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/Algebra/Order/Sub/Canonical.lean
|
tsub_min
|
[
{
"state_after": "case inl\nα : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : a ≤ b\n⊢ a - min a b = a - b\n\ncase inr\nα : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : b ≤ a\n⊢ a - min a b = a - b",
"state_before": "α : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ a - min a b = a - b",
"tactic": "cases' le_total a b with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : a ≤ b\n⊢ a - min a b = a - b",
"tactic": "rw [min_eq_left h, tsub_self, tsub_eq_zero_of_le h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nh : b ≤ a\n⊢ a - min a b = a - b",
"tactic": "rw [min_eq_right h]"
}
] |
[
507,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.LocallyIntegrableOn.integrableOn_isCompact
|
[] |
[
71,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_image_left_comm
|
[] |
[
369,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.totalDegree_add
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nthis : n ∈ a.support ∪ b.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"tactic": "have := Finsupp.support_add hn"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nthis : n ∈ a.support ∨ n ∈ b.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nthis : n ∈ a.support ∪ b.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"tactic": "rw [Finset.mem_union] at this"
},
{
"state_after": "case inl\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nh : n ∈ a.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)\n\ncase inr\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nh : n ∈ b.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nthis : n ∈ a.support ∨ n ∈ b.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"tactic": "cases' this with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nh : n ∈ a.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"tactic": "exact le_max_of_le_left (le_totalDegree h)"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.397208\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a + b)\nh : n ∈ b.support\n⊢ (sum n fun x e => e) ≤ max (totalDegree a) (totalDegree b)",
"tactic": "exact le_max_of_le_right (le_totalDegree h)"
}
] |
[
652,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Data/Polynomial/HasseDeriv.lean
|
Polynomial.natDegree_hasseDeriv
|
[
{
"state_after": "case inl\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : natDegree p < n\n⊢ natDegree (↑(hasseDeriv n) p) = natDegree p - n\n\ncase inr\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ natDegree (↑(hasseDeriv n) p) = natDegree p - n",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\n⊢ natDegree (↑(hasseDeriv n) p) = natDegree p - n",
"tactic": "cases' lt_or_le p.natDegree n with hn hn"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : natDegree p < n\n⊢ natDegree (↑(hasseDeriv n) p) = natDegree p - n",
"tactic": "simpa [hasseDeriv_eq_zero_of_lt_natDegree, hn] using (tsub_eq_zero_of_le hn.le).symm"
},
{
"state_after": "case inr.refine'_1\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ ∀ (f : R[X]), natDegree f < n → ↑(hasseDeriv n) f = 0\n\ncase inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ ∀ (n_1 : ℕ) (c : R), c ≠ 0 → natDegree (↑(hasseDeriv n) (↑(monomial n_1) c)) = n_1 - n",
"state_before": "case inr\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ natDegree (↑(hasseDeriv n) p) = natDegree p - n",
"tactic": "refine' map_natDegree_eq_sub _ _"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_1\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ ∀ (f : R[X]), natDegree f < n → ↑(hasseDeriv n) f = 0",
"tactic": "exact fun h => hasseDeriv_eq_zero_of_lt_natDegree _ _"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ ∀ (n_1 : ℕ) (c : R), c ≠ 0 → natDegree (↑(hasseDeriv n) (↑(monomial n_1) c)) = n_1 - n",
"tactic": "classical\n simp only [ite_eq_right_iff, Ne.def, natDegree_monomial, hasseDeriv_monomial]\n intro k c c0 hh\n rw [← nsmul_eq_mul, smul_eq_zero, Nat.choose_eq_zero_iff] at hh\n exact (tsub_eq_zero_of_le (Or.resolve_right hh c0).le).symm"
},
{
"state_after": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ ∀ (n_1 : ℕ) (c : R), ¬c = 0 → ↑(choose n_1 n) * c = 0 → 0 = n_1 - n",
"state_before": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ ∀ (n_1 : ℕ) (c : R), c ≠ 0 → natDegree (↑(hasseDeriv n) (↑(monomial n_1) c)) = n_1 - n",
"tactic": "simp only [ite_eq_right_iff, Ne.def, natDegree_monomial, hasseDeriv_monomial]"
},
{
"state_after": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk✝ : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\nk : ℕ\nc : R\nc0 : ¬c = 0\nhh : ↑(choose k n) * c = 0\n⊢ 0 = k - n",
"state_before": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\n⊢ ∀ (n_1 : ℕ) (c : R), ¬c = 0 → ↑(choose n_1 n) * c = 0 → 0 = n_1 - n",
"tactic": "intro k c c0 hh"
},
{
"state_after": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk✝ : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\nk : ℕ\nc : R\nc0 : ¬c = 0\nhh : k < n ∨ c = 0\n⊢ 0 = k - n",
"state_before": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk✝ : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\nk : ℕ\nc : R\nc0 : ¬c = 0\nhh : ↑(choose k n) * c = 0\n⊢ 0 = k - n",
"tactic": "rw [← nsmul_eq_mul, smul_eq_zero, Nat.choose_eq_zero_iff] at hh"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_2\nR : Type u_1\ninst✝¹ : Semiring R\nk✝ : ℕ\nf : R[X]\ninst✝ : NoZeroSMulDivisors ℕ R\np : R[X]\nn : ℕ\nhn : n ≤ natDegree p\nk : ℕ\nc : R\nc0 : ¬c = 0\nhh : k < n ∨ c = 0\n⊢ 0 = k - n",
"tactic": "exact (tsub_eq_zero_of_le (Or.resolve_right hh c0).le).symm"
}
] |
[
228,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.mk_proj_snd'
|
[] |
[
372,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.comp_continuous_iff'
|
[] |
[
416,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.normSq_nonneg
|
[] |
[
484,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.comp_right_hasFDerivAt_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_3\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.203110\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : F → G\nx : E\nf' : F →L[𝕜] G\n⊢ HasFDerivAt (f ∘ ↑iso) (comp f' ↑iso) x ↔ HasFDerivAt f f' (↑iso x)",
"tactic": "simp only [← hasFDerivWithinAt_univ, ← comp_right_hasFDerivWithinAt_iff, preimage_univ]"
}
] |
[
222,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Algebra/BigOperators/Ring.lean
|
Finset.prod_powerset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nb : β\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : Finset α → β\n⊢ ∏ t in powerset s, f t = ∏ j in range (card s + 1), ∏ t in powersetLen j s, f t",
"tactic": "rw [powerset_card_disjiUnion, prod_disjiUnion]"
}
] |
[
268,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Topology/Category/Compactum.lean
|
Compactum.lim_eq_str
|
[
{
"state_after": "X : Compactum\nF : Ultrafilter X.A\n⊢ ∀ (s : Set X.A), str X F ∈ s → IsOpen s → s ∈ ↑F",
"state_before": "X : Compactum\nF : Ultrafilter X.A\n⊢ Ultrafilter.lim F = str X F",
"tactic": "rw [Ultrafilter.lim_eq_iff_le_nhds, le_nhds_iff]"
