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Mathlib/RingTheory/Derivation/Basic.lean | Derivation.mk_coe | [] | [
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Mathlib/LinearAlgebra/Projection.lean | LinearMap.isCompl_of_proj | [
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Mathlib/Data/Fin/Basic.lean | Fin.predAbove_last | [] | [
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Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | ProjectiveSpectrum.basicOpen_eq_union_of_projection | [
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},
{
"state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nH : ∀ (i : ℕ), ↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) f), ↑(↑(↑(decompose 𝒜) f) i) ∈ z.asHomogeneousIdeal",
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"tactic": "rw [← DirectSum.sum_support_decompose 𝒜 f]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nH : ∀ (i : ℕ), ↑(GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ ∑ i in Dfinsupp.support (↑(decompose 𝒜) f), ↑(↑(↑(decompose 𝒜) f) i) ∈ z.asHomogeneousIdeal",
"tactic": "apply Ideal.sum_mem _ fun i _ => H i"
},
{
"state_after": "no goals",
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"tactic": "rwa [mem_basicOpen]"
},
{
"state_after": "case mpr.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\ni : ℕ\nhz : z ∈ (fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) i\n⊢ ¬f ∈ z.asHomogeneousIdeal",
"state_before": "case mpr\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : ∃ u, (u ∈ Set.range fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) ∧ z ∈ u\n⊢ ¬f ∈ z.asHomogeneousIdeal",
"tactic": "obtain ⟨_, ⟨i, rfl⟩, hz⟩ := hz"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nR : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\ni : ℕ\nhz : z ∈ (fun i => basicOpen 𝒜 (↑(GradedAlgebra.proj 𝒜 i) f)) i\n⊢ ¬f ∈ z.asHomogeneousIdeal",
"tactic": "exact fun rid => hz (z.1.2 i rid)"
}
] | [
454,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
441,
1
] |
Mathlib/Order/CompactlyGenerated.lean | CompleteLattice.isCompactElement_iff | [
{
"state_after": "case mp\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ IsCompactElement k → ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\n\ncase mpr\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ (∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s) → IsCompactElement k",
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"tactic": "constructor"
},
{
"state_after": "case mp\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\n⊢ ∃ t, k ≤ Finset.sup t s",
"state_before": "case mp\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ IsCompactElement k → ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s",
"tactic": "intro H ι s hs"
},
{
"state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\n⊢ ∃ t, k ≤ Finset.sup t s",
"state_before": "case mp\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\n⊢ ∃ t, k ≤ Finset.sup t s",
"tactic": "obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs"
},
{
"state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nthis : ∀ (x : { x // x ∈ t }), ∃ i, s i = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s",
"state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\n⊢ ∃ t, k ≤ Finset.sup t s",
"tactic": "have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop"
},
{
"state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s",
"state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nthis : ∀ (x : { x // x ∈ t }), ∃ i, s i = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s",
"tactic": "choose f hf using this"
},
{
"state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ Finset.sup t id ≤ Finset.sup (Finset.image f Finset.univ) s",
"state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∃ t, k ≤ Finset.sup t s",
"tactic": "refine' ⟨Finset.univ.image f, ht'.trans _⟩"
},
{
"state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∀ (b : α), b ∈ t → id b ≤ Finset.sup (Finset.image f Finset.univ) s",
"state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ Finset.sup t id ≤ Finset.sup (Finset.image f Finset.univ) s",
"tactic": "rw [Finset.sup_le_iff]"
},
{
"state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ id b ≤ Finset.sup (Finset.image f Finset.univ) s",
"state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\n⊢ ∀ (b : α), b ∈ t → id b ≤ Finset.sup (Finset.image f Finset.univ) s",
"tactic": "intro b hb"
},
{
"state_after": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ s (f { val := b, property := hb }) ≤ Finset.sup (Finset.image f Finset.univ) s",
"state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ id b ≤ Finset.sup (Finset.image f Finset.univ) s",
"tactic": "rw [← show s (f ⟨b, hb⟩) = id b from hf _]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro\nι✝ : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf✝ : ι✝ → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : IsCompactElement k\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ Set.range s\nht' : k ≤ Finset.sup t id\nf : { x // x ∈ t } → ι\nhf : ∀ (x : { x // x ∈ t }), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ s (f { val := b, property := hb }) ≤ Finset.sup (Finset.image f Finset.univ) s",
"tactic": "exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))"
},
{
"state_after": "case mpr\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ ∃ t, ↑t ⊆ s ∧ k ≤ Finset.sup t id",
"state_before": "case mpr\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\n⊢ (∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s) → IsCompactElement k",
"tactic": "intro H s hs"
},
{
"state_after": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ∃ t, ↑t ⊆ s ∧ k ≤ Finset.sup t id",
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"tactic": "obtain ⟨t, ht⟩ :=\n H s Subtype.val\n (by\n delta iSup\n rwa [Subtype.range_coe])"
},
{
"state_after": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ Finset.sup t Subtype.val ≤ Finset.sup (Finset.image Subtype.val t) id",
