↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₃ j₂)) x') * ↑(F.map γ) (↑(F.map (IsFiltered.rightToMax j₃ j₂)) y)",
"tactic": "change (F.map _ ≫ F.map _) _ * (F.map _ ≫ F.map _) _ =\n (F.map _ ≫ F.map _) _ * (F.map _ ≫ F.map _) _"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map f ≫ F.map β) x * ↑(F.map (IsFiltered.rightToMax j₃ j₂) ≫ F.map γ) y =\n ↑(F.map g ≫ F.map β) x' * ↑(F.map (IsFiltered.rightToMax j₃ j₂) ≫ F.map γ) y",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₁ j₂) ≫ F.map α) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂) ≫ F.map α) y =\n ↑(F.map (IsFiltered.leftToMax j₃ j₂) ≫ F.map γ) x' * ↑(F.map (IsFiltered.rightToMax j₃ j₂) ≫ F.map γ) y",
"tactic": "simp_rw [← F.map_comp, h₁, h₂, h₃, F.map_comp]"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map f ≫ F.map β) x = ↑(F.map g ≫ F.map β) x'",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map f ≫ F.map β) x * ↑(F.map (IsFiltered.rightToMax j₃ j₂) ≫ F.map γ) y =\n ↑(F.map g ≫ F.map β) x' * ↑(F.map (IsFiltered.rightToMax j₃ j₂) ≫ F.map γ) y",
"tactic": "congr 1"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map β) (↑(F.map f) x) = ↑(F.map β) (↑(F.map g) x')",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map f ≫ F.map β) x = ↑(F.map g ≫ F.map β) x'",
"tactic": "change F.map _ (F.map _ _) = F.map _ (F.map _ _)"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\nx : ↑(F.obj j₁)\nj₂ : J\ny : ↑(F.obj j₂)\nj₃ : J\nx' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := x }.fst ⟶ l\ng : { fst := j₃, snd := x' }.fst ⟶ l\nhfg : ↑(F.map f) x = ↑(F.map g) x'\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₃ j₂ ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ\nh₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β\n⊢ ↑(F.map β) (↑(F.map f) x) = ↑(F.map β) (↑(F.map g) x')",
"tactic": "rw [hfg]"
}
] |
[
153,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_eq_zero_iff
|
[
{
"state_after": "θ : Angle\n⊢ sin θ = sin 0 ↔ θ = 0 ∨ θ = ↑π",
"state_before": "θ : Angle\n⊢ sin θ = 0 ↔ θ = 0 ∨ θ = ↑π",
"tactic": "nth_rw 1 [← sin_zero]"
},
{
"state_after": "θ : Angle\n⊢ θ = 0 ∨ θ + 0 = ↑π ↔ θ = 0 ∨ θ = ↑π",
"state_before": "θ : Angle\n⊢ sin θ = sin 0 ↔ θ = 0 ∨ θ = ↑π",
"tactic": "rw [sin_eq_iff_eq_or_add_eq_pi]"
},
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ θ = 0 ∨ θ + 0 = ↑π ↔ θ = 0 ∨ θ = ↑π",
"tactic": "simp"
}
] |
[
368,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
Prod.snd_inv
|
[] |
[
144,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
Fin.sum_pow_mul_eq_add_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.35983\nβ : Type ?u.35986\nn : ℕ\nR : Type u_1\ninst✝ : CommSemiring R\na b : R\n⊢ ∑ s : Finset (Fin n), a ^ card s * b ^ (n - card s) = (a + b) ^ n",
"tactic": "simpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b"
}
] |
[
165,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Analysis/NormedSpace/MazurUlam.lean
|
IsometryEquiv.map_midpoint
|
[
{
"state_after": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"tactic": "set e : PE ≃ᵢ PE :=\n ((f.trans <| (pointReflection ℝ <| midpoint ℝ (f x) (f y)).toIsometryEquiv).trans f.symm).trans\n (pointReflection ℝ <| midpoint ℝ x y).toIsometryEquiv"
},
{
"state_after": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"tactic": "have hx : e x = x := by simp"
},
{
"state_after": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\nhy : ↑e y = y\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"tactic": "have hy : e y = y := by simp"
},
{
"state_after": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\nhy : ↑e y = y\nhm : ↑e (midpoint ℝ x y) = midpoint ℝ x y\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\nhy : ↑e y = y\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"tactic": "have hm := e.midpoint_fixed hx hy"
},
{
"state_after": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\nhy : ↑e y = y\nhm :\n ↑(toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\n (↑(IsometryEquiv.symm f)\n (↑(toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))) (↑f (midpoint ℝ x y)))) =\n midpoint ℝ x y\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\nhy : ↑e y = y\nhm : ↑e (midpoint ℝ x y) = midpoint ℝ x y\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"tactic": "simp only [trans_apply] at hm"
},
{
"state_after": "no goals",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\nhy : ↑e y = y\nhm :\n ↑(toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\n (↑(IsometryEquiv.symm f)\n (↑(toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))) (↑f (midpoint ℝ x y)))) =\n midpoint ℝ x y\n⊢ ↑f (midpoint ℝ x y) = midpoint ℝ (↑f x) (↑f y)",
"tactic": "rwa [← eq_symm_apply, toIsometryEquiv_symm, pointReflection_symm, coe_toIsometryEquiv,\n coe_toIsometryEquiv, pointReflection_self, symm_apply_eq, @pointReflection_fixed_iff] at hm"
},
{
"state_after": "no goals",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\n⊢ ↑e x = x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "E : Type u_3\nPE : Type u_1\nF : Type u_4\nPF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : PE ≃ᵢ PF\nx y : PE\ne : PE ≃ᵢ PE :=\n IsometryEquiv.trans\n (IsometryEquiv.trans (IsometryEquiv.trans f (toIsometryEquiv (pointReflection ℝ (midpoint ℝ (↑f x) (↑f y)))))\n (IsometryEquiv.symm f))\n (toIsometryEquiv (pointReflection ℝ (midpoint ℝ x y)))\nhx : ↑e x = x\n⊢ ↑e y = y",
"tactic": "simp"
}
] |
[
101,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Measure.lean
|
BoxIntegral.Box.measurableSet_Ioo
|
[] |
[
70,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.WOppSide.trans_wSameSide
|
[] |
[
579,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Std/Data/Int/DivMod.lean
|
Int.mod_add_div
|
[
{
"state_after": "m n : Nat\n⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m",
"state_before": "m n : Nat\n⊢ mod (ofNat m) -[n+1] + -[n+1] * div (ofNat m) -[n+1] = ofNat m",
"tactic": "show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m"
},
{
"state_after": "m n : Nat\n⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m",
"state_before": "m n : Nat\n⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m",
"tactic": "rw [Int.neg_mul_neg]"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m",
"tactic": "exact congrArg ofNat (Nat.mod_add_div ..)"
