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/Fin/Basic.lean
|
Fin.orderEmbedding_eq
|
[] |
[
611,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
610,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/UV.lean
|
UV.compress_mem_compression_of_mem_compression
|
[
{
"state_after": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\n⊢ compress u v a ∈ s ∧ compress u v (compress u v a) ∈ s ∨\n ¬compress u v a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = compress u v a",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ 𝓒 u v s\n⊢ compress u v a ∈ 𝓒 u v s",
"tactic": "rw [mem_compression] at ha⊢"
},
{
"state_after": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\n⊢ compress u v a ∈ s ∧ compress u v a ∈ s ∨ ¬compress u v a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = compress u v a",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\n⊢ compress u v a ∈ s ∧ compress u v (compress u v a) ∈ s ∨\n ¬compress u v a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = compress u v a",
"tactic": "simp only [compress_idem, exists_prop]"
},
{
"state_after": "case inl.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nleft✝ : a ∈ s\nha : compress u v a ∈ s\n⊢ compress u v a ∈ s ∧ compress u v a ∈ s ∨ ¬compress u v a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = compress u v a\n\ncase inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ b : α\nhb : b ∈ s\nleft✝ : ¬compress u v b ∈ s\n⊢ compress u v (compress u v b) ∈ s ∧ compress u v (compress u v b) ∈ s ∨\n ¬compress u v (compress u v b) ∈ s ∧ ∃ b_1, b_1 ∈ s ∧ compress u v b_1 = compress u v (compress u v b)",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\n⊢ compress u v a ∈ s ∧ compress u v a ∈ s ∨ ¬compress u v a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = compress u v a",
"tactic": "obtain ⟨_, ha⟩ | ⟨_, b, hb, rfl⟩ := ha"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nleft✝ : a ∈ s\nha : compress u v a ∈ s\n⊢ compress u v a ∈ s ∧ compress u v a ∈ s ∨ ¬compress u v a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = compress u v a",
"tactic": "exact Or.inl ⟨ha, ha⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ b : α\nhb : b ∈ s\nleft✝ : ¬compress u v b ∈ s\n⊢ compress u v (compress u v b) ∈ s ∧ compress u v (compress u v b) ∈ s ∨\n ¬compress u v (compress u v b) ∈ s ∧ ∃ b_1, b_1 ∈ s ∧ compress u v b_1 = compress u v (compress u v b)",
"tactic": "exact Or.inr ⟨by rwa [compress_idem], b, hb, (compress_idem _ _ _).symm⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v b✝ b : α\nhb : b ∈ s\nleft✝ : ¬compress u v b ∈ s\n⊢ ¬compress u v (compress u v b) ∈ s",
"tactic": "rwa [compress_idem]"
}
] |
[
190,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Data/Vector/Mem.lean
|
Vector.mem_cons_self
|
[] |
[
61,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.smul_subset_smul_right
|
[] |
[
183,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.isBounded_iff_eventually
|
[] |
[
701,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
697,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.union_empty_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u s t : Set α\n⊢ s ∪ t ⊆ ∅ ↔ s ⊆ ∅ ∧ t ⊆ ∅",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u s t : Set α\n⊢ s ∪ t = ∅ ↔ s = ∅ ∧ t = ∅",
"tactic": "simp only [← subset_empty_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u s t : Set α\n⊢ s ∪ t ⊆ ∅ ↔ s ⊆ ∅ ∧ t ⊆ ∅",
"tactic": "exact union_subset_iff"
}
] |
[
873,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean
|
Doset.out_eq'
|
[] |
[
138,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.norm_inv_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : DecidableEq α\nx : FreeGroup α\n⊢ norm x⁻¹ = norm x",
"tactic": "simp only [norm, toWord_inv, invRev_length]"
}
] |
[
1411,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1410,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
IsFractionRing.ideal_span_singleton_map_subset
|
[
{
"state_after": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\nx : L\nhx : x ∈ ↑(Ideal.span {↑(algebraMap S L) a})\n⊢ x ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\n⊢ ↑(Ideal.span {↑(algebraMap S L) a}) ⊆ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "intro x hx"
},
{
"state_after": "case intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\nx' : L\nhx : ↑(algebraMap S L) a * x' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\n⊢ ↑(algebraMap S L) a * x' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\nx : L\nhx : x ∈ ↑(Ideal.span {↑(algebraMap S L) a})\n⊢ x ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "obtain ⟨x', rfl⟩ := Ideal.mem_span_singleton.mp hx"
},
{
"state_after": "case intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "case intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\nx' : L\nhx : ↑(algebraMap S L) a * x' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\n⊢ ↑(algebraMap S L) a * x' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "obtain ⟨y', z', rfl⟩ := IsLocalization.mk'_surjective S⁰ x'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "case intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "obtain ⟨y, z, hz0, yz_eq⟩ :=\n IsIntegralClosure.exists_smul_eq_mul alg inj y' (nonZeroDivisors.coe_ne_zero z')"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "have injRS : Function.Injective (algebraMap R S) := by\n refine'\n Function.Injective.of_comp (show Function.Injective (algebraMap S L ∘ algebraMap R S) from _)\n rwa [← RingHom.coe_comp, ← IsScalarTower.algebraMap_eq]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "have hz0' : algebraMap R S z ∈ S⁰ :=\n map_mem_nonZeroDivisors (algebraMap R S) injRS (mem_nonZeroDivisors_of_ne_zero hz0)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "have mk_yz_eq : IsLocalization.mk' L y' z' = IsLocalization.mk' L y ⟨_, hz0'⟩ := by\n rw [Algebra.smul_def, mul_comm _ y, mul_comm _ y'] at yz_eq\n exact IsLocalization.mk'_eq_of_eq (by rw [mul_comm _ y, mul_comm _ y', yz_eq])"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\nhy : ↑(algebraMap S L) (a * y) ∈ Submodule.span K (↑(algebraMap S L) '' b)\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))\n\ncase hy\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) (a * y) ∈ Submodule.span K (↑(algebraMap S L) '' b)",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "suffices hy : algebraMap S L (a * y) ∈ Submodule.span K ((algebraMap S L) '' b)"
},
{
"state_after": "case hy\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) (a * y) ∈ ↑(Submodule.span R (↑(algebraMap S L) '' b))",
"state_before": "case hy\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) (a * y) ∈ Submodule.span K (↑(algebraMap S L) '' b)",
"tactic": "refine' Submodule.span_subset_span R K _ _"
},
{
"state_after": "case hy\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) (a * y) ∈ ↑(Submodule.map (↑R (Algebra.linearMap S L)) (Submodule.span R b))",
"state_before": "case hy\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) (a * y) ∈ ↑(Submodule.span R (↑(algebraMap S L) '' b))",
"tactic": "rw [Submodule.span_algebraMap_image_of_tower]"
},
{
"state_after": "no goals",
"state_before": "case hy\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\n⊢ ↑(algebraMap S L) (a * y) ∈ ↑(Submodule.map (↑R (Algebra.linearMap S L)) (Submodule.span R b))",
"tactic": "exact Submodule.mem_map_of_mem (h (Ideal.mem_span_singleton.mpr ⟨y, rfl⟩))"
},
{
"state_after": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\n⊢ Function.Injective (↑(algebraMap S L) ∘ ↑(algebraMap R S))",
"state_before": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\n⊢ Function.Injective ↑(algebraMap R S)",
"tactic": "refine'\n Function.Injective.of_comp (show Function.Injective (algebraMap S L ∘ algebraMap R S) from _)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\n⊢ Function.Injective (↑(algebraMap S L) ∘ ↑(algebraMap R S))",
"tactic": "rwa [← RingHom.coe_comp, ← IsScalarTower.algebraMap_eq]"
},
{
"state_after": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : y' * ↑(algebraMap R S) z = y * ↑z'\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\n⊢ mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }",
"state_before": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\n⊢ mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }",
"tactic": "rw [Algebra.smul_def, mul_comm _ y, mul_comm _ y'] at yz_eq"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : y' * ↑(algebraMap R S) z = y * ↑z'\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\n⊢ mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }",
"tactic": "exact IsLocalization.mk'_eq_of_eq (by rw [mul_comm _ y, mul_comm _ y', yz_eq])"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : y' * ↑(algebraMap R S) z = y * ↑z'\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\n⊢ ↑z' * y = ↑{ val := ↑(algebraMap R S) z, property := hz0' } * y'",
"tactic": "rw [mul_comm _ y, mul_comm _ y', yz_eq]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\nhy : ↑(algebraMap S L) (a * y) ∈ Submodule.span K (↑(algebraMap S L) '' b)\n⊢ (↑(algebraMap R K) z)⁻¹ • ↑(algebraMap S L) (a * y) ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\nhy : ↑(algebraMap S L) (a * y) ∈ Submodule.span K (↑(algebraMap S L) '' b)\n⊢ ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "rw [mk_yz_eq, IsFractionRing.mk'_eq_div, ← IsScalarTower.algebraMap_apply,\n IsScalarTower.algebraMap_apply R K L, div_eq_mul_inv, ← mul_assoc, mul_comm, ← map_inv₀, ←\n Algebra.smul_def, ← _root_.map_mul]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹⁷ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\nP : Type ?u.667790\ninst✝¹⁴ : CommRing P\nA : Type ?u.667796\nK : Type u_4\ninst✝¹³ : CommRing A\ninst✝¹² : IsDomain A\nL : Type u_1\ninst✝¹¹ : IsDomain R\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : Algebra R K\ninst✝⁶ : Algebra R L\ninst✝⁵ : Algebra S L\ninst✝⁴ : IsIntegralClosure S R L\ninst✝³ : IsFractionRing S L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R S L\ninst✝ : IsScalarTower R K L\na : S\nb : Set S\nalg : Algebra.IsAlgebraic R L\ninj : Function.Injective ↑(algebraMap R L)\nh : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)\ny' : S\nz' : { x // x ∈ S⁰ }\nhx : ↑(algebraMap S L) a * mk' L y' z' ∈ ↑(Ideal.span {↑(algebraMap S L) a})\ny : S\nz : R\nhz0 : z ≠ 0\nyz_eq : z • y' = ↑z' * y\ninjRS : Function.Injective ↑(algebraMap R S)\nhz0' : ↑(algebraMap R S) z ∈ S⁰\nmk_yz_eq : mk' L y' z' = mk' L y { val := ↑(algebraMap R S) z, property := hz0' }\nhy : ↑(algebraMap S L) (a * y) ∈ Submodule.span K (↑(algebraMap S L) '' b)\n⊢ (↑(algebraMap R K) z)⁻¹ • ↑(algebraMap S L) (a * y) ∈ ↑(Submodule.span K (↑(algebraMap S L) '' b))",
