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/Topology/Homeomorph.lean
|
Homeomorph.normalSpace
|
[] |
[
340,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
11
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.comp_eq_mk
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.349637\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ng : β → γ\nhg : Continuous g\nf : α →ₘ[μ] β\n⊢ comp g hg f = mk (g ∘ ↑f) (_ : AEStronglyMeasurable (fun x => g (↑f x)) μ)",
"tactic": "rw [← comp_mk g hg f f.aestronglyMeasurable, mk_coeFn]"
}
] |
[
225,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Std/Data/String/Lemmas.lean
|
String.singleton_eq
|
[] |
[
483,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
483,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_le_of_submodule
|
[] |
[
187,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.type_eq_one_of_unique
|
[] |
[
251,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.AddValuation.map_one
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ addValuationDef 1 = 0",
"tactic": "rw [addValuationDef, if_neg one_ne_zero, valuation_one, WithTop.coe_zero]"
}
] |
[
1117,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Data/Sum/Order.lean
|
Sum.inl_lt_inl_iff
|
[] |
[
143,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
MulMemClass.coe_subtype
|
[] |
[
578,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Algebra/ModEq.lean
|
AddCommGroup.ModEq.of_nsmul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\nx✝ : a ≡ b [PMOD n • p]\nm : ℤ\nhm : b - a = m • n • p\n⊢ b - a = (m * ↑n) • p",
"tactic": "rwa [mul_smul, coe_nat_zsmul]"
}
] |
[
156,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
11
] |
Mathlib/Algebra/Module/LinearMap.lean
|
DistribMulActionHom.toLinearMap_injective
|
[
{
"state_after": "case a\nR : Type u_1\nR₁ : Type ?u.321615\nR₂ : Type ?u.321618\nR₃ : Type ?u.321621\nk : Type ?u.321624\nS : Type ?u.321627\nS₃ : Type ?u.321630\nT : Type ?u.321633\nM : Type u_2\nM₁ : Type ?u.321639\nM₂ : Type u_3\nM₃ : Type ?u.321645\nN₁ : Type ?u.321648\nN₂ : Type ?u.321651\nN₃ : Type ?u.321654\nι : Type ?u.321657\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nf g : M →+[R] M₂\nh : ↑f = ↑g\nm : M\n⊢ ↑f m = ↑g m",
"state_before": "R : Type u_1\nR₁ : Type ?u.321615\nR₂ : Type ?u.321618\nR₃ : Type ?u.321621\nk : Type ?u.321624\nS : Type ?u.321627\nS₃ : Type ?u.321630\nT : Type ?u.321633\nM : Type u_2\nM₁ : Type ?u.321639\nM₂ : Type u_3\nM₃ : Type ?u.321645\nN₁ : Type ?u.321648\nN₂ : Type ?u.321651\nN₃ : Type ?u.321654\nι : Type ?u.321657\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nf g : M →+[R] M₂\nh : ↑f = ↑g\n⊢ f = g",
"tactic": "ext m"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_1\nR₁ : Type ?u.321615\nR₂ : Type ?u.321618\nR₃ : Type ?u.321621\nk : Type ?u.321624\nS : Type ?u.321627\nS₃ : Type ?u.321630\nT : Type ?u.321633\nM : Type u_2\nM₁ : Type ?u.321639\nM₂ : Type u_3\nM₃ : Type ?u.321645\nN₁ : Type ?u.321648\nN₂ : Type ?u.321651\nN₃ : Type ?u.321654\nι : Type ?u.321657\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nf g : M →+[R] M₂\nh : ↑f = ↑g\nm : M\n⊢ ↑f m = ↑g m",
"tactic": "exact LinearMap.congr_fun h m"
}
] |
[
667,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
ciInf_le
|
[] |
[
818,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
817,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_infDist
|
[] |
[
1534,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1533,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.one_mul
|
[
{
"state_after": "case c\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ 1 * (r /ₒ s) = r /ₒ s",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nx : OreLocalization R S\n⊢ 1 * x = x",
"tactic": "induction' x using OreLocalization.ind with r s"
},
{
"state_after": "no goals",
"state_before": "case c\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ 1 * (r /ₒ s) = r /ₒ s",
"tactic": "simp [OreLocalization.one_def, oreDiv_mul_char (1 : R) r (1 : S) s r 1 (by simp)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ r * ↑1 = ↑1 * r",
"tactic": "simp"
}
] |
[
297,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
11
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.toMeasure_toPmf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.194253\nγ : Type ?u.194256\ninst✝³ : Countable α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\np : Pmf α\nμ : MeasureTheory.Measure α\ninst✝ : IsProbabilityMeasure μ\nx : α\n⊢ ↑(Measure.toPmf (toMeasure p)) x = ↑p x",
"tactic": "rw [← p.toMeasure_apply_singleton x (measurableSet_singleton x), p.toMeasure.toPmf_apply]"
}
] |
[
388,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
MulHom.srange_eq_map
|
[] |
[
772,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
771,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_zero_of_column_eq
|
[
{
"state_after": "m : Type ?u.1635426\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\ni j : n\ni_ne_j : i ≠ j\nhij : ∀ (k : n), M k i = M k j\n⊢ Mᵀ i = Mᵀ j",
"state_before": "m : Type ?u.1635426\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\ni j : n\ni_ne_j : i ≠ j\nhij : ∀ (k : n), M k i = M k j\n⊢ det M = 0",
"tactic": "rw [← det_transpose, det_zero_of_row_eq i_ne_j]"
},
{
"state_after": "no goals",
"state_before": "m : Type ?u.1635426\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\ni j : n\ni_ne_j : i ≠ j\nhij : ∀ (k : n), M k i = M k j\n⊢ Mᵀ i = Mᵀ j",
"tactic": "exact funext hij"
}
] |
[
386,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Topology/UnitInterval.lean
|
iccHomeoI_apply_coe
|
[] |
[
236,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_apply₂₂
|
[] |
[
74,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.zpow_add_of_nonpos
|
[
{
"state_after": "case inl.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℤ\nhm : m ≤ 0\nhn : n ≤ 0\nh : A⁻¹ ⬝ A = 1\nright✝ : A ⬝ A⁻¹ = 1\n⊢ A ^ (m + n) = A ^ m * A ^ n\n\ncase inr\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℤ\nhm : m ≤ 0\nhn : n ≤ 0\nh : A⁻¹ = 0\n⊢ A ^ (m + n) = A ^ m * A ^ n",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℤ\nhm : m ≤ 0\nhn : n ≤ 0\n⊢ A ^ (m + n) = A ^ m * A ^ n",
"tactic": "rcases nonsing_inv_cancel_or_zero A with (⟨h, _⟩ | h)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℤ\nhm : m ≤ 0\nhn : n ≤ 0\nh : A⁻¹ ⬝ A = 1\nright✝ : A ⬝ A⁻¹ = 1\n⊢ A ^ (m + n) = A ^ m * A ^ n",
"tactic": "exact zpow_add (isUnit_det_of_left_inverse h) m n"
},
{
"state_after": "case inr.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nn : ℤ\nhn : n ≤ 0\nh : A⁻¹ = 0\nk : ℕ\nhm : -↑k ≤ 0\n⊢ A ^ (-↑k + n) = A ^ (-↑k) * A ^ n",
"state_before": "case inr\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nm n : ℤ\nhm : m ≤ 0\nhn : n ≤ 0\nh : A⁻¹ = 0\n⊢ A ^ (m + n) = A ^ m * A ^ n",
"tactic": "obtain ⟨k, rfl⟩ := exists_eq_neg_ofNat hm"
},
{
"state_after": "case inr.intro.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : A⁻¹ = 0\nk : ℕ\nhm : -↑k ≤ 0\nl : ℕ\nhn : -↑l ≤ 0\n⊢ A ^ (-↑k + -↑l) = A ^ (-↑k) * A ^ (-↑l)",
"state_before": "case inr.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nn : ℤ\nhn : n ≤ 0\nh : A⁻¹ = 0\nk : ℕ\nhm : -↑k ≤ 0\n⊢ A ^ (-↑k + n) = A ^ (-↑k) * A ^ n",
"tactic": "obtain ⟨l, rfl⟩ := exists_eq_neg_ofNat hn"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : A⁻¹ = 0\nk : ℕ\nhm : -↑k ≤ 0\nl : ℕ\nhn : -↑l ≤ 0\n⊢ A ^ (-↑k + -↑l) = A ^ (-↑k) * A ^ (-↑l)",
"tactic": "simp_rw [← neg_add, ← Int.ofNat_add, zpow_neg_coe_nat, ← inv_pow', h, pow_add]"
}
] |
[
183,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
finrank_le_one
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nv : V\nh : ∀ (w : V), ∃ c, c • v = w\nthis : Nontrivial K\n⊢ finrank K V ≤ 1",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nv : V\nh : ∀ (w : V), ∃ c, c • v = w\n⊢ finrank K V ≤ 1",
"tactic": "haveI := nontrivial_of_invariantBasisNumber K"
},
{
"state_after": "case inl\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\n⊢ finrank K V ≤ 1\n\ncase inr\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nv : V\nh : ∀ (w : V), ∃ c, c • v = w\nthis : Nontrivial K\nhn : v ≠ 0\n⊢ finrank K V ≤ 1",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nv : V\nh : ∀ (w : V), ∃ c, c • v = w\nthis : Nontrivial K\n⊢ finrank K V ≤ 1",
"tactic": "rcases eq_or_ne v 0 with (rfl | hn)"
},
{
"state_after": "case inl\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis✝ : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\nthis : Subsingleton V\n⊢ finrank K V ≤ 1",
"state_before": "case inl\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\n⊢ finrank K V ≤ 1",
"tactic": "haveI :=\n subsingleton_of_forall_eq (0 : V) fun w => by\n obtain ⟨c, rfl⟩ := h w\n simp"
},
{
