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/Analysis/Asymptotics/Asymptotics.lean
Asymptotics.isLittleO_iff_nat_mul_le
[]
[ 266, 64 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 265, 1 ]
Mathlib/Analysis/Convex/Combination.lean
Finset.centroid_eq_centerMass
[]
[ 256, 82 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 254, 1 ]
Mathlib/Algebra/AddTorsor.lean
Prod.snd_vadd
[]
[ 311, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 310, 1 ]
Mathlib/Computability/Ackermann.lean
ack_three
[ { "state_after": "case zero\n\n⊢ ack 3 zero = 2 ^ (zero + 3) - 3\n\ncase succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ ack 3 (succ n) = 2 ^ (succ n + 3) - 3", "state_before": "n : ℕ\n⊢ ack 3 n = 2 ^ (n + 3) - 3", "tactic": "induction' n with n IH" }, { "state_after": "no goals", "state_before": "case zero\n\n⊢ ack 3 zero = 2 ^ (zero + 3) - 3", "tactic": "rfl" }, { "state_after": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)", "state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ ack 3 (succ n) = 2 ^ (succ n + 3) - 3", "tactic": "rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,\n Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]" }, { "state_after": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)", "state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)", "tactic": "have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_num" }, { "state_after": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 2 ^ 3 ≤ 2 * 2 ^ (n + 3)", "state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)", "tactic": "apply H.trans" }, { "state_after": "no goals", "state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 2 ^ 3 ≤ 2 * 2 ^ (n + 3)", "tactic": "simp [pow_le_pow]" }, { "state_after": "no goals", "state_before": "n : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ 2 * 3 ≤ 2 * 2 ^ 3", "tactic": "norm_num" } ]
[ 106, 22 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 99, 1 ]
Mathlib/NumberTheory/Padics/PadicIntegers.lean
PadicInt.p_nonnunit
[ { "state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nthis : (↑p)⁻¹ < 1\n⊢ ↑p ∈ nonunits ℤ_[p]", "state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ↑p ∈ nonunits ℤ_[p]", "tactic": "have : (p : ℝ)⁻¹ < 1 := inv_lt_one <| by exact_mod_cast hp.1.one_lt" }, { "state_after": "no goals", "state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nthis : (↑p)⁻¹ < 1\n⊢ ↑p ∈ nonunits ℤ_[p]", "tactic": "rwa [← norm_p, ← mem_nonunits] at this" }, { "state_after": "no goals", "state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ 1 < ↑p", "tactic": "exact_mod_cast hp.1.one_lt" } ]
[ 596, 41 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 594, 1 ]
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
Matrix.separatingLeft_toLinearMap₂_iff
[ { "state_after": "no goals", "state_before": "R : Type ?u.2748126\nR₁ : Type u_2\nR₂ : Type ?u.2748132\nM✝ : Type ?u.2748135\nM₁ : Type u_3\nM₂ : Type ?u.2748141\nM₁' : Type ?u.2748144\nM₂' : Type ?u.2748147\nn : Type ?u.2748150\nm : Type ?u.2748153\nn' : Type ?u.2748156\nm' : Type ?u.2748159\nι : Type u_1\ninst✝⁴ : CommRing R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nB : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁\nM : Matrix ι ι R₁\nb : Basis ι R₁ M₁\n⊢ SeparatingLeft (↑(toLinearMap₂ b b) M) ↔ Matrix.Nondegenerate M", "tactic": "rw [← Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂,\n Matrix.separatingLeft_toLinearMap₂'_iff]" } ]
[ 712, 45 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 709, 1 ]
Mathlib/Topology/Algebra/Module/Basic.lean
ContinuousLinearMap.coe_smul
[]
[ 592, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 590, 1 ]
Mathlib/Init/Algebra/Order.lean
le_refl
[]
[ 55, 19 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 54, 1 ]
Mathlib/Topology/Order/Hom/Esakia.lean
EsakiaHom.comp_id
[]
[ 342, 19 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 341, 1 ]
Mathlib/Deprecated/Group.lean
IsMonoidHom.map_mul'
[]
[ 173, 17 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 172, 1 ]
Mathlib/SetTheory/ZFC/Basic.lean
ZFSet.toSet_range
[ { "state_after": "case h\nα : Type u\nf : α → ZFSet\nx✝ : ZFSet\n⊢ x✝ ∈ toSet (range f) ↔ x✝ ∈ Set.range f", "state_before": "α : Type u\nf : α → ZFSet\n⊢ toSet (range f) = Set.range f", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h\nα : Type u\nf : α → ZFSet\nx✝ : ZFSet\n⊢ x✝ ∈ toSet (range f) ↔ x✝ ∈ Set.range f", "tactic": "simp" } ]
[ 1282, 7 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1279, 1 ]
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae
[]
[ 468, 99 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 464, 1 ]
Mathlib/Topology/Instances/ENNReal.lean
ENNReal.tsum_mono_subtype
[]
[ 996, 59 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 994, 1 ]
Mathlib/RingTheory/DedekindDomain/Ideal.lean
FractionalIdeal.spanSingleton_div_self
[ { "state_after": "no goals", "state_before": "R : Type ?u.45600\nA : Type ?u.45603\nK : Type u_1\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing A\ninst✝⁸ : Field K\ninst✝⁷ : IsDomain A\nR₁ : Type u_2\ninst✝⁶ : CommRing R₁\ninst✝⁵ : IsDomain R₁\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nI J : FractionalIdeal R₁⁰ K\nK' : Type ?u.46614\ninst✝² : Field K'\ninst✝¹ : Algebra R₁ K'\ninst✝ : IsFractionRing R₁ K'\nx : K\nhx : x ≠ 0\n⊢ spanSingleton R₁⁰ x / spanSingleton R₁⁰ x = 1", "tactic": "rw [spanSingleton_div_spanSingleton, div_self hx, spanSingleton_one]" } ]
[ 164, 71 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 162, 1 ]
Mathlib/Topology/Basic.lean
tendsto_atTop_nhds
[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝² : TopologicalSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nf : β → α\na : α\n⊢ (∀ (ib : Set α), a ∈ ib ∧ IsOpen ib → ∃ ia, True ∧ ∀ (x : β), x ∈ Ici ia → f x ∈ ib) ↔\n ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : β), N ≤ n → f n ∈ U", "tactic": "simp only [and_imp, exists_prop, true_and_iff, mem_Ici, ge_iff_le]" } ]
[ 1033, 71 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1030, 1 ]
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
hasStrictFDerivAt_pi'
[ { "state_after": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type ?u.395005\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type ?u.395100\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type ?u.395195\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : 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\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\nφ' : (i : ι) → E →L[𝕜] F' i\nΦ : E → (i : ι) → F' i\nΦ' : E →L[𝕜] (i : ι) → F' i\n⊢ ((fun p => Φ p.fst - Φ p.snd - ↑Φ' (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd) ↔\n ∀ (i : ι),\n (fun p => Φ p.fst i - Φ p.snd i - ↑(comp (proj i) Φ') (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd", "state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type ?u.395005\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type ?u.395100\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type ?u.395195\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : 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\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\nφ' : (i : ι) → E →L[𝕜] F' i\nΦ : E → (i : ι) → F' i\nΦ' : E →L[𝕜] (i : ι) → F' i\n⊢ HasStrictFDerivAt Φ Φ' x ↔ ∀ (i : ι), HasStrictFDerivAt (fun x => Φ x i) (comp (proj i) Φ') x", "tactic": "simp only [HasStrictFDerivAt, ContinuousLinearMap.coe_pi]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type ?u.395005\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type ?u.395100\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type ?u.395195\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : 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\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\nφ' : (i : ι) → E →L[𝕜] F' i\nΦ : E → (i : ι) → F' i\nΦ' : E →L[𝕜] (i : ι) → F' i\n⊢ ((fun p => Φ p.fst - Φ p.snd - ↑Φ' (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd) ↔\n ∀ (i : ι),\n (fun p => Φ p.fst i - Φ p.snd i - ↑(comp (proj i) Φ') (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd", "tactic": "exact isLittleO_pi" } ]