"tactic": "exact (Submodule.span K _).smul_mem _ hy"
}
] |
[
464,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Order/Directed.lean
|
directed_of_sup
|
[] |
[
113,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
adjoin_le_integralClosure
|
[
{
"state_after": "R : Type u_1\nA : Type u_2\nB : Type ?u.694032\nS : Type ?u.694035\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nhx : IsIntegral R x\n⊢ {x} ⊆ ↑(integralClosure R A)",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.694032\nS : Type ?u.694035\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nhx : IsIntegral R x\n⊢ Algebra.adjoin R {x} ≤ integralClosure R A",
"tactic": "rw [Algebra.adjoin_le_iff]"
},
{
"state_after": "R : Type u_1\nA : Type u_2\nB : Type ?u.694032\nS : Type ?u.694035\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nhx : IsIntegral R x\n⊢ x ∈ integralClosure R A",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.694032\nS : Type ?u.694035\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nhx : IsIntegral R x\n⊢ {x} ⊆ ↑(integralClosure R A)",
"tactic": "simp only [SetLike.mem_coe, Set.singleton_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.694032\nS : Type ?u.694035\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nhx : IsIntegral R x\n⊢ x ∈ integralClosure R A",
"tactic": "exact hx"
}
] |
[
571,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.reachable_comm
|
[] |
[
1881,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1880,
1
] |
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean
|
Asymptotics.IsEquivalent.tendsto_atBot
|
[
{
"state_after": "case h.e'_3\nα : Type u_2\nβ : Type u_1\ninst✝¹ : NormedLinearOrderedField β\nu v : α → β\nl : Filter α\ninst✝ : OrderTopology β\nhuv : u ~[l] v\nhu : Tendsto u l atBot\n⊢ v = Neg.neg ∘ fun x => -v x",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝¹ : NormedLinearOrderedField β\nu v : α → β\nl : Filter α\ninst✝ : OrderTopology β\nhuv : u ~[l] v\nhu : Tendsto u l atBot\n⊢ Tendsto v l atBot",
"tactic": "convert tendsto_neg_atTop_atBot.comp (huv.neg.tendsto_atTop <| tendsto_neg_atBot_atTop.comp hu)"
},
{
"state_after": "case h.e'_3.h\nα : Type u_2\nβ : Type u_1\ninst✝¹ : NormedLinearOrderedField β\nu v : α → β\nl : Filter α\ninst✝ : OrderTopology β\nhuv : u ~[l] v\nhu : Tendsto u l atBot\nx✝ : α\n⊢ v x✝ = (Neg.neg ∘ fun x => -v x) x✝",
"state_before": "case h.e'_3\nα : Type u_2\nβ : Type u_1\ninst✝¹ : NormedLinearOrderedField β\nu v : α → β\nl : Filter α\ninst✝ : OrderTopology β\nhuv : u ~[l] v\nhu : Tendsto u l atBot\n⊢ v = Neg.neg ∘ fun x => -v x",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nα : Type u_2\nβ : Type u_1\ninst✝¹ : NormedLinearOrderedField β\nu v : α → β\nl : Filter α\ninst✝ : OrderTopology β\nhuv : u ~[l] v\nhu : Tendsto u l atBot\nx✝ : α\n⊢ v x✝ = (Neg.neg ∘ fun x => -v x) x✝",
"tactic": "simp"
}
] |
[
325,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Data/Polynomial/DenomsClearable.lean
|
DenomsClearable.add
|
[
{
"state_after": "R : Type u_1\nK : Type u_2\ninst✝¹ : Semiring R\ninst✝ : CommSemiring K\ni : R →+* K\na b : R\nbi : K\nN : ℕ\nf g : R[X]\nx✝¹ : DenomsClearable a b N f i\nx✝ : DenomsClearable a b N g i\nDf : R\nbf : K\nbfu : bf * ↑i b = 1\nHf : ↑i Df = ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f)\nDg : R\nbg : K\nbgu : bg * ↑i b = 1\nHg : ↑i Dg = ↑i b ^ N * eval (↑i a * bg) (Polynomial.map i g)\n⊢ ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f) + ↑i b ^ N * eval (↑i a * bg) (Polynomial.map i g) =\n ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f) + ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i g)",
"state_before": "R : Type u_1\nK : Type u_2\ninst✝¹ : Semiring R\ninst✝ : CommSemiring K\ni : R →+* K\na b : R\nbi : K\nN : ℕ\nf g : R[X]\nx✝¹ : DenomsClearable a b N f i\nx✝ : DenomsClearable a b N g i\nDf : R\nbf : K\nbfu : bf * ↑i b = 1\nHf : ↑i Df = ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f)\nDg : R\nbg : K\nbgu : bg * ↑i b = 1\nHg : ↑i Dg = ↑i b ^ N * eval (↑i a * bg) (Polynomial.map i g)\n⊢ ↑i (Df + Dg) = ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i (f + g))",
"tactic": "rw [RingHom.map_add, Polynomial.map_add, eval_add, mul_add, Hf, Hg]"
},
{
"state_after": "case e_a.e_a.e_a.e_a\nR : Type u_1\nK : Type u_2\ninst✝¹ : Semiring R\ninst✝ : CommSemiring K\ni : R →+* K\na b : R\nbi : K\nN : ℕ\nf g : R[X]\nx✝¹ : DenomsClearable a b N f i\nx✝ : DenomsClearable a b N g i\nDf : R\nbf : K\nbfu : bf * ↑i b = 1\nHf : ↑i Df = ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f)\nDg : R\nbg : K\nbgu : bg * ↑i b = 1\nHg : ↑i Dg = ↑i b ^ N * eval (↑i a * bg) (Polynomial.map i g)\n⊢ bg = bf",
"state_before": "R : Type u_1\nK : Type u_2\ninst✝¹ : Semiring R\ninst✝ : CommSemiring K\ni : R →+* K\na b : R\nbi : K\nN : ℕ\nf g : R[X]\nx✝¹ : DenomsClearable a b N f i\nx✝ : DenomsClearable a b N g i\nDf : R\nbf : K\nbfu : bf * ↑i b = 1\nHf : ↑i Df = ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f)\nDg : R\nbg : K\nbgu : bg * ↑i b = 1\nHg : ↑i Dg = ↑i b ^ N * eval (↑i a * bg) (Polynomial.map i g)\n⊢ ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f) + ↑i b ^ N * eval (↑i a * bg) (Polynomial.map i g) =\n ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f) + ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i g)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_a.e_a.e_a\nR : Type u_1\nK : Type u_2\ninst✝¹ : Semiring R\ninst✝ : CommSemiring K\ni : R →+* K\na b : R\nbi : K\nN : ℕ\nf g : R[X]\nx✝¹ : DenomsClearable a b N f i\nx✝ : DenomsClearable a b N g i\nDf : R\nbf : K\nbfu : bf * ↑i b = 1\nHf : ↑i Df = ↑i b ^ N * eval (↑i a * bf) (Polynomial.map i f)\nDg : R\nbg : K\nbgu : bg * ↑i b = 1\nHg : ↑i Dg = ↑i b ^ N * eval (↑i a * bg) (Polynomial.map i g)\n⊢ bg = bf",
"tactic": "refine' @inv_unique K _ (i b) bg bf _ _ <;> rwa [mul_comm]"
}
] |
[
68,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.add_lt_add_iff_left
|
[
{
"state_after": "case h\nx y b c : ℝ\ny✝ : CauSeq ℚ abs\n⊢ c + mk y✝ < c + b ↔ mk y✝ < b",
"state_before": "x y a b c : ℝ\n⊢ c + a < c + b ↔ a < b",
"tactic": "induction a using Real.ind_mk"
},
{
"state_after": "case h.h\nx y c : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ c + mk y✝¹ < c + mk y✝ ↔ mk y✝¹ < mk y✝",
"state_before": "case h\nx y b c : ℝ\ny✝ : CauSeq ℚ abs\n⊢ c + mk y✝ < c + b ↔ mk y✝ < b",
"tactic": "induction b using Real.ind_mk"
},
{
"state_after": "case h.h.h\nx y : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝ + mk y✝² < mk y✝ + mk y✝¹ ↔ mk y✝² < mk y✝¹",
"state_before": "case h.h\nx y c : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ c + mk y✝¹ < c + mk y✝ ↔ mk y✝¹ < mk y✝",
"tactic": "induction c using Real.ind_mk"
},
{
"state_after": "case h.h.h\nx y : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝ + y✝² < y✝ + y✝¹ ↔ y✝² < y✝¹",
"state_before": "case h.h.h\nx y : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝ + mk y✝² < mk y✝ + mk y✝¹ ↔ mk y✝² < mk y✝¹",
"tactic": "simp only [mk_lt, ← mk_add]"
},
{
"state_after": "case h.h.h\nx y : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ Pos (y✝ + y✝¹ - (y✝ + y✝²)) ↔ Pos (y✝¹ - y✝²)",
"state_before": "case h.h.h\nx y : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝ + y✝² < y✝ + y✝¹ ↔ y✝² < y✝¹",
"tactic": "show Pos _ ↔ Pos _"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nx y : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ Pos (y✝ + y✝¹ - (y✝ + y✝²)) ↔ Pos (y✝¹ - y✝²)",
"tactic": "rw [add_sub_add_left_eq_sub]"
}
] |
[
374,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.le_op_norm
|
[
{
"state_after": "case intro.intro\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ∀ (x : E), ‖↑f x‖ ≤ C * ‖x‖\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"tactic": "obtain ⟨C, _, hC⟩ := f.bound"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ∀ (x : E), ‖↑f x‖ ≤ C * ‖x‖\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"tactic": "replace hC := hC x"
},
{
"state_after": "case pos\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh✝ : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\nh : ‖x‖ = 0\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖\n\ncase neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh✝ : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\nh : ¬‖x‖ = 0\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"tactic": "by_cases h : ‖x‖ = 0"
},
{
"state_after": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh✝ : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\nh : ¬‖x‖ = 0\nhlt : 0 < ‖x‖\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh✝ : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\nh : ¬‖x‖ = 0\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"tactic": "have hlt : 0 < ‖x‖ := lt_of_le_of_ne (norm_nonneg x) (Ne.symm h)"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh✝ : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\nh : ¬‖x‖ = 0\nhlt : 0 < ‖x‖\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"tactic": "exact (div_le_iff hlt).mp\n (le_csInf bounds_nonempty fun c ⟨_, hc⟩ => (div_le_iff hlt).mpr <| by apply hc)"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh✝ : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\nh : ‖x‖ = 0\n⊢ ‖↑f x‖ ≤ ‖f‖ * ‖x‖",
"tactic": "rwa [h, MulZeroClass.mul_zero] at hC⊢"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.300329\nE : Type u_2\nEₗ : Type ?u.300335\nF : Type u_1\nFₗ : Type ?u.300341\nG : Type ?u.300344\nGₗ : Type ?u.300347\n𝓕 : Type ?u.300350\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh✝ : F →SL[σ₂₃] G\nx : E\nC : ℝ\nleft✝¹ : 0 < C\nhC : ‖↑f x‖ ≤ C * ‖x‖\nh : ¬‖x‖ = 0\nhlt : 0 < ‖x‖\nc : ℝ\nx✝ : c ∈ {c | 0 ≤ c ∧ ∀ (x : E), ‖↑f x‖ ≤ c * ‖x‖}\nleft✝ : 0 ≤ c\nhc : ∀ (x : E), ‖↑f x‖ ≤ c * ‖x‖\n⊢ ‖↑f x‖ ≤ c * ‖x‖",
