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/Finset/LocallyFinite.lean
|
Finset.Ioc_eq_cons_Ioo
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.108656\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b c : α\nh : a < b\n⊢ Ioc a b = cons b (Ioo a b) (_ : ¬b ∈ Ioo a b)",
"tactic": "classical rw [cons_eq_insert, Ioo_insert_right h]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.108656\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b c : α\nh : a < b\n⊢ Ioc a b = cons b (Ioo a b) (_ : ¬b ∈ Ioo a b)",
"tactic": "rw [cons_eq_insert, Ioo_insert_right h]"
}
] |
[
625,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.C_1
|
[] |
[
214,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.pi_univ_Ico_subset
|
[] |
[
84,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.toFinset_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs : Set.Finite s\nht : Set.Finite t\ninst✝ : Fintype α\nh : Set.Finite univ\n⊢ Finite.toFinset h = Finset.univ",
"tactic": "simp"
}
] |
[
277,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
11
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degrees_one
|
[] |
[
137,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeRing.coe_surjective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nx : FreeCommRing α\n⊢ ∃ a, ↑a = x",
"tactic": "induction x using FreeCommRing.induction_on with\n| hn1 =>\n use -1\n rfl\n| hb b =>\n exact ⟨FreeRing.of b, rfl⟩\n| ha _ _ hx hy =>\n rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩\n exact ⟨x + y, (FreeRing.lift _).map_add _ _⟩\n| hm _ _ hx hy =>\n rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩\n exact ⟨x * y, (FreeRing.lift _).map_mul _ _⟩"
},
{
"state_after": "case hn1\nα : Type u\n⊢ ↑(-1) = -1",
"state_before": "case hn1\nα : Type u\n⊢ ∃ a, ↑a = -1",
"tactic": "use -1"
},
{
"state_after": "no goals",
"state_before": "case hn1\nα : Type u\n⊢ ↑(-1) = -1",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case hb\nα : Type u\nb : α\n⊢ ∃ a, ↑a = FreeCommRing.of b",
"tactic": "exact ⟨FreeRing.of b, rfl⟩"
},
{
"state_after": "case ha.intro\nα : Type u\ny✝ : FreeCommRing α\nhy : ∃ a, ↑a = y✝\nx : FreeRing α\n⊢ ∃ a, ↑a = ↑x + y✝",
"state_before": "case ha\nα : Type u\nx✝ y✝ : FreeCommRing α\nhx : ∃ a, ↑a = x✝\nhy : ∃ a, ↑a = y✝\n⊢ ∃ a, ↑a = x✝ + y✝",
"tactic": "rcases hx with ⟨x, rfl⟩"
},
{
"state_after": "case ha.intro.intro\nα : Type u\nx y : FreeRing α\n⊢ ∃ a, ↑a = ↑x + ↑y",
"state_before": "case ha.intro\nα : Type u\ny✝ : FreeCommRing α\nhy : ∃ a, ↑a = y✝\nx : FreeRing α\n⊢ ∃ a, ↑a = ↑x + y✝",
"tactic": "rcases hy with ⟨y, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case ha.intro.intro\nα : Type u\nx y : FreeRing α\n⊢ ∃ a, ↑a = ↑x + ↑y",
"tactic": "exact ⟨x + y, (FreeRing.lift _).map_add _ _⟩"
},
{
"state_after": "case hm.intro\nα : Type u\ny✝ : FreeCommRing α\nhy : ∃ a, ↑a = y✝\nx : FreeRing α\n⊢ ∃ a, ↑a = ↑x * y✝",
"state_before": "case hm\nα : Type u\nx✝ y✝ : FreeCommRing α\nhx : ∃ a, ↑a = x✝\nhy : ∃ a, ↑a = y✝\n⊢ ∃ a, ↑a = x✝ * y✝",
"tactic": "rcases hx with ⟨x, rfl⟩"
},
{
"state_after": "case hm.intro.intro\nα : Type u\nx y : FreeRing α\n⊢ ∃ a, ↑a = ↑x * ↑y",
"state_before": "case hm.intro\nα : Type u\ny✝ : FreeCommRing α\nhy : ∃ a, ↑a = y✝\nx : FreeRing α\n⊢ ∃ a, ↑a = ↑x * y✝",
"tactic": "rcases hy with ⟨y, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case hm.intro.intro\nα : Type u\nx y : FreeRing α\n⊢ ∃ a, ↑a = ↑x * ↑y",
"tactic": "exact ⟨x * y, (FreeRing.lift _).map_mul _ _⟩"
}
] |
[
377,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
11
] |
Mathlib/RingTheory/Polynomial/Dickson.lean
|
Polynomial.map_dickson
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nf : R →+* S\n⊢ map f (dickson k a 0) = dickson k (↑f a) 0",
"tactic": "simp_rw [dickson_zero, Polynomial.map_sub, Polynomial.map_nat_cast, Polynomial.map_ofNat]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nf : R →+* S\n⊢ map f (dickson k a 1) = dickson k (↑f a) 1",
"tactic": "simp only [dickson_one, map_X]"
},
{
"state_after": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nf : R →+* S\nn : ℕ\n⊢ X * map f (dickson k a (n + 1)) - ↑C (↑f a) * map f (dickson k a n) =\n X * dickson k (↑f a) (n + 1) - ↑C (↑f a) * dickson k (↑f a) n",
"state_before": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nf : R →+* S\nn : ℕ\n⊢ map f (dickson k a (n + 2)) = dickson k (↑f a) (n + 2)",
"tactic": "simp only [dickson_add_two, Polynomial.map_sub, Polynomial.map_mul, map_X, map_C]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nk : ℕ\na : R\nf : R →+* S\nn : ℕ\n⊢ X * map f (dickson k a (n + 1)) - ↑C (↑f a) * map f (dickson k a n) =\n X * dickson k (↑f a) (n + 1) - ↑C (↑f a) * dickson k (↑f a) n",
"tactic": "rw [map_dickson f n, map_dickson f (n + 1)]"
}
] |
[
104,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Algebra/Opposites.lean
|
MulOpposite.op_smul
|
[] |
[
319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/Data/List/Count.lean
|
List.count_cons_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\na : α\nl : List α\n⊢ count a (a :: l) = count a l + 1",
"tactic": "simp [count_cons']"
}
] |
[
192,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.ext'_iff
|
[] |
[
150,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
Int.comap_cast_atTop
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.2256\nR : Type u_1\ninst✝¹ : StrictOrderedRing R\ninst✝ : Archimedean R\nr : R\nn : ℕ\nhn : r ≤ ↑n\n⊢ r ≤ ↑↑n",
"tactic": "exact_mod_cast hn"
}
] |
[
48,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
9
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.eq_replicate_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.30399\nn n' m : ℕ\ns : Sym α n\na b : α\n⊢ (↑Multiset.card ↑s = n ∧ ∀ (b : α), b ∈ ↑s → b = a) ↔ ∀ (b : α), b ∈ s → b = a",
"state_before": "α : Type u_1\nβ : Type ?u.30399\nn n' m : ℕ\ns : Sym α n\na b : α\n⊢ s = replicate n a ↔ ∀ (b : α), b ∈ s → b = a",
"tactic": "erw [Subtype.ext_iff, Multiset.eq_replicate]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30399\nn n' m : ℕ\ns : Sym α n\na b : α\n⊢ (↑Multiset.card ↑s = n ∧ ∀ (b : α), b ∈ ↑s → b = a) ↔ ∀ (b : α), b ∈ s → b = a",
"tactic": "exact and_iff_right s.2"
}
] |
[
295,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.stronglyMeasurable_of_isEmpty
|
[] |
[
157,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.disjSups_disjSups_disjSups_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : DistribLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns t u v : Finset α\n⊢ s ○ t ○ (u ○ v) = s ○ u ○ (t ○ v)",
"tactic": "simp_rw [← disjSups_assoc, disjSups_right_comm]"
}
] |
[
554,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.mapQ_mkQ
|
[
{
"state_after": "case h\nR : Type u_1\nM : Type u_3\nr : R\nx✝ y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ comap f q\nx : M\n⊢ ↑(comp (mapQ p q f h) (mkQ p)) x = ↑(comp (mkQ q) f) x",
"state_before": "R : Type u_1\nM : Type u_3\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ comap f q\n⊢ comp (mapQ p q f h) (mkQ p) = comp (mkQ q) f",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nM : Type u_3\nr : R\nx✝ y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type u_2\nM₂ : Type u_4\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ comap f q\nx : M\n⊢ ↑(comp (mapQ p q f h) (mkQ p)) x = ↑(comp (mkQ q) f) x",
"tactic": "rfl"
}
] |
[
419,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
inv_smul_smul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : Group α\ninst✝ : MulAction α β\nc : α\nx : β\n⊢ c⁻¹ • c • x = x",
"tactic": "rw [smul_smul, mul_left_inv, one_smul]"
}
] |
[
39,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
|
BoxIntegral.hasIntegral_GP_divergence_of_forall_hasDerivWithinAt
|
[
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni✝ : Fin (n + 1)\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHs : ∀ (x : Fin (n + 1) → ℝ), x ∈ s → ContinuousWithinAt f (↑Box.Icc I) x\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ ↑Box.Icc I \\ s → HasFDerivWithinAt f (f' x) (↑Box.Icc I) x\ni : Fin (n + 1)\nx✝ : i ∈ Finset.univ\n⊢ HasIntegral I GP (fun x => ↑(f' x) (Pi.single i 1) i) BoxAdditiveMap.volume\n (integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i) BoxAdditiveMap.volume -\n integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i) BoxAdditiveMap.volume)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni : Fin (n + 1)\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHs : ∀ (x : Fin (n + 1) → ℝ), x ∈ s → ContinuousWithinAt f (↑Box.Icc I) x\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ ↑Box.Icc I \\ s → HasFDerivWithinAt f (f' x) (↑Box.Icc I) x\n⊢ HasIntegral I GP (fun x => ∑ i : Fin (n + 1), ↑(f' x) (Pi.single i 1) i) BoxAdditiveMap.volume\n (∑ i : Fin (n + 1),\n (integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i) BoxAdditiveMap.volume -\n integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i) BoxAdditiveMap.volume))",
"tactic": "refine HasIntegral.sum fun i _ => ?_"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni✝ : Fin (n + 1)\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\ni : Fin (n + 1)\nx✝ : i ∈ Finset.univ\nHd :\n ∀ (x : Fin (n + 1) → ℝ),\n x ∈ ↑Box.Icc I \\ s →\n ∀ (i : Fin (n + 1)),\n HasFDerivWithinAt (fun x => f x i) (ContinuousLinearMap.comp (ContinuousLinearMap.proj i) (f' x)) (↑Box.Icc I) x\nHs : ∀ (x : Fin (n + 1) → ℝ), x ∈ s → ∀ (i : Fin (n + 1)), ContinuousWithinAt (fun y => f y i) (↑Box.Icc I) x\n⊢ HasIntegral I GP (fun x => ↑(f' x) (Pi.single i 1) i) BoxAdditiveMap.volume\n (integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i) BoxAdditiveMap.volume -\n integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i) BoxAdditiveMap.volume)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni✝ : Fin (n + 1)\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHs : ∀ (x : Fin (n + 1) → ℝ), x ∈ s → ContinuousWithinAt f (↑Box.Icc I) x\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ ↑Box.Icc I \\ s → HasFDerivWithinAt f (f' x) (↑Box.Icc I) x\ni : Fin (n + 1)\nx✝ : i ∈ Finset.univ\n⊢ HasIntegral I GP (fun x => ↑(f' x) (Pi.single i 1) i) BoxAdditiveMap.volume\n (integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i) BoxAdditiveMap.volume -\n integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i) BoxAdditiveMap.volume)",