},
{
"state_after": "no goals",
"state_before": "X : Compactum\nF : Ultrafilter X.A\n⊢ ∀ (s : Set X.A), str X F ∈ s → IsOpen s → s ∈ ↑F",
"tactic": "tauto"
}
] |
[
374,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.mk₂'ₛₗ_apply
|
[] |
[
97,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/CategoryTheory/Endofunctor/Algebra.lean
|
CategoryTheory.Endofunctor.Adjunction.Coalgebra.homEquiv_naturality_str_symm
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nF G : C ⥤ C\nadj : F ⊣ G\nV₁ V₂ : Coalgebra G\nf : V₁ ⟶ V₂\n⊢ F.map f.f ≫ ↑(Adjunction.homEquiv adj V₂.V V₂.V).symm V₂.str = ↑(Adjunction.homEquiv adj V₁.V V₁.V).symm V₁.str ≫ f.f",
"tactic": "rw [← Adjunction.homEquiv_naturality_left_symm, ← Adjunction.homEquiv_naturality_right_symm,\n f.h]"
}
] |
[
466,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
zero_pow_eq_zero
|
[
{
"state_after": "case mp\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nn : ℕ\nh : 0 ^ n = 0\n⊢ 0 < n\n\ncase mpr\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nn : ℕ\nh : 0 < n\n⊢ 0 ^ n = 0",
"state_before": "R : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nn : ℕ\n⊢ 0 ^ n = 0 ↔ 0 < n",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nn : ℕ\nh : 0 ^ n = 0\n⊢ n ≠ 0",
"state_before": "case mp\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nn : ℕ\nh : 0 ^ n = 0\n⊢ 0 < n",
"tactic": "rw [pos_iff_ne_zero]"
},
{
"state_after": "case mp\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nh : 0 ^ 0 = 0\n⊢ False",
"state_before": "case mp\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nn : ℕ\nh : 0 ^ n = 0\n⊢ n ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nh : 0 ^ 0 = 0\n⊢ False",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type ?u.47966\nS : Type ?u.47969\nM : Type u_1\ninst✝¹ : MonoidWithZero M\ninst✝ : Nontrivial M\nn : ℕ\nh : 0 < n\n⊢ 0 ^ n = 0",
"tactic": "exact zero_pow' n h.ne.symm"
}
] |
[
105,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.X_pow_add_C_ne_zero
|
[
{
"state_after": "R : Type u\nS : Type v\na✝ b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\n⊢ ↑n ≠ ⊥",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\n⊢ degree (X ^ n + ↑C a) ≠ ⊥",
"tactic": "rw [degree_X_pow_add_C hn a]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\n⊢ ↑n ≠ ⊥",
"tactic": "exact WithBot.coe_ne_bot"
}
] |
[
1388,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1385,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
Monoid.mem_closure_union_iff
|
[
{
"state_after": "no goals",
"state_before": "M✝ : Type ?u.75380\ninst✝² : Monoid M✝\ns✝ : Set M✝\nA : Type ?u.75389\ninst✝¹ : AddMonoid A\nt✝ : Set A\nM : Type u_1\ninst✝ : CommMonoid M\ns t : Set M\nx : M\nhx : x ∈ Closure (s ∪ t)\nL : List M\nHL1✝ : ∀ (x : M), x ∈ L → x ∈ s ∪ t\nHL2 : List.prod L = x\nhd : M\ntl : List M\nih : (∀ (x : M), x ∈ tl → x ∈ s ∪ t) → ∃ y, y ∈ Closure s ∧ ∃ z, z ∈ Closure t ∧ y * z = List.prod tl\nHL1 : ∀ (x : M), x ∈ hd :: tl → x ∈ s ∪ t\ny : M\nhy : y ∈ Closure s\nz : M\nhz : z ∈ Closure t\nhyzx : y * z = List.prod tl\nhs : hd ∈ s\n⊢ hd * y * z = List.prod (hd :: tl)",
"tactic": "rw [mul_assoc, List.prod_cons, ← hyzx]"
},
{
"state_after": "no goals",
"state_before": "M✝ : Type ?u.75380\ninst✝² : Monoid M✝\ns✝ : Set M✝\nA : Type ?u.75389\ninst✝¹ : AddMonoid A\nt✝ : Set A\nM : Type u_1\ninst✝ : CommMonoid M\ns t : Set M\nx : M\nhx : x ∈ Closure (s ∪ t)\nL : List M\nHL1✝ : ∀ (x : M), x ∈ L → x ∈ s ∪ t\nHL2 : List.prod L = x\nhd : M\ntl : List M\nih : (∀ (x : M), x ∈ tl → x ∈ s ∪ t) → ∃ y, y ∈ Closure s ∧ ∃ z, z ∈ Closure t ∧ y * z = List.prod tl\nHL1 : ∀ (x : M), x ∈ hd :: tl → x ∈ s ∪ t\ny : M\nhy : y ∈ Closure s\nz : M\nhz : z ∈ Closure t\nhyzx : y * z = List.prod tl\nht : hd ∈ t\n⊢ y * (z * hd) = List.prod (hd :: tl)",
"tactic": "rw [← mul_assoc, List.prod_cons, ← hyzx, mul_comm hd]"
}
] |
[
416,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.top_snoc
|
[] |
[
560,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/Order/Partition/Finpartition.lean
|
Finpartition.card_atomise_le
|
[] |
[
576,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
575,
1
] |
Mathlib/Algebra/BigOperators/Intervals.lean
|
Finset.prod_range_mul_prod_Ico
|
[] |
[
94,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Order/SupIndep.lean
|
CompleteLattice.Independent.disjoint_biSup
|
[] |
[
357,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.prime_pow_dvd_iff_le
|
[
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ p ^ k ≤ Associates.mk a ↔ k ≤ count p (factors (Associates.mk a))",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\nm p : Associates α\nh₁ : m ≠ 0\nh₂ : Irreducible p\nk : ℕ\n⊢ p ^ k ≤ m ↔ k ≤ count p (factors m)",
"tactic": "obtain ⟨a, nz, rfl⟩ := Associates.exists_non_zero_rep h₁"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ some (replicate k { val := p, property := h₂ }) ≤ ↑(factors' a) ↔\n replicate k { val := p, property := ?intro.intro.hp } ≤ factors' a\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p",
"state_before": "case intro.intro\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ p ^ k ≤ Associates.mk a ↔ k ≤ count p (factors (Associates.mk a))",
"tactic": "rw [factors_mk _ nz, ← WithTop.some_eq_coe, count_some, Multiset.le_count_iff_replicate_le, ←\n factors_le, factors_prime_pow h₂, factors_mk _ nz]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ some (replicate k { val := p, property := h₂ }) ≤ ↑(factors' a) ↔\n replicate k { val := p, property := ?intro.intro.hp } ≤ factors' a\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p\n\ncase intro.intro.hp\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝¹ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\ninst✝ : Nontrivial α\np : Associates α\nh₂ : Irreducible p\nk : ℕ\na : α\nnz : a ≠ 0\nh₁ : Associates.mk a ≠ 0\n⊢ Irreducible p",
"tactic": "exact WithTop.coe_le_coe"
}
] |
[
1690,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1685,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.rTensor_comp_map
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommSemiring R\nR' : Type ?u.1289962\ninst✝¹³ : Monoid R'\nR'' : Type ?u.1289968\ninst✝¹² : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_2\nQ : Type u_6\nS : Type u_3\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : AddCommMonoid S\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝² : Module R S\ninst✝¹ : DistribMulAction R' M\ninst✝ : Module R'' M\ng✝ : P →ₗ[R] Q\nf✝ : N →ₗ[R] P\nf' : P →ₗ[R] S\nf : M →ₗ[R] P\ng : N →ₗ[R] Q\n⊢ comp (rTensor Q f') (map f g) = map (comp f' f) g",
"tactic": "simp only [lTensor, rTensor, ← map_comp, id_comp, comp_id]"
}
] |
[
1143,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1141,
1
] |
Mathlib/Algebra/Order/Module.lean
|
smul_nonneg_of_nonpos_of_nonpos
|
[] |
[
98,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
|
DifferentiableOn.cexp
|
[] |
[
150,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Topology/Bornology/Hom.lean
|
LocallyBoundedMap.comp_apply
|
[] |
[
175,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
isUnit_exp
|
[] |
[
476,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.comp₂