"state_after": "case inl\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis✝ : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\nthis : Subsingleton V\n⊢ 0 ≤ 1",
"state_before": "case inl\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis✝ : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\nthis : Subsingleton V\n⊢ finrank K V ≤ 1",
"tactic": "rw [finrank_zero_of_subsingleton]"
},
{
"state_after": "no goals",
"state_before": "case inl\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis✝ : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\nthis : Subsingleton V\n⊢ 0 ≤ 1",
"tactic": "exact zero_le_one"
},
{
"state_after": "case intro\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\nc : K\n⊢ c • 0 = 0",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\nw : V\n⊢ w = 0",
"tactic": "obtain ⟨c, rfl⟩ := h w"
},
{
"state_after": "no goals",
"state_before": "case intro\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nthis : Nontrivial K\nh : ∀ (w : V), ∃ c, c • 0 = w\nc : K\n⊢ c • 0 = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nK : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : NoZeroSMulDivisors K V\ninst✝ : StrongRankCondition K\nv : V\nh : ∀ (w : V), ∃ c, c • v = w\nthis : Nontrivial K\nhn : v ≠ 0\n⊢ finrank K V ≤ 1",
"tactic": "exact (finrank_eq_one v hn h).le"
}
] |
[
517,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_ofReal_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : 0 ≤ x\n⊢ arg ↑x = 0",
"tactic": "simp [arg, hx]"
}
] |
[
221,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
AbsoluteValue.map_sub_eq_zero_iff
|
[] |
[
161,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean
|
fixingSubgroup_antitone
|
[] |
[
136,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.exists_of_compat
|
[
{
"state_after": "α : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\nH₁ : ∀ (Q₁ Q₂ : DiscreteQuotient X), Q₁ ≤ Q₂ → proj Q₁ ⁻¹' {Qs Q₁} ⊆ proj Q₂ ⁻¹' {Qs Q₂}\n⊢ ∃ x, ∀ (Q : DiscreteQuotient X), proj Q x = Qs Q",
"state_before": "α : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\n⊢ ∃ x, ∀ (Q : DiscreteQuotient X), proj Q x = Qs Q",
"tactic": "have H₁ : ∀ Q₁ Q₂, Q₁ ≤ Q₂ → proj Q₁ ⁻¹' {Qs Q₁} ⊆ proj Q₂ ⁻¹' {Qs Q₂} := fun _ _ h => by\n rw [← compat _ _ h]\n exact fiber_subset_ofLE _ _"
},
{
"state_after": "case intro\nα : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\nH₁ : ∀ (Q₁ Q₂ : DiscreteQuotient X), Q₁ ≤ Q₂ → proj Q₁ ⁻¹' {Qs Q₁} ⊆ proj Q₂ ⁻¹' {Qs Q₂}\nx : X\nhx : x ∈ ⋂ (Q : DiscreteQuotient X), proj Q ⁻¹' {Qs Q}\n⊢ ∃ x, ∀ (Q : DiscreteQuotient X), proj Q x = Qs Q",
"state_before": "α : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\nH₁ : ∀ (Q₁ Q₂ : DiscreteQuotient X), Q₁ ≤ Q₂ → proj Q₁ ⁻¹' {Qs Q₁} ⊆ proj Q₂ ⁻¹' {Qs Q₂}\n⊢ ∃ x, ∀ (Q : DiscreteQuotient X), proj Q x = Qs Q",
"tactic": "obtain ⟨x, hx⟩ : Set.Nonempty (⋂ Q, proj Q ⁻¹' {Qs Q}) :=\n IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed\n (fun Q : DiscreteQuotient X => Q.proj ⁻¹' {Qs _}) (directed_of_inf H₁)\n (fun Q => (singleton_nonempty _).preimage Q.proj_surjective)\n (fun Q => (Q.isClosed_preimage {Qs _}).isCompact) fun Q => Q.isClosed_preimage _"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\nH₁ : ∀ (Q₁ Q₂ : DiscreteQuotient X), Q₁ ≤ Q₂ → proj Q₁ ⁻¹' {Qs Q₁} ⊆ proj Q₂ ⁻¹' {Qs Q₂}\nx : X\nhx : x ∈ ⋂ (Q : DiscreteQuotient X), proj Q ⁻¹' {Qs Q}\n⊢ ∃ x, ∀ (Q : DiscreteQuotient X), proj Q x = Qs Q",
"tactic": "exact ⟨x, mem_iInter.1 hx⟩"
},
{
"state_after": "α : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\nx✝¹ x✝ : DiscreteQuotient X\nh : x✝¹ ≤ x✝\n⊢ proj x✝¹ ⁻¹' {Qs x✝¹} ⊆ proj x✝ ⁻¹' {ofLE h (Qs x✝¹)}",
"state_before": "α : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\nx✝¹ x✝ : DiscreteQuotient X\nh : x✝¹ ≤ x✝\n⊢ proj x✝¹ ⁻¹' {Qs x✝¹} ⊆ proj x✝ ⁻¹' {Qs x✝}",
"tactic": "rw [← compat _ _ h]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.38375\nX : Type u_1\nY : Type ?u.38381\nZ : Type ?u.38384\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nS : DiscreteQuotient X\ninst✝ : CompactSpace X\nQs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid\ncompat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B\nx✝¹ x✝ : DiscreteQuotient X\nh : x✝¹ ≤ x✝\n⊢ proj x✝¹ ⁻¹' {Qs x✝¹} ⊆ proj x✝ ⁻¹' {ofLE h (Qs x✝¹)}",
"tactic": "exact fiber_subset_ofLE _ _"
}
] |
[
393,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/Topology/Order/Basic.lean
|
interior_Ici'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b : α\ns : Set α\na : α\nha : Set.Nonempty (Iio a)\n⊢ interior (Ici a) = Ioi a",
"tactic": "rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]"
}
] |
[
2273,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2272,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.map_erase
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.118066\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : α ↪ β\ns : Finset α\na : α\n⊢ image (↑f) (erase s a) = erase (image (↑f) s) (↑f a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.118066\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : α ↪ β\ns : Finset α\na : α\n⊢ map f (erase s a) = erase (map f s) (↑f a)",
"tactic": "simp_rw [map_eq_image]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.118066\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq α\nf : α ↪ β\ns : Finset α\na : α\n⊢ image (↑f) (erase s a) = erase (image (↑f) s) (↑f a)",
"tactic": "exact s.image_erase f.2 a"
}
] |
[
624,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.comap_iInf
|
[] |
[
866,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.pow_comp
|
[] |
[
142,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.roots_of_cyclotomic
|
[
{
"state_after": "R✝ : Type ?u.78925\ninst✝³ : CommRing R✝\ninst✝² : IsDomain R✝\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ roots (∏ μ in primitiveRoots n R, (X - ↑C μ)) = (primitiveRoots n R).val",
"state_before": "R✝ : Type ?u.78925\ninst✝³ : CommRing R✝\ninst✝² : IsDomain R✝\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ roots (cyclotomic' n R) = (primitiveRoots n R).val",
"tactic": "rw [cyclotomic']"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.78925\ninst✝³ : CommRing R✝\ninst✝² : IsDomain R✝\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ roots (∏ μ in primitiveRoots n R, (X - ↑C μ)) = (primitiveRoots n R).val",
"tactic": "exact roots_prod_X_sub_C (primitiveRoots n R)"
}
] |
[
131,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.pred_self_lt
|
[] |
[
108,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/List/Rdrop.lean
|
List.rdropWhile_idempotent
|
[] |
[
187,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
Submonoid.mem_smul_pointwise_iff_exists
|
[] |
[
249,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
Even.zpow_pos_iff
|
[
{
"state_after": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nk : ℤ\nh : k + k ≠ 0\n⊢ 0 < a ^ (k + k) ↔ a ≠ 0",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhn : Even n\nh : n ≠ 0\n⊢ 0 < a ^ n ↔ a ≠ 0",
"tactic": "obtain ⟨k, rfl⟩ := hn"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nk : ℤ\nh : k + k ≠ 0\n⊢ 0 < a ^ (k + k) ↔ a ≠ 0",
"tactic": "exact zpow_bit0_pos_iff (by rintro rfl; simp at h)"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nh : 0 + 0 ≠ 0\n⊢ False",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nk : ℤ\nh : k + k ≠ 0\n⊢ k ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nh : 0 + 0 ≠ 0\n⊢ False",
"tactic": "simp at h"
}
] |
[
186,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.fromEdgeSet_mono
|
[
{
"state_after": "ι : Sort ?u.91344\n𝕜 : Type ?u.91347\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\ns✝ s t : Set (Sym2 V)\nh : s ⊆ t\nv w : V\n⊢ Adj (fromEdgeSet s) v w → Adj (fromEdgeSet t) v w",
"state_before": "ι : Sort ?u.91344\n𝕜 : Type ?u.91347\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\ns✝ s t : Set (Sym2 V)\nh : s ⊆ t\n⊢ fromEdgeSet s ≤ fromEdgeSet t",
"tactic": "rintro v w"
},
{
"state_after": "ι : Sort ?u.91344\n𝕜 : Type ?u.91347\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\ns✝ s t : Set (Sym2 V)\nh : s ⊆ t\nv w : V\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ s → ¬v = w → Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ t",
"state_before": "ι : Sort ?u.91344\n𝕜 : Type ?u.91347\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\ns✝ s t : Set (Sym2 V)\nh : s ⊆ t\nv w : V\n⊢ Adj (fromEdgeSet s) v w → Adj (fromEdgeSet t) v w",
"tactic": "simp (config := { contextual := true }) only [fromEdgeSet_adj, Ne.def, not_false_iff,\n and_true_iff, and_imp]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.91344\n𝕜 : Type ?u.91347\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\ns✝ s t : Set (Sym2 V)\nh : s ⊆ t\nv w : V\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ s → ¬v = w → Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ t",