[ 383, 21 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 380, 1 ]
Mathlib/CategoryTheory/Subobject/Lattice.lean
CategoryTheory.Subobject.inf_eq_map_pullback'
[ { "state_after": "case h\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA : C\nf₁ f₂ : MonoOver A\n⊢ (inf.obj (Quotient.mk'' f₁)).obj (Quotient.mk'' f₂) =\n (map (MonoOver.arrow f₁)).obj ((pullback (MonoOver.arrow f₁)).obj (Quotient.mk'' f₂))", "state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA : C\nf₁ : MonoOver A\nf₂ : Subobject A\n⊢ (inf.obj (Quotient.mk'' f₁)).obj f₂ = (map (MonoOver.arrow f₁)).obj ((pullback (MonoOver.arrow f₁)).obj f₂)", "tactic": "induction' f₂ using Quotient.inductionOn' with f₂" }, { "state_after": "no goals", "state_before": "case h\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA : C\nf₁ f₂ : MonoOver A\n⊢ (inf.obj (Quotient.mk'' f₁)).obj (Quotient.mk'' f₂) =\n (map (MonoOver.arrow f₁)).obj ((pullback (MonoOver.arrow f₁)).obj (Quotient.mk'' f₂))", "tactic": "rfl" } ]
[ 467, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 463, 1 ]
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
Finset.summable
[]
[ 185, 24 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 183, 11 ]
Mathlib/RingTheory/Valuation/Basic.lean
Valuation.map_add_eq_of_lt_left
[ { "state_after": "K : Type ?u.2215047\nF : Type ?u.2215050\nR : Type u_2\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type ?u.2215062\nΓ''₀ : Type ?u.2215065\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : ↑v y < ↑v x\n⊢ ↑v (y + x) = ↑v x", "state_before": "K : Type ?u.2215047\nF : Type ?u.2215050\nR : Type u_2\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type ?u.2215062\nΓ''₀ : Type ?u.2215065\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : ↑v y < ↑v x\n⊢ ↑v (x + y) = ↑v x", "tactic": "rw [add_comm]" }, { "state_after": "no goals", "state_before": "K : Type ?u.2215047\nF : Type ?u.2215050\nR : Type u_2\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type ?u.2215062\nΓ''₀ : Type ?u.2215065\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : ↑v y < ↑v x\n⊢ ↑v (y + x) = ↑v x", "tactic": "exact map_add_eq_of_lt_right _ h" } ]
[ 331, 50 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 330, 1 ]
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
fderiv_snd
[]
[ 316, 25 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 315, 1 ]
Mathlib/Analysis/Complex/Basic.lean
Complex.norm_eq_abs
[]
[ 54, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 53, 1 ]
Mathlib/Algebra/GroupWithZero/Basic.lean
mul_self_div_self
[ { "state_after": "no goals", "state_before": "α : Type ?u.25591\nM₀ : Type ?u.25594\nG₀ : Type u_1\nM₀' : Type ?u.25600\nG₀' : Type ?u.25603\nF : Type ?u.25606\nF' : Type ?u.25609\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\n⊢ a * a / a = a", "tactic": "rw [div_eq_mul_inv, mul_self_mul_inv a]" } ]
[ 375, 97 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 375, 1 ]
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
CategoryTheory.Limits.Trident.IsLimit.hom_ext
[]
[ 304, 42 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 302, 1 ]
Mathlib/Data/Matrix/Basic.lean
Matrix.of_sub_of
[]
[ 333, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 332, 1 ]
Mathlib/Algebra/Star/Subalgebra.lean
StarSubalgebra.ext
[]
[ 87, 16 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 86, 1 ]
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
SimpleGraph.Subgraph.neighborSet_sup
[]
[ 498, 71 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 497, 1 ]
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
Real.cosh_le_cosh
[]
[ 731, 86 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 730, 1 ]
Mathlib/Algebra/Order/ToIntervalMod.lean
toIcoMod_apply_left
[ { "state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a < a + p ∧ ∃ z, a = a + z • p", "state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ toIcoMod hp a a = a", "tactic": "rw [toIcoMod_eq_iff hp, Set.left_mem_Ico]" }, { "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 : α\n⊢ a < a + p ∧ ∃ z, a = a + z • p", "tactic": "exact ⟨lt_add_of_pos_right _ hp, 0, by simp⟩" }, { "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 : α\n⊢ a = a + 0 • p", "tactic": "simp" } ]
[ 207, 47 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 205, 1 ]
Mathlib/Algebra/Lie/Free.lean
FreeLieAlgebra.lift_comp_of
[ { "state_after": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ ↑(lift R) (↑(lift R).symm F) = F", "state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ ↑(lift R) (↑F ∘ of R) = F", "tactic": "rw [← lift_symm_apply]" }, { "state_after": "no goals", "state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ ↑(lift R) (↑(lift R).symm F) = F", "tactic": "exact (lift R).apply_symm_apply F" } ]
[ 266, 60 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 265, 1 ]
Mathlib/Data/Finset/Lattice.lean
Finset.inf_lt_iff
[]
[ 729, 37 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 728, 11 ]
Mathlib/Topology/Instances/RatLemmas.lean
Rat.dense_compl_compact
[]
[ 50, 69 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 49, 1 ]
Mathlib/Topology/MetricSpace/Basic.lean
Subtype.nndist_eq
[]
[ 1647, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1646, 1 ]
Mathlib/Topology/LocalHomeomorph.lean
LocalHomeomorph.eventually_right_inverse'
[]
[ 253, 47 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 251, 1 ]
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
bddBelow_range_of_tendsto_atTop_atTop
[]
[ 128, 61 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 126, 1 ]
Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean
AlgebraicGeometry.Polynomial.imageOfDf_eq_comap_C_compl_zeroLocus