"tactic": "simp only [hasFDerivWithinAt_pi', continuousWithinAt_pi] at Hd Hs"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\ni✝ : Fin (n + 1)\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\ni : Fin (n + 1)\nx✝ : i ∈ Finset.univ\nHd :\n ∀ (x : Fin (n + 1) → ℝ),\n x ∈ ↑Box.Icc I \\ s →\n ∀ (i : Fin (n + 1)),\n HasFDerivWithinAt (fun x => f x i) (ContinuousLinearMap.comp (ContinuousLinearMap.proj i) (f' x)) (↑Box.Icc I) x\nHs : ∀ (x : Fin (n + 1) → ℝ), x ∈ s → ∀ (i : Fin (n + 1)), ContinuousWithinAt (fun y => f y i) (↑Box.Icc I) x\n⊢ HasIntegral I GP (fun x => ↑(f' x) (Pi.single i 1) i) BoxAdditiveMap.volume\n (integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.upper I i) x) i) BoxAdditiveMap.volume -\n integral (Box.face I i) GP (fun x => f (Fin.insertNth i (Box.lower I i) x) i) BoxAdditiveMap.volume)",
"tactic": "exact hasIntegral_GP_pderiv I _ _ s hs (fun x hx => Hs x hx i) (fun x hx => Hd x hx i) i"
}
] |
[
287,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/RingTheory/IsTensorProduct.lean
|
IsTensorProduct.equiv_symm_apply
|
[
{
"state_after": "case a\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.53163\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.55402\nN₂ : Type ?u.55405\nN : Type ?u.55408\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(equiv h) (↑(LinearEquiv.symm (equiv h)) (↑(↑f x₁) x₂)) = ↑(equiv h) (x₁ ⊗ₜ[R] x₂)",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.53163\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.55402\nN₂ : Type ?u.55405\nN : Type ?u.55408\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(LinearEquiv.symm (equiv h)) (↑(↑f x₁) x₂) = x₁ ⊗ₜ[R] x₂",
"tactic": "apply h.equiv.injective"
},
{
"state_after": "case a\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.53163\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.55402\nN₂ : Type ?u.55405\nN : Type ?u.55408\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(↑f x₁) x₂ = ↑(equiv h) (x₁ ⊗ₜ[R] x₂)",
"state_before": "case a\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.53163\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.55402\nN₂ : Type ?u.55405\nN : Type ?u.55408\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(equiv h) (↑(LinearEquiv.symm (equiv h)) (↑(↑f x₁) x₂)) = ↑(equiv h) (x₁ ⊗ₜ[R] x₂)",
"tactic": "refine' (h.equiv.apply_symm_apply _).trans _"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_1\ninst✝¹⁴ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM : Type u_4\nM' : Type ?u.53163\ninst✝¹³ : AddCommMonoid M₁\ninst✝¹² : AddCommMonoid M₂\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R M₂\ninst✝⁷ : Module R M\ninst✝⁶ : Module R M'\nf : M₁ →ₗ[R] M₂ →ₗ[R] M\nN₁ : Type ?u.55402\nN₂ : Type ?u.55405\nN : Type ?u.55408\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N\ng : N₁ →ₗ[R] N₂ →ₗ[R] N\nh : IsTensorProduct f\nx₁ : M₁\nx₂ : M₂\n⊢ ↑(↑f x₁) x₂ = ↑(equiv h) (x₁ ⊗ₜ[R] x₂)",
"tactic": "simp"
}
] |
[
97,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.order_X
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ order X = 1",
"tactic": "simpa only [Nat.cast_one] using order_monomial_of_ne_zero 1 (1 : R) one_ne_zero"
}
] |
[
2492,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2491,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
Orthonormal.exists_orthonormalBasis_extension_of_card_eq
|
[
{
"state_after": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"tactic": "have hsv : Injective (s.restrict v) := hv.linearIndependent.injective"
},
{
"state_after": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX : Orthonormal 𝕜 Subtype.val\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"tactic": "have hX : Orthonormal 𝕜 ((↑) : Set.range (s.restrict v) → E) := by\n rwa [orthonormal_subtype_range hsv]"
},
{
"state_after": "case intro.intro.intro\nι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX : Orthonormal 𝕜 Subtype.val\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"tactic": "obtain ⟨Y, b₀, hX, hb₀⟩ := hX.exists_orthonormalBasis_extension"
},
{
"state_after": "case intro.intro.intro\nι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"state_before": "case intro.intro.intro\nι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"tactic": "have hιY : Fintype.card ι = Y.card := by\n refine' card_ι.symm.trans _\n exact FiniteDimensional.finrank_eq_card_finset_basis b₀.toBasis"
},
{
"state_after": "case intro.intro.intro\nι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\nhvsY : MapsTo v s ↑Y\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"state_before": "case intro.intro.intro\nι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\n⊢ ∃ b, ∀ (i : ι), i ∈ s → ↑b i = v i",
"tactic": "have hvsY : s.MapsTo v Y := (s.mapsTo_image v).mono_right (by rwa [← range_restrict])"
},
{
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"tactic": "have hsv' : Set.InjOn v s := by\n rw [Set.injOn_iff_injective]\n exact hsv"
},
{
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"tactic": "obtain ⟨g, hg⟩ := hvsY.exists_equiv_extend_of_card_eq hιY hsv'"
},
{
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"tactic": "use b₀.reindex g.symm"
},
{
"state_after": "case intro.intro.intro.intro\nι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\nhvsY : MapsTo v s ↑Y\nhsv' : InjOn v s\ng : ι ≃ { x // x ∈ Y }\nhg : ∀ (i : ι), i ∈ s → ↑(↑g i) = v i\ni : ι\nhi : i ∈ s\n⊢ ↑(OrthonormalBasis.reindex b₀ g.symm) i = v i",
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"tactic": "intro i hi"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\n⊢ Orthonormal 𝕜 Subtype.val",
"tactic": "rwa [orthonormal_subtype_range hsv]"
},
{
"state_after": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\n⊢ finrank 𝕜 E = Finset.card Y",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\n⊢ Fintype.card ι = Finset.card Y",
"tactic": "refine' card_ι.symm.trans _"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\n⊢ finrank 𝕜 E = Finset.card Y",
"tactic": "exact FiniteDimensional.finrank_eq_card_finset_basis b₀.toBasis"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\n⊢ v '' s ⊆ ↑Y",
"tactic": "rwa [← range_restrict]"
},
{
"state_after": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\nhvsY : MapsTo v s ↑Y\n⊢ Injective (restrict s v)",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\nhvsY : MapsTo v s ↑Y\n⊢ InjOn v s",
"tactic": "rw [Set.injOn_iff_injective]"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\nhvsY : MapsTo v s ↑Y\n⊢ Injective (restrict s v)",
"tactic": "exact hsv"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nι✝ : Type ?u.1635810\nι' : Type ?u.1635813\n𝕜 : Type u_2\ninst✝¹¹ : IsROrC 𝕜\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace 𝕜 E\nE' : Type ?u.1635842\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : InnerProductSpace 𝕜 E'\nF : Type ?u.1635860\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ F\nF' : Type ?u.1635880\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : InnerProductSpace ℝ F'\ninst✝² : Fintype ι✝\nv✝ : Set E\nA : ι✝ → Submodule 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_1\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (restrict s v)\nhsv : Injective (restrict s v)\nhX✝ : Orthonormal 𝕜 Subtype.val\nY : Finset E\nb₀ : OrthonormalBasis { x // x ∈ Y } 𝕜 E\nhX : range (restrict s v) ⊆ ↑Y\nhb₀ : ↑b₀ = Subtype.val\nhιY : Fintype.card ι = Finset.card Y\nhvsY : MapsTo v s ↑Y\nhsv' : InjOn v s\ng : ι ≃ { x // x ∈ Y }\nhg : ∀ (i : ι), i ∈ s → ↑(↑g i) = v i\ni : ι\nhi : i ∈ s\n⊢ ↑(OrthonormalBasis.reindex b₀ g.symm) i = v i",
"tactic": "simp [hb₀, hg i hi]"
}
] |
[
802,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
785,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
uniformEmbedding_comap
|
[] |
[
410,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Analysis/NormedSpace/Complemented.lean
|
Subspace.coe_prodEquivOfClosedCompl
|
[] |
[
107,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.comap_le_comap_iff_of_surjective
|
[] |
[
930,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
929,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.shiftl_neg
|
[] |
[
372,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.isSheafFor_iso
|
[
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\n⊢ IsSheafFor P R → IsSheafFor P' R",
"tactic": "intro h x hx"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"tactic": "let x' := x.compPresheafMap i.inv"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"tactic": "have : x'.Compatible := FamilyOfElements.Compatible.compPresheafMap i.inv hx"
},
{
"state_after": "case intro.intro\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"tactic": "obtain ⟨t, ht1, ht2⟩ := h x' this"
},
{
"state_after": "case intro.intro\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (i.hom.app X.op t) ∧\n ∀ (y : P'.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = i.hom.app X.op t",
"state_before": "case intro.intro\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ ∃! t, FamilyOfElements.IsAmalgamation x t",
"tactic": "use i.hom.app _ t"
},
{
"state_after": "case intro.intro.left\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (i.hom.app X.op t)\n\ncase intro.intro.right\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ ∀ (y : P'.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = i.hom.app X.op t",
"state_before": "case intro.intro\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (i.hom.app X.op t) ∧\n ∀ (y : P'.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = i.hom.app X.op t",
"tactic": "fconstructor"
},
{
"state_after": "case h.e.h.e'_6.h.h.h\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ x = FamilyOfElements.compPresheafMap i.hom x'",
"state_before": "case intro.intro.left\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ (fun t => FamilyOfElements.IsAmalgamation x t) (i.hom.app X.op t)",
"tactic": "convert FamilyOfElements.IsAmalgamation.compPresheafMap i.hom ht1"