},
{
"state_after": "m n : Nat\n⊢ -↑(succ m % n) + ↑n * -↑(succ m / n) = -↑(succ m)",
"state_before": "m n : Nat\n⊢ mod -[m+1] (ofNat n) + ofNat n * div -[m+1] (ofNat n) = -[m+1]",
"tactic": "show -(↑((succ m) % n) : Int) + ↑n * -↑(succ m / n) = -↑(succ m)"
},
{
"state_after": "m n : Nat\n⊢ -(↑(succ m % n) + ↑n * ↑(succ m / n)) = -↑(succ m)",
"state_before": "m n : Nat\n⊢ -↑(succ m % n) + ↑n * -↑(succ m / n) = -↑(succ m)",
"tactic": "rw [Int.mul_neg, ← Int.neg_add]"
},
{
"state_after": "m n : Nat\n⊢ -↑(succ m % succ n) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m)",
"state_before": "m n : Nat\n⊢ mod -[m+1] -[n+1] + -[n+1] * div -[m+1] -[n+1] = -[m+1]",
"tactic": "show -(↑(succ m % succ n) : Int) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m)"
},
{
"state_after": "m n : Nat\n⊢ -(↑(succ m % succ n) + ↑(succ n) * ↑(succ m / succ n)) = -↑(succ m)",
"state_before": "m n : Nat\n⊢ -↑(succ m % succ n) + -↑(succ n) * ↑(succ m / succ n) = -↑(succ m)",
"tactic": "rw [Int.neg_mul, ← Int.neg_add]"
}
] |
[
280,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
267,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_eq_measure_of_null_diff
|
[] |
[
265,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
Homeomorph.shearMulRight_symm_coe
|
[] |
[
669,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/RingTheory/Ideal/AssociatedPrime.lean
|
IsAssociatedPrime.annihilator_le
|
[
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\nJ : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nM' : Type ?u.70734\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nx : M\nhI : Ideal.IsPrime (Submodule.annihilator (Submodule.span R {x}))\n⊢ Submodule.annihilator ⊤ ≤ Submodule.annihilator (Submodule.span R {x})",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\nI J : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nM' : Type ?u.70734\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nh : IsAssociatedPrime I M\n⊢ Submodule.annihilator ⊤ ≤ I",
"tactic": "obtain ⟨hI, x, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\nJ : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nM' : Type ?u.70734\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nx : M\nhI : Ideal.IsPrime (Submodule.annihilator (Submodule.span R {x}))\n⊢ Submodule.annihilator ⊤ ≤ Submodule.annihilator (Submodule.span R {x})",
"tactic": "exact Submodule.annihilator_mono le_top"
}
] |
[
141,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Combinatorics/Quiver/Symmetric.lean
|
Quiver.Symmetrify.lift_unique
|
[
{
"state_after": "U : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\n⊢ Φ = lift (of ⋙q Φ)",
"state_before": "U : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nφ : V ⥤q V'\nΦ : Symmetrify V ⥤q V'\nhΦ : of ⋙q Φ = φ\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\n⊢ Φ = lift φ",
"tactic": "subst_vars"
},
{
"state_after": "case h_obj\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\n⊢ ∀ (X : Symmetrify V), Φ.obj X = (lift (of ⋙q Φ)).obj X\n\ncase h_map\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\n⊢ ∀ (X Y : Symmetrify V) (f : X ⟶ Y),\n Φ.map f =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map f))",
"state_before": "U : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\n⊢ Φ = lift (of ⋙q Φ)",
"tactic": "fapply Prefunctor.ext"
},
{
"state_after": "case h_obj\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX : Symmetrify V\n⊢ Φ.obj X = (lift (of ⋙q Φ)).obj X",
"state_before": "case h_obj\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\n⊢ ∀ (X : Symmetrify V), Φ.obj X = (lift (of ⋙q Φ)).obj X",
"tactic": "rintro X"
},
{
"state_after": "no goals",
"state_before": "case h_obj\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX : Symmetrify V\n⊢ Φ.obj X = (lift (of ⋙q Φ)).obj X",
"tactic": "rfl"
},
{
"state_after": "case h_map\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX Y : Symmetrify V\nf : X ⟶ Y\n⊢ Φ.map f =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map f))",
"state_before": "case h_map\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\n⊢ ∀ (X Y : Symmetrify V) (f : X ⟶ Y),\n Φ.map f =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map f))",
"tactic": "rintro X Y f"
},
{
"state_after": "case h_map.inl\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX Y : Symmetrify V\nval✝ : X ⟶ Y\n⊢ Φ.map (Sum.inl val✝) =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map (Sum.inl val✝)))\n\ncase h_map.inr\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX Y : Symmetrify V\nval✝ : Y ⟶ X\n⊢ Φ.map (Sum.inr val✝) =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map (Sum.inr val✝)))",
"state_before": "case h_map\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX Y : Symmetrify V\nf : X ⟶ Y\n⊢ Φ.map f =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map f))",
"tactic": "cases f"
},
{
"state_after": "no goals",
"state_before": "case h_map.inl\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX Y : Symmetrify V\nval✝ : X ⟶ Y\n⊢ Φ.map (Sum.inl val✝) =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map (Sum.inl val✝)))",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h_map.inr\nU : Type ?u.8739\nV : Type u_2\nW : Type ?u.8745\ninst✝⁴ : Quiver U\ninst✝³ : Quiver V\ninst✝² : Quiver W\nV' : Type u_1\ninst✝¹ : Quiver V'\ninst✝ : HasReverse V'\nΦ : Symmetrify V ⥤q V'\nhΦinv : ∀ {X Y : Symmetrify V} (f : X ⟶ Y), Φ.map (reverse f) = reverse (Φ.map f)\nX Y : Symmetrify V\nval✝ : Y ⟶ X\n⊢ Φ.map (Sum.inr val✝) =\n Eq.recOn (_ : (lift (of ⋙q Φ)).obj Y = Φ.obj Y)\n (Eq.recOn (_ : (lift (of ⋙q Φ)).obj X = Φ.obj X) ((lift (of ⋙q Φ)).map (Sum.inr val✝)))",
"tactic": "exact hΦinv (Sum.inl _)"
}
] |
[
224,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.multiplicity_sub_of_gt
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p b < multiplicity p a\n⊢ multiplicity p b < multiplicity p a",
"state_before": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p b < multiplicity p a\n⊢ multiplicity p (a - b) = multiplicity p b",
"tactic": "rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p b < multiplicity p a\n⊢ multiplicity p b < multiplicity p a",
"tactic": "assumption"
}
] |
[
457,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.coe_dualDual
|
[] |
[
956,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
955,
1
] |
Mathlib/CategoryTheory/Sites/Plus.lean
|
CategoryTheory.GrothendieckTopology.toPlus_naturality
|
[
{
"state_after": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ (η ≫ toPlus J Q).app x✝ = (toPlus J P ≫ plusMap J η).app x✝",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\n⊢ η ≫ toPlus J Q = toPlus J P ≫ plusMap J η",
"tactic": "ext"
},
{
"state_after": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫ Cover.toMultiequalizer ⊤ Q ≫ colimit.ι (diagram J Q x✝.unop) ⊤.op =\n (Cover.toMultiequalizer ⊤ P ≫ colimit.ι (diagram J P x✝.unop) ⊤.op) ≫ colimMap (diagramNatTrans J η x✝.unop)",
"state_before": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ (η ≫ toPlus J Q).app x✝ = (toPlus J P ≫ plusMap J η).app x✝",
"tactic": "dsimp [toPlus, plusMap]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op =\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n colimit.ι (diagram J P x✝.unop) ⊤.op) ≫\n colimMap (diagramNatTrans J η x✝.unop)",
"state_before": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫ Cover.toMultiequalizer ⊤ Q ≫ colimit.ι (diagram J Q x✝.unop) ⊤.op =\n (Cover.toMultiequalizer ⊤ P ≫ colimit.ι (diagram J P x✝.unop) ⊤.op) ≫ colimMap (diagramNatTrans J η x✝.unop)",
"tactic": "delta Cover.toMultiequalizer"
},
{
"state_after": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op =\n Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagramNatTrans J η x✝.unop).app ⊤.op ≫ colimit.ι (diagram J Q x✝.unop) ⊤.op",