[ { "state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ imageOfDf f ↔ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ", "state_before": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ imageOfDf f = ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ", "tactic": "ext x" }, { "state_after": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } ∈ zeroLocus {f}ᶜ\n\ncase h.refine'_2\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } =\n x\n\ncase h.refine'_3\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ → x ∈ imageOfDf f", "state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ imageOfDf f ↔ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ", "tactic": "refine' ⟨fun hx => ⟨⟨map C x.asIdeal, isPrime_map_C_of_isPrime x.IsPrime⟩, ⟨_, _⟩⟩, _⟩" }, { "state_after": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal", "state_before": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } ∈ zeroLocus {f}ᶜ", "tactic": "rw [mem_compl_iff, mem_zeroLocus, singleton_subset_iff]" }, { "state_after": "case h.refine'_1.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\ni : ℕ\nhi : ¬coeff f i ∈ x.asIdeal\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal", "state_before": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal", "tactic": "cases' hx with i hi" }, { "state_after": "no goals", "state_before": "case h.refine'_1.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\ni : ℕ\nhi : ¬coeff f i ∈ x.asIdeal\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal", "tactic": "exact fun a => hi (mem_map_C_iff.mp a i)" }, { "state_after": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\n⊢ x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal ↔\n x ∈ x✝.asIdeal", "state_before": "case h.refine'_2\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } =\n x", "tactic": "ext x" }, { "state_after": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ x ∈ x✝.asIdeal", "state_before": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\n⊢ x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal ↔\n x ∈ x✝.asIdeal", "tactic": "refine' ⟨fun h => _, fun h => subset_span (mem_image_of_mem C.1 h)⟩" }, { "state_after": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ coeff (↑C x) 0 ∈ x✝.asIdeal", "state_before": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ x ∈ x✝.asIdeal", "tactic": "rw [← @coeff_C_zero R x _]" }, { "state_after": "no goals", "state_before": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ coeff (↑C x) 0 ∈ x✝.asIdeal", "tactic": "exact mem_map_C_iff.mp h 0" }, { "state_after": "case h.refine'_3.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nxli : PrimeSpectrum R[X]\ncomplement : xli ∈ zeroLocus {f}ᶜ\n⊢ ↑(PrimeSpectrum.comap C) xli ∈ imageOfDf f", "state_before": "case h.refine'_3\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ → x ∈ imageOfDf f", "tactic": "rintro ⟨xli, complement, rfl⟩" }, { "state_after": "no goals", "state_before": "case h.refine'_3.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nxli : PrimeSpectrum R[X]\ncomplement : xli ∈ zeroLocus {f}ᶜ\n⊢ ↑(PrimeSpectrum.comap C) xli ∈ imageOfDf f", "tactic": "exact comap_C_mem_imageOfDf complement" } ]
[ 69, 43 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 57, 1 ]
Mathlib/Algebra/Algebra/Basic.lean
NoZeroSMulDivisors.trans
[ { "state_after": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : r • m = 0\n⊢ r = 0 ∨ m = 0", "state_before": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\n⊢ NoZeroSMulDivisors R M", "tactic": "refine' ⟨fun {r m} h => _⟩" }, { "state_after": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\n⊢ r = 0 ∨ m = 0", "state_before": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : r • m = 0\n⊢ r = 0 ∨ m = 0", "tactic": "rw [algebra_compatible_smul A r m] at h" }, { "state_after": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\n⊢ r = 0 ∨ m = 0\n\ncase inr\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ r = 0 ∨ m = 0", "state_before": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\n⊢ r = 0 ∨ m = 0", "tactic": "cases' smul_eq_zero.1 h with H H" }, { "state_after": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0 ∨ m = 0", "state_before": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\n⊢ r = 0 ∨ m = 0", "tactic": "have : Function.Injective (algebraMap R A) :=\n NoZeroSMulDivisors.iff_algebraMap_injective.1 inferInstance" }, { "state_after": "case inl.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0", "state_before": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0 ∨ m = 0", "tactic": "left" }, { "state_after": "no goals", "state_before": "case inl.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0", "tactic": "exact (injective_iff_map_eq_zero _).1 this _ H" }, { "state_after": "case inr.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ m = 0", "state_before": "case inr\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ r = 0 ∨ m = 0", "tactic": "right" }, { "state_after": "no goals", "state_before": "case inr.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ m = 0", "tactic": "exact H" } ]
[ 892, 12 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 881, 1 ]
Mathlib/FieldTheory/IntermediateField.lean
IntermediateField.sum_mem
[]
[ 236, 12 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 234, 11 ]
Mathlib/Data/Finset/Basic.lean
Finset.filter_disj_union
[]
[ 2794, 41 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 2792, 1 ]
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
[ { "state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ f - g =ᵐ[μ] 0", "state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ f =ᵐ[μ] g", "tactic": "rw [← sub_ae_eq_zero]" }, { "state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\n⊢ f - g =ᵐ[μ] 0", "state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ f - g =ᵐ[μ] 0", "tactic": "have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by\n intro s hs hμs\n rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),\n sub_eq_zero.mpr (hfg_eq s hs hμs)]" }, { "state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\nhfg_int : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (f - g) s\n⊢ f - g =ᵐ[μ] 0", "state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\n⊢ f - g =ᵐ[μ] 0", "tactic": "have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>\n (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)" }, { "state_after": "no goals", "state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\nhfg_int : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (f - g) s\n⊢ f - g =ᵐ[μ] 0", "tactic": "exact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg" }, { "state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ (∫ (x : α) in s, (f - g) x ∂μ) = 0", "state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0", "tactic": "intro s hs hμs" }, { "state_after": "no goals", "state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ (∫ (x : α) in s, (f - g) x ∂μ) = 0", "tactic": "rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),\n sub_eq_zero.mpr (hfg_eq s hs hμs)]" } ]
[ 471, 74 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 459, 1 ]
Mathlib/LinearAlgebra/Lagrange.lean
Lagrange.basisDivisor_ne_zero_iff
[ { "state_after": "no goals", "state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\n⊢ basisDivisor x y ≠ 0 ↔ x ≠ y", "tactic": "rw [Ne.def, basisDivisor_eq_zero_iff]" } ]
[ 148, 40 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 147, 1 ]
Mathlib/Order/SupIndep.lean
CompleteLattice.independent_pair