|
[] |
[
479,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContinuousLinearMap.iteratedFDerivWithin_comp_left
|
[] |
[
244,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_subset_Ioc
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.12902\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nha : a₂ ≤ a₁\nhb : b₁ ≤ b₂\n⊢ Ioc a₁ b₁ ⊆ Ioc a₂ b₂",
"tactic": "simpa [← coe_subset] using Set.Ioc_subset_Ioc ha hb"
}
] |
[
171,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Order/Partition/Finpartition.lean
|
Finpartition.biUnion_filter_atomise
|
[
{
"state_after": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\n⊢ a ∈ Finset.biUnion (filter (fun u => u ⊆ t ∧ Finset.Nonempty u) (atomise s F).parts) id ↔ a ∈ t",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\n⊢ Finset.biUnion (filter (fun u => u ⊆ t ∧ Finset.Nonempty u) (atomise s F).parts) id = t",
"tactic": "ext a"
},
{
"state_after": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\n⊢ ∃ a_1, a_1 ∈ filter (fun u => u ⊆ t ∧ Finset.Nonempty u) (atomise s F).parts ∧ a ∈ id a_1",
"state_before": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\n⊢ a ∈ Finset.biUnion (filter (fun u => u ⊆ t ∧ Finset.Nonempty u) (atomise s F).parts) id ↔ a ∈ t",
"tactic": "refine' mem_biUnion.trans ⟨fun ⟨u, hu, ha⟩ ↦ (mem_filter.1 hu).2.1 ha, fun ha ↦ _⟩"
},
{
"state_after": "case a.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nu : Finset α\nhu : u ∈ (atomise s F).parts\nhau : a ∈ u\n⊢ ∃ a_1, a_1 ∈ filter (fun u => u ⊆ t ∧ Finset.Nonempty u) (atomise s F).parts ∧ a ∈ id a_1",
"state_before": "case a\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\n⊢ ∃ a_1, a_1 ∈ filter (fun u => u ⊆ t ∧ Finset.Nonempty u) (atomise s F).parts ∧ a ∈ id a_1",
"tactic": "obtain ⟨u, hu, hau⟩ := (atomise s F).exists_mem (hts ha)"
},
{
"state_after": "case a.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nu : Finset α\nhu : u ∈ (atomise s F).parts\nhau : a ∈ u\nb : α\nhb : b ∈ u\n⊢ b ∈ t",
"state_before": "case a.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nu : Finset α\nhu : u ∈ (atomise s F).parts\nhau : a ∈ u\n⊢ ∃ a_1, a_1 ∈ filter (fun u => u ⊆ t ∧ Finset.Nonempty u) (atomise s F).parts ∧ a ∈ id a_1",
"tactic": "refine' ⟨u, mem_filter.2 ⟨hu, fun b hb ↦ _, _, hau⟩, hau⟩"
},
{
"state_after": "case a.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nb : α\nQ : Finset (Finset α)\n_hQ : Q ⊆ F\nhu : filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s ∈ (atomise s F).parts\nhau : a ∈ filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s\nhb : b ∈ filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s\n⊢ b ∈ t",
"state_before": "case a.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nu : Finset α\nhu : u ∈ (atomise s F).parts\nhau : a ∈ u\nb : α\nhb : b ∈ u\n⊢ b ∈ t",
"tactic": "obtain ⟨Q, _hQ, rfl⟩ := (mem_atomise.1 hu).2"
},
{
"state_after": "case a.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nb : α\nQ : Finset (Finset α)\n_hQ : Q ⊆ F\nhu : filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s ∈ (atomise s F).parts\nhau : a ∈ s ∧ ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ a ∈ u)\nhb : b ∈ s ∧ ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ b ∈ u)\n⊢ b ∈ t",
"state_before": "case a.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nb : α\nQ : Finset (Finset α)\n_hQ : Q ⊆ F\nhu : filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s ∈ (atomise s F).parts\nhau : a ∈ filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s\nhb : b ∈ filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s\n⊢ b ∈ t",
"tactic": "rw [mem_filter] at hau hb"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nF : Finset (Finset α)\nht : t ∈ F\nhts : t ⊆ s\na : α\nha : a ∈ t\nb : α\nQ : Finset (Finset α)\n_hQ : Q ⊆ F\nhu : filter (fun i => ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ i ∈ u)) s ∈ (atomise s F).parts\nhau : a ∈ s ∧ ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ a ∈ u)\nhb : b ∈ s ∧ ∀ (u : Finset α), u ∈ F → (u ∈ Q ↔ b ∈ u)\n⊢ b ∈ t",
"tactic": "rwa [← hb.2 _ ht, hau.2 _ ht]"
}
] |
[
587,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.top_toSubmodule
|
[] |
[
790,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.countable_not_continuousWithinAt_Ioi
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"tactic": "nontriviality α"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"tactic": "let s := { x | ¬ContinuousWithinAt f (Ioi x) x }"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nthis : ∀ (x : α), x ∈ s → ∃ z, f x < z ∧ ∀ (y : α), x < y → z ≤ f y\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"tactic": "choose! z hz using this"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"tactic": "have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nfs_count : Set.Countable (f '' s)\n⊢ Set.Countable {x | ¬ContinuousWithinAt f (Ioi x) x}",
"tactic": "exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ¬ContinuousWithinAt f (Ioi x) x\n⊢ ∃ z, f x < z ∧ ∀ (y : α), x < y → z ≤ f y",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\n⊢ ∀ (x : α), x ∈ s → ∃ z, f x < z ∧ ∀ (y : α), x < y → z ≤ f y",
"tactic": "rintro x (hx : ¬ContinuousWithinAt f (Ioi x) x)"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\n⊢ ContinuousWithinAt f (Ioi x) x",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ¬ContinuousWithinAt f (Ioi x) x\n⊢ ∃ z, f x < z ∧ ∀ (y : α), x < y → z ≤ f y",
"tactic": "contrapose! hx"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nm : β\nhm : m < f x\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, m < f b\n\ncase refine'_2\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, f b < u",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\n⊢ ContinuousWithinAt f (Ioi x) x",
"tactic": "refine' tendsto_order.2 ⟨fun m hm => _, fun u hu => _⟩"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\nv : α\nxv : x < v\nfvu : f v < u\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, f b < u",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, f b < u",
"tactic": "rcases hx u hu with ⟨v, xv, fvu⟩"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\nv : α\nxv : x < v\nfvu : f v < u\nthis : Ioo x v ∈ 𝓝[Ioi x] x\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, f b < u",
"state_before": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\nv : α\nxv : x < v\nfvu : f v < u\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, f b < u",
"tactic": "have : Ioo x v ∈ 𝓝[>] x := Ioo_mem_nhdsWithin_Ioi ⟨le_refl _, xv⟩"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y✝ : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\nv : α\nxv : x < v\nfvu : f v < u\nthis : Ioo x v ∈ 𝓝[Ioi x] x\ny : α\nhy : y ∈ Ioo x v\n⊢ f y < u",
"state_before": "case refine'_2.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\nv : α\nxv : x < v\nfvu : f v < u\nthis : Ioo x v ∈ 𝓝[Ioi x] x\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, f b < u",
"tactic": "filter_upwards [this]with y hy"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y✝ : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nu : β\nhu : u > f x\nv : α\nxv : x < v\nfvu : f v < u\nthis : Ioo x v ∈ 𝓝[Ioi x] x\ny : α\nhy : y ∈ Ioo x v\n⊢ f y < u",