"tactic": "apply hc"
}
] |
[
207,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
Submodule.exists_lieSubalgebra_coe_eq_iff
|
[
{
"state_after": "case mp\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\np : Submodule R L\n⊢ (∃ K, K.toSubmodule = p) → ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p\n\ncase mpr\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\np : Submodule R L\n⊢ (∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p) → ∃ K, K.toSubmodule = p",
"state_before": "R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\np : Submodule R L\n⊢ (∃ K, K.toSubmodule = p) ↔ ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nK : LieSubalgebra R L\nx✝ y✝ : L\n⊢ x✝ ∈ K.toSubmodule → y✝ ∈ K.toSubmodule → ⁅x✝, y✝⁆ ∈ K.toSubmodule",
"state_before": "case mp\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\np : Submodule R L\n⊢ (∃ K, K.toSubmodule = p) → ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p",
"tactic": "rintro ⟨K, rfl⟩ _ _"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nK : LieSubalgebra R L\nx✝ y✝ : L\n⊢ x✝ ∈ K.toSubmodule → y✝ ∈ K.toSubmodule → ⁅x✝, y✝⁆ ∈ K.toSubmodule",
"tactic": "exact K.lie_mem'"
},
{
"state_after": "case mpr\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\np : Submodule R L\nh : ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p\n⊢ ∃ K, K.toSubmodule = p",
"state_before": "case mpr\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\np : Submodule R L\n⊢ (∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p) → ∃ K, K.toSubmodule = p",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\np : Submodule R L\nh : ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p\n⊢ ∃ K, K.toSubmodule = p",
"tactic": "use { p with lie_mem' := h _ _ }"
}
] |
[
367,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
round_natCast
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.251851\nα : Type u_1\nβ : Type ?u.251857\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nn : ℕ\n⊢ round ↑n = ↑n",
"tactic": "simp [round]"
}
] |
[
1347,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1347,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.map_subtype_span_singleton
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₂ : Type ?u.205557\nK : Type ?u.205560\nM : Type u_2\nM₂ : Type ?u.205566\nV : Type ?u.205569\nS : Type ?u.205572\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np✝ p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\np : Submodule R M\nx : { x // x ∈ p }\n⊢ map (Submodule.subtype p) (span R {x}) = span R {↑x}",
"tactic": "simp [← span_image]"
}
] |
[
588,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.BicartesianSq.of_has_biproduct₂
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproduct X Y\n⊢ BicartesianSq 0 0 biprod.inl biprod.inr",
"tactic": "convert of_is_biproduct₂ (BinaryBiproduct.isBilimit X Y)"
}
] |
[
987,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
985,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.div_nonempty
|
[] |
[
595,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
594,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieModuleHom.coe_restrictLie
|
[] |
[
268,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
CharP.char_ne_zero_of_finite
|
[
{
"state_after": "R : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Finite R\ninst✝ : CharP R 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : NonAssocRing R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Finite R\n⊢ p ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "R : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Finite R\ninst✝ : CharP R 0\nthis : CharZero R\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Finite R\ninst✝ : CharP R 0\n⊢ False",
"tactic": "haveI : CharZero R := charP_to_charZero R"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Finite R\ninst✝ : CharP R 0\nthis : CharZero R\nval✝ : Fintype R\n⊢ False",
"state_before": "R : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Finite R\ninst✝ : CharP R 0\nthis : CharZero R\n⊢ False",
"tactic": "cases nonempty_fintype R"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Finite R\ninst✝ : CharP R 0\nthis : CharZero R\nval✝ : Fintype R\n⊢ False",
"tactic": "exact absurd Nat.cast_injective (not_injective_infinite_finite ((↑) : ℕ → R))"
}
] |
[
483,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.mkRat_def
|
[] |
[
89,
88
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
89,
1
] |
Mathlib/Data/Nat/Totient.lean
|
Nat.totient_dvd_of_dvd
|
[
{
"state_after": "case inl\nb : ℕ\nh : 0 ∣ b\n⊢ φ 0 ∣ φ b\n\ncase inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\n⊢ φ a ∣ φ b",
"state_before": "a b : ℕ\nh : a ∣ b\n⊢ φ a ∣ φ b",
"tactic": "rcases eq_or_ne a 0 with (rfl | ha0)"
},
{
"state_after": "case inr.inl\na : ℕ\nha0 : a ≠ 0\nh : a ∣ 0\n⊢ φ a ∣ φ 0\n\ncase inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ φ a ∣ φ b",
"state_before": "case inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\n⊢ φ a ∣ φ b",
"tactic": "rcases eq_or_ne b 0 with (rfl | hb0)"
},
{
"state_after": "case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : (factorization a).support ⊆ (factorization b).support\n⊢ φ a ∣ φ b",
"state_before": "case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ φ a ∣ φ b",
"tactic": "have hab' : a.factorization.support ⊆ b.factorization.support := by\n intro p\n simp only [support_factorization, List.mem_toFinset]\n apply factors_subset_of_dvd h hb0"
},
{
"state_after": "case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : (factorization a).support ⊆ (factorization b).support\n⊢ (Finsupp.prod (factorization a) fun p k => p ^ (k - 1) * (p - 1)) ∣\n Finsupp.prod (factorization b) fun p k => p ^ (k - 1) * (p - 1)",
"state_before": "case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : (factorization a).support ⊆ (factorization b).support\n⊢ φ a ∣ φ b",
"tactic": "rw [totient_eq_prod_factorization ha0, totient_eq_prod_factorization hb0]"
},
{
"state_after": "case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : (factorization a).support ⊆ (factorization b).support\np : ℕ\nx✝ : p ∈ (factorization a).support\n⊢ p ^ (↑(factorization a) p - 1) ∣ p ^ (↑(factorization b) p - 1)",
"state_before": "case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : (factorization a).support ⊆ (factorization b).support\n⊢ (Finsupp.prod (factorization a) fun p k => p ^ (k - 1) * (p - 1)) ∣\n Finsupp.prod (factorization b) fun p k => p ^ (k - 1) * (p - 1)",
"tactic": "refine' Finsupp.prod_dvd_prod_of_subset_of_dvd hab' fun p _ => mul_dvd_mul _ dvd_rfl"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\na b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\nhab' : (factorization a).support ⊆ (factorization b).support\np : ℕ\nx✝ : p ∈ (factorization a).support\n⊢ p ^ (↑(factorization a) p - 1) ∣ p ^ (↑(factorization b) p - 1)",
"tactic": "exact pow_dvd_pow p (tsub_le_tsub_right ((factorization_le_iff_dvd ha0 hb0).2 h p) 1)"
},
{
"state_after": "no goals",
"state_before": "case inl\nb : ℕ\nh : 0 ∣ b\n⊢ φ 0 ∣ φ b",
"tactic": "simp [zero_dvd_iff.1 h]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\na : ℕ\nha0 : a ≠ 0\nh : a ∣ 0\n⊢ φ a ∣ φ 0",
"tactic": "simp"
},
{
"state_after": "a b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\np : ℕ\n⊢ p ∈ (factorization a).support → p ∈ (factorization b).support",
"state_before": "a b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ (factorization a).support ⊆ (factorization b).support",
"tactic": "intro p"
},
{
"state_after": "a b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\np : ℕ\n⊢ p ∈ factors a → p ∈ factors b",
"state_before": "a b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\np : ℕ\n⊢ p ∈ (factorization a).support → p ∈ (factorization b).support",
"tactic": "simp only [support_factorization, List.mem_toFinset]"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nh : a ∣ b\nha0 : a ≠ 0\nhb0 : b ≠ 0\np : ℕ\n⊢ p ∈ factors a → p ∈ factors b",
"tactic": "apply factors_subset_of_dvd h hb0"
}
] |
[
373,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.add_apply
|
[] |
[
197,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.nfp_mul_eq_opow_omega
|
[
{
"state_after": "case inl\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : a = 0\n⊢ nfp (fun x => a * x) b = a ^ ω\n\ncase inr\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ nfp (fun x => a * x) b = a ^ ω",
"state_before": "a b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\n⊢ nfp (fun x => a * x) b = a ^ ω",
"tactic": "cases' eq_zero_or_pos a with ha ha"
},
{
"state_after": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ nfp (fun x => a * x) b ≤ a ^ ω\n\ncase inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ a ^ ω ≤ nfp (fun x => a * x) b",
"state_before": "case inr\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ nfp (fun x => a * x) b = a ^ ω",
"tactic": "apply le_antisymm"
},
{
"state_after": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ nfp (fun x => a * x) 1 ≤ nfp (fun x => a * x) b",
"state_before": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ a ^ ω ≤ nfp (fun x => a * x) b",
"tactic": "rw [← nfp_mul_one ha]"
},
{
"state_after": "no goals",
"state_before": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ nfp (fun x => a * x) 1 ≤ nfp (fun x => a * x) b",
"tactic": "exact nfp_monotone (mul_isNormal ha).monotone (one_le_iff_pos.2 hb)"
},
{
"state_after": "case inl\na b : Ordinal\nhb : 0 < b\nhba : b ≤ 0\nha : a = 0\n⊢ nfp (fun x => 0 * x) b = 0",
"state_before": "case inl\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : a = 0\n⊢ nfp (fun x => a * x) b = a ^ ω",
"tactic": "rw [ha, zero_opow omega_ne_zero] at hba ⊢"
},
{
"state_after": "case inl\na b : Ordinal\nhb : 0 < b\nhba : b ≤ 0\nha : a = 0\n⊢ id 0 = 0",