[ { "state_after": "case mp\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Independent t → Disjoint (t i) (t j)\n\ncase mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Disjoint (t i) (t j) → Independent t", "state_before": "α : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Independent t ↔ Disjoint (t i) (t j)", "tactic": "constructor" }, { "state_after": "no goals", "state_before": "case mp\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Independent t → Disjoint (t i) (t j)", "tactic": "exact fun h => h.pairwiseDisjoint hij" }, { "state_after": "case mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\nh : Disjoint (t i) (t j)\nk : ι\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)", "state_before": "case mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Disjoint (t i) (t j) → Independent t", "tactic": "rintro h k" }, { "state_after": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)\n\ncase mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)", "state_before": "case mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\nh : Disjoint (t i) (t j)\nk : ι\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)", "tactic": "obtain rfl | rfl := huniv k" }, { "state_after": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\ni : ι\nhi : i ≠ k\n⊢ t i = t j", "state_before": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)", "tactic": "refine' h.mono_right (iSup_le fun i => iSup_le fun hi => Eq.le _)" }, { "state_after": "no goals", "state_before": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\ni : ι\nhi : i ≠ k\n⊢ t i = t j", "tactic": "rw [(huniv i).resolve_left hi]" }, { "state_after": "case mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\nj : ι\nhj : j ≠ k\n⊢ t j = t i", "state_before": "case mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)", "tactic": "refine' h.symm.mono_right (iSup_le fun j => iSup_le fun hj => Eq.le _)" }, { "state_after": "no goals", "state_before": "case mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\nj : ι\nhj : j ≠ k\n⊢ t j = t i", "tactic": "rw [(huniv j).resolve_right hj]" } ]
[ 334, 38 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 325, 1 ]
Mathlib/Data/Ordmap/Ordset.lean
Ordnode.Bounded.trans_left
[]
[ 958, 58 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 955, 1 ]
Mathlib/Tactic/Ring/Basic.lean
Mathlib.Tactic.Ring.atom_pf
[ { "state_after": "no goals", "state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na : R\n⊢ a = a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0", "tactic": "simp" } ]
[ 865, 91 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 865, 1 ]
Mathlib/Data/Polynomial/Eval.lean
Polynomial.eval₂_multiset_prod
[]
[ 1109, 41 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1107, 1 ]
Mathlib/CategoryTheory/Sites/Adjunction.lean
CategoryTheory.Sheaf.adjunctionToTypes_unit_app_val
[ { "state_after": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ((adjunction J adj).unit.app ((sheafEquivSheafOfTypes J).inverse.obj Y)).val ≫\n 𝟙 (sheafify J (Y.val ⋙ G) ⋙ forget D) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)", "state_before": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((adjunctionToTypes J adj).unit.app Y).val =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app ((sheafOfTypesToPresheaf J).obj Y) ≫\n whiskerRight\n (toSheafify J (((whiskeringRight Cᵒᵖ (Type (max v u)) D).obj G).obj ((sheafOfTypesToPresheaf J).obj Y)))\n (forget D)", "tactic": "dsimp [adjunctionToTypes, Adjunction.comp]" }, { "state_after": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) Y.val (sheafify J (Y.val ⋙ G)))\n (toSheafify J (Y.val ⋙ G)) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)", "state_before": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ((adjunction J adj).unit.app ((sheafEquivSheafOfTypes J).inverse.obj Y)).val ≫\n 𝟙 (sheafify J (Y.val ⋙ G) ⋙ forget D) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)", "tactic": "simp" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) Y.val (sheafify J (Y.val ⋙ G)))\n (toSheafify J (Y.val ⋙ G)) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)", "tactic": "rfl" } ]
[ 136, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 129, 1 ]
Mathlib/Order/WithBot.lean
WithTop.lt_iff_exists_coe_btwn
[]
[ 1354, 39 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1348, 1 ]
Mathlib/Data/Polynomial/Coeff.lean
Polynomial.finset_sum_coeff
[]
[ 109, 23 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 107, 1 ]
Mathlib/Analysis/NormedSpace/Multilinear.lean
ContinuousMultilinearMap.mkPiField_apply_one_eq_self
[]
[ 902, 75 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 900, 1 ]
Mathlib/Data/Setoid/Partition.lean
Finpartition.isPartition_parts
[]
[ 316, 35 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 312, 1 ]
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
LinearMap.toMatrix'_toLinearMapₛₗ₂'
[]
[ 259, 44 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 257, 1 ]
Mathlib/Data/Finset/Pointwise.lean
Finset.smul_finset_subset_smul_finset_iff
[]
[ 1970, 50 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1969, 1 ]
Mathlib/Algebra/Order/Field/Basic.lean
div_le_one_of_ge
[ { "state_after": "no goals", "state_before": "ι : Type ?u.148913\nα : Type u_1\nβ : Type ?u.148919\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nh : b ≤ a\nhb : b ≤ 0\n⊢ a / b ≤ 1", "tactic": "simpa only [neg_div_neg_eq] using div_le_one_of_le (neg_le_neg h) (neg_nonneg_of_nonpos hb)" } ]
[ 753, 94 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 752, 1 ]
Mathlib/Data/Finset/Pointwise.lean
Finset.mul_eq_one_iff
[ { "state_after": "no goals", "state_before": "F : Type ?u.557283\nα : Type u_1\nβ : Type ?u.557289\nγ : Type ?u.557292\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DivisionMonoid α\ns t : Finset α\n⊢ s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1", "tactic": "simp_rw [← coe_inj, coe_mul, coe_one, Set.mul_eq_one_iff, coe_singleton]" } ]
[ 1005, 75 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1004, 11 ]
Mathlib/LinearAlgebra/Lagrange.lean
Polynomial.eq_zero_of_degree_lt_of_eval_index_eq_zero
[ { "state_after": "no goals", "state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card s)\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0", "tactic": "classical\n rw [← card_image_of_injOn hvs] at degree_f_lt\n refine' eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_f_lt _\n intro x hx\n rcases mem_image.mp hx with ⟨_, hj, rfl⟩\n exact eval_f _ hj" }, { "state_after": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0", "state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card s)\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0", "tactic": "rw [← card_image_of_injOn hvs] at degree_f_lt" }, { "state_after": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ ∀ (x : R), x ∈ image v s → eval x f = 0", "state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0", "tactic": "refine' eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_f_lt _" }, { "state_after": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nx : R\nhx : x ∈ image v s\n⊢ eval x f = 0", "state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ ∀ (x : R), x ∈ image v s → eval x f = 0", "tactic": "intro x hx" }, { "state_after": "case intro.intro\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nw✝ : ι\nhj : w✝ ∈ s\nhx : v w✝ ∈ image v s\n⊢ eval (v w✝) f = 0", "state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nx : R\nhx : x ∈ image v s\n⊢ eval x f = 0", "tactic": "rcases mem_image.mp hx with ⟨_, hj, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro.intro\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nw✝ : ι\nhj : w✝ ∈ s\nhx : v w✝ ∈ image v s\n⊢ eval (v w✝) f = 0", "tactic": "exact eval_f _ hj" } ]
[ 88, 22 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 81, 1 ]
Mathlib/Computability/Language.lean
Language.map_id
[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.42797\nγ : Type ?u.42800\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ ↑(map id) l = l", "tactic": "simp [map]" } ]