"tactic": "apply (hf hy.2.le).trans_lt fvu"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝ y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nx : α\nhx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z\nm : β\nhm : m < f x\n⊢ ∀ᶠ (b : α) in 𝓝[Ioi x] x, m < f b",
"tactic": "filter_upwards [@self_mem_nhdsWithin _ _ x (Ioi x)] with y hy using hm.trans_le\n (hf (le_of_lt hy))"
},
{
"state_after": "case H\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\n⊢ StrictMonoOn f s",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\n⊢ InjOn f s",
"tactic": "apply StrictMonoOn.injOn"
},
{
"state_after": "case H\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝¹ y✝ : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nx : α\nhx : x ∈ s\ny : α\nx✝ : y ∈ s\nhxy : x < y\n⊢ f x < f y",
"state_before": "case H\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\n⊢ StrictMonoOn f s",
"tactic": "intro x hx y _ hxy"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx✝¹ y✝ : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nx : α\nhx : x ∈ s\ny : α\nx✝ : y ∈ s\nhxy : x < y\n⊢ f x < f y",
"tactic": "calc\n f x < z x := (hz x hx).1\n _ ≤ f y := (hz x hx).2 y hxy"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nA : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x))\n⊢ ∀ (x : β), x ∈ f '' s → x < z (invFunOn f s x)",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nA : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x))\n⊢ Set.Countable (f '' s)",
"tactic": "apply Set.PairwiseDisjoint.countable_of_Ioo A"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y✝ : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nA : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x))\ny : α\nys : y ∈ s\n⊢ f y < z (invFunOn f s (f y))",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nA : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x))\n⊢ ∀ (x : β), x ∈ f '' s → x < z (invFunOn f s x)",
"tactic": "rintro _ ⟨y, ys, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y✝ : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nA : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x))\ny : α\nys : y ∈ s\n⊢ f y < z (invFunOn f s (f y))",
"tactic": "simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\n⊢ PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x))",
"tactic": "rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv"
},
{
"state_after": "case intro.intro.intro.intro.inr\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nthis :\n ∀ (u : α),\n u ∈ s → ∀ (v : α), v ∈ s → f u ≠ f v → u ≤ v → (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)\nhle : ¬u ≤ v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)\n\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)",
"tactic": "wlog hle : u ≤ v generalizing u v"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)",
"tactic": "have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv)"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\n⊢ ∀ ⦃a : β⦄,\n a ∈ (fun x => Ioo x (z (invFunOn f s x))) (f u) → ∀ ⦃b : β⦄, b ∈ (fun x => Ioo x (z (invFunOn f s x))) (f v) → a ≠ b",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)",
"tactic": "apply disjoint_iff_forall_ne.2"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\na : β\nha : a ∈ (fun x => Ioo x (z (invFunOn f s x))) (f u)\nhb : a ∈ (fun x => Ioo x (z (invFunOn f s x))) (f v)\n⊢ False",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\n⊢ ∀ ⦃a : β⦄,\n a ∈ (fun x => Ioo x (z (invFunOn f s x))) (f u) → ∀ ⦃b : β⦄, b ∈ (fun x => Ioo x (z (invFunOn f s x))) (f v) → a ≠ b",
"tactic": "rintro a ha b hb rfl"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\na : β\nha : a ∈ Ioo (f u) (z u)\nhb : a ∈ Ioo (f v) (z v)\n⊢ False",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\na : β\nha : a ∈ (fun x => Ioo x (z (invFunOn f s x))) (f u)\nhb : a ∈ (fun x => Ioo x (z (invFunOn f s x))) (f v)\n⊢ False",
"tactic": "simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nhle : u ≤ v\nhlt : u < v\na : β\nha : a ∈ Ioo (f u) (z u)\nhb : a ∈ Ioo (f v) (z v)\n⊢ False",
"tactic": "exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.inr\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : ConditionallyCompleteLinearOrder β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : TopologicalSpace.SecondCountableTopology β\n✝ : Nontrivial α\ns : Set α := {x | ¬ContinuousWithinAt f (Ioi x) x}\nz : α → β\nhz : ∀ (x : α), x ∈ s → f x < z x ∧ ∀ (y : α), x < y → z x ≤ f y\nI : InjOn f s\nu : α\nus : u ∈ s\nv : α\nvs : v ∈ s\nhuv : f u ≠ f v\nthis :\n ∀ (u : α),\n u ∈ s → ∀ (v : α), v ∈ s → f u ≠ f v → u ≤ v → (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)\nhle : ¬u ≤ v\n⊢ (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v)",
"tactic": "exact (this v vs u us huv.symm (le_of_not_le hle)).symm"
}
] |
[
269,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Data/List/Basic.lean
|
List.bind_singleton'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.21722\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\n⊢ (List.bind l fun x => [x]) = l",
"tactic": "induction l <;> simp [*]"
}
] |
[
504,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
9
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.toSubring_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.95871\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\n⊢ ↑(frange 0) ⊆ ↑T",
"tactic": "simp [frange_zero]"
},
{
"state_after": "case a.a\nR : Type u\nS : Type ?u.95871\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ ↑(coeff (toSubring 0 T (_ : ↑∅ ⊆ ↑T)) i) = ↑(coeff 0 i)",
"state_before": "R : Type u\nS : Type ?u.95871\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\n⊢ toSubring 0 T (_ : ↑∅ ⊆ ↑T) = 0",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case a.a\nR : Type u\nS : Type ?u.95871\ninst✝¹ : Ring R\ninst✝ : Semiring S\nf : R →+* S\nx : S\np : R[X]\nT : Subring R\nhp : ↑(frange p) ⊆ ↑T\ni : ℕ\n⊢ ↑(coeff (toSubring 0 T (_ : ↑∅ ⊆ ↑T)) i) = ↑(coeff 0 i)",
"tactic": "simp"
}
] |
[
395,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.MapsTo.mem_iff
|
[] |
[
542,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
541,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.cos_arccos
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ cos (arccos x) = x",
"tactic": "rw [arccos, cos_pi_div_two_sub, sin_arcsin hx₁ hx₂]"
}
] |
[
362,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Algebra/EuclideanDomain/Defs.lean
|
EuclideanDomain.div_add_mod'
|
[
{
"state_after": "R : Type u\ninst✝ : EuclideanDomain R\nm k : R\n⊢ k * (m / k) + m % k = m",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\nm k : R\n⊢ m / k * k + m % k = m",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\nm k : R\n⊢ k * (m / k) + m % k = m",
"tactic": "exact div_add_mod _ _"
}
] |
[
139,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Order/Antichain.lean
|
IsAntichain.image_relIso
|
[] |
[
136,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/NumberTheory/SumFourSquares.lean