"tactic": "exact fun vws _ => h vws"
}
] |
[
675,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
671,
1
] |
Mathlib/Init/Algebra/Order.lean
|
compare_gt_iff_gt
|
[
{
"state_after": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.gt ↔ a > b",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ compare a b = Ordering.gt ↔ a > b",
"tactic": "rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]"
},
{
"state_after": "case inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ True ↔ a > b",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\n⊢ (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.gt ↔ a > b",
"tactic": "split_ifs <;> simp only [*, lt_irrefl, not_lt_of_gt]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u\ninst✝ : LinearOrder α\na b : α\nh✝¹ : ¬a < b\nh✝ : ¬a = b\n⊢ True ↔ a > b",
"tactic": "case _ h₁ h₂ =>\n have h : b < a := lt_trichotomy a b |>.resolve_left h₁ |>.resolve_left h₂\n exact true_iff_iff.2 h"
},
{
"state_after": "α : Type u\ninst✝ : LinearOrder α\na b : α\nh₁ : ¬a < b\nh₂ : ¬a = b\nh : b < a\n⊢ True ↔ a > b",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\nh₁ : ¬a < b\nh₂ : ¬a = b\n⊢ True ↔ a > b",
"tactic": "have h : b < a := lt_trichotomy a b |>.resolve_left h₁ |>.resolve_left h₂"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : LinearOrder α\na b : α\nh₁ : ¬a < b\nh₂ : ¬a = b\nh : b < a\n⊢ True ↔ a > b",
"tactic": "exact true_iff_iff.2 h"
}
] |
[
433,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.prod_divisorsAntidiagonal
|
[
{
"state_after": "n✝ : ℕ\nM : Type u_1\ninst✝ : CommMonoid M\nf : ℕ → ℕ → M\nn : ℕ\n⊢ ∏ x in divisors n,\n f\n (↑{ toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n x).fst\n (↑{ toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n x).snd =\n ∏ i in divisors n, f i (n / i)",
"state_before": "n✝ : ℕ\nM : Type u_1\ninst✝ : CommMonoid M\nf : ℕ → ℕ → M\nn : ℕ\n⊢ ∏ i in divisorsAntidiagonal n, f i.fst i.snd = ∏ i in divisors n, f i (n / i)",
"tactic": "rw [← map_div_right_divisors, Finset.prod_map]"
},
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nM : Type u_1\ninst✝ : CommMonoid M\nf : ℕ → ℕ → M\nn : ℕ\n⊢ ∏ x in divisors n,\n f\n (↑{ toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n x).fst\n (↑{ toFun := fun d => (d, n / d),\n inj' :=\n (_ :\n ∀ (p₁ p₂ : ℕ),\n (fun d => (d, n / d)) p₁ = (fun d => (d, n / d)) p₂ →\n ((fun d => (d, n / d)) p₁).fst = ((fun d => (d, n / d)) p₂).fst) }\n x).snd =\n ∏ i in divisors n, f i (n / i)",
"tactic": "rfl"
}
] |
[
472,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
Real.geom_mean_le_arith_mean3_weighted
|
[] |
[
231,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Order/Cover.lean
|
Set.covby_insert
|
[] |
[
482,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Data/Real/Pi/Bounds.lean
|
Real.pi_upper_bound_start
|
[
{
"state_after": "n : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n ≤ a",
"state_before": "n : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ π < a",
"tactic": "refine' lt_of_lt_of_le (pi_lt_sqrtTwoAddSeries n) _"
},
{
"state_after": "n : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries 0 n\n\nn : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 0 ≤ (a - 1 / 4 ^ n) / 2 ^ (n + 1)\n\nn : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 0 < 2 ^ (n + 1)",
"state_before": "n : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n ≤ a",
"tactic": "rw [← le_sub_iff_add_le, ← le_div_iff', sqrt_le_left, sub_le_comm]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries 0 n",
"tactic": "rwa [Nat.cast_zero, zero_div] at h"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 0 ≤ (a - 1 / 4 ^ n) / 2 ^ (n + 1)",
"tactic": "exact div_nonneg (sub_nonneg.2 h₂) (pow_nonneg (le_of_lt zero_lt_two) _)"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nh : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries (↑0 / ↑1) n\nh₂ : 1 / 4 ^ n ≤ a\n⊢ 0 < 2 ^ (n + 1)",
"tactic": "exact pow_pos zero_lt_two _"
}
] |
[
134,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.mem_map_of_mem
|
[] |
[
154,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
154,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.mem_boundedContinuousFunction_iff
|
[] |
[
1590,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1587,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
BoundedContinuousFunction.toLp_inj
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nh : ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g\n⊢ f = g",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\n⊢ ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g ↔ f = g",
"tactic": "refine' ⟨fun h => _, by tauto⟩"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nh : ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g\n⊢ ↑f =ᵐ[μ] ↑g",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nh : ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g\n⊢ f = g",
"tactic": "rw [← FunLike.coe_fn_eq, ← (map_continuous f).ae_eq_iff_eq μ (map_continuous g)]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nh : ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g\n⊢ ↑↑(↑(toLp p μ 𝕜) f) =ᵐ[μ] ↑↑(↑(toLp p μ 𝕜) g)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nh : ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g\n⊢ ↑f =ᵐ[μ] ↑g",
"tactic": "refine' (coeFn_toLp p μ 𝕜 f).symm.trans (EventuallyEq.trans _ <| coeFn_toLp p μ 𝕜 g)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nh : ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g\n⊢ ↑↑(↑(toLp p μ 𝕜) f) =ᵐ[μ] ↑↑(↑(toLp p μ 𝕜) g)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.12183215\nG : Type ?u.12183218\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : SecondCountableTopologyEither α E\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\nf g : α →ᵇ E\ninst✝² : Measure.IsOpenPosMeasure μ\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\n⊢ f = g → ↑(toLp p μ 𝕜) f = ↑(toLp p μ 𝕜) g",
"tactic": "tauto"
}
] |
[
1679,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1674,
1
] |
Mathlib/Topology/Constructions.lean
|
Continuous.subtype_mk
|
[] |
[
1044,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1042,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
DistribMulActionHom.comp_id
|
[
{
"state_after": "no goals",
"state_before": "M' : Type ?u.178346\nX : Type ?u.178349\ninst✝²³ : SMul M' X\nY : Type ?u.178356\ninst✝²² : SMul M' Y\nZ : Type ?u.178363\ninst✝²¹ : SMul M' Z\nM : Type u_1\ninst✝²⁰ : Monoid M\nA : Type u_2\ninst✝¹⁹ : AddMonoid A\ninst✝¹⁸ : DistribMulAction M A\nA' : Type ?u.178404\ninst✝¹⁷ : AddGroup A'\ninst✝¹⁶ : DistribMulAction M A'\nB : Type u_3\ninst✝¹⁵ : AddMonoid B\ninst✝¹⁴ : DistribMulAction M B\nB' : Type ?u.178706\ninst✝¹³ : AddGroup B'\ninst✝¹² : DistribMulAction M B'\nC : Type ?u.178982\ninst✝¹¹ : AddMonoid C\ninst✝¹⁰ : DistribMulAction M C\nR : Type ?u.179008\ninst✝⁹ : Semiring R\ninst✝⁸ : MulSemiringAction M R\nR' : Type ?u.179035\ninst✝⁷ : Ring R'\ninst✝⁶ : MulSemiringAction M R'\nS : Type ?u.179231\ninst✝⁵ : Semiring S\ninst✝⁴ : MulSemiringAction M S\nS' : Type ?u.179258\ninst✝³ : Ring S'\ninst✝² : MulSemiringAction M S'\nT : Type ?u.179454\ninst✝¹ : Semiring T\ninst✝ : MulSemiringAction M T\nf : A →+[M] B\nx : A\n⊢ ↑(comp f (DistribMulActionHom.id M)) x = ↑f x",
"tactic": "rw [comp_apply, id_apply]"
}
] |
[
383,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.LHom.realize_onFormula
|
[] |
[
704,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.SummableFamily.smul_apply
|
[] |
[
1595,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1594,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sInf_lt_iff
|
[] |
[
594,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
MonoidHom.snd_comp_inr
|
[] |
[
536,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Order/Chain.lean
|
isChain_of_trichotomous
|
[] |
[
90,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.smulRightₗ_apply
|
[] |
[
558,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.angle_eq_zero_iff_eq_and_ne_or_sbtw
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\n⊢ ∠ p₁ p₂ p₃ = 0 ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "rw [angle_eq_zero_iff_ne_and_wbtw]"
},
{
"state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "by_cases hp₁p₂ : p₁ = p₂"
},
{
"state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : p₁ = p₃\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : ¬p₁ = p₃\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "by_cases hp₁p₃ : p₁ = p₃"
},
{
"state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : ¬p₁ = p₃\nhp₃p₂ : p₃ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : ¬p₁ = p₃\nhp₃p₂ : ¬p₃ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : ¬p₁ = p₃\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "by_cases hp₃p₂ : p₃ = p₂"