"state_before": "case inl\na b : Ordinal\nhb : 0 < b\nhba : b ≤ 0\nha : a = 0\n⊢ nfp (fun x => 0 * x) b = 0",
"tactic": "rw [Ordinal.le_zero.1 hba, nfp_zero_mul]"
},
{
"state_after": "no goals",
"state_before": "case inl\na b : Ordinal\nhb : 0 < b\nhba : b ≤ 0\nha : a = 0\n⊢ id 0 = 0",
"tactic": "rfl"
},
{
"state_after": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ (fun x x_1 => x * x_1) a (a ^ ω) ≤ a ^ ω",
"state_before": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ nfp (fun x => a * x) b ≤ a ^ ω",
"tactic": "apply nfp_le_fp (mul_isNormal ha).monotone hba"
},
{
"state_after": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ a * a ^ ω ≤ a ^ ω",
"state_before": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ (fun x x_1 => x * x_1) a (a ^ ω) ≤ a ^ ω",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case inr.a\na b : Ordinal\nhb : 0 < b\nhba : b ≤ a ^ ω\nha : 0 < a\n⊢ a * a ^ ω ≤ a ^ ω",
"tactic": "rw [← opow_one_add, one_add_omega]"
}
] |
[
663,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Data/List/Indexes.lean
|
List.length_mapIdx
|
[
{
"state_after": "case nil\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ f : ℕ → α → β\n⊢ length (mapIdx f []) = length []\n\ncase cons\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ : ℕ → α → β\nhd : α\ntl : List α\nIH : ∀ (f : ℕ → α → β), length (mapIdx f tl) = length tl\nf : ℕ → α → β\n⊢ length (mapIdx f (hd :: tl)) = length (hd :: tl)",
"state_before": "α✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nl : List α\nf : ℕ → α → β\n⊢ length (mapIdx f l) = length l",
"tactic": "induction' l with hd tl IH generalizing f"
},
{
"state_after": "no goals",
"state_before": "case nil\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ f : ℕ → α → β\n⊢ length (mapIdx f []) = length []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα✝ : Type u\nβ✝ : Type v\nα : Type u_1\nβ : Type u_2\nf✝ : ℕ → α → β\nhd : α\ntl : List α\nIH : ∀ (f : ℕ → α → β), length (mapIdx f tl) = length tl\nf : ℕ → α → β\n⊢ length (mapIdx f (hd :: tl)) = length (hd :: tl)",
"tactic": "simp [IH]"
}
] |
[
194,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
AffineBasis.coord_apply_combination_of_not_mem
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.106545\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : Ring k\ninst✝ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\nhi : ¬i ∈ s\nw : ι → k\nhw : Finset.sum s w = 1\n⊢ ↑(coord b i) (↑(Finset.affineCombination k s ↑b) w) = 0",
"tactic": "classical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_false,\n mul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,\n s.map_affineCombination b w hw]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.106545\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : Ring k\ninst✝ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\nhi : ¬i ∈ s\nw : ι → k\nhw : Finset.sum s w = 1\n⊢ ↑(coord b i) (↑(Finset.affineCombination k s ↑b) w) = 0",
"tactic": "simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_false,\nmul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,\ns.map_affineCombination b w hw]"
}
] |
[
202,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Std/Data/Int/DivMod.lean
|
Int.dvd_mul_left
|
[] |
[
613,
79
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
613,
11
] |
Mathlib/Analysis/InnerProductSpace/Orientation.lean
|
Orientation.volumeForm_comp_linearIsometryEquiv
|
[
{
"state_after": "case zero\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\n_i : Fact (finrank ℝ E = Nat.zero)\no : Orientation ℝ E (Fin Nat.zero)\nx : Fin Nat.zero → E\n⊢ ↑(volumeForm o) (↑φ ∘ x) = ↑(volumeForm o) x\n\ncase succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\n⊢ ↑(volumeForm o) (↑φ ∘ x) = ↑(volumeForm o) x",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n)\no : Orientation ℝ E (Fin n)\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nx : Fin n → E\n⊢ ↑(volumeForm o) (↑φ ∘ x) = ↑(volumeForm o) x",
"tactic": "cases' n with n"
},
{
"state_after": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ ↑(volumeForm o) (↑φ ∘ x) = ↑(volumeForm o) x",
"state_before": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\n⊢ ↑(volumeForm o) (↑φ ∘ x) = ↑(volumeForm o) x",
"tactic": "haveI : FiniteDimensional ℝ E := fact_finiteDimensional_of_finrank_eq_succ n"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ o = ↑(map (Fin (Nat.succ n)) φ.toLinearEquiv) o\n\ncase h.e'_3.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ x = ↑(LinearIsometryEquiv.symm φ) ∘ ↑φ ∘ x",
"state_before": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ ↑(volumeForm o) (↑φ ∘ x) = ↑(volumeForm o) x",
"tactic": "convert o.volumeForm_map φ (φ ∘ x)"
},
{
"state_after": "no goals",
"state_before": "case zero\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\n_i : Fact (finrank ℝ E = Nat.zero)\no : Orientation ℝ E (Fin Nat.zero)\nx : Fin Nat.zero → E\n⊢ ↑(volumeForm o) (↑φ ∘ x) = ↑(volumeForm o) x",
"tactic": "refine' o.eq_or_eq_neg_of_isEmpty.elim _ _ <;> rintro rfl <;> simp"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ ↑(map (Fin (Nat.succ n)) φ.toLinearEquiv) o = o",
"state_before": "case h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ o = ↑(map (Fin (Nat.succ n)) φ.toLinearEquiv) o",
"tactic": "symm"
},
{
"state_after": "case h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ Fintype.card (Fin (Nat.succ n)) = finrank ℝ E",
"state_before": "case h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ ↑(map (Fin (Nat.succ n)) φ.toLinearEquiv) o = o",
"tactic": "rwa [← o.map_eq_iff_det_pos φ.toLinearEquiv] at hφ"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ Fintype.card (Fin (Nat.succ n)) = finrank ℝ E",
"tactic": "rw [_i.out, Fintype.card_fin]"
},
{
"state_after": "case h.e'_3.h.e'_6.h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\nx✝ : Fin (Nat.succ n)\n⊢ x x✝ = (↑(LinearIsometryEquiv.symm φ) ∘ ↑φ ∘ x) x✝",
"state_before": "case h.e'_3.h.e'_6\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\n⊢ x = ↑(LinearIsometryEquiv.symm φ) ∘ ↑φ ∘ x",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6.h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < ↑LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = Nat.succ n)\no : Orientation ℝ E (Fin (Nat.succ n))\nx : Fin (Nat.succ n) → E\nthis : FiniteDimensional ℝ E\nx✝ : Fin (Nat.succ n)\n⊢ x x✝ = (↑(LinearIsometryEquiv.symm φ) ∘ ↑φ ∘ x) x✝",
"tactic": "simp"
}
] |
[
349,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Algebra/Order/Module.lean
|
smul_pos_iff_of_neg
|
[
{
"state_after": "k : Type u_1\nM : Type u_2\nN : Type ?u.36780\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ -c • a < 0 ↔ a < 0",
"state_before": "k : Type u_1\nM : Type u_2\nN : Type ?u.36780\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ 0 < c • a ↔ a < 0",
"tactic": "rw [← neg_neg c, neg_smul, neg_pos]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nM : Type u_2\nN : Type ?u.36780\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n⊢ -c • a < 0 ↔ a < 0",
"tactic": "exact smul_neg_iff_of_pos (neg_pos_of_neg hc)"
}
] |
[
88,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
IsPGroup.exists_le_sylow
|
[
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nP : Subgroup G\nhP : IsPGroup p { x // x ∈ P }\nc : Set (Subgroup G)\nhc1 : c ⊆ {Q | IsPGroup p { x // x ∈ Q }}\nhc2 : IsChain (fun x x_1 => x ≤ x_1) c\nQ : Subgroup G\nhQ : Q ∈ c\nx✝ :\n { x //\n x ∈\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ : ∀ {g : G} (h : G), (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) },\n inv_mem' :=\n (_ :\n ∀ {g : G},\n g ∈\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ :\n ∀ {g : G} (h : G),\n (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) }.toSubsemigroup.carrier →\n g⁻¹ ∈\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ :\n ∀ {g : G} (h : G),\n (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) }.toSubsemigroup.carrier) } }\ng : G\nS : ↑c\nhg : g ∈ (fun R => ↑↑R) S\nk : ℕ\nhk : { val := g, property := hg } ^ p ^ k = 1\n⊢ { val := g, property := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ g ∈ t) } ^ p ^ k = 1",
"state_before": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nP : Subgroup G\nhP : IsPGroup p { x // x ∈ P }\nc : Set (Subgroup G)\nhc1 : c ⊆ {Q | IsPGroup p { x // x ∈ Q }}\nhc2 : IsChain (fun x x_1 => x ≤ x_1) c\nQ : Subgroup G\nhQ : Q ∈ c\nx✝ :\n { x //\n x ∈\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ : ∀ {g : G} (h : G), (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) },\n inv_mem' :=\n (_ :\n ∀ {g : G},\n g ∈\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ :\n ∀ {g : G} (h : G),\n (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) }.toSubsemigroup.carrier →\n g⁻¹ ∈\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ :\n ∀ {g : G} (h : G),\n (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) }.toSubsemigroup.carrier) } }\ng : G\nS : ↑c\nhg : g ∈ (fun R => ↑↑R) S\n⊢ ∃ k, { val := g, property := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ g ∈ t) } ^ p ^ k = 1",
"tactic": "refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nP : Subgroup G\nhP : IsPGroup p { x // x ∈ P }\nc : Set (Subgroup G)\nhc1 : c ⊆ {Q | IsPGroup p { x // x ∈ Q }}\nhc2 : IsChain (fun x x_1 => x ≤ x_1) c\nQ : Subgroup G\nhQ : Q ∈ c\nx✝ :\n { x //\n x ∈\n {\n toSubmonoid :=\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ : ∀ {g : G} (h : G), (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) },\n inv_mem' :=\n (_ :\n ∀ {g : G},\n g ∈\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ :\n ∀ {g : G} (h : G),\n (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) }.toSubsemigroup.carrier →\n g⁻¹ ∈\n {\n toSubsemigroup :=\n { carrier := ⋃ (R : ↑c), ↑↑R,\n mul_mem' :=\n (_ :\n ∀ {g : G} (h : G),\n (g ∈ ⋃ (R : ↑c), ↑↑R) → (h ∈ ⋃ (R : ↑c), ↑↑R) → g * h ∈ ⋃ (R : ↑c), ↑↑R) },\n one_mem' := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ 1 ∈ t) }.toSubsemigroup.carrier) } }\ng : G\nS : ↑c\nhg : g ∈ (fun R => ↑↑R) S\nk : ℕ\nhk : { val := g, property := hg } ^ p ^ k = 1\n⊢ { val := g, property := (_ : ∃ t, (t ∈ Set.range fun R => ↑↑R) ∧ g ∈ t) } ^ p ^ k = 1",