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"tactic": "refine' ⟨t.image Subtype.val, by simp, ht.trans _⟩"
},
{
"state_after": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ∀ (b : ↑s), b ∈ t → ↑b ≤ Finset.sup (Finset.image Subtype.val t) id",
"state_before": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ Finset.sup t Subtype.val ≤ Finset.sup (Finset.image Subtype.val t) id",
"tactic": "rw [Finset.sup_le_iff]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ∀ (b : ↑s), b ∈ t → ↑b ≤ Finset.sup (Finset.image Subtype.val t) id",
"tactic": "exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem Subtype.val hx)"
},
{
"state_after": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ k ≤ sSup (Set.range Subtype.val)",
"state_before": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ k ≤ iSup Subtype.val",
"tactic": "delta iSup"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\n⊢ k ≤ sSup (Set.range Subtype.val)",
"tactic": "rwa [Subtype.range_coe]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.1219\nα✝ : Type ?u.1222\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ Finset.sup t s\ns : Set α\nhs : k ≤ sSup s\nt : Finset ↑s\nht : k ≤ Finset.sup t Subtype.val\n⊢ ↑(Finset.image Subtype.val t) ⊆ s",
"tactic": "simp"
}
] | [
105,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.map_val_Ioi_atTop | [] | [
1551,
53
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1548,
1
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Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean | IsBoundedBilinearMap.hasFDerivWithinAt | [] | [
94,
38
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92,
1
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Mathlib/Algebra/Hom/Aut.lean | AddAut.inv_def | [] | [
216,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
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Mathlib/Computability/TuringMachine.lean | Turing.reaches₀_eq | [] | [
806,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
805,
1
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Mathlib/Data/Polynomial/RingDivision.lean | Polynomial.eq_zero_of_dvd_of_natDegree_lt | [
{
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{
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"tactic": "exact (lt_iff_not_ge _ _).mp h₂ (natDegree_le_of_dvd h₁ hc)"
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] | [
188,
62
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185,
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Mathlib/GroupTheory/Submonoid/Pointwise.lean | Submonoid.pointwise_smul_le_iff₀ | [] | [
334,
26
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333,
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Mathlib/Algebra/Group/Units.lean | Units.mul_eq_one_iff_inv_eq | [
{
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395,
99
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395,
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Mathlib/Analysis/NormedSpace/LinearIsometry.lean | SemilinearIsometryClass.nnnorm_map | [] | [
97,
28
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96,
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Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.sum_centroidWeights_eq_one_of_card_eq_add_one | [] | [
834,
79
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832,
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Mathlib/RingTheory/Subring/Basic.lean | Subring.mem_top | [] | [
550,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
549,
1
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Mathlib/Topology/UniformSpace/Basic.lean | comp3_mem_uniformity | [] | [
571,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
568,
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Mathlib/Data/Real/ENNReal.lean | ENNReal.mul_div_cancel' | [
{
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1396,
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1395,
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Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.completeTheory.isComplete | [] | [
513,
44
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512,
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Mathlib/SetTheory/Ordinal/Exponential.lean | Ordinal.log_mono_right | [
{
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},
{
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364,
67
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358,
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Mathlib/Data/MvPolynomial/Division.lean | MvPolynomial.divMonomial_add_modMonomial_single | [] | [
205,
34
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203,
1
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Mathlib/Order/Hom/Lattice.lean | LatticeHom.coe_comp_sup_hom' | [] | [
1120,
6
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1118,
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Mathlib/Data/Finset/Basic.lean | Finset.mem_union_left | [] | [
1350,
26
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1349,
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Mathlib/Logic/Lemmas.lean | ite_ite_distrib_left | [] | [
67,
25
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66,
1
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Mathlib/Analysis/Complex/Basic.lean | Complex.dist_eq_re_im | [
{
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},
{
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] | [
97,
6
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Mathlib/LinearAlgebra/Pi.lean | LinearMap.pi_ext' | [
{
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{
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192,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
190,
1
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Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | BilinForm.nondegenerate_of_anisotropic | [] | [