},
{
"state_after": "no goals",
"state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : ¬p₁ = p₃\nhp₃p₂ : ¬p₃ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "simp [hp₁p₂, hp₁p₃, Ne.symm hp₁p₃, Sbtw, hp₃p₂]"
},
{
"state_after": "no goals",
"state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "simp [hp₁p₂]"
},
{
"state_after": "no goals",
"state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : p₁ = p₃\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "simp [hp₁p₃]"
},
{
"state_after": "no goals",
"state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nhp₁p₂ : ¬p₁ = p₂\nhp₁p₃ : ¬p₁ = p₃\nhp₃p₂ : p₃ = p₂\n⊢ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁",
"tactic": "simp [hp₃p₂]"
}
] |
[
413,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.dfinsupp_sum_apply
|
[] |
[
1158,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1156,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
quotient_norm_mk_le
|
[] |
[
155,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
Prod.swap_one
|
[] |
[
122,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
src/lean/Init/Core.lean
|
dif_pos
|
[] |
[
810,
33
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
807,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Iso.connectedComponentEquiv_symm
|
[
{
"state_after": "case H.mk\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nφ : G ≃g G'\nx✝ : ConnectedComponent G'\na✝ : V'\n⊢ ↑(connectedComponentEquiv (symm φ)) (Quot.mk (Reachable G') a✝) =\n ↑(connectedComponentEquiv φ).symm (Quot.mk (Reachable G') a✝)",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nφ : G ≃g G'\n⊢ connectedComponentEquiv (symm φ) = (connectedComponentEquiv φ).symm",
"tactic": "ext ⟨_⟩"
},
{
"state_after": "no goals",
"state_before": "case H.mk\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nφ : G ≃g G'\nx✝ : ConnectedComponent G'\na✝ : V'\n⊢ ↑(connectedComponentEquiv (symm φ)) (Quot.mk (Reachable G') a✝) =\n ↑(connectedComponentEquiv φ).symm (Quot.mk (Reachable G') a✝)",
"tactic": "rfl"
}
] |
[
2117,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2114,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sin_neg
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sin (-x) = -sin x",
"tactic": "simp [sin, sub_eq_add_neg, exp_neg, (neg_div _ _).symm, add_mul]"
}
] |
[
801,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
800,
1
] |
Mathlib/Order/Cover.lean
|
Covby.Ico_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.36866\ninst✝ : PartialOrder α\na b c : α\nh : a ⋖ b\n⊢ Ico a b = {a}",
"tactic": "rw [← Ioo_union_left h.lt, h.Ioo_eq, empty_union]"
}
] |
[
405,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.IsBasis.mem_filterBasis_iff
|
[] |
[
157,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Order/Filter/NAry.lean
|
Filter.map_map₂_right_comm
|
[] |
[
353,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.SignedMeasure.subset_positive_null_set
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\nthis : ↑s v + ↑s (w \\ v) = 0\n⊢ ↑s v = 0",
"state_before": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\n⊢ ↑s v = 0",
"tactic": "have : s v + s (w \\ v) = 0 := by\n rw [← hw₁, ← of_union Set.disjoint_sdiff_right hv (hw.diff hv), Set.union_diff_self,\n Set.union_eq_self_of_subset_left hwt]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\nthis : ↑s v + ↑s (w \\ v) = 0\nh₁ : 0 ≤ ↑s v\n⊢ ↑s v = 0",
"state_before": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\nthis : ↑s v + ↑s (w \\ v) = 0\n⊢ ↑s v = 0",
"tactic": "have h₁ := nonneg_of_zero_le_restrict _ (restrict_le_restrict_subset _ _ hu hsu (hwt.trans hw₂))"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\nthis : ↑s v + ↑s (w \\ v) = 0\nh₁ : 0 ≤ ↑s v\nh₂ : 0 ≤ ↑s (w \\ v)\n⊢ ↑s v = 0",
"state_before": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\nthis : ↑s v + ↑s (w \\ v) = 0\nh₁ : 0 ≤ ↑s v\n⊢ ↑s v = 0",
"tactic": "have h₂ :=\n nonneg_of_zero_le_restrict _\n (restrict_le_restrict_subset _ _ hu hsu ((w.diff_subset v).trans hw₂))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\nthis : ↑s v + ↑s (w \\ v) = 0\nh₁ : 0 ≤ ↑s v\nh₂ : 0 ≤ ↑s (w \\ v)\n⊢ ↑s v = 0",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.41858\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhw₁ : ↑s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\n⊢ ↑s v + ↑s (w \\ v) = 0",
"tactic": "rw [← hw₁, ← of_union Set.disjoint_sdiff_right hv (hw.diff hv), Set.union_diff_self,\n Set.union_eq_self_of_subset_left hwt]"
}
] |
[
293,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.range_comp_le_range
|
[] |
[
1236,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1234,
1
] |
Mathlib/CategoryTheory/Functor/EpiMono.lean
|
CategoryTheory.Functor.preservesMonomorphisms.of_iso
|
[
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G : C ⥤ D\ninst✝ : PreservesMonomorphisms F\nα : F ≅ G\nX Y : C\nf : X ⟶ Y\nh : Mono f\nthis : Mono (F.map f ≫ (α.app Y).hom)\n⊢ Mono (G.map f)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G : C ⥤ D\ninst✝ : PreservesMonomorphisms F\nα : F ≅ G\nX Y : C\nf : X ⟶ Y\nh : Mono f\n⊢ Mono (G.map f)",
"tactic": "haveI : Mono (F.map f ≫ (α.app Y).hom) := mono_comp _ _"
},
{
"state_after": "case h.e'_5\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G : C ⥤ D\ninst✝ : PreservesMonomorphisms F\nα : F ≅ G\nX Y : C\nf : X ⟶ Y\nh : Mono f\nthis : Mono (F.map f ≫ (α.app Y).hom)\n⊢ G.map f = (α.app X).inv ≫ F.map f ≫ (α.app Y).hom",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G : C ⥤ D\ninst✝ : PreservesMonomorphisms F\nα : F ≅ G\nX Y : C\nf : X ⟶ Y\nh : Mono f\nthis : Mono (F.map f ≫ (α.app Y).hom)\n⊢ Mono (G.map f)",
"tactic": "convert (mono_comp _ _ : Mono ((α.app X).inv ≫ F.map f ≫ (α.app Y).hom))"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nC : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nE : Type u₃\ninst✝¹ : Category E\nF G : C ⥤ D\ninst✝ : PreservesMonomorphisms F\nα : F ≅ G\nX Y : C\nf : X ⟶ Y\nh : Mono f\nthis : Mono (F.map f ≫ (α.app Y).hom)\n⊢ G.map f = (α.app X).inv ≫ F.map f ≫ (α.app Y).hom",
"tactic": "rw [Iso.eq_inv_comp, Iso.app_hom, Iso.app_hom, NatTrans.naturality]"
}
] |
[
129,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.map_iSup
|
[] |
[
1015,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1013,
1
] |
Mathlib/LinearAlgebra/ProjectiveSpace/Basic.lean
|
Projectivization.map_id
|
[
{
"state_after": "case h.mk\nK : Type u_2\nV : Type u_1\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nL : Type ?u.156698\nW : Type ?u.156701\ninst✝² : DivisionRing L\ninst✝¹ : AddCommGroup W\ninst✝ : Module L W\nx✝ : ℙ K V\nv : { v // v ≠ 0 }\n⊢ map LinearMap.id (_ : Function.Injective ↑(LinearEquiv.refl K V)) (Quot.mk Setoid.r v) = id (Quot.mk Setoid.r v)",
"state_before": "K : Type u_2\nV : Type u_1\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nL : Type ?u.156698\nW : Type ?u.156701\ninst✝² : DivisionRing L\ninst✝¹ : AddCommGroup W\ninst✝ : Module L W\n⊢ map LinearMap.id (_ : Function.Injective ↑(LinearEquiv.refl K V)) = id",
"tactic": "ext ⟨v⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nK : Type u_2\nV : Type u_1\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nL : Type ?u.156698\nW : Type ?u.156701\ninst✝² : DivisionRing L\ninst✝¹ : AddCommGroup W\ninst✝ : Module L W\nx✝ : ℙ K V\nv : { v // v ≠ 0 }\n⊢ map LinearMap.id (_ : Function.Injective ↑(LinearEquiv.refl K V)) (Quot.mk Setoid.r v) = id (Quot.mk Setoid.r v)",
"tactic": "rfl"
}
] |
[
223,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhp : 1 ≤ p\nhf : ∀ (i : ι), i ∈ s → 0 ≤ f i\n⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p",
"tactic": "convert rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>\n simp only [abs_of_nonneg, hf i hi]"
}
] |
[
655,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
652,
1
] |
Mathlib/RingTheory/Filtration.lean
|
Ideal.iInf_pow_eq_bot_of_isDomain
|
[
{
"state_after": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\n⊢ (⨅ (i : ℕ), I ^ i) ≤ ⊥",
"state_before": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\n⊢ (⨅ (i : ℕ), I ^ i) = ⊥",
"tactic": "rw [eq_bot_iff]"
},
{
"state_after": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\n⊢ x ∈ ⊥",
"state_before": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\n⊢ (⨅ (i : ℕ), I ^ i) ≤ ⊥",
"tactic": "intro x hx"
},
{
"state_after": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\n⊢ False",
"state_before": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\n⊢ x ∈ ⊥",
"tactic": "by_contra hx'"
},
{
"state_after": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis : (x ∈ ⨅ (i : ℕ), I ^ i • ⊤) ↔ ∃ r, ↑r • x = x\n⊢ False",