|
euler_four_squares
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\na b c d x y z w : R\n⊢ (a * x - b * y - c * z - d * w) ^ 2 + (a * y + b * x + c * w - d * z) ^ 2 + (a * z - b * w + c * x + d * y) ^ 2 +\n (a * w + b * z - c * y + d * x) ^ 2 =\n (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2)",
"tactic": "ring"
}
] |
[
39,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.quotientMulEquivOfEq_mk
|
[] |
[
450,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
normalize_eq_normalize
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na b : α\nhab : a ∣ b\nhba : b ∣ a\n✝ : Nontrivial α\n⊢ ↑normalize a = ↑normalize b",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na b : α\nhab : a ∣ b\nhba : b ∣ a\n⊢ ↑normalize a = ↑normalize b",
"tactic": "nontriviality α"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\n⊢ ↑normalize a = ↑normalize (a * ↑u)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na b : α\nhab : a ∣ b\nhba : b ∣ a\n✝ : Nontrivial α\n⊢ ↑normalize a = ↑normalize b",
"tactic": "rcases associated_of_dvd_dvd hab hba with ⟨u, rfl⟩"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\nha : a ≠ 0\n⊢ ↑normalize a = ↑normalize (a * ↑u)",
"state_before": "case intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\n⊢ ↑normalize a = ↑normalize (a * ↑u)",
"tactic": "refine' by_cases (by rintro rfl; simp only [zero_mul]) fun ha : a ≠ 0 => _"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\nha : a ≠ 0\n⊢ a * ↑(normUnit a) = a * ↑u * ↑(normUnit a) * ↑u⁻¹",
"state_before": "case intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\nha : a ≠ 0\n⊢ ↑normalize a = ↑normalize (a * ↑u)",
"tactic": "suffices a * ↑(normUnit a) = a * ↑u * ↑(normUnit a) * ↑u⁻¹ by\n simpa only [normalize_apply, mul_assoc, normUnit_mul ha u.ne_zero, normUnit_coe_units]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\nha : a ≠ 0\n⊢ a * ↑(normUnit a) = a * ↑u * ↑(normUnit a) * ↑u⁻¹",
"tactic": "calc\n a * ↑(normUnit a) = a * ↑(normUnit a) * ↑u * ↑u⁻¹ := (Units.mul_inv_cancel_right _ _).symm\n _ = a * ↑u * ↑(normUnit a) * ↑u⁻¹ := by rw [mul_right_comm a]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\n✝ : Nontrivial α\nu : αˣ\nhab : 0 ∣ 0 * ↑u\nhba : 0 * ↑u ∣ 0\n⊢ ↑normalize 0 = ↑normalize (0 * ↑u)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\n⊢ a = 0 → ↑normalize a = ↑normalize (a * ↑u)",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\n✝ : Nontrivial α\nu : αˣ\nhab : 0 ∣ 0 * ↑u\nhba : 0 * ↑u ∣ 0\n⊢ ↑normalize 0 = ↑normalize (0 * ↑u)",
"tactic": "simp only [zero_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\nha : a ≠ 0\nthis : a * ↑(normUnit a) = a * ↑u * ↑(normUnit a) * ↑u⁻¹\n⊢ ↑normalize a = ↑normalize (a * ↑u)",
"tactic": "simpa only [normalize_apply, mul_assoc, normUnit_mul ha u.ne_zero, normUnit_coe_units]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nu : αˣ\nhab : a ∣ a * ↑u\nhba : a * ↑u ∣ a\nha : a ≠ 0\n⊢ a * ↑(normUnit a) * ↑u * ↑u⁻¹ = a * ↑u * ↑(normUnit a) * ↑u⁻¹",
"tactic": "rw [mul_right_comm a]"
}
] |
[
184,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/NormedSpace/Ray.lean
|
SameRay.norm_eq_iff
|
[] |
[
120,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Topology/Bases.lean
|
QuotientMap.secondCountableTopology
|
[
{
"state_after": "case intro.intro.intro\nα : Type u\nt : TopologicalSpace α\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nπ : X → Y\ninst✝ : SecondCountableTopology X\nh' : QuotientMap π\nh : IsOpenMap π\nV : Set (Set X)\nV_countable : Set.Countable V\nV_generates : IsTopologicalBasis V\n⊢ ∃ b, Set.Countable b ∧ inst✝¹ = generateFrom b",
"state_before": "α : Type u\nt : TopologicalSpace α\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nπ : X → Y\ninst✝ : SecondCountableTopology X\nh' : QuotientMap π\nh : IsOpenMap π\n⊢ ∃ b, Set.Countable b ∧ inst✝¹ = generateFrom b",
"tactic": "obtain ⟨V, V_countable, -, V_generates⟩ := exists_countable_basis X"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u\nt : TopologicalSpace α\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\nπ : X → Y\ninst✝ : SecondCountableTopology X\nh' : QuotientMap π\nh : IsOpenMap π\nV : Set (Set X)\nV_countable : Set.Countable V\nV_generates : IsTopologicalBasis V\n⊢ ∃ b, Set.Countable b ∧ inst✝¹ = generateFrom b",
"tactic": "exact ⟨Set.image π '' V, V_countable.image (Set.image π),\n (V_generates.quotientMap h' h).eq_generateFrom⟩"
}
] |
[
860,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
Associates.out_mk
|
[] |
[
223,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.write_nth
|
[] |
[
674,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
670,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_real_eq_id
|
[] |
[
864,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
863,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.le_zero_iff
|
[
{
"state_after": "n✝ m n : ℕ\ninst✝ : NeZero n\nk : Fin n\nh : k ≤ 0\n⊢ 0 = ↑0",
"state_before": "n✝ m n : ℕ\ninst✝ : NeZero n\nk : Fin n\nh : k ≤ 0\n⊢ ↑k = ↑0",
"tactic": "rw [Nat.eq_zero_of_le_zero h]"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\ninst✝ : NeZero n\nk : Fin n\nh : k ≤ 0\n⊢ 0 = ↑0",
"tactic": "rfl"
},
{
"state_after": "n✝ m n : ℕ\ninst✝ : NeZero n\n⊢ 0 ≤ 0",
"state_before": "n✝ m n : ℕ\ninst✝ : NeZero n\nk : Fin n\n⊢ k = 0 → k ≤ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\ninst✝ : NeZero n\n⊢ 0 ≤ 0",
"tactic": "exact le_refl _"
}
] |
[
989,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
988,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.inv_comp_eq_id
|
[] |
[
490,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/AlgebraicGeometry/Stalks.lean
|
AlgebraicGeometry.PresheafedSpace.stalkMap.congr
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα β : X ⟶ Y\nh₁ : α = β\nx x' : ↑↑X\nh₂ : x = x'\n⊢ stalk X x = stalk X x'",
"tactic": "rw [h₂]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα β : X ⟶ Y\nh₁ : α = β\nx x' : ↑↑X\nh₂ : x = x'\n⊢ stalk Y ((CategoryTheory.forget TopCat).map α.base x) = stalk Y ((CategoryTheory.forget TopCat).map β.base x')",
"tactic": "rw [h₁, h₂]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα : X ⟶ Y\nx x' : ↑↑X\nh₂ : x = x'\nU : Opens ↑↑Y\nhx : (CategoryTheory.forget TopCat).map α.base x ∈ U\n⊢ germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x') =\n germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n eqToHom\n (_ :\n stalk Y ((CategoryTheory.forget TopCat).map α.base x) =\n stalk Y ((CategoryTheory.forget TopCat).map α.base x')) ≫\n stalkMap α x'",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα β : X ⟶ Y\nh₁ : α = β\nx x' : ↑↑X\nh₂ : x = x'\nU : Opens ↑↑Y\nhx : (CategoryTheory.forget TopCat).map α.base x ∈ U\n⊢ germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x') =\n germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n eqToHom\n (_ :\n stalk Y ((CategoryTheory.forget TopCat).map α.base x) =\n stalk Y ((CategoryTheory.forget TopCat).map β.base x')) ≫\n stalkMap β x'",