[ 171, 64 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 171, 1 ]
Mathlib/Analysis/Normed/Group/Seminorm.lean
NonarchAddGroupNorm.ext
[]
[ 924, 18 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 923, 1 ]
Mathlib/Data/Polynomial/Degree/Definitions.lean
Polynomial.eq_C_of_degree_eq_zero
[]
[ 627, 38 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 626, 1 ]
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
CategoryTheory.Limits.HasBinaryBiproduct.of_hasBinaryCoproduct
[]
[ 433, 91 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 429, 1 ]
Mathlib/Data/Set/Image.lean
Subtype.preimage_coe_compl
[]
[ 1477, 47 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1476, 1 ]
Mathlib/Logic/Equiv/Basic.lean
Equiv.swap_self
[]
[ 1551, 33 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1550, 1 ]
Mathlib/CategoryTheory/Abelian/RightDerived.lean
CategoryTheory.NatTrans.rightDerived_id
[ { "state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Functor.Additive F\nn : ℕ\n⊢ 𝟙\n (injectiveResolutions C ⋙\n Functor.mapHomotopyCategory F (ComplexShape.up ℕ) ⋙ HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n) =\n 𝟙 (Functor.rightDerived F n)", "state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Functor.Additive F\nn : ℕ\n⊢ rightDerived (𝟙 F) n = 𝟙 (Functor.rightDerived F n)", "tactic": "simp [NatTrans.rightDerived]" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Functor.Additive F\nn : ℕ\n⊢ 𝟙\n (injectiveResolutions C ⋙\n Functor.mapHomotopyCategory F (ComplexShape.up ℕ) ⋙ HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n) =\n 𝟙 (Functor.rightDerived F n)", "tactic": "rfl" } ]
[ 142, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 139, 1 ]
Mathlib/Algebra/Module/Basic.lean
rat_cast_smul_eq
[]
[ 545, 48 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 543, 1 ]
Mathlib/Algebra/Group/Semiconj.lean
SemiconjBy.units_inv_symm_left_iff
[]
[ 147, 45 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 146, 1 ]
Mathlib/Analysis/InnerProductSpace/Basic.lean
InnerProductSpace.Core.inner_smul_right
[ { "state_after": "𝕜 : Type u_1\nE : Type ?u.527762\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ ↑(starRingEnd 𝕜) (↑(starRingEnd 𝕜) r * inner y x) = r * inner x y", "state_before": "𝕜 : Type u_1\nE : Type ?u.527762\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ inner x (r • y) = r * inner x y", "tactic": "rw [← inner_conj_symm, inner_smul_left]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\nE : Type ?u.527762\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ ↑(starRingEnd 𝕜) (↑(starRingEnd 𝕜) r * inner y x) = r * inner x y", "tactic": "simp only [conj_conj, inner_conj_symm, RingHom.map_mul]" } ]
[ 246, 60 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 244, 1 ]
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
MeasureTheory.AEStronglyMeasurable.nnnorm
[]
[ 1471, 49 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1469, 11 ]
Mathlib/MeasureTheory/Function/LpSpace.lean
MeasureTheory.Lp.norm_def
[]
[ 263, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 262, 1 ]
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
MeasureTheory.OuterMeasure.smul_apply
[]
[ 317, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 316, 1 ]
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
div_mul_left
[]
[ 120, 32 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 119, 1 ]
Mathlib/Algebra/Ring/BooleanRing.lean
mul_self
[]
[ 65, 25 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 64, 1 ]
Mathlib/MeasureTheory/Group/Measure.lean
MeasureTheory.forall_measure_preimage_mul_right_iff
[ { "state_after": "no goals", "state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G), Measure.map (fun x => x * g) μ = μ) ↔ IsMulRightInvariant μ", "tactic": "exact ⟨fun h => ⟨h⟩, fun h => h.1⟩" }, { "state_after": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔\n ∀ (g : G) (s : Set G), MeasurableSet s → ↑↑(Measure.map (fun x => x * g) μ) s = ↑↑μ s", "state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔\n ∀ (g : G), Measure.map (fun x => x * g) μ = μ", "tactic": "simp_rw [Measure.ext_iff]" }, { "state_after": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\ng : G\nA : Set G\nhA : MeasurableSet A\n⊢ ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A ↔ ↑↑(Measure.map (fun x => x * g) μ) A = ↑↑μ A", "state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔\n ∀ (g : G) (s : Set G), MeasurableSet s → ↑↑(Measure.map (fun x => x * g) μ) s = ↑↑μ s", "tactic": "refine' forall_congr' fun g => forall_congr' fun A => forall_congr' fun hA => _" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\ng : G\nA : Set G\nhA : MeasurableSet A\n⊢ ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A ↔ ↑↑(Measure.map (fun x => x * g) μ) A = ↑↑μ A", "tactic": "rw [map_apply (measurable_mul_const g) hA]" } ]
[ 201, 37 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 194, 1 ]
Mathlib/Topology/UniformSpace/Basic.lean
uniformContinuous_def
[]
[ 1107, 10 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1105, 1 ]
Mathlib/Topology/MetricSpace/EMetricSpace.lean
Prod.edist_eq
[]
[ 464, 6 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 462, 1 ]
Mathlib/Topology/Bornology/Constructions.lean
Bornology.isBounded_pi
[ { "state_after": "case pos\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)\n\ncase neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)", "state_before": "α : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)", "tactic": "by_cases hne : ∃ i, S i = ∅" }, { "state_after": "no goals", "state_before": "case pos\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)", "tactic": "simp [hne, univ_pi_eq_empty_iff.2 hne]" }, { "state_after": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)", "state_before": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)", "tactic": "simp only [hne, false_or_iff]" }, { "state_after": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : Set.Nonempty (Set.pi univ fun i => S i)\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)", "state_before": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)", "tactic": "simp only [not_exists, ← Ne.def, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne" }, { "state_after": "no goals", "state_before": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : Set.Nonempty (Set.pi univ fun i => S i)\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)", "tactic": "exact isBounded_pi_of_nonempty hne" } ]
[ 126, 39 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 121, 1 ]
Mathlib/Topology/Algebra/Polynomial.lean
Polynomial.exists_forall_norm_le
[ { "state_after": "α : Type ?u.177157\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\np : R[X]\nhp0 : ¬0 < degree p\n⊢ ∀ (y : R), ‖eval (coeff (↑C (coeff p 0)) 0) (↑C (coeff p 0))‖ ≤ ‖eval y (↑C (coeff p 0))‖", "state_before": "α : Type ?u.177157\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\np : R[X]\nhp0 : ¬0 < degree p\n⊢ ∀ (y : R), ‖eval (coeff p 0) p‖ ≤ ‖eval y p‖", "tactic": "rw [eq_C_of_degree_le_zero (le_of_not_gt hp0)]" }, { "state_after": "no goals", "state_before": "α : Type ?u.177157\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\np : R[X]\nhp0 : ¬0 < degree p\n⊢ ∀ (y : R), ‖eval (coeff (↑C (coeff p 0)) 0) (↑C (coeff p 0))‖ ≤ ‖eval y (↑C (coeff p 0))‖", "tactic": "simp" } ]