"state_before": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op =\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n colimit.ι (diagram J P x✝.unop) ⊤.op) ≫\n colimMap (diagramNatTrans J η x✝.unop)",
"tactic": "simp only [ι_colimMap, Category.assoc]"
},
{
"state_after": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ (η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I)) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op =\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagramNatTrans J η x✝.unop).app ⊤.op) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op",
"state_before": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op =\n Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagramNatTrans J η x✝.unop).app ⊤.op ≫ colimit.ι (diagram J Q x✝.unop) ⊤.op",
"tactic": "simp_rw [← Category.assoc]"
},
{
"state_after": "case w.h.e_a\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I) =\n Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagramNatTrans J η x✝.unop).app ⊤.op",
"state_before": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ (η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I)) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op =\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagramNatTrans J η x✝.unop).app ⊤.op) ≫\n colimit.ι (diagram J Q x✝.unop) ⊤.op",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case w.h.e_a\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\n⊢ η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I) =\n Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagramNatTrans J η x✝.unop).app ⊤.op",
"tactic": "exact Multiequalizer.hom_ext _ _ _ (fun I => by dsimp ; simp)"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\nI : (Cover.index ⊤ Q).L\n⊢ (η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I)) ≫\n Multiequalizer.ι (Cover.index ⊤ Q) I =\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (multiequalizer (Cover.index ⊤ P))\n (fun i => Multiequalizer.ι (Cover.index ⊤ P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index ⊤.op.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index ⊤.op.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index ⊤.op.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop Q) i)) ≫\n Multiequalizer.ι (Cover.index ⊤ Q) I",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\nI : (Cover.index ⊤ Q).L\n⊢ (η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝.unop.op) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I)) ≫\n Multiequalizer.ι (Cover.index ⊤ Q) I =\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝.unop.op) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n (diagramNatTrans J η x✝.unop).app ⊤.op) ≫\n Multiequalizer.ι (Cover.index ⊤ Q) I",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nx✝ : Cᵒᵖ\nI : (Cover.index ⊤ Q).L\n⊢ (η.app x✝ ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (Q.obj x✝) (fun I => Q.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ Q).R),\n (fun I => Q.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ Q) I =\n (fun I => Q.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ Q) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ Q) I)) ≫\n Multiequalizer.ι (Cover.index ⊤ Q) I =\n (Multiequalizer.lift (Cover.index ⊤ P) (P.obj x✝) (fun I => P.map I.f.op)\n (_ :\n ∀ (I : (Cover.index ⊤ P).R),\n (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.fst (Cover.index ⊤ P) I =\n (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index ⊤ P) I) ≫\n MulticospanIndex.snd (Cover.index ⊤ P) I) ≫\n Multiequalizer.lift (Cover.index ⊤ Q) (multiequalizer (Cover.index ⊤ P))\n (fun i => Multiequalizer.ι (Cover.index ⊤ P) i ≫ η.app i.Y.op)\n (_ :\n ∀ (i : (Cover.index ⊤.op.unop Q).R),\n (fun i => Multiequalizer.ι (Cover.index ⊤.op.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.fstTo (Cover.index ⊤.op.unop Q) i) ≫\n MulticospanIndex.fst (Cover.index ⊤.op.unop Q) i =\n (fun i => Multiequalizer.ι (Cover.index ⊤.op.unop P) i ≫ η.app i.Y.op)\n (MulticospanIndex.sndTo (Cover.index ⊤.op.unop Q) i) ≫\n MulticospanIndex.snd (Cover.index ⊤.op.unop Q) i)) ≫\n Multiequalizer.ι (Cover.index ⊤ Q) I",
"tactic": "simp"
}
] |
[
234,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Hom.apply_mem_neighborSet
|
[] |
[
1694,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1693,
1
] |
Std/Tactic/Ext.lean
|
PUnit.ext
|
[] |
[
205,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
205,
18
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_mul_of_le_of_one_lt
|
[] |
[
672,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_def
|
[] |
[
83,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.mapFun.pow
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_2\nS : Type u_1\nT : Type ?u.460757\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.460772\nβ : Type ?u.460775\nf : R →+* S\nx y : 𝕎 R\nn : ℕ\n⊢ mapFun (↑f) (x ^ n) = mapFun (↑f) x ^ n",
"tactic": "map_fun_tac"
}
] |
[
133,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.Nonempty.subset_one_iff
|
[] |
[
138,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
BumpCovering.IsSubordinate.mono
|
[] |
[
285,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Products.lean
|
CategoryTheory.Limits.PreservesProduct.iso_hom
|
[] |
[
104,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.castLT_succAbove
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\nx : Fin n\ny : Fin (n + 1)\nh : ↑castSucc x < y\nh' : optParam (↑(↑(succAbove y) x) < n) (_ : ↑(↑(succAbove y) x) < n)\n⊢ castLT (↑(succAbove y) x) h' = x",
"tactic": "simp only [succAbove_below _ _ h, castLT_castSucc]"
}
] |
[
2147,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2145,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toNNReal_top_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.818755\nβ : Type ?u.818758\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\n⊢ ENNReal.toNNReal (⊤ * a) = 0",
"tactic": "simp"
}
] |
[
2203,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2203,
1
] |
Mathlib/Analysis/Complex/OperatorNorm.lean
|
Complex.imClm_norm
|
[
{
"state_after": "no goals",
"state_before": "⊢ 1 = ‖↑imClm I‖",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "⊢ ‖I‖ ≤ 1",
"tactic": "simp"
}
] |
[
58,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.liftOn_div
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] → K[X] → P\nf0 : ∀ (p : K[X]), f p 0 = f 0 1\nH' : ∀ {p q p' q' : K[X]}, q ≠ 0 → q' ≠ 0 → q' * p = q * p' → f p q = f p' q'\nH :\n optParam (∀ {p q p' q' : K[X]}, q ∈ K[X]⁰ → q' ∈ K[X]⁰ → q' * p = q * p' → f p q = f p' q')\n (_ : ∀ {p q p' q' : K[X]}, q ∈ K[X]⁰ → q' ∈ K[X]⁰ → q' * p = q * p' → f p q = f p' q')\n⊢ RatFunc.liftOn (↑(algebraMap K[X] (RatFunc K)) p / ↑(algebraMap K[X] (RatFunc K)) q) f H = f p q",
"tactic": "rw [← mk_eq_div, liftOn_mk _ _ f f0 @H']"
}
] |
[
1013,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1008,
1
] |
Mathlib/Analysis/Convex/Normed.lean
|
convexOn_norm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.38\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nhs : Convex ℝ s\nx : E\nx✝² : x ∈ s\ny : E\nx✝¹ : y ∈ s\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nx✝ : a + b = 1\n⊢ ‖a • x‖ + ‖b • y‖ = a * ‖x‖ + b * ‖y‖",
"tactic": "rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]"
}
] |
[
47,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
Antitone.monovary
|
[] |
[
329,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
11
] |
Mathlib/RingTheory/Coprime/Lemmas.lean
|
exists_sum_eq_one_iff_pairwise_coprime'
|
[
{
"state_after": "case h.e'_2.a\nR : Type u\nI : Type v\ninst✝³ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\ninst✝² : Fintype I\ninst✝¹ : Nonempty I\ninst✝ : DecidableEq I\n⊢ Pairwise (IsCoprime on s) ↔ Pairwise (IsCoprime on fun i => s ↑i)",