1060,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1059,
1
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Mathlib/Algebra/Associated.lean | IsSquare.not_irreducible | [] | [
305,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
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Mathlib/Algebra/Quaternion.lean | Quaternion.rank_eq_four | [] | [
1086,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1085,
1
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Mathlib/Logic/Basic.lean | heq_of_cast_eq | [] | [
552,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
551,
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Mathlib/Algebra/BigOperators/Ring.lean | Finset.sum_range_succ_mul_sum_range_succ | [
{
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"tactic": "simp only [add_mul, mul_add, add_assoc, sum_range_succ]"
}
] | [
277,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
272,
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Mathlib/RingTheory/Algebraic.lean | Algebra.isAlgebraic_of_larger_base | [] | [
250,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
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Mathlib/Algebra/DirectSum/Ring.lean | DirectSum.of_zero_one | [] | [
428,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
427,
1
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Mathlib/Data/Polynomial/Derivative.lean | Polynomial.iterate_derivative_zero | [
{
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"tactic": "induction' k with k ih"
},
{
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{
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] | [
87,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
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Mathlib/Algebra/Star/Pointwise.lean | Set.star_subset_star | [] | [
107,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
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Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | fderivWithin_cosh | [] | [
1092,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1090,
1
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Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_toAdd | [] | [
1460,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1459,
1
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Mathlib/LinearAlgebra/Dual.lean | Subspace.dualRestrict_leftInverse | [
{
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"tactic": "rw [dualRestrict_comp_dualLift]"
},
{
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}
] | [
1039,
8
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1035,
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Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.le_div_iff_mul_le | [
{
"state_after": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ I * J' ≤ J",
"state_before": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ J / J' ↔ I * J' ≤ J",
"tactic": "rw [div_nonzero hJ']"
},
{
"state_after": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ ↑I * ↑J' ≤ ↑J",
"state_before": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ I * J' ≤ J",
"tactic": "rw [← coe_le_coe (I := I * J') (J := J), coe_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1213100\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1213307\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI✝ J✝ I J J' : FractionalIdeal R₁⁰ K\nhJ' : J' ≠ 0\n⊢ I ≤ { val := ↑J / ↑J', property := (_ : IsFractional R₁⁰ (↑J / ↑J')) } ↔ ↑I * ↑J' ≤ ↑J",
"tactic": "exact Submodule.le_div_iff_mul_le"
}
] | [
1150,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1145,
1
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Mathlib/Topology/Sets/Compacts.lean | TopologicalSpace.PositiveCompacts.nonempty | [] | [
341,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
11
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Mathlib/Data/Real/Basic.lean | Real.lt_cauchy | [
{
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"state_before": "x y : ℝ\nf g : CauSeq ℚ abs\n⊢ Real.lt { cauchy := Quotient.mk equiv f } { cauchy := Quotient.mk equiv g } ↔ f < g",
"tactic": "rw [lt_def]"
},
{
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320,
38
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Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | GeneralizedContinuedFraction.of_terminatedAt_n_iff_succ_nth_intFractPair_stream_eq_none | [
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215,
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Mathlib/Data/Real/Irrational.lean | Irrational.div_int | [
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Mathlib/Topology/Basic.lean | eventually_eventuallyEq_nhds | [] | [
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Mathlib/Data/Matrix/Kronecker.lean | Matrix.kroneckerMap_diagonal_diagonal | [
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},
{
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132,
83
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Mathlib/Data/Finite/Defs.lean | finite_iff_exists_equiv_fin | [] | [
66,
45
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Mathlib/Topology/Order/Hom/Esakia.lean | PseudoEpimorphism.coe_comp | [] | [
184,
38
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183,
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Mathlib/Data/Set/Basic.lean | Set.mem_symmDiff | [] | [
2093,
10
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2092,
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Mathlib/Data/Pi/Algebra.lean | Pi.const_pow | [] | [
135,
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134,
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Mathlib/Data/Finset/LocallyFinite.lean | Finset.Icc_subset_Icc_right | [] | [
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Mathlib/Data/Real/EReal.lean | EReal.coe_ennreal_eq_coe_ennreal_iff | [] | [
512,
31
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Mathlib/FieldTheory/PerfectClosure.lean | PerfectClosure.eq_iff | [