},
{
"state_after": "no goals",
"state_before": "case h.e.h.e'_6.h.h.h\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ x = FamilyOfElements.compPresheafMap i.hom x'",
"tactic": "simp"
},
{
"state_after": "case intro.intro.right\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\ny : P'.obj X.op\nhy : FamilyOfElements.IsAmalgamation x y\n⊢ y = i.hom.app X.op t",
"state_before": "case intro.intro.right\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\n⊢ ∀ (y : P'.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x t) y → y = i.hom.app X.op t",
"tactic": "intro y hy"
},
{
"state_after": "case intro.intro.right\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\ny : P'.obj X.op\nhy : FamilyOfElements.IsAmalgamation x y\n⊢ (i.inv.app X.op ≫ i.hom.app X.op) y = i.hom.app X.op t",
"state_before": "case intro.intro.right\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\ny : P'.obj X.op\nhy : FamilyOfElements.IsAmalgamation x y\n⊢ y = i.hom.app X.op t",
"tactic": "rw [show y = (i.inv.app (op X) ≫ i.hom.app (op X)) y by simp]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.right\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\ny : P'.obj X.op\nhy : FamilyOfElements.IsAmalgamation x y\n⊢ (i.inv.app X.op ≫ i.hom.app X.op) y = i.hom.app X.op t",
"tactic": "simp [ht2 (i.inv.app _ y) (FamilyOfElements.IsAmalgamation.compPresheafMap i.inv hy)]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nP' : Cᵒᵖ ⥤ Type w\ni : P ≅ P'\nh : IsSheafFor P R\nx : FamilyOfElements P' R\nhx : FamilyOfElements.Compatible x\nx' : FamilyOfElements P R := FamilyOfElements.compPresheafMap i.inv x\nthis : FamilyOfElements.Compatible x'\nt : P.obj X.op\nht1 : FamilyOfElements.IsAmalgamation x' t\nht2 : ∀ (y : P.obj X.op), (fun t => FamilyOfElements.IsAmalgamation x' t) y → y = t\ny : P'.obj X.op\nhy : FamilyOfElements.IsAmalgamation x y\n⊢ y = (i.inv.app X.op ≫ i.hom.app X.op) y",
"tactic": "simp"
}
] |
[
681,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
670,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
gramSchmidt_inv_triangular
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\n⊢ inner (gramSchmidt 𝕜 v j)\n (gramSchmidt 𝕜 v i +\n ∑ i_1 in Iio i, (inner (gramSchmidt 𝕜 v i_1) (v i) / ↑‖gramSchmidt 𝕜 v i_1‖ ^ 2) • gramSchmidt 𝕜 v i_1) =\n 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\n⊢ inner (gramSchmidt 𝕜 v j) (v i) = 0",
"tactic": "rw [gramSchmidt_def'' 𝕜 v]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\n⊢ inner (gramSchmidt 𝕜 v j) (gramSchmidt 𝕜 v i) +\n ∑ x in Iio i,\n inner (gramSchmidt 𝕜 v x) (v i) / ↑‖gramSchmidt 𝕜 v x‖ ^ 2 * inner (gramSchmidt 𝕜 v j) (gramSchmidt 𝕜 v x) =\n 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\n⊢ inner (gramSchmidt 𝕜 v j)\n (gramSchmidt 𝕜 v i +\n ∑ i_1 in Iio i, (inner (gramSchmidt 𝕜 v i_1) (v i) / ↑‖gramSchmidt 𝕜 v i_1‖ ^ 2) • gramSchmidt 𝕜 v i_1) =\n 0",
"tactic": "simp only [inner_add_right, inner_sum, inner_smul_right]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ inner (b j) (b i) + ∑ x in Iio i, inner (b x) (v i) / ↑‖b x‖ ^ 2 * inner (b j) (b x) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\n⊢ inner (gramSchmidt 𝕜 v j) (gramSchmidt 𝕜 v i) +\n ∑ x in Iio i,\n inner (gramSchmidt 𝕜 v x) (v i) / ↑‖gramSchmidt 𝕜 v x‖ ^ 2 * inner (gramSchmidt 𝕜 v j) (gramSchmidt 𝕜 v x) =\n 0",
"tactic": "set b : ι → E := gramSchmidt 𝕜 v"
},
{
"state_after": "case h.e'_2.h.e'_5\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ inner (b j) (b i) = 0\n\ncase h.e'_2.h.e'_6\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ ∑ x in Iio i, inner (b x) (v i) / ↑‖b x‖ ^ 2 * inner (b j) (b x) = 0",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ inner (b j) (b i) + ∑ x in Iio i, inner (b x) (v i) / ↑‖b x‖ ^ 2 * inner (b j) (b x) = 0",
"tactic": "convert zero_add (0 : 𝕜)"
},
{
"state_after": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ ∀ (x : ι), x ∈ Iio i → inner (b x) (v i) / ↑‖b x‖ ^ 2 * inner (b j) (b x) = 0",
"state_before": "case h.e'_2.h.e'_6\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ ∑ x in Iio i, inner (b x) (v i) / ↑‖b x‖ ^ 2 * inner (b j) (b x) = 0",
"tactic": "apply Finset.sum_eq_zero"
},
{
"state_after": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\n⊢ inner (b k) (v i) / ↑‖b k‖ ^ 2 * inner (b j) (b k) = 0",
"state_before": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ ∀ (x : ι), x ∈ Iio i → inner (b x) (v i) / ↑‖b x‖ ^ 2 * inner (b j) (b x) = 0",
"tactic": "rintro k hki'"
},
{
"state_after": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\nhki : k < i\n⊢ inner (b k) (v i) / ↑‖b k‖ ^ 2 * inner (b j) (b k) = 0",
"state_before": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\n⊢ inner (b k) (v i) / ↑‖b k‖ ^ 2 * inner (b j) (b k) = 0",
"tactic": "have hki : k < i := by simpa using hki'"
},
{
"state_after": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\nhki : k < i\nthis : inner (b j) (b k) = 0\n⊢ inner (b k) (v i) / ↑‖b k‖ ^ 2 * inner (b j) (b k) = 0",
"state_before": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\nhki : k < i\n⊢ inner (b k) (v i) / ↑‖b k‖ ^ 2 * inner (b j) (b k) = 0",
"tactic": "have : ⟪b j, b k⟫ = 0 := gramSchmidt_orthogonal 𝕜 v (hki.trans hij).ne'"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6.h\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\nhki : k < i\nthis : inner (b j) (b k) = 0\n⊢ inner (b k) (v i) / ↑‖b k‖ ^ 2 * inner (b j) (b k) = 0",
"tactic": "simp [this]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\n⊢ inner (b j) (b i) = 0",
"tactic": "exact gramSchmidt_orthogonal 𝕜 v hij.ne'"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nv : ι → E\ni j : ι\nhij : i < j\nb : ι → E := gramSchmidt 𝕜 v\nk : ι\nhki' : k ∈ Iio i\n⊢ k < i",
"tactic": "simpa using hki'"
}
] |
[
134,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Algebra/Group/Pi.lean
|
Pi.mulSingle_commute
|
[
{
"state_after": "ι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\n⊢ Commute (mulSingle i x) (mulSingle j y)",
"state_before": "ι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx y : (i : I) → f i\ni j : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\n⊢ Pairwise fun i j => ∀ (x : f i) (y : f j), Commute (mulSingle i x) (mulSingle j y)",
"tactic": "intro i j hij x y"
},
{
"state_after": "case h\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"state_before": "ι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\n⊢ Commute (mulSingle i x) (mulSingle j y)",
"tactic": "ext k"
},
{
"state_after": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : i = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k\n\ncase neg\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : ¬i = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"state_before": "case h\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"tactic": "by_cases h1 : i = k"
},
{
"state_after": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : ¬i = k\nh2 : j = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k\n\ncase neg\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : ¬i = k\nh2 : ¬j = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"state_before": "case neg\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : ¬i = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"tactic": "by_cases h2 : j = k"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : ¬i = k\nh2 : ¬j = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"tactic": "simp [h1, h2]"
},
{
"state_after": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\n⊢ (mulSingle i x * mulSingle j y) i = (mulSingle j y * mulSingle i x) i",
"state_before": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : i = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"tactic": "subst h1"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\n⊢ (mulSingle i x * mulSingle j y) i = (mulSingle j y * mulSingle i x) i",
"tactic": "simp [hij]"
},
{
"state_after": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nh1 : ¬i = j\n⊢ (mulSingle i x * mulSingle j y) j = (mulSingle j y * mulSingle i x) j",
"state_before": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nk : I\nh1 : ¬i = k\nh2 : j = k\n⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k",
"tactic": "subst h2"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.68984\nα : Type ?u.68987\nI : Type u\nf : I → Type v\nx✝ y✝ : (i : I) → f i\ni✝ j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → MulOneClass (f i)\ni j : I\nhij : i ≠ j\nx : f i\ny : f j\nh1 : ¬i = j\n⊢ (mulSingle i x * mulSingle j y) j = (mulSingle j y * mulSingle i x) j",
"tactic": "simp [hij]"
}
] |
[
552,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Std/Data/Int/DivMod.lean
|
Int.le_of_dvd
|
[] |
[
847,
97
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
844,
1
] |
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
isPrimePow_iff_minFac_pow_factorization_eq
|
[] |
[
42,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
OrthogonalFamily.hasSum_linearIsometry
|
[] |
[
232,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
11
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.eq_zero_of_le_zero
|
[] |
[
302,
32
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
301,
1
] |
Mathlib/Analysis/Convex/Star.lean
|
starConvex_pi
|
[] |
[
151,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tendsto_nat_nhds_top
|
[] |
[
183,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Topology/UniformSpace/Pi.lean
|
uniformContinuous_pi
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u\nU : (i : ι) → UniformSpace (α i)\nβ : Type u_1\ninst✝ : UniformSpace β\nf : β → (i : ι) → α i\n⊢ UniformContinuous f ↔ ∀ (i : ι), UniformContinuous fun x => f x i",
"tactic": "simp only [UniformContinuous, Pi.uniformity, tendsto_iInf, tendsto_comap_iff, Function.comp]"
}
] |
[
47,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Std/Classes/Order.lean
|
Ordering.swap_swap
|
[
{
"state_after": "no goals",
"state_before": "o : Ordering\n⊢ swap (swap o) = o",
"tactic": "cases o <;> rfl"
}
] |
[
11,
90
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