"state_before": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\n⊢ False",
"tactic": "have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x"
},
{
"state_after": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis : (x ∈ ⨅ (i : ℕ), I ^ i) ↔ ∃ r, ↑r * x = x\n⊢ False",
"state_before": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis : (x ∈ ⨅ (i : ℕ), I ^ i • ⊤) ↔ ∃ r, ↑r • x = x\n⊢ False",
"tactic": "simp_rw [smul_eq_mul, ← Ideal.one_eq_top, mul_one] at this"
},
{
"state_after": "case intro\nR M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis : (x ∈ ⨅ (i : ℕ), I ^ i) ↔ ∃ r, ↑r * x = x\nr : { x // x ∈ I }\nhr : ↑r * x = x\n⊢ False",
"state_before": "R M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis : (x ∈ ⨅ (i : ℕ), I ^ i) ↔ ∃ r, ↑r * x = x\n⊢ False",
"tactic": "obtain ⟨r, hr⟩ := this.mp hx"
},
{
"state_after": "case intro\nR M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis✝ : (x ∈ ⨅ (i : ℕ), I ^ i) ↔ ∃ r, ↑r * x = x\nr : { x // x ∈ I }\nhr : ↑r * x = x\nthis : ↑r = 1\n⊢ False",
"state_before": "case intro\nR M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis : (x ∈ ⨅ (i : ℕ), I ^ i) ↔ ∃ r, ↑r * x = x\nr : { x // x ∈ I }\nhr : ↑r * x = x\n⊢ False",
"tactic": "have := mul_right_cancel₀ hx' (hr.trans (one_mul x).symm)"
},
{
"state_after": "no goals",
"state_before": "case intro\nR M : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsDomain R\nh : I ≠ ⊤\nx : R\nhx : x ∈ ⨅ (i : ℕ), I ^ i\nhx' : ¬x ∈ ⊥\nthis✝ : (x ∈ ⨅ (i : ℕ), I ^ i) ↔ ∃ r, ↑r * x = x\nr : { x // x ∈ I }\nhr : ↑r * x = x\nthis : ↑r = 1\n⊢ False",
"tactic": "exact I.eq_top_iff_one.not.mp h (this ▸ r.prop)"
}
] |
[
490,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.lowerBounds_u_image
|
[] |
[
122,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.mem_nsmul
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.47121\nγ : Type ?u.47124\na : α\ns : Multiset α\nn : ℕ\nh0 : n ≠ 0\nh : a ∈ s\n⊢ a ∈ n • s",
"state_before": "α : Type u_1\nβ : Type ?u.47121\nγ : Type ?u.47124\na : α\ns : Multiset α\nn : ℕ\nh0 : n ≠ 0\n⊢ a ∈ n • s ↔ a ∈ s",
"tactic": "refine' ⟨mem_of_mem_nsmul, fun h => _⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.47121\nγ : Type ?u.47124\na : α\ns : Multiset α\nh : a ∈ s\nn : ℕ\nh0 : succ n ≠ 0\n⊢ a ∈ succ n • s",
"state_before": "α : Type u_1\nβ : Type ?u.47121\nγ : Type ?u.47124\na : α\ns : Multiset α\nn : ℕ\nh0 : n ≠ 0\nh : a ∈ s\n⊢ a ∈ n • s",
"tactic": "obtain ⟨n, rfl⟩ := exists_eq_succ_of_ne_zero h0"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.47121\nγ : Type ?u.47124\na : α\ns : Multiset α\nh : a ∈ s\nn : ℕ\nh0 : succ n ≠ 0\n⊢ a ∈ s ∨ a ∈ n • s",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.47121\nγ : Type ?u.47124\na : α\ns : Multiset α\nh : a ∈ s\nn : ℕ\nh0 : succ n ≠ 0\n⊢ a ∈ succ n • s",
"tactic": "rw [succ_nsmul, mem_add]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.47121\nγ : Type ?u.47124\na : α\ns : Multiset α\nh : a ∈ s\nn : ℕ\nh0 : succ n ≠ 0\n⊢ a ∈ s ∨ a ∈ n • s",
"tactic": "exact Or.inl h"
}
] |
[
700,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
696,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
fderivWithin_const_sub
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.574896\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.574991\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhxs : UniqueDiffWithinAt 𝕜 s x\nc : F\n⊢ fderivWithin 𝕜 (fun y => c - f y) s x = -fderivWithin 𝕜 f s x",
"tactic": "simp only [sub_eq_add_neg, fderivWithin_const_add, fderivWithin_neg, hxs]"
}
] |
[
659,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
657,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.union_iInter
|
[] |
[
557,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Data/Finset/Sigma.lean
|
Finset.disjiUnion_map_sigma_mk
|
[] |
[
91,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Data/List/AList.lean
|
AList.mk_cons_eq_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nc : Sigma β\nl : List (Sigma β)\nh : NodupKeys (c :: l)\n⊢ { entries := c :: l, nodupKeys := h } = insert c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }",
"tactic": "simpa [insert] using (kerase_of_not_mem_keys <| not_mem_keys_of_nodupKeys_cons h).symm"
}
] |
[
350,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/RingTheory/Filtration.lean
|
Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
|
[
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))) ↔\n ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\n\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ≤\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ), ↑(single R i) '' ↑(N F i))",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ Filtration.submodule F =\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ↔\n ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)",
"tactic": "rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ≤\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ), ↑(single R i) '' ↑(N F i))\n\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))) ↔\n ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))) ↔\n ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\n\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ≤\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ), ↑(single R i) '' ↑(N F i))",
"tactic": "swap"
},
{
"state_after": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))) →\n ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\n\ncase mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)) →\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))) ↔\n ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ≤\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ), ↑(single R i) '' ↑(N F i))",
"tactic": "exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _)"
},
{
"state_after": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\n⊢ I • N F n = N F (n + 1)",
"state_before": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))) →\n ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)",
"tactic": "intro H n hn"
},
{
"state_after": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\n⊢ N F (n + 1) ≤ I • N F n",
"state_before": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\n⊢ I • N F n = N F (n + 1)",
"tactic": "refine' (F.smul_le n).antisymm _"
},
{
"state_after": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\n⊢ x ∈ I • N F n",
"state_before": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\n⊢ N F (n + 1) ≤ I • N F n",
"tactic": "intro x hx"
},
{
"state_after": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l =\n ↑(single R (n + 1)) x\n⊢ x ∈ I • N F n",
"state_before": "case mp\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\n⊢ x ∈ I • N F n",
"tactic": "obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩)"
},
{
"state_after": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n (fun f => ↑f (n + 1))\n (↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l) =\n (fun f => ↑f (n + 1)) (↑(single R (n + 1)) x)\n⊢ x ∈ I • N F n",
"state_before": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l =\n ↑(single R (n + 1)) x\n⊢ x ∈ I • N F n",
"tactic": "replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl"
},
{
"state_after": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n ↑(↑(single R (n + 1)) x) (n + 1)\n⊢ x ∈ I • N F n",
"state_before": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n (fun f => ↑f (n + 1))\n (↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l) =\n (fun f => ↑f (n + 1)) (↑(single R (n + 1)) x)\n⊢ x ∈ I • N F n",
"tactic": "dsimp only at hl"
},
{
"state_after": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\n⊢ x ∈ I • N F n",
"state_before": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n ↑(↑(single R (n + 1)) x) (n + 1)\n⊢ x ∈ I • N F n",
"tactic": "erw [Finsupp.single_eq_same] at hl"
},
{
"state_after": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\n⊢ (Finsupp.sum l fun a₁ b => ↑(b • ↑a₁) (n + 1)) ∈ I • N F n",
"state_before": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\n⊢ x ∈ I • N F n",
"tactic": "rw [← hl, Finsupp.total_apply, Finsupp.sum_apply]"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\n⊢ ∀ (c : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))),\n c ∈ l.support → (fun a₁ b => ↑(b • ↑a₁) (n + 1)) c (↑l c) ∈ I • N F n",
"state_before": "case mp.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\n⊢ (Finsupp.sum l fun a₁ b => ↑(b • ↑a₁) (n + 1)) ∈ I • N F n",
"tactic": "apply Submodule.sum_mem _ _"
},
{
"state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ (fun a₁ b => ↑(b • ↑a₁) (n + 1))\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }\n (↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }) ∈\n I • N F n",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\n⊢ ∀ (c : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))),\n c ∈ l.support → (fun a₁ b => ↑(b • ↑a₁) (n + 1)) c (↑l c) ∈ I • N F n",