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Mathlib/RingTheory/Ideal/QuotientOperations.lean | DoubleQuot.quotQuotEquivCommₐ_toRingEquiv | [] | [
791,
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Mathlib/GroupTheory/Perm/Support.lean | Equiv.Perm.ofSubtype_swap_eq | [
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"tactic": "split_ifs with hzx hzy"
},
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},
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"tactic": "rw [swap_apply_of_ne_of_ne] <;>\nsimp [Subtype.ext_iff, *]"
},
{
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},
{
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{
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{
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{
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{
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"tactic": "exact Subtype.prop y"
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] | [
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27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
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Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.cyclotomic_zero | [
{
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"tactic": "simp only [cyclotomic, dif_pos]"
}
] | [
308,
34
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Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometry.toAffineMap_injective | [
{
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Mathlib/GroupTheory/FreeAbelianGroup.lean | FreeAbelianGroup.lift_neg' | [] | [
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74
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187,
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Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_floor_atBot | [] | [
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68
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Mathlib/Order/SuccPred/Basic.lean | Pred.rec_iff | [] | [
1428,
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1426,
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Mathlib/GroupTheory/PGroup.lean | IsPGroup.bot_lt_center | [
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264,
96
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Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.constantCoeff_one | [] | [
1557,
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Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.toSubalgebra_eq_iff | [
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Mathlib/Order/Ideal.lean | Order.Ideal.mem_compl_of_ge | [] | [
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Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.list_prod_mem | [] | [
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Mathlib/CategoryTheory/EqToHom.lean | CategoryTheory.Functor.hcongr_hom | [
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259,
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Mathlib/Logic/Equiv/Defs.lean | Equiv.forall_congr_left | [] | [
916,
41
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Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | HasSum.map | [] | [
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34
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Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | ContDiffAt.exp | [] | [
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Mathlib/Algebra/Group/Conj.lean | conj_zpow | [
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{
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"tactic": "simp [zpow_ofNat]"
},
{
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"tactic": "simp [zpow_negSucc, conj_pow]"
},
{
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Mathlib/Algebra/BigOperators/Finprod.lean | finprod_dmem | [] | [
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67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/Nat/EvenOddRec.lean | Nat.evenOddRec_zero | [] | [
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Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_const_preimage_eq_union | [
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{
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"tactic": "split_ifs <;> simp [← compl_eq_univ_diff]"
}
] | [
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Mathlib/ModelTheory/Definability.lean | Set.empty_definable_iff | [
{
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"tactic": "rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]"
},
{
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"tactic": "simp [-constantsOn]"
}
] | [
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Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.closure_withConstants_eq | [
{
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"tactic": "refine' closure_eq_of_le ((A.subset_union_right s).trans subset_closure) _"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ ↑(LHom.substructureReduct (lhomWithConstants L ↑A))\n (withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s)))) ≤\n ↑(LHom.substructureReduct (lhomWithConstants L ↑A)) (LowerAdjoint.toFun (closure (L[[↑A]])) s)",
"state_before": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s))) ≤\n LowerAdjoint.toFun (closure (L[[↑A]])) s",
"tactic": "rw [← (L.lhomWithConstants A).substructureReduct.le_iff_le]"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ A ⊆ ↑(LowerAdjoint.toFun (closure (L[[↑A]])) s)",
"state_before": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ ↑(LHom.substructureReduct (lhomWithConstants L ↑A))\n (withConstants (LowerAdjoint.toFun (closure L) (A ∪ s)) (_ : A ⊆ ↑(LowerAdjoint.toFun (closure L) (A ∪ s)))) ≤\n ↑(LHom.substructureReduct (lhomWithConstants L ↑A)) (LowerAdjoint.toFun (closure (L[[↑A]])) s)",