"tactic": "subst h₁"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα : X ⟶ Y\nx : ↑↑X\nU : Opens ↑↑Y\nhx : (CategoryTheory.forget TopCat).map α.base x ∈ U\n⊢ germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x) =\n germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n eqToHom\n (_ :\n stalk Y ((CategoryTheory.forget TopCat).map α.base x) =\n stalk Y ((CategoryTheory.forget TopCat).map α.base x)) ≫\n stalkMap α x",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα : X ⟶ Y\nx x' : ↑↑X\nh₂ : x = x'\nU : Opens ↑↑Y\nhx : (CategoryTheory.forget TopCat).map α.base x ∈ U\n⊢ germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x') =\n germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n eqToHom\n (_ :\n stalk Y ((CategoryTheory.forget TopCat).map α.base x) =\n stalk Y ((CategoryTheory.forget TopCat).map α.base x')) ≫\n stalkMap α x'",
"tactic": "subst h₂"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nα : X ⟶ Y\nx : ↑↑X\nU : Opens ↑↑Y\nhx : (CategoryTheory.forget TopCat).map α.base x ∈ U\n⊢ germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x) =\n germ Y.presheaf { val := (CategoryTheory.forget TopCat).map α.base x, property := hx } ≫\n eqToHom\n (_ :\n stalk Y ((CategoryTheory.forget TopCat).map α.base x) =\n stalk Y ((CategoryTheory.forget TopCat).map α.base x)) ≫\n stalkMap α x",
"tactic": "simp"
}
] |
[
169,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.total_unique
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_3\nN : Type ?u.323475\nP : Type ?u.323478\nR : Type u_2\nS : Type ?u.323484\ninst✝¹³ : Semiring R\ninst✝¹² : Semiring S\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R N\ninst✝⁷ : AddCommMonoid P\ninst✝⁶ : Module R P\nα' : Type ?u.323576\nM' : Type ?u.323579\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M'\ninst✝¹ : Module R M\nv✝ : α → M\nv' : α' → M'\ninst✝ : Unique α\nl : α →₀ R\nv : α → M\n⊢ ↑(Finsupp.total α M R v) l = ↑l default • v default",
"tactic": "rw [← total_single, ← unique_single l]"
}
] |
[
585,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.lineMap_mem
|
[] |
[
481,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.inducedOuterMeasure_exists_set
|
[
{
"state_after": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nh : ↑(inducedOuterMeasure m P0 m0) s < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"tactic": "have h := ENNReal.lt_add_right hs hε"
},
{
"state_after": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nh : (⨅ (t : Set α) (ht : P t) (_ : s ⊆ t), m t ht) < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nh : ↑(inducedOuterMeasure m P0 m0) s < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"tactic": "conv at h =>\n lhs\n rw [inducedOuterMeasure_eq_iInf _ msU m_mono]"
},
{
"state_after": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nh : ∃ i h i_1, m i (_ : P i) < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nh : (⨅ (t : Set α) (ht : P t) (_ : s ⊆ t), m t ht) < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"tactic": "simp only [iInf_lt_iff] at h"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nt : Set α\nh1t : P t\nh2t : s ⊆ t\nh3t : m t (_ : P t) < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nh : ∃ i h i_1, m i (_ : P i) < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"tactic": "rcases h with ⟨t, h1t, h2t, h3t⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nhs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nt : Set α\nh1t : P t\nh2t : s ⊆ t\nh3t : m t (_ : P t) < ↑(inducedOuterMeasure m P0 m0) s + ε\n⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε",
"tactic": "exact\n ⟨t, h1t, h2t, le_trans (le_of_eq <| inducedOuterMeasure_eq' _ msU m_mono h1t) (le_of_lt h3t)⟩"
}
] |
[
1513,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1502,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
NNReal.pow_nat_rpow_nat_inv
|
[
{
"state_after": "x : ℝ≥0\nn : ℕ\nhn : n ≠ 0\n⊢ (↑x ^ n) ^ (↑n)⁻¹ = ↑x",
"state_before": "x : ℝ≥0\nn : ℕ\nhn : n ≠ 0\n⊢ (x ^ n) ^ (↑n)⁻¹ = x",
"tactic": "rw [← NNReal.coe_eq, coe_rpow, NNReal.coe_pow]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ≥0\nn : ℕ\nhn : n ≠ 0\n⊢ (↑x ^ n) ^ (↑n)⁻¹ = ↑x",
"tactic": "exact Real.pow_nat_rpow_nat_inv x.2 hn"
}
] |
[
268,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/CategoryTheory/Conj.lean
|
CategoryTheory.Iso.homCongr_apply
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y X₁ Y₁ : C\nα : X ≅ X₁\nβ : Y ≅ Y₁\nf : X ⟶ Y\n⊢ ↑(homCongr α β) f = α.inv ≫ f ≫ β.hom",
"tactic": "rfl"
}
] |
[
54,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.mk''_eq_coe
|
[] |
[
478,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
codisjoint_hnot_right
|
[] |
[
1037,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1036,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.rTensor_smul
|
[] |
[
1063,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/MeasureTheory/Function/Jacobian.lean
|
MeasureTheory.measurableEmbedding_of_fderivWithin
|
[] |
[
793,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
789,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter₂_comm
|
[] |
[
803,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_le_one_iff
|
[
{
"state_after": "case inl\nι : Type ?u.173722\nα : Type u_1\nβ : Type ?u.173728\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : b < 0\n⊢ a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a\n\ncase inr.inl\nι : Type ?u.173722\nα : Type u_1\nβ : Type ?u.173728\ninst✝ : LinearOrderedField α\na c d : α\nn : ℤ\n⊢ a / 0 ≤ 1 ↔ 0 < 0 ∧ a ≤ 0 ∨ 0 = 0 ∨ 0 < 0 ∧ 0 ≤ a\n\ncase inr.inr\nι : Type ?u.173722\nα : Type u_1\nβ : Type ?u.173728\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : 0 < b\n⊢ a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a",
"state_before": "ι : Type ?u.173722\nα : Type u_1\nβ : Type ?u.173728\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\n⊢ a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a",
"tactic": "rcases lt_trichotomy b 0 with (hb | rfl | hb)"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.173722\nα : Type u_1\nβ : Type ?u.173728\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : b < 0\n⊢ a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a",
"tactic": "simp [hb, hb.not_lt, hb.ne, div_le_one_of_neg]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nι : Type ?u.173722\nα : Type u_1\nβ : Type ?u.173728\ninst✝ : LinearOrderedField α\na c d : α\nn : ℤ\n⊢ a / 0 ≤ 1 ↔ 0 < 0 ∧ a ≤ 0 ∨ 0 = 0 ∨ 0 < 0 ∧ 0 ≤ a",
"tactic": "simp [zero_le_one]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nι : Type ?u.173722\nα : Type u_1\nβ : Type ?u.173728\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhb : 0 < b\n⊢ a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a",
"tactic": "simp [hb, hb.not_lt, div_le_one, hb.ne.symm]"
}
] |
[
860,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
856,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Iso.eq_comp_inv
|
[] |
[
234,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_add
|
[
{
"state_after": "case intro\nα : Type ?u.789991\nβ : Type ?u.789994\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nhb : b ≠ ⊤\na : ℝ≥0\n⊢ ENNReal.toReal (↑a + b) = ENNReal.toReal ↑a + ENNReal.toReal b",