"tactic": "rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ -"
},
{
"state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ ↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) } •\n ↑(single R n') m)\n (n + 1) ∈\n I • N F n",
"state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ (fun a₁ b => ↑(b • ↑a₁) (n + 1))\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }\n (↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }) ∈\n I • N F n",
"tactic": "dsimp only [Subtype.coe_mk]"
},
{
"state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ ↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) } •\n ↑(single R n') m)\n (n + 1) ∈\n I • N F n",
"tactic": "rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by linarith)]"
},
{
"state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\ne : n' ≤ n\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"tactic": "have e : n' ≤ n := by linarith"
},
{
"state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\ne : n' ≤ n\nthis : I ^ (n - n' + 1) • N F n' ≤ I ^ 1 • N F (n - n' + n')\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\ne : n' ≤ n\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"tactic": "have := F.pow_smul_le_pow_smul (n - n') n' 1"
},
{
"state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\ne : n' ≤ n\nthis : I ^ (n + 1 - n') • N F n' ≤ I • N F n\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\ne : n' ≤ n\nthis : I ^ (n - n' + 1) • N F n' ≤ I ^ 1 • N F (n - n' + n')\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"tactic": "rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, ← add_tsub_assoc_of_le e, add_comm] at this"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\ne : n' ≤ n\nthis : I ^ (n + 1 - n') • N F n' ≤ I • N F n\n⊢ coeff\n (↑(↑l\n { val := ↑(single R n') m,\n property :=\n (_ : ∃ t, (t ∈ Set.range fun i => ⋃ (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) ∧ ↑(single R n') m ∈ t) }))\n (n + 1 - n') •\n m ∈\n I • N F n",
"tactic": "exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)"
},
{
"state_after": "no goals",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ n' ≤ n + 1",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\nH :\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nn : ℕ\nhn : n ≥ n₀\nx : M\nhx : x ∈ N F (n + 1)\nl : ↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)) →₀ { x // x ∈ reesAlgebra I }\nhl :\n ↑(↑(Finsupp.total (↑(⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))) (PolynomialModule R M)\n { x // x ∈ reesAlgebra I } Subtype.val)\n l)\n (n + 1) =\n x\nn' : ℕ\nhn' : n' ≤ n₀\nm : M\nhm : m ∈ ↑(N F n')\n⊢ n' ≤ n",
"tactic": "linarith"
},
{
"state_after": "case mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\n⊢ (∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)) →\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn₀ : ℕ\n⊢ (∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)) →\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M))"
},
{
"state_after": "case mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni : ℕ\n⊢ ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\n⊢ (∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)) →\n ∀ (i : ℕ),\n ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "intro hF i"
},
{
"state_after": "case mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni : ℕ\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\n⊢ ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni : ℕ\n⊢ ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by\n intro i hi\n refine Set.Subset.trans ?_ Submodule.subset_span\n refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_\n exact hi"
},
{
"state_after": "case mpr.zero\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\n⊢ ↑(single R Nat.zero) '' ↑(N F Nat.zero) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n\ncase mpr.succ\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case mpr\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni : ℕ\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\n⊢ ↑(single R i) '' ↑(N F i) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "induction' i with j hj"
},
{
"state_after": "case pos\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : Nat.succ j ≤ n₀\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n\ncase neg\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : ¬Nat.succ j ≤ n₀\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case mpr.succ\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "by_cases hj' : j.succ ≤ n₀"
},
{
"state_after": "case neg\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case neg\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : ¬Nat.succ j ≤ n₀\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "simp only [not_le, Nat.lt_succ_iff] at hj'"
},
{
"state_after": "case neg\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\n⊢ ↑(single R (j + 1)) '' ↑(I • N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case neg\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "rw [Nat.succ_eq_add_one, ← hF _ hj']"
},
{
"state_after": "case neg.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\n⊢ ↑(single R (j + 1)) m ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case neg\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\n⊢ ↑(single R (j + 1)) '' ↑(I • N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "rintro _ ⟨m, hm, rfl⟩"
},
{
"state_after": "case neg.intro.intro.refine'_1\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nr : R\nhr : r ∈ I\nm' : M\nhm' : m' ∈ N F j\n⊢ ↑(single R (j + 1)) (r • m') ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n\ncase neg.intro.intro.refine'_2\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nx y : M\nhx :\n ↑(single R (j + 1)) x ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhy :\n ↑(single R (j + 1)) y ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n⊢ ↑(single R (j + 1)) (x + y) ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case neg.intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\n⊢ ↑(single R (j + 1)) m ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "refine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni✝ i : ℕ\nhi : i ≤ n₀\n⊢ ↑(single R i) '' ↑(N F i) ⊆ ↑F'",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni : ℕ\n⊢ ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'",
"tactic": "intro i hi"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni✝ i : ℕ\nhi : i ≤ n₀\n⊢ ↑(single R i) '' ↑(N F i) ⊆ ⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni✝ i : ℕ\nhi : i ≤ n₀\n⊢ ↑(single R i) '' ↑(N F i) ⊆ ↑F'",
"tactic": "refine Set.Subset.trans ?_ Submodule.subset_span"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni✝ i : ℕ\nhi : i ≤ n₀\n⊢ i ≤ n₀",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni✝ i : ℕ\nhi : i ≤ n₀\n⊢ ↑(single R i) '' ↑(N F i) ⊆ ⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)",
"tactic": "refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_"
},
{
"state_after": "no goals",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\ni✝ i : ℕ\nhi : i ≤ n₀\n⊢ i ≤ n₀",
"tactic": "exact hi"
},
{
"state_after": "no goals",
"state_before": "case mpr.zero\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\n⊢ ↑(single R Nat.zero) '' ↑(N F Nat.zero) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "exact this _ (zero_le _)"
},
{
"state_after": "no goals",
"state_before": "case pos\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : Nat.succ j ≤ n₀\n⊢ ↑(single R (Nat.succ j)) '' ↑(N F (Nat.succ j)) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "exact this _ hj'"
},
{
"state_after": "case neg.intro.intro.refine'_1\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nr : R\nhr : r ∈ I\nm' : M\nhm' : m' ∈ N F j\n⊢ ↑(monomial 1) r • ↑(single R j) m' ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case neg.intro.intro.refine'_1\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nr : R\nhr : r ∈ I\nm' : M\nhm' : m' ∈ N F j\n⊢ ↑(single R (j + 1)) (r • m') ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "rw [add_comm, ← monomial_smul_single]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.refine'_1\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nr : R\nhr : r ∈ I\nm' : M\nhm' : m' ∈ N F j\n⊢ ↑(monomial 1) r • ↑(single R j) m' ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "exact F'.smul_mem\n ⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm')"
},
{
"state_after": "no goals",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nr : R\nhr : r ∈ I\nm' : M\nhm' : m' ∈ N F j\n⊢ r ∈ I ^ 1",