"tactic": "simp only [subset_closure, reduct_withConstants, closure_le, LHom.coe_substructureReduct,\n Set.union_subset_iff, and_true_iff]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\nN : Type ?u.683008\nP : Type ?u.683011\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ A ⊆ ↑(LowerAdjoint.toFun (closure (L[[↑A]])) s)",
"tactic": "exact subset_closure_withConstants"
}
] | [
784,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
777,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.wf_subsequent | [] | [
260,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
259,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ioo_inter_Ioo | [
{
"state_after": "α : Type u_1\nβ : Type ?u.187524\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioi a₁ ∩ Iio b₁ ∩ (Ioi a₂ ∩ Iio b₂) = Ioi a₁ ∩ Ioi a₂ ∩ (Iio b₁ ∩ Iio b₂)",
"state_before": "α : Type u_1\nβ : Type ?u.187524\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Ioo a₁ b₁ ∩ Ioo a₂ b₂ = Ioo (a₁ ⊔ a₂) (b₁ ⊓ b₂)",
"tactic": "simp only [Ioi_inter_Iio.symm, Ioi_inter_Ioi.symm, Iio_inter_Iio.symm]"
},
{
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"tactic": "ac_rfl"
}
] | [
1792,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1791,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean | FractionalIdeal.mul_inv_cancel_of_le_one | [
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "R : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "by_cases hI1 : I = ⊤"
},
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "by_cases hNF : IsField A"
},
{
"state_after": "case neg.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=\n le_one_iff_exists_coeIdeal.mp mul_one_div_le_one"
},
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "case neg.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "by_cases hJ0 : J = ⊥"
},
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1\n\ncase neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "by_cases hJ1 : J = ⊤"
},
{
"state_after": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nhx1 : ¬x ∈ 1\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "case neg\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "obtain ⟨x, hx, hx1⟩ :\n ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=\n exists_not_mem_one_of_ne_bot hNF hJ0 hJ1"
},
{
"state_after": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1",
"state_before": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nhx1 : ¬x ∈ 1\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "contrapose! hx1 with h_abs"
},
{
"state_after": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑I * (↑I)⁻¹)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1",
"state_before": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑J)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1",
"tactic": "rw [hJ] at hx"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : ¬J = ⊤\nx : K\nhx : x ∈ (↑I * (↑I)⁻¹)⁻¹\nh_abs : ↑I * (↑I)⁻¹ ≠ 1\n⊢ x ∈ 1",
"tactic": "exact hI hx"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : I = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "rw [hI1, coeIdeal_top, one_mul, inv_one]"
},
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\nthis : Field A := IsField.toField hNF\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "letI := hNF.toField"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : IsField A\nthis : Field A := IsField.toField hNF\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩"
},
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : J = ⊥\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "subst hJ0"
},
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ I = ⊥",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "refine' absurd _ hI0"
},
{
"state_after": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I ≤ ↑I * (↑I)⁻¹",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ I = ⊥",
"tactic": "rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nhJ : ↑⊥ = ↑I * (↑I)⁻¹\n⊢ ↑I ≤ ↑I * (↑I)⁻¹",
"tactic": "exact coe_ideal_le_self_mul_inv K I"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.278737\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nh : IsDedekindDomain A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1\nhI1 : ¬I = ⊤\nhNF : ¬IsField A\nJ : Ideal A\nhJ : ↑J = ↑I * (↑I)⁻¹\nhJ0 : ¬J = ⊥\nhJ1 : J = ⊤\n⊢ ↑I * (↑I)⁻¹ = 1",
"tactic": "rw [← hJ, hJ1, coeIdeal_top]"
}
] | [
485,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
462,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.degree_pow_le | [
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\n⊢ degree 1 ≤ 0",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\n⊢ degree (p ^ 0) ≤ 0 • degree p",
"tactic": "rw [pow_zero, zero_nsmul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\n⊢ degree 1 ≤ 0",
"tactic": "exact degree_one_le"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree (p * p ^ n) ≤ degree p + degree (p ^ n)",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree (p ^ (n + 1)) ≤ degree p + degree (p ^ n)",
"tactic": "rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree (p * p ^ n) ≤ degree p + degree (p ^ n)",
"tactic": "exact degree_mul_le _ _"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree p + degree (p ^ n) ≤ degree p + n • degree p",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree p + degree (p ^ n) ≤ (n + 1) • degree p",
"tactic": "rw [succ_nsmul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.598995\np : R[X]\nn : ℕ\n⊢ degree p + degree (p ^ n) ≤ degree p + n • degree p",
"tactic": "exact add_le_add le_rfl (degree_pow_le _ _)"
}
] | [
791,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
785,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean | Complex.UnitDisc.conj_zero | [] | [
223,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.aleph'_succ | [
{
"state_after": "o : Ordinal\n⊢ alephIdx (aleph' (succ o)) ≤ alephIdx (succ (aleph' o))",
"state_before": "o : Ordinal\n⊢ aleph' (succ o) = succ (aleph' o)",
"tactic": "apply (succ_le_of_lt <| aleph'_lt.2 <| lt_succ o).antisymm' (Cardinal.alephIdx_le.1 <| _)"
},
{
"state_after": "o : Ordinal\n⊢ aleph' o < succ (aleph' o)",