11,
9
] |
Mathlib/Algebra/Order/Ring/Cone.lean
|
Ring.PositiveCone.one_pos
|
[] |
[
53,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
snd_himp
|
[] |
[
118,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Submodule.dualAnnihilator_sup_eq
|
[] |
[
903,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
901,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
tendsto_rpow_neg_atTop
|
[] |
[
54,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.biUnion_congr
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.506363\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nhs : s₁ = s₂\nht : ∀ (a : α), a ∈ s₁ → t₁ a = t₂ a\nx : β\n⊢ (∃ a, a ∈ s₁ ∧ x ∈ t₁ a) ↔ ∃ a, a ∈ s₂ ∧ x ∈ t₂ a",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.506363\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nhs : s₁ = s₂\nht : ∀ (a : α), a ∈ s₁ → t₁ a = t₂ a\nx : β\n⊢ x ∈ Finset.biUnion s₁ t₁ ↔ x ∈ Finset.biUnion s₂ t₂",
"tactic": "simp_rw [mem_biUnion]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.506363\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nhs : s₁ = s₂\nht : ∀ (a : α), a ∈ s₁ → t₁ a = t₂ a\nx : β\n⊢ ∀ (a : α), a ∈ s₁ ∧ x ∈ t₁ a ↔ a ∈ s₂ ∧ x ∈ t₂ a",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.506363\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nhs : s₁ = s₂\nht : ∀ (a : α), a ∈ s₁ → t₁ a = t₂ a\nx : β\n⊢ (∃ a, a ∈ s₁ ∧ x ∈ t₁ a) ↔ ∃ a, a ∈ s₂ ∧ x ∈ t₂ a",
"tactic": "apply exists_congr"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.506363\ninst✝ : DecidableEq β\ns s₁ s₂ : Finset α\nt t₁ t₂ : α → Finset β\nhs : s₁ = s₂\nht : ∀ (a : α), a ∈ s₁ → t₁ a = t₂ a\nx : β\n⊢ ∀ (a : α), a ∈ s₁ ∧ x ∈ t₁ a ↔ a ∈ s₂ ∧ x ∈ t₂ a",
"tactic": "simp (config := { contextual := true }) only [hs, and_congr_right_iff, ht, implies_true]"
}
] |
[
3561,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3557,
1
] |
Mathlib/Data/List/Forall2.lean
|
Relator.RightUnique.forall₂
|
[] |
[
155,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Topology/Connected.lean
|
connectedComponentIn_mem_nhds
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.113849\nπ : ι → Type ?u.113854\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\ninst✝ : LocallyConnectedSpace α\nF : Set α\nx : α\nh : ∃ i, (IsOpen i ∧ x ∈ i ∧ IsConnected i) ∧ id i ⊆ F\n⊢ connectedComponentIn F x ∈ 𝓝 x",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.113849\nπ : ι → Type ?u.113854\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\ninst✝ : LocallyConnectedSpace α\nF : Set α\nx : α\nh : F ∈ 𝓝 x\n⊢ connectedComponentIn F x ∈ 𝓝 x",
"tactic": "rw [(LocallyConnectedSpace.open_connected_basis x).mem_iff] at h"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.113849\nπ : ι → Type ?u.113854\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : LocallyConnectedSpace α\nF : Set α\nx : α\ns : Set α\nhsF : id s ⊆ F\nh1s : IsOpen s\nhxs : x ∈ s\nh2s : IsConnected s\n⊢ connectedComponentIn F x ∈ 𝓝 x",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.113849\nπ : ι → Type ?u.113854\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\ninst✝ : LocallyConnectedSpace α\nF : Set α\nx : α\nh : ∃ i, (IsOpen i ∧ x ∈ i ∧ IsConnected i) ∧ id i ⊆ F\n⊢ connectedComponentIn F x ∈ 𝓝 x",
"tactic": "rcases h with ⟨s, ⟨h1s, hxs, h2s⟩, hsF⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.113849\nπ : ι → Type ?u.113854\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : LocallyConnectedSpace α\nF : Set α\nx : α\ns : Set α\nhsF : id s ⊆ F\nh1s : IsOpen s\nhxs : x ∈ s\nh2s : IsConnected s\n⊢ connectedComponentIn F x ∈ 𝓝 x",
"tactic": "exact mem_nhds_iff.mpr ⟨s, h2s.isPreconnected.subset_connectedComponentIn hxs hsF, h1s, hxs⟩"
}
] |
[
1146,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1142,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.coe_cons_eq_coe_append
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\na : α\n⊢ rotate (a :: l) 1 = l ++ [a]",
"tactic": "rw [rotate_cons_succ, rotate_zero]"
}
] |
[
483,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.degree_pos_iff_exists_adj
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.269390\n𝕜 : Type ?u.269393\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\ninst✝ : Fintype ↑(neighborSet G v)\n⊢ 0 < degree G v ↔ ∃ w, Adj G v w",
"tactic": "simp only [degree, card_pos, Finset.Nonempty, mem_neighborFinset]"
}
] |
[
1387,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1386,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieHom.coe_id
|
[] |
[
342,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Std/Data/List/Lemmas.lean
|
List.not_mem_nil
|
[] |
[
58,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
58,
9
] |
Mathlib/GroupTheory/GroupAction/BigOperators.lean
|
Finset.smul_sum
|
[] |
[
57,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Algebra/Opposites.lean
|
MulOpposite.op_unop
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_sub'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoMod hp (a - p) b = toIcoMod hp a b - p",
"tactic": "simpa only [one_zsmul] using toIcoMod_sub_zsmul' hp a b 1"
}
] |
[
519,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.natAbs_zero
|
[] |
[
140,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
140,
9
] |
Mathlib/Algebra/Order/Archimedean.lean
|
existsUnique_zsmul_near_of_pos
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "let s : Set ℤ := { n : ℤ | n • a ≤ g }"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk : ℕ\nhk : -g ≤ k • a\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "obtain ⟨k, hk : -g ≤ k • a⟩ := Archimedean.arch (-g) ha"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk : ℕ\nhk : -g ≤ k • a\nh_ne : Set.Nonempty s\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"state_before": "case intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk : ℕ\nhk : -g ≤ k • a\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "have h_ne : s.Nonempty := ⟨-k, by simpa using neg_le_neg hk⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"state_before": "case intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk : ℕ\nhk : -g ≤ k • a\nh_ne : Set.Nonempty s\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "obtain ⟨k, hk⟩ := Archimedean.arch g ha"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "have h_bdd : ∀ n ∈ s, n ≤ (k : ℤ) := by\n intro n hn\n apply (zsmul_le_zsmul_iff ha).mp\n rw [← coe_nat_zsmul] at hk\n exact le_trans hn hk"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "obtain ⟨m, hm, hm'⟩ := Int.exists_greatest_of_bdd ⟨k, h_bdd⟩ h_ne"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\nhm'' : g < (m + 1) • a\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "have hm'' : g < (m + 1) • a := by\n contrapose! hm'\n exact ⟨m + 1, hm', lt_add_one _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\nhm'' : g < (m + 1) • a\nn : ℤ\nhn : (fun k => k • a ≤ g ∧ g < (k + 1) • a) n\n⊢ m < n + 1",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\nhm'' : g < (m + 1) • a\n⊢ ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
"tactic": "refine' ⟨m, ⟨hm, hm''⟩, fun n hn => (hm' n hn.1).antisymm <| Int.le_of_lt_add_one _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\nhm'' : g < (m + 1) • a\nn : ℤ\nhn : (fun k => k • a ≤ g ∧ g < (k + 1) • a) n\n⊢ m • a < (n + 1) • a",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\nhm'' : g < (m + 1) • a\nn : ℤ\nhn : (fun k => k • a ≤ g ∧ g < (k + 1) • a) n\n⊢ m < n + 1",
"tactic": "rw [← zsmul_lt_zsmul_iff ha]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\nhm'' : g < (m + 1) • a\nn : ℤ\nhn : (fun k => k • a ≤ g ∧ g < (k + 1) • a) n\n⊢ m • a < (n + 1) • a",
"tactic": "exact lt_of_le_of_lt hm hn.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk : ℕ\nhk : -g ≤ k • a\n⊢ -↑k ∈ s",
"tactic": "simpa using neg_le_neg hk"
},
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nn : ℤ\nhn : n ∈ s\n⊢ n ≤ ↑k",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\n⊢ ∀ (n : ℤ), n ∈ s → n ≤ ↑k",
"tactic": "intro n hn"
},
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nn : ℤ\nhn : n ∈ s\n⊢ n • a ≤ ↑k • a",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nn : ℤ\nhn : n ∈ s\n⊢ n ≤ ↑k",
"tactic": "apply (zsmul_le_zsmul_iff ha).mp"
},
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ ↑k • a\nn : ℤ\nhn : n ∈ s\n⊢ n • a ≤ ↑k • a",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nn : ℤ\nhn : n ∈ s\n⊢ n • a ≤ ↑k • a",
"tactic": "rw [← coe_nat_zsmul] at hk"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ ↑k • a\nn : ℤ\nhn : n ∈ s\n⊢ n • a ≤ ↑k • a",
"tactic": "exact le_trans hn hk"
},
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : (m + 1) • a ≤ g\n⊢ ∃ z, z ∈ {n | n • a ≤ g} ∧ m < z",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : ∀ (z : ℤ), z ∈ s → z ≤ m\n⊢ g < (m + 1) • a",
"tactic": "contrapose! hm'"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedAddCommGroup α\ninst✝ : Archimedean α\na : α\nha : 0 < a\ng : α\ns : Set ℤ := {n | n • a ≤ g}\nk✝ : ℕ\nhk✝ : -g ≤ k✝ • a\nh_ne : Set.Nonempty s\nk : ℕ\nhk : g ≤ k • a\nh_bdd : ∀ (n : ℤ), n ∈ s → n ≤ ↑k\nm : ℤ\nhm : m ∈ s\nhm' : (m + 1) • a ≤ g\n⊢ ∃ z, z ∈ {n | n • a ≤ g} ∧ m < z",
"tactic": "exact ⟨m + 1, hm', lt_add_one _⟩"
}
] |
[
76,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Algebra.algebraMap_pUnit
|
[] |
[
478,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/FieldTheory/Laurent.lean
|
RatFunc.laurent_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nhdomain : IsDomain R\nr s : R\np q : R[X]\nf : RatFunc R\nx : R\n⊢ ↑(laurent r) (↑C x) = ↑C x",
"tactic": "rw [← algebraMap_C, laurent_algebraMap, taylor_C]"
}
] |
[
107,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.commute_X_pow
|
[] |
[
619,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_of_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.38067\nα : Type u_1\nβ : Type ?u.38073\nπ : ι → Type ?u.38078\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c d a b : α\nh : a ≤ b\n⊢ a ⇔ b = b ⇨ a",
"tactic": "rw [bihimp, himp_eq_top_iff.2 h, inf_top_eq]"
}
] |
[