"tactic": "rwa [Subtype.ext_iff, coe_pow] at hk⊢"
}
] |
[
164,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.dist_eq_of_L2
|
[
{
"state_after": "p : ℝ≥0∞\n𝕜 : Type ?u.289391\n𝕜' : Type ?u.289394\nι : Type u_2\nα : ι → Type ?u.289402\nβ✝ : ι → Type ?u.289407\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx y : PiLp 2 β\n⊢ Real.sqrt (∑ i : ι, ‖(x - y) i‖ ^ 2) = Real.sqrt (∑ x_1 : ι, ‖x x_1 - y x_1‖ ^ 2)",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.289391\n𝕜' : Type ?u.289394\nι : Type u_2\nα : ι → Type ?u.289402\nβ✝ : ι → Type ?u.289407\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx y : PiLp 2 β\n⊢ dist x y = Real.sqrt (∑ i : ι, dist (x i) (y i) ^ 2)",
"tactic": "simp_rw [dist_eq_norm, norm_eq_of_L2, Pi.sub_apply]"
},
{
"state_after": "no goals",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.289391\n𝕜' : Type ?u.289394\nι : Type u_2\nα : ι → Type ?u.289402\nβ✝ : ι → Type ?u.289407\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\nβ : ι → Type u_1\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx y : PiLp 2 β\n⊢ Real.sqrt (∑ i : ι, ‖(x - y) i‖ ^ 2) = Real.sqrt (∑ x_1 : ι, ‖x x_1 - y x_1‖ ^ 2)",
"tactic": "rfl"
}
] |
[
606,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/GroupTheory/Coset.lean
|
Subgroup.card_dvd_of_injective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : Group α\ns t : Subgroup α\nH : Type u_2\ninst✝² : Group H\ninst✝¹ : Fintype α\ninst✝ : Fintype H\nf : α →* H\nhf : Injective ↑f\n⊢ card α ∣ card H",
"tactic": "classical calc\n card α = card (f.range : Subgroup H) := card_congr (Equiv.ofInjective f hf)\n _ ∣ card H := card_subgroup_dvd_card _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : Group α\ns t : Subgroup α\nH : Type u_2\ninst✝² : Group H\ninst✝¹ : Fintype α\ninst✝ : Fintype H\nf : α →* H\nhf : Injective ↑f\n⊢ card α ∣ card H",
"tactic": "calc\ncard α = card (f.range : Subgroup H) := card_congr (Equiv.ofInjective f hf)\n_ ∣ card H := card_subgroup_dvd_card _"
}
] |
[
814,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
810,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_neg_of_neg_of_pos
|
[
{
"state_after": "a b : Int\nha : a < 0\nhb : 0 < b\nh : a * b < 0 * b\n⊢ a * b < 0",
"state_before": "a b : Int\nha : a < 0\nhb : 0 < b\n⊢ a * b < 0",
"tactic": "have h : a * b < 0 * b := Int.mul_lt_mul_of_pos_right ha hb"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nha : a < 0\nhb : 0 < b\nh : a * b < 0 * b\n⊢ a * b < 0",
"tactic": "rwa [Int.zero_mul] at h"
}
] |
[
1202,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1200,
11
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.coe_comap
|
[] |
[
293,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
|
MeasureTheory.SignedMeasure.restrictNonposSeq_subset
|
[
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.19730\ninst✝³ : MeasurableSpace α\nM : Type ?u.19736\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nn✝ : ℕ\n⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s\n (i \\\n ⋃ (k : ℕ) (H : k ≤ n✝),\n let_fun this := (_ : k < Nat.succ n✝);\n MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊆\n i",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.19730\ninst✝³ : MeasurableSpace α\nM : Type ?u.19736\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nn✝ : ℕ\n⊢ MeasureTheory.SignedMeasure.restrictNonposSeq s i (Nat.succ n✝) ⊆ i",
"tactic": "rw [restrictNonposSeq]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.19730\ninst✝³ : MeasurableSpace α\nM : Type ?u.19736\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nn✝ : ℕ\n⊢ MeasureTheory.SignedMeasure.someExistsOneDivLT s\n (i \\\n ⋃ (k : ℕ) (H : k ≤ n✝),\n let_fun this := (_ : k < Nat.succ n✝);\n MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ⊆\n i",
"tactic": "exact someExistsOneDivLT_subset'"
}
] |
[
180,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
9
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
bit1_pos
|
[] |
[
303,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Order/CompleteLatticeIntervals.lean
|
subset_sSup_of_within
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Set α\ninst✝¹ : SupSet α\ninst✝ : Inhabited ↑s\nt : Set ↑s\nh : sSup (Subtype.val '' t) ∈ s\n⊢ sSup (Subtype.val '' t) = ↑(sSup t)",
"tactic": "simp [dif_pos h]"
}
] |
[
60,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.mem_closure_range_iff_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.304312\nι : Type ?u.304315\ninst✝ : PseudoMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ\ns : Set α\ne : β → α\na : α\n⊢ (∀ (i : ℕ), True → ∃ y, y ∈ range e ∧ y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)",
"tactic": "simp only [mem_ball, dist_comm, exists_range_iff, forall_const]"
}
] |
[
1910,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1907,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix'_apply
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj : n\n⊢ ↑f (↑(stdBasis R (fun x => R) j) 1) i = ↑f (fun j' => if j' = j then 1 else 0) i",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj : n\n⊢ ↑toMatrix' f i j = ↑f (fun j' => if j' = j then 1 else 0) i",
"tactic": "simp only [LinearMap.toMatrix', LinearEquiv.coe_mk, of_apply]"
},
{
"state_after": "R : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj : n\n⊢ ↑f (↑(stdBasis R (fun x => R) j) 1) = ↑f fun j' => if j' = j then 1 else 0",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj : n\n⊢ ↑f (↑(stdBasis R (fun x => R) j) 1) i = ↑f (fun j' => if j' = j then 1 else 0) i",
"tactic": "refine congr_fun ?_ _"
},
{
"state_after": "case h.e_6.h\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj : n\n⊢ ↑(stdBasis R (fun x => R) j) 1 = fun j' => if j' = j then 1 else 0",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj : n\n⊢ ↑f (↑(stdBasis R (fun x => R) j) 1) = ↑f fun j' => if j' = j then 1 else 0",
"tactic": "congr"
},
{
"state_after": "case h.e_6.h.h\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj j' : n\n⊢ ↑(stdBasis R (fun x => R) j) 1 j' = if j' = j then 1 else 0",
"state_before": "case h.e_6.h\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj : n\n⊢ ↑(stdBasis R (fun x => R) j) 1 = fun j' => if j' = j then 1 else 0",
"tactic": "ext j'"
},
{
"state_after": "case h.e_6.h.h.inl\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj j' : n\nh : j' = j\n⊢ ↑(stdBasis R (fun x => R) j) 1 j' = 1\n\ncase h.e_6.h.h.inr\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj j' : n\nh : ¬j' = j\n⊢ ↑(stdBasis R (fun x => R) j) 1 j' = 0",
"state_before": "case h.e_6.h.h\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj j' : n\n⊢ ↑(stdBasis R (fun x => R) j) 1 j' = if j' = j then 1 else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case h.e_6.h.h.inr\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj j' : n\nh : ¬j' = j\n⊢ ↑(stdBasis R (fun x => R) j) 1 j' = 0",
"tactic": "apply stdBasis_ne _ _ _ _ h"
},
{
"state_after": "no goals",
"state_before": "case h.e_6.h.h.inl\nR : Type u_1\ninst✝² : CommSemiring R\nk : Type ?u.456748\nl : Type ?u.456751\nm : Type u_3\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nf : (n → R) →ₗ[R] m → R\ni : m\nj j' : n\nh : j' = j\n⊢ ↑(stdBasis R (fun x => R) j) 1 j' = 1",
"tactic": "rw [h, stdBasis_same]"
}
] |
[
348,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.ker_restrict
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.1239960\nR₂ : Type ?u.1239963\nR₃ : Type ?u.1239966\nR₄ : Type ?u.1239969\nS : Type ?u.1239972\nK : Type ?u.1239975\nK₂ : Type ?u.1239978\nM : Type u_3\nM' : Type ?u.1239984\nM₁ : Type u_1\nM₂ : Type ?u.1239990\nM₃ : Type ?u.1239993\nM₄ : Type ?u.1239996\nN : Type ?u.1239999\nN₂ : Type ?u.1240002\nι : Type ?u.1240005\nV : Type ?u.1240008\nV₂ : Type ?u.1240011\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring R₂\ninst✝¹⁰ : Semiring R₃\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₂\ninst✝⁷ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁶ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝² : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type ?u.1240421\nsc : SemilinearMapClass F τ₁₂ M M₂\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\np : Submodule R M\nq : Submodule R M₁\nf : M →ₗ[R] M₁\nhf : ∀ (x : M), x ∈ p → ↑f x ∈ q\n⊢ ker (restrict f hf) = ker (domRestrict f p)",
"tactic": "rw [restrict_eq_codRestrict_domRestrict, ker_codRestrict]"
}
] |
[
1384,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1381,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.compl_infty
|
[] |
[
140,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.germ_stalkSpecializes
|
[] |
[
331,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.head_cons
|
[] |
[
149,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
src/lean/Init/Control/Lawful.lean
|
ReaderT.run_seqRight
|
[] |
[
207,
57
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
206,
9
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
volume_preimage_coe
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1870561\nγ : Type ?u.1870564\nδ : Type ?u.1870567\nι : Type ?u.1870570\nR : Type ?u.1870573\nR' : Type ?u.1870576\ninst✝ : MeasureSpace α\ns t : Set α\nhs : NullMeasurableSet s\nht : MeasurableSet t\n⊢ ↑↑volume (Subtype.val ⁻¹' t) = ↑↑volume (t ∩ s)",
"tactic": "rw [volume_set_coe_def,\n comap_apply₀ _ _ Subtype.coe_injective\n (fun h => MeasurableSet.nullMeasurableSet_subtype_coe hs)\n (measurable_subtype_coe ht).nullMeasurableSet,\n image_preimage_eq_inter_range, Subtype.range_coe]"
}
] |
[
4276,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4270,
1
] |
Mathlib/Order/Lattice.lean
|
Antitone.map_inf
|
[] |
[
1207,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1205,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.comap_sup_eq
|
[] |
[
3159,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3156,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.nth_succ_cons
|
[] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
div_le_self_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝³ : Group α\ninst✝² : LE α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na✝ b✝ c d a b : α\n⊢ a / b ≤ a ↔ 1 ≤ b",