"state_before": "o : Ordinal\n⊢ alephIdx (aleph' (succ o)) ≤ alephIdx (succ (aleph' o))",
"tactic": "rw [alephIdx_aleph', succ_le_iff, ← aleph'_lt, aleph'_alephIdx]"
},
{
"state_after": "no goals",
"state_before": "o : Ordinal\n⊢ aleph' o < succ (aleph' o)",
"tactic": "apply lt_succ"
}
] | [
206,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
] |
Std/Data/List/Lemmas.lean | List.diff_subset | [] | [
1533,
83
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1533,
1
] |
Mathlib/Deprecated/Subgroup.lean | Group.normalClosure.isSubgroup | [] | [
713,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
712,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean | Polynomial.div_def | [] | [
195,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/GroupTheory/Nilpotent.lean | mem_lowerCentralSeries_succ_iff | [] | [
301,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
298,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.sum_div_factorial_le | [
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ n ≤ m",
"tactic": "simp at hm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m",
"tactic": "tauto"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ m < j",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ m < j",
"tactic": "simp at hm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ m < j",
"tactic": "tauto"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ n ≤ m",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ 1 / ↑(Nat.factorial m) = 1 / ↑(Nat.factorial ((fun m x => m - n) m hm + n))",
"tactic": "rw [tsub_add_cancel_of_le]"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m ∈ filter (fun k => n ≤ k) (range j)\n⊢ n ≤ m",
"tactic": "simp at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nm : ℕ\nhm : m < j ∧ n ≤ m\n⊢ n ≤ m",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\na₁ a₂ : ℕ\nha₁ : a₁ ∈ filter (fun k => n ≤ k) (range j)\nha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)\nh : (fun m x => m - n) a₁ ha₁ = (fun m x => m - n) a₂ ha₂\n⊢ a₁ = a₂",
"tactic": "rwa [tsub_eq_iff_eq_add_of_le, tsub_add_eq_add_tsub, eq_comm, tsub_eq_iff_eq_add_of_le,\n add_left_inj, eq_comm] at h <;>\nsimp at * <;> aesop"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nb : ℕ\nhb : b ∈ range (j - n)\n⊢ b = b + n - n",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nb : ℕ\nhb : b ∈ range (j - n)\n⊢ b = (fun m x => m - n) (b + n) (_ : b + n ∈ filter (fun k => n ≤ k) (range j))",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nb : ℕ\nhb : b ∈ range (j - n)\n⊢ b = b + n - n",
"tactic": "rw [add_tsub_cancel_right]"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ x in range (j - n), (↑(Nat.factorial (x + n)))⁻¹ ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ m in range (j - n), 1 / ↑(Nat.factorial (m + n)) ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹",
"tactic": "simp_rw [one_div]"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ ↑(Nat.factorial n) * ↑(Nat.succ n) ^ i✝ ≤ ↑(Nat.factorial (i✝ + n))",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ x in range (j - n), (↑(Nat.factorial (x + n)))⁻¹ ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹",
"tactic": "gcongr"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ Nat.factorial n * Nat.succ n ^ i✝ ≤ Nat.factorial (n + i✝)",
"state_before": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ ↑(Nat.factorial n) * ↑(Nat.succ n) ^ i✝ ≤ ↑(Nat.factorial (i✝ + n))",
"tactic": "rw [← Nat.cast_pow, ← Nat.cast_mul, Nat.cast_le, add_comm]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\ni✝ : ℕ\na✝ : i✝ ∈ range (j - n)\n⊢ Nat.factorial n * Nat.succ n ^ i✝ ≤ Nat.factorial (n + i✝)",
"tactic": "exact Nat.factorial_mul_pow_le_factorial"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹ =\n (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m",
"tactic": "simp [mul_inv, mul_sum.symm, sum_mul.symm, -Nat.factorial_succ, mul_comm, inv_pow]"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"tactic": "have h₁ : (n.succ : α) ≠ 1 :=\n @Nat.cast_one α _ ▸ mt Nat.cast_inj.1 (mt Nat.succ.inj (pos_iff_ne_zero.1 hn))"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"tactic": "have h₂ : (n.succ : α) ≠ 0 := by positivity"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"tactic": "have h₃ : (n.factorial * n : α) ≠ 0 := by positivity"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\nh₄ : ↑(Nat.succ n) - 1 = ↑n\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nn j : ℕ\nhn : 0 < n\nh₁ : ↑(Nat.succ n) ≠ 1\nh₂ : ↑(Nat.succ n) ≠ 0\nh₃ : ↑(Nat.factorial n) * ↑n ≠ 0\n⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =\n (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑n)",
"tactic": "have h₄ : (n.succ - 1 : α) = n := by simp"
},
{
"state_after": "no goals",
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"tactic": "rw [geom_sum_inv h₁ h₂, eq_div_iff_mul_eq h₃, mul_comm _ (n.factorial * n : α),\n ← mul_assoc (n.factorial⁻¹ : α), ← mul_inv_rev, h₄, ← mul_assoc (n.factorial * n : α),\n mul_comm (n : α) n.factorial, mul_inv_cancel h₃, one_mul, mul_comm]"
},
{
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{
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{
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{
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{
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{
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1608,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1573,
1
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Mathlib/Analysis/Convex/Hull.lean | mem_convexHull_iff | [
{
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}
] | [
72,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Algebra/Module/LocalizedModule.lean | IsLocalizedModule.mk'_cancel_right | [
{
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{
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"tactic": "rw [LocalizedModule.mk_cancel_common_right]"
}
] | [