[ 145, 73 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 141, 1 ]
Mathlib/Data/Bitvec/Lemmas.lean
Bitvec.ofNat_toNat
[ { "state_after": "case mk\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h }) = { val := xs, property := h }", "state_before": "n : ℕ\nv : Bitvec n\n⊢ Bitvec.ofNat n (Bitvec.toNat v) = v", "tactic": "cases' v with xs h" }, { "state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ ↑(Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = ↑{ val := xs, property := h }", "state_before": "case mk\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h }) = { val := xs, property := h }", "tactic": "apply Subtype.ext" }, { "state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = xs", "state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ ↑(Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = ↑{ val := xs, property := h }", "tactic": "change Vector.toList _ = xs" }, { "state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs", "state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = xs", "tactic": "dsimp [Bitvec.toNat, bitsToNat]" }, { "state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs", "state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs", "tactic": "rw [← List.length_reverse] at h" }, { "state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (List.reverse xs))) =\n List.reverse (List.reverse xs)", "state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs", "tactic": "rw [← List.reverse_reverse xs, List.foldl_reverse]" }, { "state_after": "case mk.a\nn : ℕ\nxs ys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys", "state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (List.reverse xs))) =\n List.reverse (List.reverse xs)", "tactic": "generalize xs.reverse = ys at h⊢" }, { "state_after": "case mk.a\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys", "state_before": "case mk.a\nn : ℕ\nxs ys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys", "tactic": "clear xs" }, { "state_after": "case mk.a.nil\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nn : ℕ\nh : List.length [] = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []\n\ncase mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : List.length (ys_head :: ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)", "state_before": "case mk.a\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys", "tactic": "induction' ys with ys_head ys_tail ys_ih generalizing n" }, { "state_after": "case mk.a.nil.refl\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat (List.length []) (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []", "state_before": "case mk.a.nil\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nn : ℕ\nh : List.length [] = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []", "tactic": "cases h" }, { "state_after": "no goals", "state_before": "case mk.a.nil.refl\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat (List.length []) (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []", "tactic": "simp [Bitvec.ofNat]" }, { "state_after": "case mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : succ (List.length ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)", "state_before": "case mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : List.length (ys_head :: ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)", "tactic": "simp only [← Nat.succ_eq_add_one, List.length] at h" }, { "state_after": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (succ (List.length ys_tail)) (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)", "state_before": "case mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : succ (List.length ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)", "tactic": "subst n" }, { "state_after": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (List.length ys_tail) (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head / 2)) ++\n [decide (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head % 2 = 1)] =\n List.reverse ys_tail ++ [ys_head]", "state_before": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (succ (List.length ys_tail)) (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)", "tactic": "simp only [Bitvec.ofNat, Vector.toList_cons, Vector.toList_nil, List.reverse_cons,\n Vector.toList_append, List.foldr]" }, { "state_after": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) ++ [ys_head] =\n List.reverse ys_tail ++ [ys_head]", "state_before": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (List.length ys_tail) (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head / 2)) ++\n [decide (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head % 2 = 1)] =\n List.reverse ys_tail ++ [ys_head]", "tactic": "erw [addLsb_div_two, decide_addLsb_mod_two]" }, { "state_after": "case mk.a.cons.e_a\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) =\n List.reverse ys_tail", "state_before": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) ++ [ys_head] =\n List.reverse ys_tail ++ [ys_head]", "tactic": "congr" }, { "state_after": "case mk.a.cons.e_a.h\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ List.length ys_tail = List.length ys_tail", "state_before": "case mk.a.cons.e_a\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) =\n List.reverse ys_tail", "tactic": "apply ys_ih" }, { "state_after": "no goals", "state_before": "case mk.a.cons.e_a.h\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ List.length ys_tail = List.length ys_tail", "tactic": "rfl" } ]
[ 147, 8 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 128, 1 ]
Mathlib/Topology/Algebra/Order/Field.lean
tendsto_inv_zero_atTop
[ { "state_after": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b", "state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\n⊢ Tendsto (fun x => x⁻¹) (𝓝[Ioi 0] 0) atTop", "tactic": "refine' (atTop_basis' 1).tendsto_right_iff.2 fun b hb => _" }, { "state_after": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\nhb' : 0 < b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b", "state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b", "tactic": "have hb' : 0 < b := by positivity" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\nhb' : 0 < b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b", "tactic": "filter_upwards [Ioc_mem_nhdsWithin_Ioi\n ⟨le_rfl, inv_pos.2 hb'⟩]with x hx using(le_inv hx.1 hb').1 hx.2" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\n⊢ 0 < b", "tactic": "positivity" } ]
[ 128, 70 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 124, 1 ]
Mathlib/ModelTheory/Syntax.lean
FirstOrder.Language.BoundedFormula.sum_elim_comp_relabelAux