263,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_mem_eq_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.195249\nι : Type ?u.195252\nG : Type ?u.195255\nM : Type u_2\nN : Type ?u.195261\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\ns : Set α\nhf : Set.Finite (s ∩ mulSupport f)\n⊢ (s ∩ mulSupport fun i => f i) = ↑(Finite.toFinset hf) ∩ mulSupport fun i => f i",
"tactic": "simp [inter_assoc]"
}
] |
[
492,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Data/MvPolynomial/CommRing.lean
|
MvPolynomial.hom_C
|
[] |
[
147,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/Convex/Hull.lean
|
LinearMap.convexHull_image
|
[] |
[
181,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Topology/MetricSpace/Baire.lean
|
eventually_residual
|
[
{
"state_after": "case h.e'_2.h.e'_2.h.a\nα : Type u_1\nβ : Type ?u.15155\nγ : Type ?u.15158\nι : Type ?u.15161\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\np : α → Prop\nx✝ : Set α\n⊢ (IsGδ x✝ ∧ Dense x✝ ∧ ∀ (x : α), x ∈ x✝ → p x) ↔ ∃ x, IsGδ x✝ ∧ Dense x✝",
"state_before": "α : Type u_1\nβ : Type ?u.15155\nγ : Type ?u.15158\nι : Type ?u.15161\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\np : α → Prop\n⊢ (∀ᶠ (x : α) in residual α, p x) ↔ ∃ t, IsGδ t ∧ Dense t ∧ ∀ (x : α), x ∈ t → p x",
"tactic": "convert@mem_residual _ _ _ p"
},
{
"state_after": "case h.e'_2.h.e'_2.h.a\nα : Type u_1\nβ : Type ?u.15155\nγ : Type ?u.15158\nι : Type ?u.15161\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\np : α → Prop\nx✝ : Set α\n⊢ (IsGδ x✝ ∧ Dense x✝ ∧ ∀ (x : α), x ∈ x✝ → p x) ↔ IsGδ x✝ ∧ Dense x✝ ∧ x✝ ⊆ p",
"state_before": "case h.e'_2.h.e'_2.h.a\nα : Type u_1\nβ : Type ?u.15155\nγ : Type ?u.15158\nι : Type ?u.15161\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\np : α → Prop\nx✝ : Set α\n⊢ (IsGδ x✝ ∧ Dense x✝ ∧ ∀ (x : α), x ∈ x✝ → p x) ↔ ∃ x, IsGδ x✝ ∧ Dense x✝",
"tactic": "simp_rw [exists_prop, @and_comm ((_ : Set α) ⊆ p), and_assoc]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_2.h.a\nα : Type u_1\nβ : Type ?u.15155\nγ : Type ?u.15158\nι : Type ?u.15161\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\np : α → Prop\nx✝ : Set α\n⊢ (IsGδ x✝ ∧ Dense x✝ ∧ ∀ (x : α), x ∈ x✝ → p x) ↔ IsGδ x✝ ∧ Dense x✝ ∧ x✝ ⊆ p",
"tactic": "rfl"
}
] |
[
248,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.ord_proj_dvd_ord_proj_iff_dvd
|
[
{
"state_after": "a b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\n⊢ a ∣ b",
"state_before": "a b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ (∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p) ↔ a ∣ b",
"tactic": "refine' ⟨fun h => _, fun hab p => ord_proj_dvd_ord_proj_of_dvd hb0 hab p⟩"
},
{
"state_after": "a b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\n⊢ factorization a ≤ factorization b",
"state_before": "a b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\n⊢ a ∣ b",
"tactic": "rw [← factorization_le_iff_dvd ha0 hb0]"
},
{
"state_after": "a b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\nq : ℕ\n⊢ ↑(factorization a) q ≤ ↑(factorization b) q",
"state_before": "a b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\n⊢ factorization a ≤ factorization b",
"tactic": "intro q"
},
{
"state_after": "case inl\na b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\nq : ℕ\nhq_le : q ≤ 1\n⊢ ↑(factorization a) q ≤ ↑(factorization b) q\n\ncase inr\na b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\nq : ℕ\nhq1 : 1 < q\n⊢ ↑(factorization a) q ≤ ↑(factorization b) q",
"state_before": "a b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\nq : ℕ\n⊢ ↑(factorization a) q ≤ ↑(factorization b) q",
"tactic": "rcases le_or_lt q 1 with (hq_le | hq1)"
},
{
"state_after": "no goals",
"state_before": "case inr\na b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\nq : ℕ\nhq1 : 1 < q\n⊢ ↑(factorization a) q ≤ ↑(factorization b) q",
"tactic": "exact (pow_dvd_pow_iff_le_right hq1).1 (h q)"
},
{
"state_after": "no goals",
"state_before": "case inl\na b : ℕ\nha0 : a ≠ 0\nhb0 : b ≠ 0\nh : ∀ (p : ℕ), p ^ ↑(factorization a) p ∣ p ^ ↑(factorization b) p\nq : ℕ\nhq_le : q ≤ 1\n⊢ ↑(factorization a) q ≤ ↑(factorization b) q",
"tactic": "interval_cases q <;> simp"
}
] |
[
609,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/Data/Real/Cardinality.lean
|
Cardinal.mk_univ_real
|
[
{
"state_after": "no goals",
"state_before": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ (#↑Set.univ) = 𝔠",
"tactic": "rw [mk_univ, mk_real]"
}
] |
[
217,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Analysis/NormedSpace/ENorm.lean
|
ENorm.finite_norm_eq
|
[] |
[
237,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix'_comp
|
[
{
"state_after": "R : Type u_2\ninst✝⁴ : CommSemiring R\nk : Type ?u.752944\nl : Type u_1\nm : Type u_4\nn : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Fintype l\ninst✝ : DecidableEq l\nf : (n → R) →ₗ[R] m → R\ng : (l → R) →ₗ[R] n → R\n⊢ comp f g = ↑toLin' (↑toMatrix' f ⬝ ↑toMatrix' g)",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nk : Type ?u.752944\nl : Type u_1\nm : Type u_4\nn : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Fintype l\ninst✝ : DecidableEq l\nf : (n → R) →ₗ[R] m → R\ng : (l → R) →ₗ[R] n → R\n⊢ ↑toMatrix' (comp f g) = ↑toMatrix' f ⬝ ↑toMatrix' g",
"tactic": "suffices f.comp g = Matrix.toLin' (LinearMap.toMatrix' f ⬝ LinearMap.toMatrix' g) by\n rw [this, LinearMap.toMatrix'_toLin']"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nk : Type ?u.752944\nl : Type u_1\nm : Type u_4\nn : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Fintype l\ninst✝ : DecidableEq l\nf : (n → R) →ₗ[R] m → R\ng : (l → R) →ₗ[R] n → R\n⊢ comp f g = ↑toLin' (↑toMatrix' f ⬝ ↑toMatrix' g)",
"tactic": "rw [Matrix.toLin'_mul, Matrix.toLin'_toMatrix', Matrix.toLin'_toMatrix']"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nk : Type ?u.752944\nl : Type u_1\nm : Type u_4\nn : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Fintype l\ninst✝ : DecidableEq l\nf : (n → R) →ₗ[R] m → R\ng : (l → R) →ₗ[R] n → R\nthis : comp f g = ↑toLin' (↑toMatrix' f ⬝ ↑toMatrix' g)\n⊢ ↑toMatrix' (comp f g) = ↑toMatrix' f ⬝ ↑toMatrix' g",
"tactic": "rw [this, LinearMap.toMatrix'_toLin']"
}
] |
[
401,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.mem_prod
|
[] |
[
1548,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1547,
1
] |
Mathlib/Topology/UrysohnsLemma.lean
|
Urysohns.CU.approx_le_succ
|
[
{
"state_after": "case zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nc : CU X\n⊢ approx Nat.zero c x ≤ approx (Nat.zero + 1) c x\n\ncase succ\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nn : ℕ\nihn : ∀ (c : CU X), approx n c x ≤ approx (n + 1) c x\nc : CU X\n⊢ approx (Nat.succ n) c x ≤ approx (Nat.succ n + 1) c x",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc : CU X\nn : ℕ\nx : X\n⊢ approx n c x ≤ approx (n + 1) c x",
"tactic": "induction' n with n ihn generalizing c"
},
{
"state_after": "case zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nc : CU X\n⊢ indicator (c.Uᶜ) 1 x ≤ indicator ((left c).Uᶜ) 1 x",
"state_before": "case zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nc : CU X\n⊢ approx Nat.zero c x ≤ approx (Nat.zero + 1) c x",
"tactic": "simp only [approx, right_U, right_le_midpoint]"
},
{
"state_after": "no goals",
"state_before": "case zero\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nc : CU X\n⊢ indicator (c.Uᶜ) 1 x ≤ indicator ((left c).Uᶜ) 1 x",
"tactic": "exact (approx_mem_Icc_right_left c 0 x).2"
},
{
"state_after": "case succ\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nn : ℕ\nihn : ∀ (c : CU X), approx n c x ≤ approx (n + 1) c x\nc : CU X\n⊢ midpoint ℝ (approx n (left c) x) (approx n (right c) x) ≤\n midpoint ℝ (approx (n + 1) (left c) x) (approx (n + 1) (right c) x)",
"state_before": "case succ\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nn : ℕ\nihn : ∀ (c : CU X), approx n c x ≤ approx (n + 1) c x\nc : CU X\n⊢ approx (Nat.succ n) c x ≤ approx (Nat.succ n + 1) c x",
"tactic": "rw [approx, approx]"
},
{
"state_after": "no goals",
"state_before": "case succ\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nc✝ : CU X\nx : X\nn : ℕ\nihn : ∀ (c : CU X), approx n c x ≤ approx (n + 1) c x\nc : CU X\n⊢ midpoint ℝ (approx n (left c) x) (approx n (right c) x) ≤\n midpoint ℝ (approx (n + 1) (left c) x) (approx (n + 1) (right c) x)",
"tactic": "exact midpoint_le_midpoint (ihn _) (ihn _)"
}
] |
[
205,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift'_bot
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.36509\nι : Sort ?u.36512\nf f₁ f₂ : Filter α\nh h₁ h₂ : Set α → Set β\nhh : Monotone h\n⊢ Filter.lift' ⊥ h = 𝓟 (h ∅)",
"tactic": "rw [← principal_empty, lift'_principal hh]"
}
] |
[
328,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
iteratedFDerivWithin_succ_apply_right
|
[
{
"state_after": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n + 1) → E\nx : E\nhx : x ∈ s\nm : Fin (Nat.zero + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.zero + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 Nat.zero (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last Nat.zero))\n\ncase succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last (Nat.succ n)))",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nm : Fin (n + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))",
"tactic": "induction' n with n IH generalizing x"
},
{
"state_after": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n + 1) → E\nx : E\nhx : x ∈ s\nm : Fin (Nat.zero + 1) → E\n⊢ ↑(↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F)\n (ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)).toLinearEquiv))\n (fderivWithin 𝕜 f s x)))\n m =\n ↑(fderivWithin 𝕜 f s x) (m (last Nat.zero))",
"state_before": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n + 1) → E\nx : E\nhx : x ∈ s\nm : Fin (Nat.zero + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.zero + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 Nat.zero (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last Nat.zero))",
"tactic": "rw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp,\n iteratedFDerivWithin_zero_apply, Function.comp_apply,\n LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]"