"state_before": "R : Type u\nI : Type v\ninst✝³ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\ninst✝² : Fintype I\ninst✝¹ : Nonempty I\ninst✝ : DecidableEq I\n⊢ (∃ μ, ∑ i : I, μ i * ∏ j in {i}ᶜ, s j = 1) ↔ Pairwise (IsCoprime on s)",
"tactic": "convert exists_sum_eq_one_iff_pairwise_coprime Finset.univ_nonempty (s := s) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.a\nR : Type u\nI : Type v\ninst✝³ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\ninst✝² : Fintype I\ninst✝¹ : Nonempty I\ninst✝ : DecidableEq I\n⊢ Pairwise (IsCoprime on s) ↔ Pairwise (IsCoprime on fun i => s ↑i)",
"tactic": "simp only [Function.onFun, pairwise_subtype_iff_pairwise_finset', coe_univ, Set.pairwise_univ]"
}
] |
[
174,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.tendsto_iff_tendstoUniformly
|
[] |
[
277,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_div_assoc'
|
[] |
[
314,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.braid_natural
|
[
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nP Q : C\ninst✝ : HasBinaryBiproducts C\nW X Y Z : C\nf : X ⟶ Y\ng : Z ⟶ W\n⊢ map f g ≫ (braiding Y W).hom = (braiding X Z).hom ≫ map g f",
"tactic": "aesop_cat"
}
] |
[
1838,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1836,
1
] |
Mathlib/Algebra/CharP/Two.lean
|
CharTwo.add_sq
|
[] |
[
90,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.eq_zero_of_le_half
|
[] |
[
424,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
inv_eq_of_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.20829\nM₀ : Type ?u.20832\nG₀ : Type u_1\nM₀' : Type ?u.20838\nG₀' : Type ?u.20841\nF : Type ?u.20844\nF' : Type ?u.20847\ninst✝ : GroupWithZero G₀\na b c g h✝ x : G₀\nh : a * b = 1\n⊢ a⁻¹ = b",
"tactic": "rw [← inv_mul_cancel_left₀ (left_ne_zero_of_mul_eq_one h) b, h, mul_one]"
}
] |
[
295,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackSymmetry_inv_comp_fst
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\n⊢ (pullbackSymmetry f g).inv ≫ pullback.fst = pullback.snd",
"tactic": "simp [Iso.inv_comp_eq]"
}
] |
[
1534,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1533,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.mem_comap
|
[] |
[
832,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
831,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Monotone.map_iInf_of_continuousAt
|
[] |
[
2702,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2700,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.tendsto_nhds_atTop
|
[
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type ?u.427366\n𝕝 : Type ?u.427369\n𝕝₂ : Type ?u.427372\nE : Type u_2\nF : Type u_4\nG : Type ?u.427381\nι : Type u_3\nι' : Type ?u.427387\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\ninst✝¹ : SemilatticeSup F\ninst✝ : Nonempty F\nhp : WithSeminorms p\nu : F → E\ny₀ : E\n⊢ (∀ (i : ι) (ε : ℝ), 0 < ε → ∀ᶠ (x : F) in Filter.atTop, ↑(p i) (u x - y₀) < ε) ↔\n ∀ (i : ι) (ε : ℝ), 0 < ε → ∃ x₀, ∀ (x : F), x₀ ≤ x → ↑(p i) (u x - y₀) < ε",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.427366\n𝕝 : Type ?u.427369\n𝕝₂ : Type ?u.427372\nE : Type u_2\nF : Type u_4\nG : Type ?u.427381\nι : Type u_3\nι' : Type ?u.427387\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\ninst✝¹ : SemilatticeSup F\ninst✝ : Nonempty F\nhp : WithSeminorms p\nu : F → E\ny₀ : E\n⊢ Filter.Tendsto u Filter.atTop (𝓝 y₀) ↔ ∀ (i : ι) (ε : ℝ), 0 < ε → ∃ x₀, ∀ (x : F), x₀ ≤ x → ↑(p i) (u x - y₀) < ε",
"tactic": "rw [hp.tendsto_nhds u y₀]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.427366\n𝕝 : Type ?u.427369\n𝕝₂ : Type ?u.427372\nE : Type u_2\nF : Type u_4\nG : Type ?u.427381\nι : Type u_3\nι' : Type ?u.427387\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nonempty ι\ninst✝² : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\ninst✝¹ : SemilatticeSup F\ninst✝ : Nonempty F\nhp : WithSeminorms p\nu : F → E\ny₀ : E\n⊢ (∀ (i : ι) (ε : ℝ), 0 < ε → ∀ᶠ (x : F) in Filter.atTop, ↑(p i) (u x - y₀) < ε) ↔\n ∀ (i : ι) (ε : ℝ), 0 < ε → ∃ x₀, ∀ (x : F), x₀ ≤ x → ↑(p i) (u x - y₀) < ε",
"tactic": "exact forall₃_congr fun _ _ _ => Filter.eventually_atTop"
}
] |
[
405,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.not_lt_of_mem_argmax
|
[] |
[
132,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.pure_div
|
[] |
[
495,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_union_Ici
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : a ≤ b\nh : c ≤ b\n⊢ Icc a b ∪ Ici c = Ici (min a c)\n\ncase inr\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ a ∨ c ≤ b\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"state_before": "α : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : c ≤ max a b\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"tactic": "cases' le_or_lt a b with hab hab <;> simp [hab] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : a ≤ b\nh : c ≤ b\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"tactic": "exact Icc_union_Ici' h"
},
{
"state_after": "case inr.inl\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ a\n⊢ Icc a b ∪ Ici c = Ici (min a c)\n\ncase inr.inr\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ b\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ a ∨ c ≤ b\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"tactic": "cases' h with h h"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ a\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"tactic": "simp [*]"
},
{
"state_after": "case inr.inr\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ b\nhca : c ≤ a\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"state_before": "case inr.inr\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ b\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"tactic": "have hca : c ≤ a := h.trans hab.le"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nβ : Type ?u.92706\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b < a\nh : c ≤ b\nhca : c ≤ a\n⊢ Icc a b ∪ Ici c = Ici (min a c)",
"tactic": "simp [*]"
}
] |
[
1357,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1351,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.X_pow_comp
|
[
{
"state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\n⊢ comp (X ^ Nat.zero) p = p ^ Nat.zero\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\nk : ℕ\nih : comp (X ^ k) p = p ^ k\n⊢ comp (X ^ Nat.succ k) p = p ^ Nat.succ k",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\nk : ℕ\n⊢ comp (X ^ k) p = p ^ k",
"tactic": "induction' k with k ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\n⊢ comp (X ^ Nat.zero) p = p ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\nk : ℕ\nih : comp (X ^ k) p = p ^ k\n⊢ comp (X ^ Nat.succ k) p = p ^ Nat.succ k",
"tactic": "simp [pow_succ', mul_X_comp, ih]"
}
] |
[
603,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/Algebra/Module/Torsion.lean
|
Submodule.torsionBySet_isInternal
|
[
{
"state_after": "R : Type u_1\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_2\ninst✝ : DecidableEq ι\nS : Finset ι\np : ι → Ideal R\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\nhM : Module.IsTorsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n⊢ (⨆ (t : ι) (_ : t ∈ ↑S), torsionBySet R M ↑(p t)) = ⊤",
"state_before": "R : Type u_1\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_2\ninst✝ : DecidableEq ι\nS : Finset ι\np : ι → Ideal R\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\nhM : Module.IsTorsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n⊢ (⨆ (i : { x // x ∈ S }), torsionBySet R M ↑(p ↑i)) = ⊤",
"tactic": "apply (iSup_subtype'' ↑S fun i => torsionBySet R M <| p i).trans"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_2\ninst✝ : DecidableEq ι\nS : Finset ι\np : ι → Ideal R\nhp : Set.Pairwise ↑S fun i j => p i ⊔ p j = ⊤\nhM : Module.IsTorsionBySet R M ↑(⨅ (i : ι) (_ : i ∈ S), p i)\n⊢ (⨆ (t : ι) (_ : t ∈ ↑S), torsionBySet R M ↑(p t)) = ⊤",