"tactic": "simp only [div_eq_mul_inv, mul_le_iff_le_one_right', Left.inv_le_one_iff]"
}
] |
[
365,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
VitaliFamily.eventually_measure_lt_top
|
[
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.594552\ninst✝¹ : NormedAddCommGroup E\ninst✝ : IsLocallyFiniteMeasure μ\nx : α\nε : ℝ\nεpos : 0 < ε\nμε : ↑↑μ (closedBall x ε) < ⊤\n⊢ ∀ᶠ (a : Set α) in filterAt v x, ↑↑μ a < ⊤",
"state_before": "α : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.594552\ninst✝¹ : NormedAddCommGroup E\ninst✝ : IsLocallyFiniteMeasure μ\nx : α\n⊢ ∀ᶠ (a : Set α) in filterAt v x, ↑↑μ a < ⊤",
"tactic": "obtain ⟨ε, εpos, με⟩ : ∃ (ε : ℝ), 0 < ε ∧ μ (closedBall x ε) < ∞ :=\n (μ.finiteAt_nhds x).exists_mem_basis nhds_basis_closedBall"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝² : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.594552\ninst✝¹ : NormedAddCommGroup E\ninst✝ : IsLocallyFiniteMeasure μ\nx : α\nε : ℝ\nεpos : 0 < ε\nμε : ↑↑μ (closedBall x ε) < ⊤\n⊢ ∀ᶠ (a : Set α) in filterAt v x, ↑↑μ a < ⊤",
"tactic": "exact v.eventually_filterAt_iff.2 ⟨ε, εpos, fun a _ haε => (measure_mono haε).trans_lt με⟩"
}
] |
[
128,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.add_smul_sub_mem
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.228490\nβ : Type ?u.228493\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nh : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ t • (y - x) + x ∈ s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.228490\nβ : Type ?u.228493\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nh : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ x + t • (y - x) ∈ s",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.228490\nβ : Type ?u.228493\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nh : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ t • (y - x) + x ∈ s",
"tactic": "exact h.lineMap_mem hx hy ht"
}
] |
[
486,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.continuum_mul_aleph0
|
[] |
[
162,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.curry0_norm
|
[] |
[
1666,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1664,
1
] |
Mathlib/Algebra/Star/Unitary.lean
|
unitary.coe_div
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : GroupWithZero R\ninst✝ : StarSemigroup R\nU₁ U₂ : { x // x ∈ unitary R }\n⊢ ↑(U₁ / U₂) = ↑U₁ / ↑U₂",
"tactic": "simp only [div_eq_mul_inv, coe_inv, Submonoid.coe_mul]"
}
] |
[
175,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Analysis/Seminorm.lean
|
absorbent_ball
|
[
{
"state_after": "R : Type ?u.1455685\nR' : Type ?u.1455688\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1455694\n𝕜₃ : Type ?u.1455697\n𝕝 : Type ?u.1455700\nE : Type u_1\nE₂ : Type ?u.1455706\nE₃ : Type ?u.1455709\nF : Type ?u.1455712\nG : Type ?u.1455715\nι : Type ?u.1455718\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nx : E\nhx : ‖x‖ < r\n⊢ Absorbent 𝕜 (Seminorm.ball (normSeminorm 𝕜 E) x r)",
"state_before": "R : Type ?u.1455685\nR' : Type ?u.1455688\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1455694\n𝕜₃ : Type ?u.1455697\n𝕝 : Type ?u.1455700\nE : Type u_1\nE₂ : Type ?u.1455706\nE₃ : Type ?u.1455709\nF : Type ?u.1455712\nG : Type ?u.1455715\nι : Type ?u.1455718\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nx : E\nhx : ‖x‖ < r\n⊢ Absorbent 𝕜 (Metric.ball x r)",
"tactic": "rw [← ball_normSeminorm 𝕜]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1455685\nR' : Type ?u.1455688\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1455694\n𝕜₃ : Type ?u.1455697\n𝕝 : Type ?u.1455700\nE : Type u_1\nE₂ : Type ?u.1455706\nE₃ : Type ?u.1455709\nF : Type ?u.1455712\nG : Type ?u.1455715\nι : Type ?u.1455718\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nx : E\nhx : ‖x‖ < r\n⊢ Absorbent 𝕜 (Seminorm.ball (normSeminorm 𝕜 E) x r)",
"tactic": "exact (normSeminorm _ _).absorbent_ball hx"
}
] |
[
1221,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1219,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.pow_def
|
[] |
[
128,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Topology/Basic.lean
|
diff_subset_closure_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t✝ : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ s \\ t ⊆ closure t ↔ s ⊆ closure t",
"tactic": "rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]"
}
] |
[
468,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.u_eq
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\n⊢ u y = z → ∀ (x : α), x ≤ z ↔ l x ≤ y\n\ncase mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\n⊢ (∀ (x : α), x ≤ z ↔ l x ≤ y) → u y = z",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\n⊢ u y = z ↔ ∀ (x : α), x ≤ z ↔ l x ≤ y",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\ny : β\nx : α\n⊢ x ≤ u y ↔ l x ≤ y",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\n⊢ u y = z → ∀ (x : α), x ≤ z ↔ l x ≤ y",
"tactic": "rintro rfl x"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\ny : β\nx : α\n⊢ x ≤ u y ↔ l x ≤ y",
"tactic": "exact (gc x y).symm"
},
{
"state_after": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\nH : ∀ (x : α), x ≤ z ↔ l x ≤ y\n⊢ u y = z",
"state_before": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\n⊢ (∀ (x : α), x ≤ z ↔ l x ≤ y) → u y = z",
"tactic": "intro H"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.12736\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\nH : ∀ (x : α), x ≤ z ↔ l x ≤ y\n⊢ u y = z",
"tactic": "exact ((H <| u y).mpr (gc.l_u_le y)).antisymm ((gc _ _).mp <| (H z).mp le_rfl)"
}
] |
[
199,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
AntitoneOn.quasiconvexOn
|
[] |
[
197,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_lt_zero
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6316\nα : Type u_1\nβ : Type ?u.6322\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\n⊢ a⁻¹ < 0 ↔ a < 0",
"tactic": "simp only [← not_le, inv_nonneg]"
}
] |
[
68,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.append1_cases_append1
|
[] |
[
143,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Algebra/CharZero/Defs.lean
|
Nat.cast_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : AddMonoidWithOne R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n = 0 ↔ n = 0",
"tactic": "rw [← cast_zero, cast_inj]"
}
] |
[
77,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.subsingleton_of_finrank_adjoin_le_one
|
[] |
[
782,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
1
] |
Mathlib/Data/List/Basic.lean
|
List.takeI_eq_take
|
[] |
[
2337,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2335,
1
] |
Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean
|
CategoryTheory.SemiadditiveOfBinaryBiproducts.add_comp
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc f g ≫ h = biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc (f ≫ h) (g ≫ h)",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ (f + g) ≫ h = f ≫ h + g ≫ h",
"tactic": "simp only [add_eq_right_addition, Category.assoc]"
},
{
"state_after": "case e_a\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ biprod.desc f g ≫ h = biprod.desc (f ≫ h) (g ≫ h)",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc f g ≫ h = biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc (f ≫ h) (g ≫ h)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_a\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ biprod.desc f g ≫ h = biprod.desc (f ≫ h) (g ≫ h)",
"tactic": "ext <;> simp"
}
] |
[
133,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.mapsTo_extend
|
[
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.131353\nM' : Type ?u.131356\nH' : Type ?u.131359\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\nhs : s ⊆ f.source\n⊢ ↑I '' (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' s) ⊆ ↑I '' (↑(LocalHomeomorph.symm f) ⁻¹' s)",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.131353\nM' : Type ?u.131356\nH' : Type ?u.131359\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\nhs : s ⊆ f.source\n⊢ MapsTo (↑(extend f I)) s (↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I)",
"tactic": "rw [mapsTo', extend_coe, extend_coe_symm, preimage_comp, ← I.image_eq, image_comp,\n f.image_eq_target_inter_inv_preimage hs]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.131353\nM' : Type ?u.131356\nH' : Type ?u.131359\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\nhs : s ⊆ f.source\n⊢ ↑I '' (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' s) ⊆ ↑I '' (↑(LocalHomeomorph.symm f) ⁻¹' s)",
"tactic": "exact image_subset _ (inter_subset_right _ _)"
}
] |
[
811,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
smul_pow'
|
[
{
"state_after": "case zero\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\n⊢ x • m ^ Nat.zero = (x • m) ^ Nat.zero\n\ncase succ\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\nn : ℕ\nih : x • m ^ n = (x • m) ^ n\n⊢ x • m ^ Nat.succ n = (x • m) ^ Nat.succ n",
"state_before": "α : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\nn : ℕ\n⊢ x • m ^ n = (x • m) ^ n",
"tactic": "induction' n with n ih"
},
{
"state_after": "case zero\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\n⊢ x • 1 = 1",
"state_before": "case zero\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\n⊢ x • m ^ Nat.zero = (x • m) ^ Nat.zero",
"tactic": "rw [pow_zero, pow_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\n⊢ x • 1 = 1",
"tactic": "exact smul_one x"
},
{
"state_after": "case succ\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\nn : ℕ\nih : x • m ^ n = (x • m) ^ n\n⊢ x • (m * m ^ n) = x • m * (x • m) ^ n",
"state_before": "case succ\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\nn : ℕ\nih : x • m ^ n = (x • m) ^ n\n⊢ x • m ^ Nat.succ n = (x • m) ^ Nat.succ n",
"tactic": "rw [pow_succ, pow_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type ?u.75902\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : MulDistribMulAction M N\nx : M\nm : N\nn : ℕ\nih : x • m ^ n = (x • m) ^ n\n⊢ x • (m * m ^ n) = x • m * (x • m) ^ n",
"tactic": "exact (smul_mul' x m (m ^ n)).trans (congr_arg _ ih)"