},
{
"state_after": "no goals",
"state_before": "case zero\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n + 1) → E\nx : E\nhx : x ∈ s\nm : Fin (Nat.zero + 1) → E\n⊢ ↑(↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F)\n (ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)).toLinearEquiv))\n (fderivWithin 𝕜 f s x)))\n m =\n ↑(fderivWithin 𝕜 f s x) (m (last Nat.zero))",
"tactic": "rfl"
},
{
"state_after": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last (Nat.succ n)))",
"state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last (Nat.succ n)))",
"tactic": "let I := continuousMultilinearCurryRightEquiv' 𝕜 n E F"
},
{
"state_after": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last (Nat.succ n)))",
"state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last (Nat.succ n)))",
"tactic": "have A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y =\n (I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y := fun y hy ↦ by\n ext m\n rw [@IH y hy m]\n rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last (Nat.succ n)))",
"tactic": "calc\n (iteratedFDerivWithin 𝕜 (n + 2) f s x : (Fin (n + 2) → E) → F) m =\n (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n.succ f s) s x : E → E[×n + 1]→L[𝕜] F) (m 0)\n (tail m) :=\n rfl\n _ = (fderivWithin 𝕜 (I ∘ iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :\n E → E[×n + 1]→L[𝕜] F) (m 0) (tail m) := by\n rw [fderivWithin_congr A (A x hx)]\n _ = (I ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :\n E → E[×n + 1]→L[𝕜] F) (m 0) (tail m) := by\n simp only [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]; rfl\n _ = (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x :\n E → E[×n]→L[𝕜] E →L[𝕜] F) (m 0) (init (tail m)) ((tail m) (last n)) := rfl\n _ = iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x (init m)\n (m (last (n + 1))) := by\n rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]\n rfl"
},
{
"state_after": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝² n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝¹ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm✝ : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\ny : E\nhy : y ∈ s\nm : Fin (Nat.succ n) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n) f s y) m =\n ↑((↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y) m",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\ny : E\nhy : y ∈ s\n⊢ iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y",
"tactic": "ext m"
},
{
"state_after": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝² n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝¹ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm✝ : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\ny : E\nhy : y ∈ s\nm : Fin (Nat.succ n) → E\n⊢ ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s y) (init m)) (m (last n)) =\n ↑((↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y) m",
"state_before": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝² n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝¹ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm✝ : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\ny : E\nhy : y ∈ s\nm : Fin (Nat.succ n) → E\n⊢ ↑(iteratedFDerivWithin 𝕜 (Nat.succ n) f s y) m =\n ↑((↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y) m",
"tactic": "rw [@IH y hy m]"
},
{
"state_after": "no goals",
"state_before": "case H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝² n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝¹ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm✝ : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\ny : E\nhy : y ∈ s\nm : Fin (Nat.succ n) → E\n⊢ ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s y) (init m)) (m (last n)) =\n ↑((↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y) m",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 (Nat.succ n) f s) s x) (m 0)) (tail m) =\n ↑(↑(fderivWithin 𝕜 (↑I ∘ iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x) (m 0)) (tail m)",
"tactic": "rw [fderivWithin_congr A (A x hx)]"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(↑(ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk (continuousMultilinearCurryRightEquiv' 𝕜 n E F).toLinearEquiv))\n (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x))\n (m 0))\n (tail m) =\n ↑((↑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘\n ↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x))\n (m 0))\n (tail m)",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(↑(fderivWithin 𝕜 (↑I ∘ iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x) (m 0)) (tail m) =\n ↑((↑I ∘ ↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x)) (m 0)) (tail m)",
"tactic": "simp only [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(↑(ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk (continuousMultilinearCurryRightEquiv' 𝕜 n E F).toLinearEquiv))\n (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x))\n (m 0))\n (tail m) =\n ↑((↑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘\n ↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x))\n (m 0))\n (tail m)",
"tactic": "rfl"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(↑(↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x) (m 0)) (init (tail m)))\n (tail m (last n)) =\n ↑(↑(↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x) (init m 0)) (init (tail m)))\n (m (last (n + 1)))",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(↑(↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x) (m 0)) (init (tail m)))\n (tail m (last n)) =\n ↑(↑(iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last (n + 1)))",
"tactic": "rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝¹ n✝¹ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn✝ : ℕ\nhs : UniqueDiffOn 𝕜 s\nhx✝ : x✝ ∈ s\nm✝ : Fin (n✝ + 1) → E\nn : ℕ\nIH :\n ∀ {x : E},\n x ∈ s →\n ∀ (m : Fin (n + 1) → E),\n ↑(iteratedFDerivWithin 𝕜 (n + 1) f s x) m =\n ↑(↑(iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))\nx : E\nhx : x ∈ s\nm : Fin (Nat.succ n + 1) → E\nI : ContinuousMultilinearMap 𝕜 (fun i => E) (E →L[𝕜] F) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun i => E) F :=\n continuousMultilinearCurryRightEquiv' 𝕜 n E F\nA :\n ∀ (y : E),\n y ∈ s →\n iteratedFDerivWithin 𝕜 (Nat.succ n) f s y = (↑I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y\n⊢ ↑(↑(↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x) (m 0)) (init (tail m)))\n (tail m (last n)) =\n ↑(↑(↑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x) (init m 0)) (init (tail m)))\n (m (last (n + 1)))",
"tactic": "rfl"
}
] |
[
850,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.658805\nG : Type ?u.658808\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace G\ninst✝ : CompleteSpace F\nA : E →L[𝕜] E\nx : E\nh : inner x (↑(↑adjoint A * A) x) = inner (↑A x) (↑A x)\n⊢ ‖↑A x‖ ^ 2 = ↑re (inner x (↑(↑adjoint A * A) x))",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.658805\nG : Type ?u.658808\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace G\ninst✝ : CompleteSpace F\nA : E →L[𝕜] E\nx : E\n⊢ ‖↑A x‖ ^ 2 = ↑re (inner x (↑(↑adjoint A * A) x))",
"tactic": "have h : ⟪x, (A† * A) x⟫ = ⟪A x, A x⟫ := by rw [← adjoint_inner_right]; rfl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.658805\nG : Type ?u.658808\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace G\ninst✝ : CompleteSpace F\nA : E →L[𝕜] E\nx : E\nh : inner x (↑(↑adjoint A * A) x) = inner (↑A x) (↑A x)\n⊢ ‖↑A x‖ ^ 2 = ↑re (inner x (↑(↑adjoint A * A) x))",
"tactic": "rw [h, ← inner_self_eq_norm_sq (𝕜 := 𝕜) _]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.658805\nG : Type ?u.658808\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace G\ninst✝ : CompleteSpace F\nA : E →L[𝕜] E\nx : E\n⊢ inner x (↑(↑adjoint A * A) x) = inner x (↑(↑adjoint A) (↑A x))",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.658805\nG : Type ?u.658808\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace G\ninst✝ : CompleteSpace F\nA : E →L[𝕜] E\nx : E\n⊢ inner x (↑(↑adjoint A * A) x) = inner (↑A x) (↑A x)",
"tactic": "rw [← adjoint_inner_right]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.658805\nG : Type ?u.658808\ninst✝⁹ : IsROrC 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpace G\ninst✝ : CompleteSpace F\nA : E →L[𝕜] E\nx : E\n⊢ inner x (↑(↑adjoint A * A) x) = inner x (↑(↑adjoint A) (↑A x))",
"tactic": "rfl"
}
] |
[
164,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
sum_add_tsum_subtype_compl
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.770476\nδ : Type ?u.770479\ninst✝⁴ : AddCommGroup α\ninst✝³ : UniformSpace α\ninst✝² : UniformAddGroup α\nf✝ g : β → α\na a₁ a₂ : α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Summable f\ns : Finset β\n⊢ (∑ x in s, f x + ∑' (x : { x // ¬x ∈ s }), f ↑x) = (∑' (x : ↑↑s), f ↑x) + ∑' (x : ↑(↑sᶜ)), f ↑x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.770476\nδ : Type ?u.770479\ninst✝⁴ : AddCommGroup α\ninst✝³ : UniformSpace α\ninst✝² : UniformAddGroup α\nf✝ g : β → α\na a₁ a₂ : α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Summable f\ns : Finset β\n⊢ (∑ x in s, f x + ∑' (x : { x // ¬x ∈ s }), f ↑x) = ∑' (x : β), f x",
"tactic": "rw [← tsum_subtype_add_tsum_subtype_compl hf s]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.770476\nδ : Type ?u.770479\ninst✝⁴ : AddCommGroup α\ninst✝³ : UniformSpace α\ninst✝² : UniformAddGroup α\nf✝ g : β → α\na a₁ a₂ : α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Summable f\ns : Finset β\n⊢ (∑' (x : { x // ¬x ∈ s }), f ↑x) = ∑' (x : ↑(↑sᶜ)), f ↑x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.770476\nδ : Type ?u.770479\ninst✝⁴ : AddCommGroup α\ninst✝³ : UniformSpace α\ninst✝² : UniformAddGroup α\nf✝ g : β → α\na a₁ a₂ : α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Summable f\ns : Finset β\n⊢ (∑ x in s, f x + ∑' (x : { x // ¬x ∈ s }), f ↑x) = (∑' (x : ↑↑s), f ↑x) + ∑' (x : ↑(↑sᶜ)), f ↑x",