"state_before": "α : Type ?u.789991\nβ : Type ?u.789994\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ ⊤\nhb : b ≠ ⊤\n⊢ ENNReal.toReal (a + b) = ENNReal.toReal a + ENNReal.toReal b",
"tactic": "lift a to ℝ≥0 using ha"
},
{
"state_after": "case intro.intro\nα : Type ?u.789991\nβ : Type ?u.789994\nc d : ℝ≥0∞\nr p q a b : ℝ≥0\n⊢ ENNReal.toReal (↑a + ↑b) = ENNReal.toReal ↑a + ENNReal.toReal ↑b",
"state_before": "case intro\nα : Type ?u.789991\nβ : Type ?u.789994\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nhb : b ≠ ⊤\na : ℝ≥0\n⊢ ENNReal.toReal (↑a + b) = ENNReal.toReal ↑a + ENNReal.toReal b",
"tactic": "lift b to ℝ≥0 using hb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type ?u.789991\nβ : Type ?u.789994\nc d : ℝ≥0∞\nr p q a b : ℝ≥0\n⊢ ENNReal.toReal (↑a + ↑b) = ENNReal.toReal ↑a + ENNReal.toReal ↑b",
"tactic": "rfl"
}
] |
[
1948,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1945,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isNat_ofScientific_of_false
|
[
{
"state_after": "α : Type u_1\ninst✝ : DivisionRing α\nσα : OfScientific α\nn✝¹ n✝ x✝ : ℕ\nσh : OfScientific.ofScientific = fun m s e => ↑(Rat.ofScientific m s e)\nh : x✝ = Nat.mul n✝¹ (10 ^ n✝)\n⊢ ↑(↑n✝¹ * 10 ^ n✝) = ↑n✝¹ * 10 ^ n✝",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\nσα : OfScientific α\nn✝¹ n✝ x✝ : ℕ\nσh : OfScientific.ofScientific = fun m s e => ↑(Rat.ofScientific m s e)\nh : x✝ = Nat.mul n✝¹ (10 ^ n✝)\n⊢ OfScientific.ofScientific (↑n✝¹) false ↑n✝ = ↑x✝",
"tactic": "simp [σh, Rat.ofScientific_false_def, h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\nσα : OfScientific α\nn✝¹ n✝ x✝ : ℕ\nσh : OfScientific.ofScientific = fun m s e => ↑(Rat.ofScientific m s e)\nh : x✝ = Nat.mul n✝¹ (10 ^ n✝)\n⊢ ↑(↑n✝¹ * 10 ^ n✝) = ↑n✝¹ * 10 ^ n✝",
"tactic": "norm_cast"
}
] |
[
603,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_of_moveRight_le
|
[] |
[
482,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_pi_eq'
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.23519\nβ : Type ?u.23522\nγ : Type ?u.23525\nδ : Type ?u.23528\ninst✝¹ : TopologicalSpace α✝\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (α i)\nI : Set ι\nhI : Set.Finite I\ns : (i : ι) → Set (α i)\nx : (i : ι) → α i\n⊢ 𝓝[Set.pi I s] x = ⨅ (i : ι), comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i))",
"tactic": "simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ←\n iInf_principal_finite hI, ← iInf_inf_eq]"
}
] |
[
319,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Tactic/LinearCombination.lean
|
Mathlib.Tactic.LinearCombination.pf_sub_c
|
[] |
[
40,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
|
MeasureTheory.MeasurePreserving.exists_mem_image_mem
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.275571\nγ : Type ?u.275574\nδ : Type ?u.275577\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\ninst✝¹ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nμ : Measure α\nf : α → α\ns : Set α\ninst✝ : IsFiniteMeasure μ\nhf : MeasurePreserving f\nhs : MeasurableSet s\nhs' : ↑↑μ s ≠ 0\nN : ℕ\nhN : ↑↑μ Set.univ < ↑N * ↑↑μ s\n⊢ ∃ x, x ∈ s ∧ ∃ m x_1, (f^[m]) x ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.275571\nγ : Type ?u.275574\nδ : Type ?u.275577\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\ninst✝¹ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nμ : Measure α\nf : α → α\ns : Set α\ninst✝ : IsFiniteMeasure μ\nhf : MeasurePreserving f\nhs : MeasurableSet s\nhs' : ↑↑μ s ≠ 0\n⊢ ∃ x, x ∈ s ∧ ∃ m x_1, (f^[m]) x ∈ s",
"tactic": "rcases ENNReal.exists_nat_mul_gt hs' (measure_ne_top μ (Set.univ : Set α)) with ⟨N, hN⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.275571\nγ : Type ?u.275574\nδ : Type ?u.275577\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\ninst✝¹ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nμ : Measure α\nf : α → α\ns : Set α\ninst✝ : IsFiniteMeasure μ\nhf : MeasurePreserving f\nhs : MeasurableSet s\nhs' : ↑↑μ s ≠ 0\nN : ℕ\nhN : ↑↑μ Set.univ < ↑N * ↑↑μ s\nx : α\nhx : x ∈ s\nm : ℕ\nhm : m ∈ Set.Ioo 0 N\nhmx : (f^[m]) x ∈ s\n⊢ ∃ x, x ∈ s ∧ ∃ m x_1, (f^[m]) x ∈ s",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.275571\nγ : Type ?u.275574\nδ : Type ?u.275577\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\ninst✝¹ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nμ : Measure α\nf : α → α\ns : Set α\ninst✝ : IsFiniteMeasure μ\nhf : MeasurePreserving f\nhs : MeasurableSet s\nhs' : ↑↑μ s ≠ 0\nN : ℕ\nhN : ↑↑μ Set.univ < ↑N * ↑↑μ s\n⊢ ∃ x, x ∈ s ∧ ∃ m x_1, (f^[m]) x ∈ s",
"tactic": "rcases hf.exists_mem_image_mem_of_volume_lt_mul_volume hs hN with ⟨x, hx, m, hm, hmx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.275571\nγ : Type ?u.275574\nδ : Type ?u.275577\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\ninst✝¹ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nμ : Measure α\nf : α → α\ns : Set α\ninst✝ : IsFiniteMeasure μ\nhf : MeasurePreserving f\nhs : MeasurableSet s\nhs' : ↑↑μ s ≠ 0\nN : ℕ\nhN : ↑↑μ Set.univ < ↑N * ↑↑μ s\nx : α\nhx : x ∈ s\nm : ℕ\nhm : m ∈ Set.Ioo 0 N\nhmx : (f^[m]) x ∈ s\n⊢ ∃ x, x ∈ s ∧ ∃ m x_1, (f^[m]) x ∈ s",
"tactic": "exact ⟨x, hx, m, hm.1.ne', hmx⟩"
}
] |
[
167,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasDerivWithinAt.cosh
|
[] |
[
867,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
865,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderOnWith.ediam_image_le
|
[] |
[
164,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Analysis/Convex/Strict.lean
|
StrictConvex.sub
|
[] |
[
391,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image_subset_image₂_left
|
[] |
[
95,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/List/Basic.lean
|
List.splitAt_eq_take_drop
|
[
{
"state_after": "case pos\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl : List α\nh : n < length l\n⊢ (if n < length l then (Array.toList #[] ++ take n l, drop n l) else (l, [])) = (take n l, drop n l)\n\ncase neg\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl : List α\nh : ¬n < length l\n⊢ (if n < length l then (Array.toList #[] ++ take n l, drop n l) else (l, [])) = (take n l, drop n l)",
"state_before": "ι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl : List α\n⊢ splitAt n l = (take n l, drop n l)",
"tactic": "by_cases h : n < l.length <;> rw [splitAt, go_eq_take_drop]"
},
{
"state_after": "case pos\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl : List α\nh : n < length l\n⊢ (Array.toList #[] ++ take n l, drop n l) = (take n l, drop n l)",
"state_before": "case pos\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl : List α\nh : n < length l\n⊢ (if n < length l then (Array.toList #[] ++ take n l, drop n l) else (l, [])) = (take n l, drop n l)",
"tactic": "rw [if_pos h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl : List α\nh : n < length l\n⊢ (Array.toList #[] ++ take n l, drop n l) = (take n l, drop n l)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl : List α\nh : ¬n < length l\n⊢ (if n < length l then (Array.toList #[] ++ take n l, drop n l) else (l, [])) = (take n l, drop n l)",
"tactic": "rw [if_neg h, take_all_of_le <| le_of_not_lt h, drop_eq_nil_of_le <| le_of_not_lt h]"
},
{