"tactic": "apply (iSup_torsionBySet_ideal_eq_torsionBySet_iInf hp).trans <|\n (Module.isTorsionBySet_iff_torsionBySet_eq_top _).mp hM"
}
] |
[
496,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Logic/Function/Iterate.lean
|
Function.Surjective.iterate
|
[] |
[
98,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieSubmodule.lowerCentralSeries_eq_lcs_comap
|
[
{
"state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ lowerCentralSeries R L { x // x ∈ ↑N } Nat.zero = comap (incl N) (lcs Nat.zero N)\n\ncase succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\n⊢ lowerCentralSeries R L { x // x ∈ ↑N } (Nat.succ k) = comap (incl N) (lcs (Nat.succ k) N)",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)",
"tactic": "induction' k with k ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\n⊢ lowerCentralSeries R L { x // x ∈ ↑N } Nat.zero = comap (incl N) (lcs Nat.zero N)",
"tactic": "simp"
},
{
"state_after": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\n⊢ ⁅⊤, lowerCentralSeries R L { x // x ∈ ↑N } k⁆ = comap (incl N) ⁅⊤, lcs k N⁆",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\n⊢ lowerCentralSeries R L { x // x ∈ ↑N } (Nat.succ k) = comap (incl N) (lcs (Nat.succ k) N)",
"tactic": "simp only [lcs_succ, lowerCentralSeries_succ] at ih ⊢"
},
{
"state_after": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\nthis : lcs k N ≤ LieModuleHom.range (incl N)\n⊢ ⁅⊤, lowerCentralSeries R L { x // x ∈ ↑N } k⁆ = comap (incl N) ⁅⊤, lcs k N⁆",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\n⊢ ⁅⊤, lowerCentralSeries R L { x // x ∈ ↑N } k⁆ = comap (incl N) ⁅⊤, lcs k N⁆",
"tactic": "have : N.lcs k ≤ N.incl.range := by\n rw [N.range_incl]\n apply lcs_le_self"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\nthis : lcs k N ≤ LieModuleHom.range (incl N)\n⊢ ⁅⊤, lowerCentralSeries R L { x // x ∈ ↑N } k⁆ = comap (incl N) ⁅⊤, lcs k N⁆",
"tactic": "rw [ih, LieSubmodule.comap_bracket_eq _ _ N.incl N.ker_incl this]"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\n⊢ lcs k N ≤ N",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\n⊢ lcs k N ≤ LieModuleHom.range (incl N)",
"tactic": "rw [N.range_incl]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\nk : ℕ\nih : lowerCentralSeries R L { x // x ∈ ↑N } k = comap (incl N) (lcs k N)\n⊢ lcs k N ≤ N",
"tactic": "apply lcs_le_self"
}
] |
[
116,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Analysis/Calculus/ParametricIntegral.lean
|
hasDerivAt_integral_of_dominated_loc_of_deriv_le
|
[
{
"state_after": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\n⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀",
"state_before": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\n⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀",
"tactic": "have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos"
},
{
"state_after": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\n⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀",
"state_before": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\n⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀",
"tactic": "have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀ :=\n h_diff.mono fun a ha => ha x₀ x₀_in"
},
{
"state_after": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\nthis : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)\n⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀",
"state_before": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\n⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀",
"tactic": "have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x : 𝕜 => F x a) (ball x₀ ε) := by\n apply (h_diff.and h_bound).mono\n rintro a ⟨ha_deriv, ha_bound⟩\n refine' (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le\n (fun x x_in => (ha_deriv x x_in).hasDerivWithinAt) fun x x_in => _\n rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]\n exact (ha_bound x x_in).trans (le_abs_self _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\nthis : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)\n⊢ Integrable (F' x₀) ∧ HasDerivAt (fun n => ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀",
"tactic": "exact\n hasDerivAt_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas this bound_integrable\n diff_x₀"
},
{
"state_after": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\n⊢ ∀ (x : α),\n ((∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1) ∧\n ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x) →\n LipschitzOnWith (↑Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)",
"state_before": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\n⊢ ∀ᵐ (a : α) ∂μ, LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)",
"tactic": "apply (h_diff.and h_bound).mono"
},
{
"state_after": "case intro\nα : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\na : α\nha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\n⊢ LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)",
"state_before": "α : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\n⊢ ∀ (x : α),\n ((∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1) ∧\n ∀ (x_1 : 𝕜), x_1 ∈ ball x₀ ε → ‖F' x_1 x‖ ≤ bound x) →\n LipschitzOnWith (↑Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)",
"tactic": "rintro a ⟨ha_deriv, ha_bound⟩"
},
{
"state_after": "case intro\nα : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\na : α\nha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nx : 𝕜\nx_in : x ∈ ball x₀ ε\n⊢ ‖F' x a‖₊ ≤ ↑Real.nnabs (bound a)",
"state_before": "case intro\nα : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\na : α\nha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\n⊢ LipschitzOnWith (↑Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)",
"tactic": "refine' (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le\n (fun x x_in => (ha_deriv x x_in).hasDerivWithinAt) fun x x_in => _"
},
{
"state_after": "case intro\nα : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\na : α\nha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nx : 𝕜\nx_in : x ∈ ball x₀ ε\n⊢ ‖F' x a‖ ≤ abs (bound a)",
"state_before": "case intro\nα : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\na : α\nha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nx : 𝕜\nx_in : x ∈ ball x₀ ε\n⊢ ‖F' x a‖₊ ≤ ↑Real.nnabs (bound a)",
"tactic": "rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type ?u.972511\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF F' : 𝕜 → α → E\nx₀ : 𝕜\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable (F' x₀) μ\nbound : α → ℝ\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nx₀_in : x₀ ∈ ball x₀ ε\ndiff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x => F x a) (F' x₀ a) x₀\na : α\nha_deriv : ∀ (x : 𝕜), x ∈ ball x₀ ε → HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : ∀ (x : 𝕜), x ∈ ball x₀ ε → ‖F' x a‖ ≤ bound a\nx : 𝕜\nx_in : x ∈ ball x₀ ε\n⊢ ‖F' x a‖ ≤ abs (bound a)",
"tactic": "exact (ha_bound x x_in).trans (le_abs_self _)"
}
] |
[
250,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/Arctan.lean
|
Measurable.arctan
|
[] |
[
37,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.EventuallyEq.div
|
[] |
[
1533,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1531,
1
] |
Mathlib/Topology/PathConnected.lean
|
IsPathConnected.preimage_coe
|
[
{
"state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\n⊢ IsPathConnected (Subtype.val ⁻¹' W)",
"state_before": "X : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.648254\nF U W : Set X\nhW : IsPathConnected W\nhWU : W ⊆ U\n⊢ IsPathConnected (Subtype.val ⁻¹' W)",
"tactic": "rcases hW with ⟨x, x_in, hx⟩"
},
{