"tactic": "simp only [Finset.tsum_subtype', add_right_inj]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.770476\nδ : Type ?u.770479\ninst✝⁴ : AddCommGroup α\ninst✝³ : UniformSpace α\ninst✝² : UniformAddGroup α\nf✝ g : β → α\na a₁ a₂ : α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Summable f\ns : Finset β\n⊢ (∑' (x : { x // ¬x ∈ s }), f ↑x) = ∑' (x : ↑(↑sᶜ)), f ↑x",
"tactic": "rfl"
}
] |
[
1241,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1237,
1
] |
Mathlib/RingTheory/Polynomial/Chebyshev.lean
|
Polynomial.Chebyshev.mul_T
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ ∀ (k : ℕ), 2 * T R 0 * T R (0 + k) = T R (2 * 0 + k) + T R k",
"tactic": "simp [two_mul, add_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ ∀ (k : ℕ), 2 * T R 1 * T R (1 + k) = T R (2 * 1 + k) + T R k",
"tactic": "simp [add_comm]"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\n⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm : ℕ\n⊢ ∀ (k : ℕ), 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"tactic": "intro k"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\n⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"tactic": "suffices 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k by\n have h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4 := by ring\n have h_nat₂ : m + 2 + k = m + k + 2 := by ring\n simpa [h_nat₁, h_nat₂] using this"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"tactic": "have H₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) := by\n have h_nat₁ : m + 1 + (k + 1) = m + k + 2 := by ring\n have h_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring\n simpa [h_nat₁, h_nat₂] using mul_T (m + 1) (k + 1)"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"tactic": "have H₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) := by\n have h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc]\n have h_nat₂ : m + (k + 2) = m + k + 2 := by simp [add_assoc]\n simpa [h_nat₁, h_nat₂] using mul_T m (k + 2)"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"tactic": "have h₁ := T_add_two R m"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m\nh₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2)\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"tactic": "have h₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2) :=\n T_add_two R (2 * m + k + 2)"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m\nh₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2)\nh₃ : T R (k + 2) = 2 * X * T R (k + 1) - T R k\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m\nh₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2)\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"tactic": "have h₃ := T_add_two R k"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH₂ : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh₁ : T R (m + 2) = 2 * X * T R (m + 1) - T R m\nh₂ : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2)\nh₃ : T R (k + 2) = 2 * X * T R (k + 1) - T R k\n⊢ 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k",
"tactic": "linear_combination 2 * T R (m + k + 2) * h₁ + 2 * (X : R[X]) * H₁ - H₂ - h₂ - h₃"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4\n⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"tactic": "have h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4 := by ring"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4\nh_nat₂ : m + 2 + k = m + k + 2\n⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4\n⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"tactic": "have h_nat₂ : m + 2 + k = m + k + 2 := by ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4\nh_nat₂ : m + 2 + k = m + k + 2\n⊢ 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k",
"tactic": "simpa [h_nat₁, h_nat₂] using this"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n⊢ 2 * (m + 2) + k = 2 * m + k + 4",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4\n⊢ m + 2 + k = m + k + 2",
"tactic": "ring"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nh_nat₁ : m + 1 + (k + 1) = m + k + 2\n⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\n⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)",
"tactic": "have h_nat₁ : m + 1 + (k + 1) = m + k + 2 := by ring"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nh_nat₁ : m + 1 + (k + 1) = m + k + 2\nh_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3\n⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nh_nat₁ : m + 1 + (k + 1) = m + k + 2\n⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)",
"tactic": "have h_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nh_nat₁ : m + 1 + (k + 1) = m + k + 2\nh_nat₂ : 2 * (m + 1) + (k + 1) = 2 * m + k + 3\n⊢ 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)",
"tactic": "simpa [h_nat₁, h_nat₂] using mul_T (m + 1) (k + 1)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\n⊢ m + 1 + (k + 1) = m + k + 2",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nh_nat₁ : m + 1 + (k + 1) = m + k + 2\n⊢ 2 * (m + 1) + (k + 1) = 2 * m + k + 3",
"tactic": "ring"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2\n⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)",
"tactic": "have h_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc]"
},
{
"state_after": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2\nh_nat₂ : m + (k + 2) = m + k + 2\n⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2\n⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)",
"tactic": "have h_nat₂ : m + (k + 2) = m + k + 2 := by simp [add_assoc]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2\nh_nat₂ : m + (k + 2) = m + k + 2\n⊢ 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)",
"tactic": "simpa [h_nat₁, h_nat₂] using mul_T m (k + 2)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n⊢ 2 * m + (k + 2) = 2 * m + k + 2",
"tactic": "simp [add_assoc]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.100205\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nm k : ℕ\nH₁ : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat₁ : 2 * m + (k + 2) = 2 * m + k + 2\n⊢ m + (k + 2) = m + k + 2",
"tactic": "simp [add_assoc]"
}
] |
[
268,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Data/MvPolynomial/Rename.lean
|
MvPolynomial.constantCoeff_rename
|
[
{
"state_after": "case h_C\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ∀ (a : R), ↑constantCoeff (↑(rename f) (↑C a)) = ↑constantCoeff (↑C a)\n\ncase h_add\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R),\n ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p →\n ↑constantCoeff (↑(rename f) q) = ↑constantCoeff q → ↑constantCoeff (↑(rename f) (p + q)) = ↑constantCoeff (p + q)\n\ncase h_X\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p →\n ↑constantCoeff (↑(rename f) (p * X n)) = ↑constantCoeff (p * X n)",
"state_before": "σ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ↑constantCoeff (↑(rename f) φ) = ↑constantCoeff φ",
"tactic": "apply φ.induction_on"
},
{
"state_after": "case h_C\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\na : R\n⊢ ↑constantCoeff (↑(rename f) (↑C a)) = ↑constantCoeff (↑C a)",
"state_before": "case h_C\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ∀ (a : R), ↑constantCoeff (↑(rename f) (↑C a)) = ↑constantCoeff (↑C a)",
"tactic": "intro a"
},
{
"state_after": "no goals",
"state_before": "case h_C\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\na : R\n⊢ ↑constantCoeff (↑(rename f) (↑C a)) = ↑constantCoeff (↑C a)",
"tactic": "simp only [constantCoeff_C, rename_C]"
},
{
"state_after": "case h_add\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ p q : MvPolynomial σ R\nhp : ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p\nhq : ↑constantCoeff (↑(rename f) q) = ↑constantCoeff q\n⊢ ↑constantCoeff (↑(rename f) (p + q)) = ↑constantCoeff (p + q)",
"state_before": "case h_add\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R),\n ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p →\n ↑constantCoeff (↑(rename f) q) = ↑constantCoeff q → ↑constantCoeff (↑(rename f) (p + q)) = ↑constantCoeff (p + q)",
"tactic": "intro p q hp hq"
},
{
"state_after": "no goals",
"state_before": "case h_add\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ p q : MvPolynomial σ R\nhp : ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p\nhq : ↑constantCoeff (↑(rename f) q) = ↑constantCoeff q\n⊢ ↑constantCoeff (↑(rename f) (p + q)) = ↑constantCoeff (p + q)",
"tactic": "simp only [hp, hq, RingHom.map_add, AlgHom.map_add]"
},
{
"state_after": "case h_X\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ p : MvPolynomial σ R\nn : σ\nhp : ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p\n⊢ ↑constantCoeff (↑(rename f) (p * X n)) = ↑constantCoeff (p * X n)",
"state_before": "case h_X\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p →\n ↑constantCoeff (↑(rename f) (p * X n)) = ↑constantCoeff (p * X n)",
"tactic": "intro p n hp"
},
{
"state_after": "no goals",
"state_before": "case h_X\nσ : Type u_2\nτ✝ : Type ?u.978943\nα : Type ?u.978946\nR : Type u_3\nS : Type ?u.978952\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nτ : Type u_1\nf : σ → τ\nφ p : MvPolynomial σ R\nn : σ\nhp : ↑constantCoeff (↑(rename f) p) = ↑constantCoeff p\n⊢ ↑constantCoeff (↑(rename f) (p * X n)) = ↑constantCoeff (p * X n)",
"tactic": "simp only [hp, rename_X, constantCoeff_X, RingHom.map_mul, AlgHom.map_mul]"
}
] |
[
336,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.finite_setOf_isRoot
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\nhp : p ≠ 0\n⊢ Set.Finite {x | IsRoot p x}",
"tactic": "simpa only [← Finset.setOf_mem, Multiset.mem_toFinset, mem_roots hp]\n using p.roots.toFinset.finite_toSet"
}
] |
[
604,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
GroupCat.ker_eq_bot_of_mono
|
[] |
[
83,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMax.add
|
[] |
[
374,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
8
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.sup_pow_eq_top
|
[
{
"state_after": "R : Type u\nι : Type ?u.280883\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn : ℕ\nh : I ⊔ J = ⊤\n⊢ I ⊔ ∏ _x in Finset.range n, J = ⊤",