[ { "state_after": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α ⊕ Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) x = Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) x", "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\n⊢ Sum.elim v xs ∘ relabelAux g m = Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n))", "tactic": "ext x" }, { "state_after": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inl x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inl x)\n\ncase h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inr x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inr x)", "state_before": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α ⊕ Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) x = Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) x", "tactic": "cases' x with x x" }, { "state_after": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ Sum.elim v xs (Sum.map id (↑finSumFinEquiv) (↑(Equiv.sumAssoc β (Fin n) (Fin m)) (Sum.inl (g x)))) =\n Sum.elim v (xs ∘ ↑(castAdd m)) (g x)", "state_before": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inl x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inl x)", "tactic": "simp only [BoundedFormula.relabelAux, Function.comp_apply, Sum.map_inl, Sum.elim_inl]" }, { "state_after": "no goals", "state_before": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ Sum.elim v xs (Sum.map id (↑finSumFinEquiv) (↑(Equiv.sumAssoc β (Fin n) (Fin m)) (Sum.inl (g x)))) =\n Sum.elim v (xs ∘ ↑(castAdd m)) (g x)", "tactic": "cases' g x with l r <;> simp" }, { "state_after": "no goals", "state_before": "case h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inr x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inr x)", "tactic": "simp [BoundedFormula.relabelAux]" } ]
[ 582, 37 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 575, 1 ]
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
ContDiffWithinAt.csin
[]
[ 434, 61 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 432, 1 ]
Mathlib/Data/Bool/Basic.lean
Bool.coe_bool_iff
[ { "state_after": "no goals", "state_before": "⊢ ∀ {a b : Bool}, (a = true ↔ b = true) ↔ a = b", "tactic": "decide" } ]
[ 142, 68 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 142, 1 ]
Mathlib/CategoryTheory/Monoidal/Preadditive.lean
CategoryTheory.monoidalPreadditive_of_faithful
[ { "state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ f✝ ⊗ 0 = 0", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f : W ⟶ X), f ⊗ 0 = 0", "tactic": "intros" }, { "state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ F.map (f✝ ⊗ 0) = F.map 0", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ f✝ ⊗ 0 = 0", "tactic": "apply F.toFunctor.map_injective" }, { "state_after": "no goals", "state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ F.map (f✝ ⊗ 0) = F.map 0", "tactic": "simp [F.map_tensor]" }, { "state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ 0 ⊗ f✝ = 0", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f : Y ⟶ Z), 0 ⊗ f = 0", "tactic": "intros" }, { "state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ F.map (0 ⊗ f✝) = F.map 0", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ 0 ⊗ f✝ = 0", "tactic": "apply F.toFunctor.map_injective" }, { "state_after": "no goals", "state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ F.map (0 ⊗ f✝) = F.map 0", "tactic": "simp [F.map_tensor]" }, { "state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ f✝ ⊗ (g✝ + h✝) = f✝ ⊗ g✝ + f✝ ⊗ h✝", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f : W ⟶ X) (g h : Y ⟶ Z), f ⊗ (g + h) = f ⊗ g + f ⊗ h", "tactic": "intros" }, { "state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ F.map (f✝ ⊗ (g✝ + h✝)) = F.map (f✝ ⊗ g✝ + f✝ ⊗ h✝)", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ f✝ ⊗ (g✝ + h✝) = f✝ ⊗ g✝ + f✝ ⊗ h✝", "tactic": "apply F.toFunctor.map_injective" }, { "state_after": "no goals", "state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ F.map (f✝ ⊗ (g✝ + h✝)) = F.map (f✝ ⊗ g✝ + f✝ ⊗ h✝)", "tactic": "simp only [F.map_tensor, Functor.map_add, Preadditive.comp_add, Preadditive.add_comp,\n MonoidalPreadditive.tensor_add]" }, { "state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ (f✝ + g✝) ⊗ h✝ = f✝ ⊗ h✝ + g✝ ⊗ h✝", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f g : W ⟶ X) (h : Y ⟶ Z), (f + g) ⊗ h = f ⊗ h + g ⊗ h", "tactic": "intros" }, { "state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ F.map ((f✝ + g✝) ⊗ h✝) = F.map (f✝ ⊗ h✝ + g✝ ⊗ h✝)", "state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ (f✝ + g✝) ⊗ h✝ = f✝ ⊗ h✝ + g✝ ⊗ h✝", "tactic": "apply F.toFunctor.map_injective" }, { "state_after": "no goals", "state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ F.map ((f✝ + g✝) ⊗ h✝) = F.map (f✝ ⊗ h✝ + g✝ ⊗ h✝)", "tactic": "simp only [F.map_tensor, Functor.map_add, Preadditive.comp_add, Preadditive.add_comp,\n MonoidalPreadditive.add_tensor]" } ]
[ 93, 42 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 73, 1 ]
Mathlib/Analysis/Convex/Function.lean
ConvexOn.le_on_segment'
[ { "state_after": "case h₁.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f x ≤ max (f x) (f y)\n\ncase h₂.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f y ≤ max (f x) (f y)", "state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ a • max (f x) (f y) + b • max (f x) (f y)", "tactic": "gcongr" }, { "state_after": "no goals", "state_before": "case h₁.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f x ≤ max (f x) (f y)", "tactic": "apply le_max_left" }, { "state_after": "no goals", "state_before": "case h₂.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f y ≤ max (f x) (f y)", "tactic": "apply le_max_right" } ]
[ 642, 51 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 634, 1 ]
Mathlib/Analysis/InnerProductSpace/l2Space.lean
HilbertBasis.orthonormal
[ { "state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\n⊢ ∀ (i j : ι),\n inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0", "state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\n⊢ Orthonormal 𝕜 fun i => ↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)", "tactic": "rw [orthonormal_iff_ite]" }, { "state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0", "state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\n⊢ ∀ (i j : ι),\n inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0", "tactic": "intro i j" }, { "state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner 1 (if h : i = j then (_ : j = i) ▸ 1 else 0) = if i = j then 1 else 0", "state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0", "tactic": "rw [← b.repr.inner_map_map (b i) (b j), b.repr_self, b.repr_self, lp.inner_single_left,\n lp.single_apply]" }, { "state_after": "no goals", "state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner 1 (if h : i = j then (_ : j = i) ▸ 1 else 0) = if i = j then 1 else 0", "tactic": "simp" } ]
[ 453, 7 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 448, 11 ]
Mathlib/Data/MvPolynomial/Equiv.lean
MvPolynomial.mapEquiv_trans
[ { "state_after": "no goals", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne✝ : ℕ\ns : σ →₀ ℕ\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S₁\ninst✝¹ : CommSemiring S₂\ninst✝ : CommSemiring S₃\ne : S₁ ≃+* S₂\nf : S₂ ≃+* S₃\np : MvPolynomial σ S₁\n⊢ ↑(RingEquiv.trans (mapEquiv σ e) (mapEquiv σ f)) p = ↑(mapEquiv σ (RingEquiv.trans e f)) p", "tactic": "simp only [RingEquiv.coe_trans, comp_apply, mapEquiv_apply, RingEquiv.coe_ringHom_trans,\n map_map]" } ]
[ 123, 15 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 119, 1 ]
Mathlib/Algebra/BigOperators/Finsupp.lean
Finsupp.comp_liftAddHom
[ { "state_after": "no goals", "state_before": "α : Type u_4\nι : Type ?u.518287\nγ : Type ?u.518290\nA : Type ?u.518293\nB : Type ?u.518296\nC : Type ?u.518299\ninst✝⁵ : AddCommMonoid A\ninst✝⁴ : AddCommMonoid B\ninst✝³ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.521429\nM : Type u_1\nM' : Type ?u.521435\nN : Type u_2\nP : Type u_3\nG : Type ?u.521444\nH : Type ?u.521447\nR : Type ?u.521450\nS : Type ?u.521453\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\ninst✝ : AddCommMonoid P\ng : N →+ P\nf : α → M →+ N\na : α\n⊢ ↑(AddEquiv.symm liftAddHom) (AddMonoidHom.comp g (↑liftAddHom f)) a = AddMonoidHom.comp g (f a)", "tactic": "rw [liftAddHom_symm_apply, AddMonoidHom.comp_assoc, liftAddHom_comp_single]" } ]