"state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\n⊢ ∀ {y : { x // x ∈ U }}, y ∈ Subtype.val ⁻¹' W → JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } y",
"state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\n⊢ IsPathConnected (Subtype.val ⁻¹' W)",
"tactic": "use ⟨x, hWU x_in⟩, by simp [x_in]"
},
{
"state_after": "case intro.intro.mk\nX : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\ny : X\nhyU : y ∈ U\nhyW : { val := y, property := hyU } ∈ Subtype.val ⁻¹' W\n⊢ JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } { val := y, property := hyU }",
"state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\n⊢ ∀ {y : { x // x ∈ U }}, y ∈ Subtype.val ⁻¹' W → JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } y",
"tactic": "rintro ⟨y, hyU⟩ hyW"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mk\nX : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\ny : X\nhyU : y ∈ U\nhyW : { val := y, property := hyU } ∈ Subtype.val ⁻¹' W\n⊢ JoinedIn (Subtype.val ⁻¹' W) { val := x, property := (_ : x ∈ U) } { val := y, property := hyU }",
"tactic": "exact ⟨(hx hyW).joined_subtype.somePath.map (continuous_inclusion hWU), by simp⟩"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\n⊢ { val := x, property := (_ : x ∈ U) } ∈ Subtype.val ⁻¹' W",
"tactic": "simp [x_in]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.648239\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type ?u.648254\nF U W : Set X\nhWU : W ⊆ U\nx : X\nx_in : x ∈ W\nhx : ∀ {y : X}, y ∈ W → JoinedIn W x y\ny : X\nhyU : y ∈ U\nhyW : { val := y, property := hyU } ∈ Subtype.val ⁻¹' W\n⊢ ∀ (t : ↑I),\n ↑(Path.map\n (Joined.somePath\n (_ :\n Joined { val := x, property := (_ : x ∈ W) }\n { val := ↑{ val := y, property := hyU }, property := (_ : ↑{ val := y, property := hyU } ∈ W) }))\n (_ : Continuous (inclusion hWU)))\n t ∈\n Subtype.val ⁻¹' W",
"tactic": "simp"
}
] |
[
1013,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1008,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.le_toIdeal_homogeneousHull
|
[
{
"state_after": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\n⊢ r ∈ toIdeal (homogeneousHull 𝒜 I)",
"state_before": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\n⊢ I ≤ toIdeal (homogeneousHull 𝒜 I)",
"tactic": "intro r hr"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\n⊢ r ∈ toIdeal (homogeneousHull 𝒜 I)",
"tactic": "classical\nrw [← DirectSum.sum_support_decompose 𝒜 r]\nrefine' Ideal.sum_mem _ _\nintro j _\napply Ideal.subset_span\nuse j\nuse ⟨r, hr⟩"
},
{
"state_after": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) r), ↑(↑(↑(decompose 𝒜) r) i) ∈ toIdeal (homogeneousHull 𝒜 I)",
"state_before": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\n⊢ r ∈ toIdeal (homogeneousHull 𝒜 I)",
"tactic": "rw [← DirectSum.sum_support_decompose 𝒜 r]"
},
{
"state_after": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\n⊢ ∀ (c : ι), c ∈ Dfinsupp.support (↑(decompose 𝒜) r) → ↑(↑(↑(decompose 𝒜) r) c) ∈ toIdeal (homogeneousHull 𝒜 I)",
"state_before": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) r), ↑(↑(↑(decompose 𝒜) r) i) ∈ toIdeal (homogeneousHull 𝒜 I)",
"tactic": "refine' Ideal.sum_mem _ _"
},
{
"state_after": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\nj : ι\na✝ : j ∈ Dfinsupp.support (↑(decompose 𝒜) r)\n⊢ ↑(↑(↑(decompose 𝒜) r) j) ∈ toIdeal (homogeneousHull 𝒜 I)",
"state_before": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\n⊢ ∀ (c : ι), c ∈ Dfinsupp.support (↑(decompose 𝒜) r) → ↑(↑(↑(decompose 𝒜) r) c) ∈ toIdeal (homogeneousHull 𝒜 I)",
"tactic": "intro j _"
},
{
"state_after": "case a\nι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\nj : ι\na✝ : j ∈ Dfinsupp.support (↑(decompose 𝒜) r)\n⊢ ↑(↑(↑(decompose 𝒜) r) j) ∈ {r | ∃ i x, ↑(↑(↑(decompose 𝒜) ↑x) i) = r}",
"state_before": "ι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\nj : ι\na✝ : j ∈ Dfinsupp.support (↑(decompose 𝒜) r)\n⊢ ↑(↑(↑(decompose 𝒜) r) j) ∈ toIdeal (homogeneousHull 𝒜 I)",
"tactic": "apply Ideal.subset_span"
},
{
"state_after": "case a\nι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\nj : ι\na✝ : j ∈ Dfinsupp.support (↑(decompose 𝒜) r)\n⊢ ∃ x, ↑(↑(↑(decompose 𝒜) ↑x) j) = ↑(↑(↑(decompose 𝒜) r) j)",
"state_before": "case a\nι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\nj : ι\na✝ : j ∈ Dfinsupp.support (↑(decompose 𝒜) r)\n⊢ ↑(↑(↑(decompose 𝒜) r) j) ∈ {r | ∃ i x, ↑(↑(↑(decompose 𝒜) ↑x) i) = r}",
"tactic": "use j"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type u_2\nσ : Type u_3\nR : Type ?u.208850\nA : Type u_1\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nr : A\nhr : r ∈ I\nj : ι\na✝ : j ∈ Dfinsupp.support (↑(decompose 𝒜) r)\n⊢ ∃ x, ↑(↑(↑(decompose 𝒜) ↑x) j) = ↑(↑(↑(decompose 𝒜) r) j)",
"tactic": "use ⟨r, hr⟩"
}
] |
[
557,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Data/Quot.lean
|
Trunc.out_eq
|
[] |
[
580,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Data/Finset/Prod.lean
|
Finset.product_eq_biUnion_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.20058\ns✝ s' : Finset α\nt✝ t' : Finset β\na : α\nb : β\ninst✝ : DecidableEq (α × β)\ns : Finset α\nt : Finset β\nx✝ : α × β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ t ↔ (x, y) ∈ Finset.biUnion t fun b => image (fun a => (a, b)) s",
"tactic": "simp only [mem_product, mem_biUnion, mem_image, exists_prop, Prod.mk.inj_iff, and_left_comm,\n exists_and_left, exists_eq_right, exists_eq_left]"
}
] |
[
131,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.moveRight_neg_symm'
|
[
{
"state_after": "no goals",
"state_before": "x : PGame\ni : RightMoves x\n⊢ moveRight x i = -moveLeft (-x) (↑toLeftMovesNeg i)",
"tactic": "simp"
}
] |
[
1301,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1300,
1
] |
Mathlib/Order/Hom/Set.lean
|
OrderIso.symm_image_image
|
[] |
[
36,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.prod_single
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.1506427\nι : Type ui\ninst✝¹ : CommSemiring k\ninst✝ : CommMonoid G\ns✝ : Finset ι\na✝ : ι → G\nb : ι → k\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : ∏ i in s, single (a✝ i) (b i) = single (∏ i in s, a✝ i) (∏ i in s, b i)\n⊢ ∏ i in cons a s has, single (a✝ i) (b i) = single (∏ i in cons a s has, a✝ i) (∏ i in cons a s has, b i)",
"tactic": "rw [prod_cons has, ih, single_mul_single, prod_cons has, prod_cons has]"
}
] |
[
1037,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1034,
1
] |
Mathlib/GroupTheory/FreeProduct.lean
|
FreeProduct.lift_word_prod_nontrivial_of_other_i
|
[
{
"state_after": "ι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\nheq1 : ↑(↑lift f) (NeWord.prod w) = 1\n⊢ False",
"state_before": "ι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\n⊢ ↑(↑lift f) (NeWord.prod w) ≠ 1",
"tactic": "intro heq1"
},
{
"state_after": "ι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\nheq1 : ↑(↑lift f) (NeWord.prod w) = 1\nthis : X k ⊆ X i\n⊢ False",
"state_before": "ι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\nheq1 : ↑(↑lift f) (NeWord.prod w) = 1\n⊢ False",
"tactic": "have : X k ⊆ X i := by simpa [heq1] using lift_word_ping_pong f X hpp w hlast.symm"
},
{
"state_after": "case intro\nι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\nheq1 : ↑(↑lift f) (NeWord.prod w) = 1\nthis : X k ⊆ X i\nx : α\nhx : x ∈ X k\n⊢ False",
"state_before": "ι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\nheq1 : ↑(↑lift f) (NeWord.prod w) = 1\nthis : X k ⊆ X i\n⊢ False",
"tactic": "obtain ⟨x, hx⟩ := hXnonempty k"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\nheq1 : ↑(↑lift f) (NeWord.prod w) = 1\nthis : X k ⊆ X i\nx : α\nhx : x ∈ X k\n⊢ False",