}
] |
[
126,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_le_aleph0
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\nc : Cardinal\n⊢ lift c ≤ ℵ₀ ↔ c ≤ ℵ₀",
"tactic": "rw [← lift_aleph0.{u,v}, lift_le]"
}
] |
[
1261,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1260,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.piecewise_insert_of_ne
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.82701\nγ : Type ?u.82704\nι : Sort ?u.82707\nπ : α → Type ?u.82712\nδ : α → Sort u_2\ns : Set α\nf g : (i : α) → δ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\ni j : α\nh : i ≠ j\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ piecewise (insert j s) f g i = piecewise s f g i",
"tactic": "simp [piecewise, h]"
}
] |
[
1429,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1428,
1
] |
Mathlib/Analysis/Fourier/FourierTransform.lean
|
Fourier.fourierIntegral_comp_add_right
|
[] |
[
225,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.update_eq_iff
|
[] |
[
596,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
594,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.sep_eq_self_iff_mem_true
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\np q : α → Prop\nx : α\n⊢ {x | x ∈ s ∧ p x} = s ↔ ∀ (x : α), x ∈ s → p x",
"tactic": "simp_rw [ext_iff, mem_sep_iff, and_iff_left_iff_imp]"
}
] |
[
1434,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1433,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
finrank_span_eq_card
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhb : LinearIndependent K b\nthis : Module.rank K { x // x ∈ span K (Set.range b) } = (#↑(Set.range b))\n⊢ Module.rank K { x // x ∈ span K (Set.range b) } = ↑(Fintype.card ι)",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhb : LinearIndependent K b\n⊢ Module.rank K { x // x ∈ span K (Set.range b) } = ↑(Fintype.card ι)",
"tactic": "have : Module.rank K (span K (Set.range b)) = (#Set.range b) := rank_span hb"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhb : LinearIndependent K b\nthis : Module.rank K { x // x ∈ span K (Set.range b) } = (#↑(Set.range b))\n⊢ Module.rank K { x // x ∈ span K (Set.range b) } = ↑(Fintype.card ι)",
"tactic": "rwa [← lift_inj, mk_range_eq_of_injective hb.injective, Cardinal.mk_fintype, lift_natCast,\n lift_eq_nat_iff] at this"
}
] |
[
337,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.divInt_ne_zero
|
[] |
[
92,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Analysis/Convex/Gauge.lean
|
gauge_def'
|
[
{
"state_after": "𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ sInf {r | 0 < r ∧ x ∈ r • s} = sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}",
"state_before": "𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ gauge s x = sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}",
"tactic": "rw [gauge]"
},
{
"state_after": "case h\n𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ {r | 0 < r ∧ x ∈ r • s} = {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}",
"state_before": "𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ sInf {r | 0 < r ∧ x ∈ r • s} = sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}",
"tactic": "apply congr_arg"
},
{
"state_after": "case h.h\n𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nx✝ : ℝ\n⊢ x✝ ∈ {r | 0 < r ∧ x ∈ r • s} ↔ x✝ ∈ {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}",
"state_before": "case h\n𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ {r | 0 < r ∧ x ∈ r • s} = {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}",
"tactic": "ext"
},
{
"state_after": "case h.h\n𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nx✝ : ℝ\n⊢ 0 < x✝ ∧ x ∈ x✝ • s ↔ 0 < x✝ ∧ x✝⁻¹ • x ∈ s",
"state_before": "case h.h\n𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nx✝ : ℝ\n⊢ x✝ ∈ {r | 0 < r ∧ x ∈ r • s} ↔ x✝ ∈ {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}",
"tactic": "simp only [mem_setOf, mem_Ioi]"
},
{
"state_after": "no goals",
"state_before": "case h.h\n𝕜 : Type ?u.5140\nE : Type u_1\nF : Type ?u.5146\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nx✝ : ℝ\n⊢ 0 < x✝ ∧ x ∈ x✝ • s ↔ 0 < x✝ ∧ x✝⁻¹ • x ∈ s",
"tactic": "exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _"
}
] |
[
75,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
HasDerivWithinAt.const_mul
|
[
{
"state_after": "case h.e'_7\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.262301\n𝔸 : Type u_1\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc✝ d : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nc : 𝔸\nhd : HasDerivWithinAt d d' s x\n⊢ c * d' = 0 * d x + c * d'",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.262301\n𝔸 : Type u_1\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc✝ d : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nc : 𝔸\nhd : HasDerivWithinAt d d' s x\n⊢ HasDerivWithinAt (fun y => c * d y) (c * d') s x",
"tactic": "convert (hasDerivWithinAt_const x s c).mul hd using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.262301\n𝔸 : Type u_1\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝔸\nc✝ d : 𝕜 → 𝔸\nc' d' : 𝔸\nu v : 𝕜 → 𝕜'\nc : 𝔸\nhd : HasDerivWithinAt d d' s x\n⊢ c * d' = 0 * d x + c * d'",
"tactic": "rw [zero_mul, zero_add]"
}
] |
[
252,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Data/Set/Pairwise/Lattice.lean
|
Set.PairwiseDisjoint.biUnion
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na : ι\nha : a ∈ ⋃ (i : ι') (_ : i ∈ s), g i\nb : ι\nhb : b ∈ ⋃ (i : ι') (_ : i ∈ s), g i\nhab : a ≠ b\n⊢ (Disjoint on f) a b",
"state_before": "α : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\n⊢ PairwiseDisjoint (⋃ (i : ι') (_ : i ∈ s), g i) f",
"tactic": "rintro a ha b hb hab"
},
{
"state_after": "α : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nha : ∃ i i_1, a ∈ g i\nhb : ∃ i i_1, b ∈ g i\n⊢ (Disjoint on f) a b",
"state_before": "α : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na : ι\nha : a ∈ ⋃ (i : ι') (_ : i ∈ s), g i\nb : ι\nhb : b ∈ ⋃ (i : ι') (_ : i ∈ s), g i\nhab : a ≠ b\n⊢ (Disjoint on f) a b",
"tactic": "simp_rw [Set.mem_iUnion] at ha hb"
},
{
"state_after": "case intro.intro\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nhb : ∃ i i_1, b ∈ g i\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\n⊢ (Disjoint on f) a b",
"state_before": "α : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nha : ∃ i i_1, a ∈ g i\nhb : ∃ i i_1, b ∈ g i\n⊢ (Disjoint on f) a b",
"tactic": "obtain ⟨c, hc, ha⟩ := ha"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\n⊢ (Disjoint on f) a b",
"state_before": "case intro.intro\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nhb : ∃ i i_1, b ∈ g i\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\n⊢ (Disjoint on f) a b",
"tactic": "obtain ⟨d, hd, hb⟩ := hb"
},
{
"state_after": "case intro.intro.intro.intro.inl\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\nhcd : g c = g d\n⊢ (Disjoint on f) a b\n\ncase intro.intro.intro.intro.inr\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\nhcd : g c ≠ g d\n⊢ (Disjoint on f) a b",
"state_before": "case intro.intro.intro.intro\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\n⊢ (Disjoint on f) a b",
"tactic": "obtain hcd | hcd := eq_or_ne (g c) (g d)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.inl\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\nhcd : g c = g d\n⊢ (Disjoint on f) a b",
"tactic": "exact hg d hd (hcd.subst ha) hb hab"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.inr\nα : Type u_3\nβ : Type ?u.1482\nγ : Type ?u.1485\nι : Type u_2\nι' : Type u_1\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι'\ng : ι' → Set ι\nf : ι → α\nhs : PairwiseDisjoint s fun i' => ⨆ (i : ι) (_ : i ∈ g i'), f i\nhg : ∀ (i : ι'), i ∈ s → PairwiseDisjoint (g i) f\na b : ι\nhab : a ≠ b\nc : ι'\nhc : c ∈ s\nha : a ∈ g c\nd : ι'\nhd : d ∈ s\nhb : b ∈ g d\nhcd : g c ≠ g d\n⊢ (Disjoint on f) a b",
"tactic": "exact (hs hc hd <| ne_of_apply_ne _ hcd).mono\n (le_iSup₂ (f := fun i (_ : i ∈ g c) => f i) a ha)\n (le_iSup₂ (f := fun i (_ : i ∈ g d) => f i) b hb)"
}
] |
[
89,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
Filter.Tendsto.neg_mul_atTop
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nα : Type u_2\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhC : C < 0\nhf : Tendsto f l (𝓝 C)\nhg : Tendsto g l atTop\n⊢ Tendsto (fun x => f x * g x) l atBot",
"tactic": "simpa only [mul_comm] using hg.atTop_mul_neg hC hf"
}
] |
[
90,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.adjugate_one
|
[
{
"state_after": "case a.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\ni✝ x✝ : n\n⊢ adjugate 1 i✝ x✝ = OfNat.ofNat 1 i✝ x✝",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\n⊢ adjugate 1 = 1",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nm : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\ni✝ x✝ : n\n⊢ adjugate 1 i✝ x✝ = OfNat.ofNat 1 i✝ x✝",
"tactic": "simp [adjugate_def, Matrix.one_apply, Pi.single_apply, eq_comm]"
}
] |
[
342,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
continuous_list_prod
|
[] |
[
542,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.extend_apply'
|
[] |
[
476,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
8
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.mem_basicOpen
|
[] |
[
772,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
771,
1
] |
Mathlib/Data/Nat/Size.lean
|
Nat.size_one
|
[
{
"state_after": "no goals",
"state_before": "⊢ size (bit1 0) = 1",
"tactic": "rw [size_bit1, size_zero]"
}
] |
[
103,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.of_zero
|
[] |
[
323,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
SimpleGraph.is3Clique_iff
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\nh : IsNClique G 3 s\n⊢ ∃ a b c, Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}\n\ncase refine'_2\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\n⊢ (∃ a b c, Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}) → IsNClique G 3 s",
"state_before": "α : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\n⊢ IsNClique G 3 s ↔ ∃ a b c, Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}",