"tactic": "rwa [pow_one]"
},
{
"state_after": "case neg.intro.intro.refine'_2\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nx y : M\nhx :\n ↑(single R (j + 1)) x ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhy :\n ↑(single R (j + 1)) y ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n⊢ ↑(single R (j + 1)) x + ↑(single R (j + 1)) y ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"state_before": "case neg.intro.intro.refine'_2\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nx y : M\nhx :\n ↑(single R (j + 1)) x ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhy :\n ↑(single R (j + 1)) y ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n⊢ ↑(single R (j + 1)) (x + y) ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "rw [map_add]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.refine'_2\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F'✝ : Filtration I M\nh : Stable F\nn₀ : ℕ\nF' : Submodule { x // x ∈ reesAlgebra I } (PolynomialModule R M) :=\n Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i))\nhF : ∀ (n : ℕ), n ≥ n₀ → I • N F n = N F (n + 1)\nthis : ∀ (i : ℕ), i ≤ n₀ → ↑(single R i) '' ↑(N F i) ⊆ ↑F'\nj : ℕ\nhj :\n ↑(single R j) '' ↑(N F j) ⊆\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhj' : n₀ ≤ j\nm : M\nhm : m ∈ ↑(I • N F j)\nx y : M\nhx :\n ↑(single R (j + 1)) x ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\nhy :\n ↑(single R (j + 1)) y ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))\n⊢ ↑(single R (j + 1)) x + ↑(single R (j + 1)) y ∈\n ↑(Submodule.span { x // x ∈ reesAlgebra I } (⋃ (i : ℕ) (_ : i ≤ n₀), ↑(single R i) '' ↑(N F i)))",
"tactic": "exact F'.add_mem hx hy"
}
] |
[
366,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/CategoryTheory/Adjunction/Limits.lean
|
CategoryTheory.Adjunction.hasLimit_comp_equivalence
|
[] |
[
254,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Topology/Separation.lean
|
TopologicalSpace.subset_trans
|
[] |
[
861,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
le_of_one_div_le_one_div
|
[] |
[
462,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Data/Int/ConditionallyCompleteOrder.lean
|
Int.csSup_of_not_bdd_above
|
[
{
"state_after": "no goals",
"state_before": "s : Set ℤ\nh : ¬BddAbove s\n⊢ ¬(Set.Nonempty s ∧ BddAbove s)",
"tactic": "simp [h]"
}
] |
[
79,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
|
MeasureTheory.MeasurableEquiv.measurePreserving_symm
|
[] |
[
176,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.piecewise_mem_Icc
|
[] |
[
54,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_left_involutive
|
[] |
[
657,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
segment_subset_uIcc
|
[
{
"state_after": "case inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : x ≤ y\n⊢ [x-[𝕜]y] ⊆ uIcc x y\n\ncase inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : y ≤ x\n⊢ [x-[𝕜]y] ⊆ uIcc x y",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\n⊢ [x-[𝕜]y] ⊆ uIcc x y",
"tactic": "cases' le_total x y with h h"
},
{
"state_after": "case inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : x ≤ y\n⊢ [x-[𝕜]y] ⊆ Icc x y",
"state_before": "case inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : x ≤ y\n⊢ [x-[𝕜]y] ⊆ uIcc x y",
"tactic": "rw [uIcc_of_le h]"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : x ≤ y\n⊢ [x-[𝕜]y] ⊆ Icc x y",
"tactic": "exact segment_subset_Icc h"
},
{
"state_after": "case inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : y ≤ x\n⊢ [y-[𝕜]x] ⊆ Icc y x",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : y ≤ x\n⊢ [x-[𝕜]y] ⊆ uIcc x y",
"tactic": "rw [uIcc_of_ge h, segment_symm]"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.269436\nG : Type ?u.269439\nι : Type ?u.269442\nπ : ι → Type ?u.269447\ninst✝³ : OrderedSemiring 𝕜\ninst✝² : LinearOrderedAddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : OrderedSMul 𝕜 E\na b : 𝕜\nx y : E\nh : y ≤ x\n⊢ [y-[𝕜]x] ⊆ Icc y x",
"tactic": "exact segment_subset_Icc h"
}
] |
[
468,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.not_top_le_coe
|
[] |
[
853,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_lt_rpow_left_iff_of_base_lt_one
|
[
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhx0 : 0 < x\nhx1 : x < 1\n⊢ x ^ y < x ^ z ↔ z < y",
"tactic": "rw [lt_iff_not_le, rpow_le_rpow_left_iff_of_base_lt_one hx0 hx1, lt_iff_not_le]"
}
] |
[
511,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.tendsto_map
|
[] |
[
2936,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2935,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.empty_ssubset_singleton
|
[] |
[
1309,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1308,
1
] |
Mathlib/Data/Set/Function.lean
|
Function.Semiconj.bijOn_image
|
[] |
[
1655,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1652,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion₂_comm
|
[] |
[
796,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.RedRed.setBlack
|
[] |
[
151,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
149,
11
] |
Mathlib/Order/Atoms.lean
|
IsAtom.le_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2208\ninst✝¹ : PartialOrder α\ninst✝ : OrderBot α\na b x : α\nh : IsAtom a\n⊢ x ≤ a ↔ x = ⊥ ∨ x = a",
"tactic": "rw [le_iff_lt_or_eq, h.lt_iff]"
}
] |
[
92,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.map_vsub
|
[] |
[
625,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.nat_cast_imJ
|
[] |
[
985,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
985,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.lift_funext
|
[
{
"state_after": "M : Type u_1\nN : Type ?u.62273\nP : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx y : M\nf✝ : M →* P\nf g : Con.Quotient c →* P\nh : ∀ (a : M), ↑f ↑a = ↑g ↑a\n⊢ lift c (MonoidHom.comp f (mk' c)) (_ : ∀ (x y : M), ↑c x y → ↑f ↑x = ↑f ↑y) =\n lift c (MonoidHom.comp g (mk' c)) (_ : ∀ (x y : M), ↑c x y → ↑g ↑x = ↑g ↑y)",
"state_before": "M : Type u_1\nN : Type ?u.62273\nP : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx y : M\nf✝ : M →* P\nf g : Con.Quotient c →* P\nh : ∀ (a : M), ↑f ↑a = ↑g ↑a\n⊢ f = g",
"tactic": "rw [← lift_apply_mk' f, ← lift_apply_mk' g]"
},
{
"state_after": "case e_f\nM : Type u_1\nN : Type ?u.62273\nP : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx y : M\nf✝ : M →* P\nf g : Con.Quotient c →* P\nh : ∀ (a : M), ↑f ↑a = ↑g ↑a\n⊢ MonoidHom.comp f (mk' c) = MonoidHom.comp g (mk' c)",
"state_before": "M : Type u_1\nN : Type ?u.62273\nP : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx y : M\nf✝ : M →* P\nf g : Con.Quotient c →* P\nh : ∀ (a : M), ↑f ↑a = ↑g ↑a\n⊢ lift c (MonoidHom.comp f (mk' c)) (_ : ∀ (x y : M), ↑c x y → ↑f ↑x = ↑f ↑y) =\n lift c (MonoidHom.comp g (mk' c)) (_ : ∀ (x y : M), ↑c x y → ↑g ↑x = ↑g ↑y)",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_f\nM : Type u_1\nN : Type ?u.62273\nP : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx y : M\nf✝ : M →* P\nf g : Con.Quotient c →* P\nh : ∀ (a : M), ↑f ↑a = ↑g ↑a\n⊢ MonoidHom.comp f (mk' c) = MonoidHom.comp g (mk' c)",
"tactic": "exact FunLike.ext_iff.2 h"
}
] |
[
967,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
964,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
|
MeasureTheory.MeasurePreserving.measure_preimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.271396\nδ : Type ?u.271399\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nf : α → β\nhf : MeasurePreserving f\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑μa (f ⁻¹' s) = ↑↑μb s",
"tactic": "rw [← hf.map_eq, map_apply hf.1 hs]"
}
] |
[
125,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.exists_retraction_subtype_of_isClosed
|
[
{
"state_after": "case intro.intro.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nf : ((n : ℕ) → E n) → (n : ℕ) → E n\nf_cont : Continuous f\nhs : IsClosed (range f)\nhne : Set.Nonempty (range f)\nfs : ∀ (x : (n : ℕ) → E n), x ∈ range f → f x = x\n⊢ ∃ f_1, (∀ (x : ↑(range f)), f_1 ↑x = x) ∧ Surjective f_1 ∧ Continuous f_1",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nhne : Set.Nonempty s\n⊢ ∃ f, (∀ (x : ↑s), f ↑x = x) ∧ Surjective f ∧ Continuous f",
"tactic": "obtain ⟨f, fs, rfl, f_cont⟩ :\n ∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ Continuous f :=\n exists_retraction_of_isClosed hs hne"
},
{
"state_after": "case intro.intro.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nf : ((n : ℕ) → E n) → (n : ℕ) → E n\nf_cont : Continuous f\nhs : IsClosed (range f)\nhne : Set.Nonempty (range f)\nfs : ∀ (x : (n : ℕ) → E n), x ∈ range f → f x = x\nA : ∀ (x : ↑(range f)), rangeFactorization f ↑x = x\n⊢ ∃ f_1, (∀ (x : ↑(range f)), f_1 ↑x = x) ∧ Surjective f_1 ∧ Continuous f_1",