"tactic": "exact (hXdisj hhead).le_bot ⟨hx, this hx⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : ι → Type ?u.747617\ninst✝⁴ : (i : ι) → Monoid (M i)\nN : Type ?u.747628\ninst✝³ : Monoid N\nhnontriv : Nontrivial ι\nG : Type u_3\ninst✝² : Group G\nH : ι → Type u_2\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nhcard : 3 ≤ (#ι) ∨ ∃ i, 3 ≤ (#H i)\nα : Type ?u.748409\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → ↑(f i) h • X j ⊆ X i\ni j k : ι\nw : NeWord H i j\nhhead : k ≠ i\nhlast : k ≠ j\nheq1 : ↑(↑lift f) (NeWord.prod w) = 1\n⊢ X k ⊆ X i",
"tactic": "simpa [heq1] using lift_word_ping_pong f X hpp w hlast.symm"
}
] |
[
745,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/Analysis/InnerProductSpace/Symmetric.lean
|
LinearMap.isSymmetric_id
|
[] |
[
92,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
balancedCore_zero_mem
|
[] |
[
131,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.coe_restrict
|
[] |
[
367,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.congr_apply
|
[] |
[
323,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.Lex.refl_right
|
[] |
[
253,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_singleton_mul_left_unit
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na✝ b a : α\nh2 : IsUnit a\nx : α\n⊢ ∃ a_1, a_1 * x = a * x\n\ncase a\nα : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na✝ b a : α\nh2 : IsUnit a\nx : α\n⊢ ∃ a_1, a_1 * (a * x) = x",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na✝ b a : α\nh2 : IsUnit a\nx : α\n⊢ span {a * x} = span {x}",
"tactic": "apply le_antisymm <;> rw [span_singleton_le_iff_mem, mem_span_singleton']"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na✝ b a : α\nh2 : IsUnit a\nx : α\n⊢ ∃ a_1, a_1 * x = a * x\n\ncase a\nα : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na✝ b a : α\nh2 : IsUnit a\nx : α\n⊢ ∃ a_1, a_1 * (a * x) = x",
"tactic": "exacts [⟨a, rfl⟩, ⟨_, h2.unit.inv_mul_cancel_left x⟩]"
}
] |
[
182,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
Subgroup.le_topologicalClosure
|
[] |
[
721,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
720,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.funMap_eq_coe_constants
|
[] |
[
377,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingIso.mk_coe
|
[] |
[
405,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Order/SupIndep.lean
|
Finset.supIndep_singleton
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.4779\nι : Type u_2\nι' : Type ?u.4785\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\ns✝ t : Finset ι\nf✝ : ι → α\ni✝ i : ι\nf : ι → α\ns : Finset ι\nhs : s ⊆ {i}\nj : ι\nhji : j ∈ {i}\nhj : ¬j ∈ s\n⊢ Disjoint (f j) ⊥",
"state_before": "α : Type u_1\nβ : Type ?u.4779\nι : Type u_2\nι' : Type ?u.4785\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\ns✝ t : Finset ι\nf✝ : ι → α\ni✝ i : ι\nf : ι → α\ns : Finset ι\nhs : s ⊆ {i}\nj : ι\nhji : j ∈ {i}\nhj : ¬j ∈ s\n⊢ Disjoint (f j) (sup s f)",
"tactic": "rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4779\nι : Type u_2\nι' : Type ?u.4785\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\ns✝ t : Finset ι\nf✝ : ι → α\ni✝ i : ι\nf : ι → α\ns : Finset ι\nhs : s ⊆ {i}\nj : ι\nhji : j ∈ {i}\nhj : ¬j ∈ s\n⊢ Disjoint (f j) ⊥",
"tactic": "exact disjoint_bot_right"
}
] |
[
84,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
Units.ne_zero
|
[] |
[
34,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.coe_add
|
[] |
[
686,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.continuous_const_section
|
[
{
"state_after": "ι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\n⊢ ContinuousAt\n (let_fun this := fun x => { fst := x, snd := v };\n this)\n x",
"state_before": "ι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\n⊢ Continuous\n (let_fun this := fun x => { fst := x, snd := v };\n this)",
"tactic": "refine continuous_iff_continuousAt.2 fun x => ?_"
},
{
"state_after": "ι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ ContinuousAt\n (let_fun this := fun x => { fst := x, snd := v };\n this)\n x",
"state_before": "ι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\n⊢ ContinuousAt\n (let_fun this := fun x => { fst := x, snd := v };\n this)\n x",
"tactic": "have A : Z.baseSet (Z.indexAt x) ∈ 𝓝 x :=\n IsOpen.mem_nhds (Z.isOpen_baseSet (Z.indexAt x)) (Z.mem_baseSet_at x)"
},
{
"state_after": "case refine_1\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ (let_fun this := fun x => { fst := x, snd := v };\n this) ⁻¹'\n (localTrivAt Z x).toLocalHomeomorph.toLocalEquiv.source ∈\n 𝓝 x\n\ncase refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ ContinuousAt\n (↑(localTrivAt Z x).toLocalHomeomorph ∘\n let_fun this := fun x => { fst := x, snd := v };\n this)\n x",
"state_before": "ι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ ContinuousAt\n (let_fun this := fun x => { fst := x, snd := v };\n this)\n x",
"tactic": "refine ((Z.localTrivAt x).toLocalHomeomorph.continuousAt_iff_continuousAt_comp_left ?_).2 ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ (let_fun this := fun x => { fst := x, snd := v };\n this) ⁻¹'\n (localTrivAt Z x).toLocalHomeomorph.toLocalEquiv.source ∈\n 𝓝 x",
"tactic": "exact A"
},
{
"state_after": "case refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ ContinuousAt\n (fun x_1 =>\n coordChange Z\n (indexAt Z\n ((let_fun this := fun x => { fst := x, snd := v };\n this)\n x_1).fst)\n (indexAt Z x)\n ((let_fun this := fun x => { fst := x, snd := v };\n this)\n x_1).fst\n ((let_fun this := fun x => { fst := x, snd := v };\n this)\n x_1).snd)\n x",
"state_before": "case refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ ContinuousAt\n (↑(localTrivAt Z x).toLocalHomeomorph ∘\n let_fun this := fun x => { fst := x, snd := v };\n this)\n x",
"tactic": "apply continuousAt_id.prod"
},
{
"state_after": "case refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\nthis : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x))\n⊢ ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x",
"state_before": "case refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\n⊢ ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x",
"tactic": "have : ContinuousOn (fun _ : B => v) (Z.baseSet (Z.indexAt x)) := continuousOn_const"
},
{
"state_after": "case refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\nthis : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x))\ny : B\nhy : y ∈ baseSet Z (indexAt Z x)\n⊢ coordChange Z (indexAt Z y) (indexAt Z x) y v = v",
"state_before": "case refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\nthis : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x))\n⊢ ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x",
"tactic": "refine (this.congr fun y hy ↦ ?_).continuousAt A"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nι : Type u_2\nB : Type u_1\nF : Type u_3\nX : Type ?u.41511\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb : B\na v : F\nh : ∀ (i j : ι) (x : B), x ∈ baseSet Z i ∩ baseSet Z j → coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) ∈ 𝓝 x\nthis : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x))\ny : B\nhy : y ∈ baseSet Z (indexAt Z x)\n⊢ coordChange Z (indexAt Z y) (indexAt Z x) y v = v",
"tactic": "exact h _ _ _ ⟨mem_baseSet_at _ _, hy⟩"
}
] |
[
653,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.symm_comp_self
|
[
{
"state_after": "case h\nR₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.1586825\ninst✝²⁴ : Semiring R₁\ninst✝²³ : Semiring R₂\ninst✝²² : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝²¹ : RingHomInvPair σ₁₂ σ₂₁\ninst✝²⁰ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁹ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁸ : RingHomInvPair σ₃₂ σ₂₃\nσ₁₃ : R₁ →+* R₃\nσ₃₁ : R₃ →+* R₁\ninst✝¹⁷ : RingHomInvPair σ₁₃ σ₃₁\ninst✝¹⁶ : RingHomInvPair σ₃₁ σ₁₃\ninst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nM₁ : Type u_3\ninst✝¹³ : TopologicalSpace M₁\ninst✝¹² : AddCommMonoid M₁\nM'₁ : Type ?u.1587679\ninst✝¹¹ : TopologicalSpace M'₁\ninst✝¹⁰ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝⁹ : TopologicalSpace M₂\ninst✝⁸ : AddCommMonoid M₂\nM₃ : Type ?u.1587697\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : AddCommMonoid M₃\nM₄ : Type ?u.1587706\ninst✝⁵ : TopologicalSpace M₄\ninst✝⁴ : AddCommMonoid M₄\ninst✝³ : Module R₁ M₁\ninst✝² : Module R₁ M'₁\ninst✝¹ : Module R₂ M₂\ninst✝ : Module R₃ M₃\ne : M₁ ≃SL[σ₁₂] M₂\nx : M₁\n⊢ (↑(ContinuousLinearEquiv.symm e) ∘ ↑e) x = id x",