969,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Analysis/InnerProductSpace/Basic.lean | Continuous.inner | [] | [
2283,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2282,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_withDensity_eq_lintegral_mul₀' | [
{
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"tactic": "let f' := hf.mk f"
},
{
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"tactic": "have : μ.withDensity f = μ.withDensity f' := withDensity_congr_ae hf.ae_eq_mk"
},
{
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"tactic": "rw [this] at hg⊢"
},
{
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"tactic": "let g' := hg.mk g"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (fun a => (f' * g') a) =ᵐ[μ] fun a => (f' * g) a",
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},
{
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"tactic": "apply ae_of_ae_restrict_of_ae_restrict_compl { x | f' x ≠ 0 }"
},
{
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"tactic": "have Z := hg.ae_eq_mk"
},
{
"state_after": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x",
"state_before": "case h.ht\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : g =ᵐ[Measure.withDensity μ f'] AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ {x | f' x ≠ 0}, (fun a => (f' * g') a) x = (fun a => (f' * g) a) x",
"tactic": "rw [EventuallyEq, ae_withDensity_iff_ae_restrict hf.measurable_mk] at Z"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\n⊢ ∀ (a : α),\n g a = AEMeasurable.mk g hg a → (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a",
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"tactic": "filter_upwards [Z]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\nx : α\nhx : g x = AEMeasurable.mk g hg x\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nZ : ∀ᵐ (x : α) ∂Measure.restrict μ {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x\n⊢ ∀ (a : α),\n g a = AEMeasurable.mk g hg a → (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a",
"tactic": "intro x hx"
},
{
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"tactic": "simp only [hx, Pi.mul_apply]"
},
{
"state_after": "case h.htc\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ ({x | f' x ≠ 0}ᶜ), (fun a => (f' * g') a) x = (fun a => (f' * g) a) x",
"state_before": "case h.htc\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ ({x | f' x ≠ 0}ᶜ), (fun a => (f' * g') a) x = (fun a => (f' * g) a) x",
"tactic": "have M : MeasurableSet ({ x : α | f' x ≠ 0 }ᶜ) :=\n (hf.measurable_mk (measurableSet_singleton 0).compl).compl"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ (a : α),\n a ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ →\n (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a",
"state_before": "case h.htc\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ᵐ (x : α) ∂Measure.restrict μ ({x | f' x ≠ 0}ᶜ), (fun a => (f' * g') a) x = (fun a => (f' * g) a) x",
"tactic": "filter_upwards [ae_restrict_mem M]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : x ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\n⊢ ∀ (a : α),\n a ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ →\n (AEMeasurable.mk f hf * AEMeasurable.mk g hg) a = (AEMeasurable.mk f hf * g) a",
"tactic": "intro x hx"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : AEMeasurable.mk f hf x = 0\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : x ∈ {x | AEMeasurable.mk f hf x ≠ 0}ᶜ\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x",
"tactic": "simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nM : MeasurableSet ({x | f' x ≠ 0}ᶜ)\nx : α\nhx : AEMeasurable.mk f hf x = 0\n⊢ (AEMeasurable.mk f hf * AEMeasurable.mk g hg) x = (AEMeasurable.mk f hf * g) x",
"tactic": "simp only [hx, MulZeroClass.zero_mul, Pi.mul_apply]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (fun a => (f' * g) a) =ᵐ[μ] fun a => (f * g) a",
"state_before": "α : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (∫⁻ (a : α), (f' * g) a ∂μ) = ∫⁻ (a : α), (f * g) a ∂μ",
"tactic": "apply lintegral_congr_ae"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ (a : α), f a = AEMeasurable.mk f hf a → (AEMeasurable.mk f hf * g) a = (f * g) a",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ (fun a => (f' * g) a) =ᵐ[μ] fun a => (f * g) a",
"tactic": "filter_upwards [hf.ae_eq_mk]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nx : α\nhx : f x = AEMeasurable.mk f hf x\n⊢ (AEMeasurable.mk f hf * g) x = (f * g) x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\n⊢ ∀ (a : α), f a = AEMeasurable.mk f hf a → (AEMeasurable.mk f hf * g) a = (f * g) a",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1801681\nγ : Type ?u.1801684\nδ : Type ?u.1801687\nm m0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g\nthis : Measure.withDensity μ f = Measure.withDensity μ f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nx : α\nhx : f x = AEMeasurable.mk f hf x\n⊢ (AEMeasurable.mk f hf * g) x = (f * g) x",
"tactic": "simp only [hx, Pi.mul_apply]"
}
] | [
1817,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1788,
1
] |
Mathlib/Control/LawfulFix.lean | Part.Fix.le_f_of_mem_approx | [
{
"state_after": "α : Type u_2\nβ : α → Type u_1\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\ni : ℕ\n⊢ ↑(approxChain f) i ≤ ↑f (↑(approxChain f) i)",
"state_before": "α : Type u_2\nβ : α → Type u_1\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\nx : (a : α) → Part (β a)\n⊢ ∀ (x_1 : ℕ), x = ↑(approxChain f) x_1 → x ≤ ↑f x",
"tactic": "rintro i rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : α → Type u_1\nf : ((a : α) → Part (β a)) →o (a : α) → Part (β a)\ni : ℕ\n⊢ ↑(approxChain f) i ≤ ↑f (↑(approxChain f) i)",
"tactic": "apply approx_mono'"
}
] | [
126,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | Complex.tan_add' | [] | [
124,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | eq_affineCombination_of_mem_affineSpan_of_fintype | [