"state_before": "case intro.intro.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nf : ((n : ℕ) → E n) → (n : ℕ) → E n\nf_cont : Continuous f\nhs : IsClosed (range f)\nhne : Set.Nonempty (range f)\nfs : ∀ (x : (n : ℕ) → E n), x ∈ range f → f x = x\n⊢ ∃ f_1, (∀ (x : ↑(range f)), f_1 ↑x = x) ∧ Surjective f_1 ∧ Continuous f_1",
"tactic": "have A : ∀ x : range f, rangeFactorization f x = x := fun x ↦ Subtype.eq <| fs x x.2"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nf : ((n : ℕ) → E n) → (n : ℕ) → E n\nf_cont : Continuous f\nhs : IsClosed (range f)\nhne : Set.Nonempty (range f)\nfs : ∀ (x : (n : ℕ) → E n), x ∈ range f → f x = x\nA : ∀ (x : ↑(range f)), rangeFactorization f ↑x = x\n⊢ ∃ f_1, (∀ (x : ↑(range f)), f_1 ↑x = x) ∧ Surjective f_1 ∧ Continuous f_1",
"tactic": "exact ⟨rangeFactorization f, A, fun x => ⟨x, A x⟩, f_cont.subtype_mk _⟩"
}
] |
[
715,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.map_subgraphOfAdj
|
[
{
"state_after": "case verts.h\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ x✝ ∈ (Subgraph.map f (subgraphOfAdj G hvw)).verts ↔ x✝ ∈ (subgraphOfAdj G' (_ : Adj G' (↑f v) (↑f w))).verts\n\ncase Adj.h.h.a\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ Subgraph.Adj (Subgraph.map f (subgraphOfAdj G hvw)) x✝¹ x✝ ↔\n Subgraph.Adj (subgraphOfAdj G' (_ : Adj G' (↑f v) (↑f w))) x✝¹ x✝",
"state_before": "ι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ Subgraph.map f (subgraphOfAdj G hvw) = subgraphOfAdj G' (_ : Adj G' (↑f v) (↑f w))",
"tactic": "ext"
},
{
"state_after": "case verts.h\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ (∃ x, (x = v ∨ x = w) ∧ ↑f x = x✝) ↔ x✝ = ↑f v ∨ x✝ = ↑f w",
"state_before": "case verts.h\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ x✝ ∈ (Subgraph.map f (subgraphOfAdj G hvw)).verts ↔ x✝ ∈ (subgraphOfAdj G' (_ : Adj G' (↑f v) (↑f w))).verts",
"tactic": "simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff,\n Set.mem_singleton_iff]"
},
{
"state_after": "case verts.h.mp\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ (∃ x, (x = v ∨ x = w) ∧ ↑f x = x✝) → x✝ = ↑f v ∨ x✝ = ↑f w\n\ncase verts.h.mpr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ x✝ = ↑f v ∨ x✝ = ↑f w → ∃ x, (x = v ∨ x = w) ∧ ↑f x = x✝",
"state_before": "case verts.h\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ (∃ x, (x = v ∨ x = w) ∧ ↑f x = x✝) ↔ x✝ = ↑f v ∨ x✝ = ↑f w",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case verts.h.mp\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ (∃ x, (x = v ∨ x = w) ∧ ↑f x = x✝) → x✝ = ↑f v ∨ x✝ = ↑f w",
"tactic": "rintro ⟨u, rfl | rfl, rfl⟩ <;> simp"
},
{
"state_after": "case verts.h.mpr.inl\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ x, (x = v ∨ x = w) ∧ ↑f x = ↑f v\n\ncase verts.h.mpr.inr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ x, (x = v ∨ x = w) ∧ ↑f x = ↑f w",
"state_before": "case verts.h.mpr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝ : W\n⊢ x✝ = ↑f v ∨ x✝ = ↑f w → ∃ x, (x = v ∨ x = w) ∧ ↑f x = x✝",
"tactic": "rintro (rfl | rfl)"
},
{
"state_after": "case verts.h.mpr.inl\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (v = v ∨ v = w) ∧ ↑f v = ↑f v",
"state_before": "case verts.h.mpr.inl\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ x, (x = v ∨ x = w) ∧ ↑f x = ↑f v",
"tactic": "use v"
},
{
"state_after": "no goals",
"state_before": "case verts.h.mpr.inl\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (v = v ∨ v = w) ∧ ↑f v = ↑f v",
"tactic": "simp"
},
{
"state_after": "case verts.h.mpr.inr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (w = v ∨ w = w) ∧ ↑f w = ↑f w",
"state_before": "case verts.h.mpr.inr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ x, (x = v ∨ x = w) ∧ ↑f x = ↑f w",
"tactic": "use w"
},
{
"state_after": "no goals",
"state_before": "case verts.h.mpr.inr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (w = v ∨ w = w) ∧ ↑f w = ↑f w",
"tactic": "simp"
},
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ (∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = x✝¹ ∧ ↑f b = x✝) ↔ ↑f v = x✝¹ ∧ ↑f w = x✝ ∨ ↑f v = x✝ ∧ ↑f w = x✝¹",
"state_before": "case Adj.h.h.a\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ Subgraph.Adj (Subgraph.map f (subgraphOfAdj G hvw)) x✝¹ x✝ ↔\n Subgraph.Adj (subgraphOfAdj G' (_ : Adj G' (↑f v) (↑f w))) x✝¹ x✝",
"tactic": "simp only [Relation.Map, Subgraph.map_Adj, subgraphOfAdj_Adj, Quotient.eq, Sym2.rel_iff]"
},
{
"state_after": "case Adj.h.h.a.mp\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ (∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = x✝¹ ∧ ↑f b = x✝) → ↑f v = x✝¹ ∧ ↑f w = x✝ ∨ ↑f v = x✝ ∧ ↑f w = x✝¹\n\ncase Adj.h.h.a.mpr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ ↑f v = x✝¹ ∧ ↑f w = x✝ ∨ ↑f v = x✝ ∧ ↑f w = x✝¹ → ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = x✝¹ ∧ ↑f b = x✝",
"state_before": "case Adj.h.h.a\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ (∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = x✝¹ ∧ ↑f b = x✝) ↔ ↑f v = x✝¹ ∧ ↑f w = x✝ ∨ ↑f v = x✝ ∧ ↑f w = x✝¹",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a.mp\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ (∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = x✝¹ ∧ ↑f b = x✝) → ↑f v = x✝¹ ∧ ↑f w = x✝ ∨ ↑f v = x✝ ∧ ↑f w = x✝¹",
"tactic": "rintro ⟨a, b, ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp"
},
{
"state_after": "case Adj.h.h.a.mpr.inl.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = ↑f v ∧ ↑f b = ↑f w\n\ncase Adj.h.h.a.mpr.inr.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = ↑f w ∧ ↑f b = ↑f v",
"state_before": "case Adj.h.h.a.mpr\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\nx✝¹ x✝ : W\n⊢ ↑f v = x✝¹ ∧ ↑f w = x✝ ∨ ↑f v = x✝ ∧ ↑f w = x✝¹ → ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = x✝¹ ∧ ↑f b = x✝",
"tactic": "rintro (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)"
},
{
"state_after": "case Adj.h.h.a.mpr.inl.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (v = v ∧ w = w ∨ v = w ∧ w = v) ∧ ↑f v = ↑f v ∧ ↑f w = ↑f w",
"state_before": "case Adj.h.h.a.mpr.inl.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = ↑f v ∧ ↑f b = ↑f w",
"tactic": "use v, w"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a.mpr.inl.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (v = v ∧ w = w ∨ v = w ∧ w = v) ∧ ↑f v = ↑f v ∧ ↑f w = ↑f w",
"tactic": "simp"
},
{
"state_after": "case Adj.h.h.a.mpr.inr.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (v = w ∧ w = v ∨ v = v ∧ w = w) ∧ ↑f w = ↑f w ∧ ↑f v = ↑f v",
"state_before": "case Adj.h.h.a.mpr.inr.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ ∃ a b, (v = a ∧ w = b ∨ v = b ∧ w = a) ∧ ↑f a = ↑f w ∧ ↑f b = ↑f v",
"tactic": "use w, v"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a.mpr.inr.intro\nι : Sort ?u.219034\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nf : G →g G'\nv w : V\nhvw : Adj G v w\n⊢ (v = w ∧ w = v ∨ v = v ∧ w = w) ∧ ↑f w = ↑f w ∧ ↑f v = ↑f v",
"tactic": "simp"
}
] |
[
933,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
914,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
atTop_hasCountableBasis_of_archimedean
|
[] |
[
111,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isIntegral_of_mem_closure
|
[] |
[
473,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
dist_prod_prod_le_of_le
|
[
{
"state_after": "𝓕 : Type ?u.739254\n𝕜 : Type ?u.739257\nα : Type ?u.739260\nι : Type u_1\nκ : Type ?u.739266\nE : Type u_2\nF : Type ?u.739272\nG : Type ?u.739275\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ns : Finset ι\nf a : ι → E\nd : ι → ℝ\nh : ∀ (b : ι), b ∈ s → ‖f b / a b‖ ≤ d b\n⊢ ‖∏ x in s, f x / a x‖ ≤ ∑ b in s, d b",
"state_before": "𝓕 : Type ?u.739254\n𝕜 : Type ?u.739257\nα : Type ?u.739260\nι : Type u_1\nκ : Type ?u.739266\nE : Type u_2\nF : Type ?u.739272\nG : Type ?u.739275\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ns : Finset ι\nf a : ι → E\nd : ι → ℝ\nh : ∀ (b : ι), b ∈ s → dist (f b) (a b) ≤ d b\n⊢ dist (∏ b in s, f b) (∏ b in s, a b) ≤ ∑ b in s, d b",
"tactic": "simp only [dist_eq_norm_div, ← Finset.prod_div_distrib] at *"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.739254\n𝕜 : Type ?u.739257\nα : Type ?u.739260\nι : Type u_1\nκ : Type ?u.739266\nE : Type u_2\nF : Type ?u.739272\nG : Type ?u.739275\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ns : Finset ι\nf a : ι → E\nd : ι → ℝ\nh : ∀ (b : ι), b ∈ s → ‖f b / a b‖ ≤ d b\n⊢ ‖∏ x in s, f x / a x‖ ≤ ∑ b in s, d b",
"tactic": "exact norm_prod_le_of_le s h"
}
] |
[
1501,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1497,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_product'
|
[] |
[
676,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
674,
1
] |
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