"state_before": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.1586825\ninst✝²⁴ : Semiring R₁\ninst✝²³ : Semiring R₂\ninst✝²² : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝²¹ : RingHomInvPair σ₁₂ σ₂₁\ninst✝²⁰ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁹ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁸ : RingHomInvPair σ₃₂ σ₂₃\nσ₁₃ : R₁ →+* R₃\nσ₃₁ : R₃ →+* R₁\ninst✝¹⁷ : RingHomInvPair σ₁₃ σ₃₁\ninst✝¹⁶ : RingHomInvPair σ₃₁ σ₁₃\ninst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nM₁ : Type u_3\ninst✝¹³ : TopologicalSpace M₁\ninst✝¹² : AddCommMonoid M₁\nM'₁ : Type ?u.1587679\ninst✝¹¹ : TopologicalSpace M'₁\ninst✝¹⁰ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝⁹ : TopologicalSpace M₂\ninst✝⁸ : AddCommMonoid M₂\nM₃ : Type ?u.1587697\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : AddCommMonoid M₃\nM₄ : Type ?u.1587706\ninst✝⁵ : TopologicalSpace M₄\ninst✝⁴ : AddCommMonoid M₄\ninst✝³ : Module R₁ M₁\ninst✝² : Module R₁ M'₁\ninst✝¹ : Module R₂ M₂\ninst✝ : Module R₃ M₃\ne : M₁ ≃SL[σ₁₂] M₂\n⊢ ↑(ContinuousLinearEquiv.symm e) ∘ ↑e = id",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.1586825\ninst✝²⁴ : Semiring R₁\ninst✝²³ : Semiring R₂\ninst✝²² : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝²¹ : RingHomInvPair σ₁₂ σ₂₁\ninst✝²⁰ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁹ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁸ : RingHomInvPair σ₃₂ σ₂₃\nσ₁₃ : R₁ →+* R₃\nσ₃₁ : R₃ →+* R₁\ninst✝¹⁷ : RingHomInvPair σ₁₃ σ₃₁\ninst✝¹⁶ : RingHomInvPair σ₃₁ σ₁₃\ninst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nM₁ : Type u_3\ninst✝¹³ : TopologicalSpace M₁\ninst✝¹² : AddCommMonoid M₁\nM'₁ : Type ?u.1587679\ninst✝¹¹ : TopologicalSpace M'₁\ninst✝¹⁰ : AddCommMonoid M'₁\nM₂ : Type u_4\ninst✝⁹ : TopologicalSpace M₂\ninst✝⁸ : AddCommMonoid M₂\nM₃ : Type ?u.1587697\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : AddCommMonoid M₃\nM₄ : Type ?u.1587706\ninst✝⁵ : TopologicalSpace M₄\ninst✝⁴ : AddCommMonoid M₄\ninst✝³ : Module R₁ M₁\ninst✝² : Module R₁ M'₁\ninst✝¹ : Module R₂ M₂\ninst✝ : Module R₃ M₃\ne : M₁ ≃SL[σ₁₂] M₂\nx : M₁\n⊢ (↑(ContinuousLinearEquiv.symm e) ∘ ↑e) x = id x",
"tactic": "exact symm_apply_apply e x"
}
] |
[
2084,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2082,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherian_iff_wellFounded
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\nP : Type ?u.112850\nN : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nthis : (WellFounded fun x x_1 => x > x_1) ↔ ∀ (k : Submodule R M), CompleteLattice.IsCompactElement k\n⊢ IsNoetherian R M ↔ WellFounded fun x x_1 => x > x_1",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type ?u.112850\nN : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\n⊢ IsNoetherian R M ↔ WellFounded fun x x_1 => x > x_1",
"tactic": "have := (CompleteLattice.wellFounded_characterisations <| Submodule R M).out 0 3"
},
{
"state_after": "R : Type u_1\nM : Type u_2\nP : Type ?u.112850\nN : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nthis : (WellFounded fun x x_1 => x > x_1) ↔ ∀ (k : Submodule R M), CompleteLattice.IsCompactElement k\n⊢ IsNoetherian R M ↔ ∀ (k : Submodule R M), CompleteLattice.IsCompactElement k",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type ?u.112850\nN : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nthis : (WellFounded fun x x_1 => x > x_1) ↔ ∀ (k : Submodule R M), CompleteLattice.IsCompactElement k\n⊢ IsNoetherian R M ↔ WellFounded fun x x_1 => x > x_1",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\nP : Type ?u.112850\nN : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nthis : (WellFounded fun x x_1 => x > x_1) ↔ ∀ (k : Submodule R M), CompleteLattice.IsCompactElement k\n⊢ IsNoetherian R M ↔ ∀ (k : Submodule R M), CompleteLattice.IsCompactElement k",
"tactic": "exact\n ⟨fun ⟨h⟩ => fun k => (fg_iff_compact k).mp (h k), fun h =>\n ⟨fun k => (fg_iff_compact k).mpr (h k)⟩⟩"
}
] |
[
308,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_right
|
[] |
[
906,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
901,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.finset_sup_mul
|
[] |
[
572,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.map_le_map_iff_of_injective
|
[] |
[
983,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
982,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.blsub_eq_lsub'
|
[] |
[
1782,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1780,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.mem_sublistsLen
|
[] |
[
336,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.multiplicity_le_multiplicity_of_dvd_left
|
[] |
[
334,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Computability/Primrec.lean
|
Nat.Primrec.swap'
|
[
{
"state_after": "no goals",
"state_before": "f : ℕ → ℕ → ℕ\nhf : Nat.Primrec (unpaired f)\nn : ℕ\n⊢ unpaired f (unpaired (swap Nat.pair) n) = unpaired (swap f) n",
"tactic": "simp"
}
] |
[
126,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.inf_subgroupOf_left
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.292171\ninst✝⁴ : Group G\ninst✝³ : Group G'\nA : Type ?u.292180\ninst✝² : AddGroup A\nH✝ K✝ : Subgroup G\nk : Set G\nN : Type ?u.292201\ninst✝¹ : Group N\nP : Type ?u.292207\ninst✝ : Group P\nH K : Subgroup G\n⊢ subgroupOf (K ⊓ H) K = subgroupOf H K",
"tactic": "rw [inf_comm, inf_subgroupOf_right]"
}
] |
[
1682,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1681,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean
|
Real.hasDerivWithinAt_arcsin_Iic
|
[
{
"state_after": "case inl\nh : -1 ≠ 1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - (-1) ^ 2)) (Iic (-1)) (-1)\n\ncase inr\nx : ℝ\nh : x ≠ 1\nh' : ¬x = -1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - x ^ 2)) (Iic x) x",
"state_before": "x : ℝ\nh : x ≠ 1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - x ^ 2)) (Iic x) x",
"tactic": "rcases em (x = -1) with (rfl | h')"
},
{
"state_after": "no goals",
"state_before": "case inl\nh : -1 ≠ 1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - (-1) ^ 2)) (Iic (-1)) (-1)",
"tactic": "convert (hasDerivWithinAt_const (-1 : ℝ) _ (-(π / 2))).congr _ _ <;>\n simp (config := { contextual := true }) [arcsin_of_le_neg_one]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nh : x ≠ 1\nh' : ¬x = -1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - x ^ 2)) (Iic x) x",
"tactic": "exact (hasDerivAt_arcsin h' h).hasDerivWithinAt"
}
] |
[
82,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.kroneckerMap_reindex_right
|
[] |
[
177,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.codisjoint_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.177622\nι : Sort ?u.177625\nκ : ι → Sort ?u.177630\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns s₁ s₂ : UpperSet α\nt t₁ t₂ : UpperSet β\nx : α × β\n⊢ Codisjoint (s₁ ×ˢ t₁) (s₂ ×ˢ t₂) ↔ Codisjoint s₁ s₂ ∨ Codisjoint t₁ t₂",
"tactic": "simp_rw [codisjoint_iff, prod_sup_prod, prod_eq_top]"
}
] |
[
1637,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1635,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
left_lt_toIocMod
|
[] |
[
103,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
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