[ 504, 82 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 498, 1 ]
Mathlib/Algebra/IndicatorFunction.lean
Set.prod_mulIndicator_subset_of_eq_one
[ { "state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∏ i in s, g i (f i) = ∏ x in s, g x (mulIndicator (↑s) f x)\n\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ t → ¬x ∈ s → g x (mulIndicator (↑s) f x) = 1", "state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∏ i in s, g i (f i) = ∏ i in t, g i (mulIndicator (↑s) f i)", "tactic": "rw [← Finset.prod_subset h _]" }, { "state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ s → g x (f x) = g x (mulIndicator (↑s) f x)", "state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∏ i in s, g i (f i) = ∏ x in s, g x (mulIndicator (↑s) f x)", "tactic": "apply Finset.prod_congr rfl" }, { "state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ g i (f i) = g i (mulIndicator (↑s) f i)", "state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ s → g x (f x) = g x (mulIndicator (↑s) f x)", "tactic": "intro i hi" }, { "state_after": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ f i = mulIndicator (↑s) f i", "state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ g i (f i) = g i (mulIndicator (↑s) f i)", "tactic": "congr" }, { "state_after": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ mulIndicator (↑s) f i = f i", "state_before": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ f i = mulIndicator (↑s) f i", "tactic": "symm" }, { "state_after": "no goals", "state_before": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ mulIndicator (↑s) f i = f i", "tactic": "exact mulIndicator_of_mem (α := α) hi f" }, { "state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ g i (mulIndicator (↑s) f i) = 1", "state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ t → ¬x ∈ s → g x (mulIndicator (↑s) f x) = 1", "tactic": "refine' fun i _ hn => _" }, { "state_after": "case h.e'_2.h.e'_2\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ mulIndicator (↑s) f i = 1", "state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ g i (mulIndicator (↑s) f i) = 1", "tactic": "convert hg i" }, { "state_after": "no goals", "state_before": "case h.e'_2.h.e'_2\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ mulIndicator (↑s) f i = 1", "tactic": "exact mulIndicator_of_not_mem (α := α) hn f" } ]
[ 597, 48 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 584, 1 ]
Mathlib/Computability/TMToPartrec.lean
Turing.PartrecToTM2.tr_init
[]
[ 1703, 28 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1701, 1 ]
Mathlib/Data/MvPolynomial/Division.lean
MvPolynomial.X_divMonomial
[]
[ 172, 44 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 171, 1 ]
Mathlib/Topology/Algebra/UniformRing.lean
UniformSpace.Completion.Continuous.mul
[]
[ 90, 64 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 88, 1 ]
Mathlib/Order/Directed.lean
directedOn_of_sup_mem
[]
[ 129, 46 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 127, 1 ]
Mathlib/Order/OmegaCompletePartialOrder.lean
CompleteLattice.inf_continuous'
[]
[ 535, 40 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 533, 1 ]
Mathlib/Analysis/Calculus/FDeriv/Add.lean
Differentiable.const_smul
[]
[ 101, 69 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 100, 1 ]
Mathlib/Order/Filter/Extr.lean
IsExtrOn.comp_mapsTo
[]
[ 441, 90 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 439, 1 ]
Mathlib/Order/BooleanAlgebra.lean
sdiff_lt
[ { "state_after": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x \\ y = x\n⊢ y = ⊥", "state_before": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\n⊢ x \\ y < x", "tactic": "refine' sdiff_le.lt_of_ne fun h => hy _" }, { "state_after": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x ⊓ y = ⊥\n⊢ y = ⊥", "state_before": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x \\ y = x\n⊢ y = ⊥", "tactic": "rw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x ⊓ y = ⊥\n⊢ y = ⊥", "tactic": "rw [← h, inf_eq_right.mpr hx]" } ]
[ 324, 32 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 321, 1 ]
Mathlib/GroupTheory/Subgroup/Basic.lean
Subgroup.le_normalClosure
[]
[ 2468, 25 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 2467, 1 ]
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
isConnected_Icc
[]
[ 482, 41 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 481, 1 ]
Mathlib/NumberTheory/Divisors.lean
Nat.image_snd_divisorsAntidiagonal
[ { "state_after": "n : ℕ\n⊢ image (Prod.snd ∘ ↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ))) (divisorsAntidiagonal n) = divisors n", "state_before": "n : ℕ\n⊢ image Prod.snd (divisorsAntidiagonal n) = divisors n", "tactic": "rw [← map_swap_divisorsAntidiagonal, map_eq_image, image_image]" }, { "state_after": "no goals", "state_before": "n : ℕ\n⊢ image (Prod.snd ∘ ↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ))) (divisorsAntidiagonal n) = divisors n", "tactic": "exact image_fst_divisorsAntidiagonal" } ]
[ 260, 39 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 258, 1 ]
Mathlib/Combinatorics/SimpleGraph/Basic.lean
SimpleGraph.map_adj
[]
[ 1244, 10 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 1242, 1 ]
Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean
SimpleGraph.incMatrix_of_mem_incidenceSet
[ { "state_after": "no goals", "state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝ : MulZeroOneClass R\na b : α\ne : Sym2 α\nh : e ∈ incidenceSet G a\n⊢ incMatrix R G a e = 1", "tactic": "rw [incMatrix_apply, Set.indicator_of_mem h, Pi.one_apply]" } ]
[ 99, 61 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 98, 1 ]
Mathlib/Topology/SubsetProperties.lean
IsCompact.inter_right
[ { "state_after": "α : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f", "state_before": "α : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\n⊢ IsCompact (s ∩ t)", "tactic": "intro f hnf hstf" }, { "state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f", "state_before": "α : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f", "tactic": "obtain ⟨a, hsa, ha⟩ : ∃ a ∈ s, ClusterPt a f :=\n hs (le_trans hstf (le_principal_iff.2 (inter_subset_left _ _)))" }, { "state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\nthis : a ∈ t\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f", "state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f", "tactic": "have : a ∈ t := ht.mem_of_nhdsWithin_neBot <|\n ha.mono <| le_trans hstf (le_principal_iff.2 (inter_subset_right _ _))" }, { "state_after": "no goals", "state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\nthis : a ∈ t\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f", "tactic": "exact ⟨a, ⟨hsa, this⟩, ha⟩" } ]
[ 118, 29 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 112, 1 ]
Mathlib/Data/Sign.lean
sign_neg
[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nha : a < 0\n⊢ ↑sign a = -1", "tactic": "rwa [sign_apply, if_neg <| asymm ha, if_pos]" } ]
[ 328, 95 ]
5a919533f110b7d76410134a237ee374f24eaaad
https://github.com/leanprover-community/mathlib4
[ 328, 1 ]