"tactic": "refine' ⟨fun h ↦ _, _⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na✝ b✝ c✝ : α\nh : IsNClique G 3 s\na b c : α\nhs : s = {a, b, c}\n⊢ ∃ a b c, Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}",
"state_before": "case refine'_1\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\nh : IsNClique G 3 s\n⊢ ∃ a b c, Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}",
"tactic": "obtain ⟨a, b, c, -, -, -, hs⟩ := card_eq_three.1 h.card_eq"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na✝ b✝ c✝ : α\nh : IsNClique G 3 s\na b c : α\nhs : s = {a, b, c}\n⊢ Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na✝ b✝ c✝ : α\nh : IsNClique G 3 s\na b c : α\nhs : s = {a, b, c}\n⊢ ∃ a b c, Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}",
"tactic": "refine' ⟨a, b, c, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na✝ b✝ c✝ : α\nh : IsNClique G 3 s\na b c : α\nhs : s = {a, b, c}\n⊢ Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}",
"tactic": "rwa [hs, eq_self_iff_true, and_true, is3Clique_triple_iff.symm, ← hs]"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\na✝ b✝ c✝ a b c : α\nhab : Adj G a b\nhbc : Adj G a c\nhca : Adj G b c\n⊢ IsNClique G 3 {a, b, c}",
"state_before": "case refine'_2\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ns : Finset α\ninst✝ : DecidableEq α\na b c : α\n⊢ (∃ a b c, Adj G a b ∧ Adj G a c ∧ Adj G b c ∧ s = {a, b, c}) → IsNClique G 3 s",
"tactic": "rintro ⟨a, b, c, hab, hbc, hca, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nG H : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\na✝ b✝ c✝ a b c : α\nhab : Adj G a b\nhbc : Adj G a c\nhca : Adj G b c\n⊢ IsNClique G 3 {a, b, c}",
"tactic": "exact is3Clique_triple_iff.2 ⟨hab, hbc, hca⟩"
}
] |
[
147,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.lift.range_eq_closure
|
[
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Subgroup.closure (Set.range f) ≤ MonoidHom.range (↑lift f)",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ MonoidHom.range (↑lift f) = Subgroup.closure (Set.range f)",
"tactic": "apply le_antisymm (lift.range_le Subgroup.subset_closure)"
},
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Set.range f ⊆ ↑(MonoidHom.range (↑lift f))",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Subgroup.closure (Set.range f) ≤ MonoidHom.range (↑lift f)",
"tactic": "rw [Subgroup.closure_le]"
},
{
"state_after": "case intro\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\na : α\n⊢ f a ∈ ↑(MonoidHom.range (↑lift f))",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\n⊢ Set.range f ⊆ ↑(MonoidHom.range (↑lift f))",
"tactic": "rintro _ ⟨a, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\na : α\n⊢ f a ∈ ↑(MonoidHom.range (↑lift f))",
"tactic": "exact ⟨FreeGroup.of a, by simp only [lift.of]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\ninst✝ : Group β\nf : α → β\nx y : FreeGroup α\na : α\n⊢ ↑(↑lift f) (FreeGroup.of a) = f a",
"tactic": "simp only [lift.of]"
}
] |
[
778,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.rel_bind
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\na : α\ns✝ t✝ : Multiset α\nf✝ g✝ : α → Multiset β\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset β\nf : α → Multiset γ\ng : β → Multiset δ\nh : (r ⇒ Rel p) f g\nhst : Rel r s t\n⊢ Rel (Rel p) (map f s) (map g t)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\na : α\ns✝ t✝ : Multiset α\nf✝ g✝ : α → Multiset β\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset β\nf : α → Multiset γ\ng : β → Multiset δ\nh : (r ⇒ Rel p) f g\nhst : Rel r s t\n⊢ Rel p (bind s f) (bind t g)",
"tactic": "apply rel_join"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\na : α\ns✝ t✝ : Multiset α\nf✝ g✝ : α → Multiset β\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset β\nf : α → Multiset γ\ng : β → Multiset δ\nh : (r ⇒ Rel p) f g\nhst : Rel r s t\n⊢ Rel (fun a b => Rel p (f a) (g b)) s t",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\na : α\ns✝ t✝ : Multiset α\nf✝ g✝ : α → Multiset β\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset β\nf : α → Multiset γ\ng : β → Multiset δ\nh : (r ⇒ Rel p) f g\nhst : Rel r s t\n⊢ Rel (Rel p) (map f s) (map g t)",
"tactic": "rw [rel_map]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\na : α\ns✝ t✝ : Multiset α\nf✝ g✝ : α → Multiset β\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset β\nf : α → Multiset γ\ng : β → Multiset δ\nh : (r ⇒ Rel p) f g\nhst : Rel r s t\n⊢ Rel (fun a b => Rel p (f a) (g b)) s t",
"tactic": "exact hst.mono fun a _ b _ hr => h hr"
}
] |
[
199,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable
|
[
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → DifferentiableAt ℂ f x\nthis :\n ∀ (z : ℂ),\n I • ↑(ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) 1 =\n ↑(ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) I\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n 0",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → DifferentiableAt ℂ f x\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n 0",
"tactic": "have : ∀ z, I • (fderiv ℂ f z).restrictScalars ℝ 1 = (fderiv ℂ f z).restrictScalars ℝ I := fun z ↦\n by rw [(fderiv ℂ f _).coe_restrictScalars', ← (fderiv ℂ f _).map_smul, smul_eq_mul, mul_one]"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nz w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHd :\n ∀ (x : ℂ), x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \\ s → DifferentiableAt ℂ f x\nthis :\n ∀ (z : ℂ),\n I • ↑(ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) 1 =\n ↑(ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) I\n⊢ ((((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * I)) +\n I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) -\n I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * I)) =\n 0",
"tactic": "refine (integral_boundary_rect_of_hasFDerivAt_real_off_countable f\n (fun z => (fderiv ℂ f z).restrictScalars ℝ) z w s hs Hc\n (fun x hx => (Hd x hx).hasFDerivAt.restrictScalars ℝ) ?_).trans ?_ <;> simp [this]"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nz✝ w : ℂ\ns : Set ℂ\nhs : Set.Countable s\nHc : ContinuousOn f ([[z✝.re, w.re]] ×ℂ [[z✝.im, w.im]])\nHd :\n ∀ (x : ℂ),\n x ∈ Ioo (min z✝.re w.re) (max z✝.re w.re) ×ℂ Ioo (min z✝.im w.im) (max z✝.im w.im) \\ s → DifferentiableAt ℂ f x\nz : ℂ\n⊢ I • ↑(ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) 1 =\n ↑(ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) I",
"tactic": "rw [(fderiv ℂ f _).coe_restrictScalars', ← (fderiv ℂ f _).map_smul, smul_eq_mul, mul_one]"
}
] |
[
265,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_neg
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.274344\nK : Type ?u.274347\nM : Type u_1\nM₂ : Type ?u.274353\nV : Type ?u.274356\nS : Type ?u.274359\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\n⊢ span R (-s) = span R (↑(-LinearMap.id) '' s)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.274344\nK : Type ?u.274347\nM : Type u_1\nM₂ : Type ?u.274353\nV : Type ?u.274356\nS : Type ?u.274359\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\n⊢ map (-LinearMap.id) (span R s) = span R s",
"tactic": "simp"
}
] |
[
825,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
821,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.RightInvOn.eqOn
|
[] |
[
1105,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1104,
1
] |
Mathlib/Logic/Basic.lean
|
not_ball
|
[] |
[
1093,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1093,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean
|
ContinuousMultilinearMap.toMultilinearMap_zero
|
[] |
[
154,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Formula.realize_relabel
|
[
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.238835\nP : Type ?u.238838\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nφ✝ ψ : Formula L α\nv✝ : α → M\nφ : Formula L α\ng : α → β\nv : β → M\n⊢ BoundedFormula.Realize φ (Sum.elim v (default ∘ ↑(Fin.cast (_ : 0 = 0))) ∘ Sum.inl ∘ g) (default ∘ ↑(natAdd 0)) =\n BoundedFormula.Realize φ (v ∘ g) default",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.238835\nP : Type ?u.238838\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nφ✝ ψ : Formula L α\nv✝ : α → M\nφ : Formula L α\ng : α → β\nv : β → M\n⊢ Realize (relabel g φ) v ↔ Realize φ (v ∘ g)",
"tactic": "rw [Realize, Realize, relabel, BoundedFormula.realize_relabel, iff_eq_eq, Fin.castAdd_zero]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.238835\nP : Type ?u.238838\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nφ✝ ψ : Formula L α\nv✝ : α → M\nφ : Formula L α\ng : α → β\nv : β → M\n⊢ BoundedFormula.Realize φ (Sum.elim v (default ∘ ↑(Fin.cast (_ : 0 = 0))) ∘ Sum.inl ∘ g) (default ∘ ↑(natAdd 0)) =\n BoundedFormula.Realize φ (v ∘ g) default",
"tactic": "exact congr rfl (funext finZeroElim)"
}
] |
[
677,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
Bornology.IsVonNBounded.of_topologicalSpace_le
|
[] |
[
113,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/CategoryTheory/Subobject/FactorThru.lean
|
CategoryTheory.Subobject.factors_comp_arrow
|
[] |
[
97,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.map_comap_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.249498\n𝕜 : Type ?u.249501\nV : Type u\nW : Type v\nX : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nf : V ↪ W\nG : SimpleGraph W\n⊢ SimpleGraph.map f (SimpleGraph.comap (↑f) G) ≤ G",
"tactic": "rw [map_le_iff_le_comap]"
}
] |
[
1296,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1295,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.add_descFactorial_eq_ascFactorial
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ descFactorial (n + 0) 0 = ascFactorial n 0",
"tactic": "rw [ascFactorial_zero, descFactorial_zero]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ descFactorial (n + succ k) (succ k) = ascFactorial n (succ k)",
"tactic": "rw [Nat.add_succ, Nat.succ_eq_add_one, Nat.succ_eq_add_one,\n succ_descFactorial_succ, ascFactorial_succ, add_descFactorial_eq_ascFactorial _ k]"
}
] |
[
413,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.