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np1 : P\np : ι → P\nh : p1 ∈ affineSpan k (Set.range p)\n⊢ ∃ w x, p1 = ↑(Finset.affineCombination k Finset.univ p) w",
"tactic": "classical\n obtain ⟨s, w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan h\n refine'\n ⟨(s : Set ι).indicator w, _, Finset.affineCombination_indicator_subset w p s.subset_univ⟩\n simp only [Finset.mem_coe, Set.indicator_apply, ← hw]\n rw [Fintype.sum_extend_by_zero s w]"
},
{
"state_after": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∃ w_1 x, ↑(Finset.affineCombination k s p) w = ↑(Finset.affineCombination k Finset.univ p) w_1",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np1 : P\np : ι → P\nh : p1 ∈ affineSpan k (Set.range p)\n⊢ ∃ w x, p1 = ↑(Finset.affineCombination k Finset.univ p) w",
"tactic": "obtain ⟨s, w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan h"
},
{
"state_after": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∑ i : ι, Set.indicator (↑s) w i = 1",
"state_before": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∃ w_1 x, ↑(Finset.affineCombination k s p) w = ↑(Finset.affineCombination k Finset.univ p) w_1",
"tactic": "refine'\n ⟨(s : Set ι).indicator w, _, Finset.affineCombination_indicator_subset w p s.subset_univ⟩"
},
{
"state_after": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ (∑ x : ι, if x ∈ s then w x else 0) = ∑ i in s, w i",
"state_before": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ ∑ i : ι, Set.indicator (↑s) w i = 1",
"tactic": "simp only [Finset.mem_coe, Set.indicator_apply, ← hw]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i in s, w i = 1\nh : ↑(Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)\n⊢ (∑ x : ι, if x ∈ s then w x else 0) = ∑ i in s, w i",
"tactic": "rw [Fintype.sum_extend_by_zero s w]"
}
] | [
1136,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1128,
1
] |
Mathlib/LinearAlgebra/Coevaluation.lean | coevaluation_apply_one | [
{
"state_after": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ↑(↑(Basis.constr (Basis.singleton Unit K) K) fun x =>\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x)\n 1 =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x",
"state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ↑(coevaluation K V) 1 =\n let bV := Basis.ofVectorSpace K V;\n ∑ i : ↑(Basis.ofVectorSpaceIndex K V), ↑bV i ⊗ₜ[K] Basis.coord bV i",
"tactic": "simp only [coevaluation, id]"
},
{
"state_after": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ∑ i : Unit,\n ↑(Basis.equivFun (Basis.singleton Unit K)) 1 i •\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x",
"state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ↑(↑(Basis.constr (Basis.singleton Unit K) K) fun x =>\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x)\n 1 =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x",
"tactic": "rw [(Basis.singleton Unit K).constr_apply_fintype K]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝³ : Field K\nV : Type v\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ ∑ i : Unit,\n ↑(Basis.equivFun (Basis.singleton Unit K)) 1 i •\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V),\n ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x =\n ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ↑(Basis.ofVectorSpace K V) x ⊗ₜ[K] Basis.coord (Basis.ofVectorSpace K V) x",
"tactic": "simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,\n Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]"
}
] | [
58,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
] |
Mathlib/Order/Compare.lean | lt_iff_lt_of_cmp_eq_cmp | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ✝ : Type ?u.20071\ninst✝¹ : LinearOrder α\nx y : α\nβ : Type u_2\ninst✝ : LinearOrder β\nx' y' : β\nh : cmp x y = cmp x' y'\n⊢ x < y ↔ x' < y'",
"tactic": "rw [← cmp_eq_lt_iff, ← cmp_eq_lt_iff, h]"
}
] | [
255,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/Order/Disjoint.lean | disjoint_top | [] | [
110,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Iio_inj | [] | [
1128,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1127,
1
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.embDomain_injective | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.227043\nι : Type ?u.227046\nM : Type u_3\nM' : Type ?u.227052\nN : Type ?u.227055\nP : Type ?u.227058\nG : Type ?u.227061\nH : Type ?u.227064\nR : Type ?u.227067\nS : Type ?u.227070\ninst✝¹ : Zero M\ninst✝ : Zero N\nf : α ↪ β\nl₁ l₂ : α →₀ M\nh : embDomain f l₁ = embDomain f l₂\na : α\n⊢ ↑l₁ a = ↑l₂ a",
"tactic": "simpa only [embDomain_apply] using FunLike.ext_iff.1 h (f a)"
}
] | [
872,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
871,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean | Multiset.inter_le_ndinter | [] | [
273,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
272,
1
] |
Mathlib/Algebra/Associated.lean | Associates.mk_mul_mk | [] | [
819,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
818,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoMod_zsmul_add' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b",
"tactic": "rw [add_comm, toIcoMod_add_zsmul', add_comm]"
}
] | [
436,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
434,
1
] |
Mathlib/Data/TypeVec.lean | TypeVec.dropFun_toSubtype | [
{
"state_after": "case a.h\nn : ℕ\nα : TypeVec (n + 1)\np : α ⟹ repeat (n + 1) Prop\ni : Fin2 n\nx✝ : drop (fun i => { x // ofRepeat (p i x) }) i\n⊢ dropFun (toSubtype p) i x✝ = toSubtype (fun i x => p (Fin2.fs i) x) i x✝",
"state_before": "n : ℕ\nα : TypeVec (n + 1)\np : α ⟹ repeat (n + 1) Prop\n⊢ dropFun (toSubtype p) = toSubtype fun i x => p (Fin2.fs i) x",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case a.h\nn : ℕ\nα : TypeVec (n + 1)\np : α ⟹ repeat (n + 1) Prop\ni : Fin2 n\nx✝ : drop (fun i => { x // ofRepeat (p i x) }) i\n⊢ dropFun (toSubtype p) i x✝ = toSubtype (fun i x => p (Fin2.fs i) x) i x✝",
"tactic": "induction i <;> simp [dropFun, *] <;> rfl"
}
] | [
709,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
706,
1
] |