"state_before": "R : Type u\nι : Type ?u.280883\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn : ℕ\nh : I ⊔ J = ⊤\n⊢ I ⊔ J ^ n = ⊤",
"tactic": "rw [← Finset.card_range n, ← Finset.prod_const]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.280883\ninst✝ : CommSemiring R\nI J K L : Ideal R\nn : ℕ\nh : I ⊔ J = ⊤\n⊢ I ⊔ ∏ _x in Finset.range n, J = ⊤",
"tactic": "exact sup_prod_eq_top fun _ _ => h"
}
] |
[
729,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.coe_zero
|
[] |
[
377,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
RingHom.FiniteType.of_surjective
|
[
{
"state_after": "A : Type u_1\nB : Type u_2\nC : Type ?u.122685\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\nhf : Surjective ↑f\n⊢ FiniteType (comp f (RingHom.id A))",
"state_before": "A : Type u_1\nB : Type u_2\nC : Type ?u.122685\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\nhf : Surjective ↑f\n⊢ FiniteType f",
"tactic": "rw [← f.comp_id]"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\nB : Type u_2\nC : Type ?u.122685\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\nhf : Surjective ↑f\n⊢ FiniteType (comp f (RingHom.id A))",
"tactic": "exact (id A).comp_surjective hf"
}
] |
[
233,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Std/Data/Int/DivMod.lean
|
Int.eq_one_of_mul_eq_one_left
|
[
{
"state_after": "no goals",
"state_before": "a b : Int\nH : 0 ≤ b\nH' : a * b = 1\n⊢ b * ?m.93792 H H' = 1",
"tactic": "rw [Int.mul_comm, H']"
}
] |
[
857,
59
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
856,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_union
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.368807\nγ : Type ?u.368810\np q : α → Prop\ninst✝² : DecidablePred p\ninst✝¹ : DecidablePred q\ninst✝ : DecidableEq α\ns₁ s₂ : Finset α\nx✝ : α\n⊢ x✝ ∈ filter p (s₁ ∪ s₂) ↔ x✝ ∈ filter p s₁ ∪ filter p s₂",
"tactic": "simp only [mem_filter, mem_union, or_and_right]"
}
] |
[
2813,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2812,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lift_type_le
|
[] |
[
737,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.nil_eq
|
[] |
[
164,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/CategoryTheory/Monoidal/Free/Basic.lean
|
CategoryTheory.FreeMonoidalCategory.mk_tensor
|
[] |
[
196,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_coe_angle_eq_iff_eq_toReal
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\nθ : Angle\n⊢ ↑(arg z) = θ ↔ arg z = Angle.toReal θ",
"tactic": "rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg]"
}
] |
[
499,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.pred_le
|
[] |
[
219,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isNat_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Semiring α\nn✝¹ n✝ : ℕ\n⊢ ↑n✝¹ ^ ↑n✝ = ↑(Nat.pow n✝¹ n✝)",
"tactic": "simp"
}
] |
[
427,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.eq_one_of_pos_of_log_eq_zero
|
[] |
[
230,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.add_isLittleO
|
[] |
[
1096,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1094,
1
] |
Std/Data/Int/DivMod.lean
|
Int.eq_ediv_of_mul_eq_right
|
[] |
[
753,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
751,
11
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.constantCoeff_X
|
[] |
[
1562,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1561,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean
|
CategoryTheory.Monad.Algebra.comp_f
|
[] |
[
134,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.centralizer_toSubsemiring
|
[] |
[
862,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
860,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.erase_subset
|
[] |
[
1079,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1078,
1
] |
Mathlib/Analysis/Normed/MulAction.lean
|
edist_smul_le
|
[] |
[
54,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Topology/Basic.lean
|
DenseRange.nonempty_iff
|
[] |
[
1854,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1853,
8
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.clm_apply
|
[] |
[
897,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
895,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioc_subset_Ioc_right
|
[] |
[
493,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.mem_vsub
|
[] |
[
1089,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1088,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_snd
|
[] |
[
1091,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1090,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.map_eq_cons
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\n⊢ (∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t) → map f s = b ::ₘ t\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\n⊢ map f s = b ::ₘ t → ∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\n⊢ (∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t) ↔ map f s = b ::ₘ t",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\na : α\nha : a ∈ s\n⊢ map f s = f a ::ₘ map f (erase s a)",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\n⊢ (∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t) → map f s = b ::ₘ t",
"tactic": "rintro ⟨a, ha, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\na : α\nha : a ∈ s\n⊢ map f s = f a ::ₘ map f (erase s a)",
"tactic": "rw [← map_cons, Multiset.cons_erase ha]"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\nh : map f s = b ::ₘ t\n⊢ ∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\n⊢ map f s = b ::ₘ t → ∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t",
"tactic": "intro h"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\nh : map f s = b ::ₘ t\nthis : b ∈ map f s\n⊢ ∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\nh : map f s = b ::ₘ t\n⊢ ∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t",
"tactic": "have : b ∈ s.map f := by\n rw [h]\n exact mem_cons_self _ _"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\na : α\nh1 : a ∈ s\nh : map f s = f a ::ₘ t\nthis : f a ∈ map f s\n⊢ ∃ a_1, a_1 ∈ s ∧ f a_1 = f a ∧ map f (erase s a_1) = t",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\nh : map f s = b ::ₘ t\nthis : b ∈ map f s\n⊢ ∃ a, a ∈ s ∧ f a = b ∧ map f (erase s a) = t",
"tactic": "obtain ⟨a, h1, rfl⟩ := mem_map.mp this"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f (a ::ₘ u) = f a ::ₘ t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ ∃ a_1, a_1 ∈ a ::ₘ u ∧ f a_1 = f a ∧ map f (erase (a ::ₘ u) a_1) = t",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\na : α\nh1 : a ∈ s\nh : map f s = f a ::ₘ t\nthis : f a ∈ map f s\n⊢ ∃ a_1, a_1 ∈ s ∧ f a_1 = f a ∧ map f (erase s a_1) = t",
"tactic": "obtain ⟨u, rfl⟩ := exists_cons_of_mem h1"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f u = t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ ∃ a_1, a_1 ∈ a ::ₘ u ∧ f a_1 = f a ∧ map f (erase (a ::ₘ u) a_1) = t",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f (a ::ₘ u) = f a ::ₘ t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ ∃ a_1, a_1 ∈ a ::ₘ u ∧ f a_1 = f a ∧ map f (erase (a ::ₘ u) a_1) = t",
"tactic": "rw [map_cons, cons_inj_right] at h"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f u = t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ map f (erase (a ::ₘ u) a) = t",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f u = t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ ∃ a_1, a_1 ∈ a ::ₘ u ∧ f a_1 = f a ∧ map f (erase (a ::ₘ u) a_1) = t",
"tactic": "refine' ⟨a, mem_cons_self _ _, rfl, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f u = t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ map f (erase (a ::ₘ u) a) = t",
"tactic": "rw [Multiset.erase_cons_head, h]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\nh : map f s = b ::ₘ t\n⊢ b ∈ b ::ₘ t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\nh : map f s = b ::ₘ t\n⊢ b ∈ map f s",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.122604\ninst✝ : DecidableEq α\nf : α → β\ns : Multiset α\nt : Multiset β\nb : β\nh : map f s = b ::ₘ t\n⊢ b ∈ b ::ₘ t",
"tactic": "exact mem_cons_self _ _"
}
] |
[
1270,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1257,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_le_nnreal_smul_snorm_of_ae_le_mul
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : p = 0\n⊢ snorm f p μ ≤ c • snorm g p μ\n\ncase neg\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\n⊢ snorm f p μ ≤ c • snorm g p μ",
"state_before": "α : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\n⊢ snorm f p μ ≤ c • snorm g p μ",
"tactic": "by_cases h0 : p = 0"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : p = ⊤\n⊢ snorm f p μ ≤ c • snorm g p μ\n\ncase neg\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : ¬p = ⊤\n⊢ snorm f p μ ≤ c • snorm g p μ",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\n⊢ snorm f p μ ≤ c • snorm g p μ",
"tactic": "by_cases h_top : p = ∞"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : ¬p = ⊤\n⊢ snorm' f (ENNReal.toReal p) μ ≤ c • snorm' g (ENNReal.toReal p) μ",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : ¬p = ⊤\n⊢ snorm f p μ ≤ c • snorm g p μ",
"tactic": "simp_rw [snorm_eq_snorm' h0 h_top]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : ¬p = ⊤\n⊢ snorm' f (ENNReal.toReal p) μ ≤ c • snorm' g (ENNReal.toReal p) μ",
"tactic": "exact snorm'_le_nnreal_smul_snorm'_of_ae_le_mul h (ENNReal.toReal_pos h0 h_top)"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : p = 0\n⊢ snorm f p μ ≤ c • snorm g p μ",
"tactic": "simp [h0]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : p = ⊤\n⊢ snorm f ⊤ μ ≤ c • snorm g ⊤ μ",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : p = ⊤\n⊢ snorm f p μ ≤ c • snorm g p μ",
"tactic": "rw [h_top]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.5059288\nF : Type u_2\nG : Type u_3\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ≥0∞\nh0 : ¬p = 0\nh_top : p = ⊤\n⊢ snorm f ⊤ μ ≤ c • snorm g ⊤ μ",
"tactic": "exact snormEssSup_le_nnreal_smul_snormEssSup_of_ae_le_mul h"
}
] |
[
1285,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1277,
1
] |
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