"state_after": "case inl\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn✝ : ℕ\nl✝ : List α\nn : ℕ\nl xs : List α\nacc : Array α\nh : n < length xs\n⊢ splitAt.go l xs n acc = (Array.toList acc ++ take n xs, drop n xs)\n\ncase inr\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn✝ : ℕ\nl✝ : List α\nn : ℕ\nl xs : List α\nacc : Array α\nh : ¬n < length xs\n⊢ splitAt.go l xs n acc = (l, [])",
"state_before": "ι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn✝ : ℕ\nl✝ : List α\nn : ℕ\nl xs : List α\nacc : Array α\n⊢ splitAt.go l xs n acc = if n < length xs then (Array.toList acc ++ take n xs, drop n xs) else (l, [])",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl.zero.x_4\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\n⊢ xs = [] → False\n\ncase inl.zero.x_5\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\n⊢ ∀ (x : α) (xs_1 : List α) (n : ℕ), xs = x :: xs_1 → zero = succ n → False",
"state_before": "case inl.zero\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\n⊢ splitAt.go l xs zero acc = (Array.toList acc ++ take zero xs, drop zero xs)",
"tactic": "rw [splitAt.go, take, drop, append_nil]"
},
{
"state_after": "case inl.zero.x_4\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\nh₁ : xs = []\n⊢ False",
"state_before": "case inl.zero.x_4\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\n⊢ xs = [] → False",
"tactic": "intros h₁"
},
{
"state_after": "case inl.zero.x_4\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length []\nh₁ : xs = []\n⊢ False",
"state_before": "case inl.zero.x_4\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\nh₁ : xs = []\n⊢ False",
"tactic": "rw [h₁] at h"
},
{
"state_after": "no goals",
"state_before": "case inl.zero.x_4\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length []\nh₁ : xs = []\n⊢ False",
"tactic": "contradiction"
},
{
"state_after": "case inl.zero.x_5\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\nx✝ : α\nxs_1✝ : List α\nn✝ : ℕ\nx_1✝ : xs = x✝ :: xs_1✝\nx_2✝ : zero = succ n✝\n⊢ False",
"state_before": "case inl.zero.x_5\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\n⊢ ∀ (x : α) (xs_1 : List α) (n : ℕ), xs = x :: xs_1 → zero = succ n → False",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case inl.zero.x_5\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : zero < length xs\nx✝ : α\nxs_1✝ : List α\nn✝ : ℕ\nx_1✝ : xs = x✝ :: xs_1✝\nx_2✝ : zero = succ n✝\n⊢ False",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case inl.succ\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih :\n ∀ (xs : List α) (acc : Array α),\n n✝ < length xs → splitAt.go l xs n✝ acc = (Array.toList acc ++ take n✝ xs, drop n✝ xs)\nxs : List α\nacc : Array α\nh : succ n✝ < length xs\n⊢ splitAt.go l xs (succ n✝) acc = (Array.toList acc ++ take (succ n✝) xs, drop (succ n✝) xs)",
"tactic": "cases xs with\n| nil => contradiction\n| cons hd tl =>\n rw [length, succ_eq_add_one] at h\n rw [splitAt.go, take, drop, append_cons, Array.toList_eq, ←Array.push_data,\n ←Array.toList_eq]\n exact ih _ _ <| lt_of_add_lt_add_right h"
},
{
"state_after": "no goals",
"state_before": "case inl.succ.nil\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih :\n ∀ (xs : List α) (acc : Array α),\n n✝ < length xs → splitAt.go l xs n✝ acc = (Array.toList acc ++ take n✝ xs, drop n✝ xs)\nacc : Array α\nh : succ n✝ < length []\n⊢ splitAt.go l [] (succ n✝) acc = (Array.toList acc ++ take (succ n✝) [], drop (succ n✝) [])",
"tactic": "contradiction"
},
{
"state_after": "case inl.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih :\n ∀ (xs : List α) (acc : Array α),\n n✝ < length xs → splitAt.go l xs n✝ acc = (Array.toList acc ++ take n✝ xs, drop n✝ xs)\nacc : Array α\nhd : α\ntl : List α\nh : n✝ + 1 < length tl + 1\n⊢ splitAt.go l (hd :: tl) (succ n✝) acc = (Array.toList acc ++ take (succ n✝) (hd :: tl), drop (succ n✝) (hd :: tl))",
"state_before": "case inl.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih :\n ∀ (xs : List α) (acc : Array α),\n n✝ < length xs → splitAt.go l xs n✝ acc = (Array.toList acc ++ take n✝ xs, drop n✝ xs)\nacc : Array α\nhd : α\ntl : List α\nh : succ n✝ < length (hd :: tl)\n⊢ splitAt.go l (hd :: tl) (succ n✝) acc = (Array.toList acc ++ take (succ n✝) (hd :: tl), drop (succ n✝) (hd :: tl))",
"tactic": "rw [length, succ_eq_add_one] at h"
},
{
"state_after": "case inl.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih :\n ∀ (xs : List α) (acc : Array α),\n n✝ < length xs → splitAt.go l xs n✝ acc = (Array.toList acc ++ take n✝ xs, drop n✝ xs)\nacc : Array α\nhd : α\ntl : List α\nh : n✝ + 1 < length tl + 1\n⊢ splitAt.go l tl n✝ (Array.push acc hd) = (Array.toList (Array.push acc hd) ++ take n✝ tl, drop n✝ tl)",
"state_before": "case inl.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih :\n ∀ (xs : List α) (acc : Array α),\n n✝ < length xs → splitAt.go l xs n✝ acc = (Array.toList acc ++ take n✝ xs, drop n✝ xs)\nacc : Array α\nhd : α\ntl : List α\nh : n✝ + 1 < length tl + 1\n⊢ splitAt.go l (hd :: tl) (succ n✝) acc = (Array.toList acc ++ take (succ n✝) (hd :: tl), drop (succ n✝) (hd :: tl))",
"tactic": "rw [splitAt.go, take, drop, append_cons, Array.toList_eq, ←Array.push_data,\n ←Array.toList_eq]"
},
{
"state_after": "no goals",
"state_before": "case inl.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih :\n ∀ (xs : List α) (acc : Array α),\n n✝ < length xs → splitAt.go l xs n✝ acc = (Array.toList acc ++ take n✝ xs, drop n✝ xs)\nacc : Array α\nhd : α\ntl : List α\nh : n✝ + 1 < length tl + 1\n⊢ splitAt.go l tl n✝ (Array.push acc hd) = (Array.toList (Array.push acc hd) ++ take n✝ tl, drop n✝ tl)",
"tactic": "exact ih _ _ <| lt_of_add_lt_add_right h"
},
{
"state_after": "case inr.zero\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : length xs = 0\n⊢ splitAt.go l xs zero acc = (l, [])",
"state_before": "case inr.zero\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : ¬zero < length xs\n⊢ splitAt.go l xs zero acc = (l, [])",
"tactic": "rw [zero_eq, not_lt, nonpos_iff_eq_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case inr.zero\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs✝ ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l xs : List α\nacc : Array α\nh : length xs = 0\n⊢ splitAt.go l xs zero acc = (l, [])",
"tactic": "rw [eq_nil_of_length_eq_zero h, splitAt.go]"
},
{
"state_after": "no goals",
"state_before": "case inr.succ.nil\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih : ∀ (xs : List α) (acc : Array α), ¬n✝ < length xs → splitAt.go l xs n✝ acc = (l, [])\nacc : Array α\nh : ¬succ n✝ < length []\n⊢ splitAt.go l [] (succ n✝) acc = (l, [])",
"tactic": "rw [splitAt.go]"
},
{
"state_after": "case inr.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih : ∀ (xs : List α) (acc : Array α), ¬n✝ < length xs → splitAt.go l xs n✝ acc = (l, [])\nacc : Array α\nhd : α\ntl : List α\nh : ¬n✝ + 1 < length tl + 1\n⊢ splitAt.go l (hd :: tl) (succ n✝) acc = (l, [])",
"state_before": "case inr.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih : ∀ (xs : List α) (acc : Array α), ¬n✝ < length xs → splitAt.go l xs n✝ acc = (l, [])\nacc : Array α\nhd : α\ntl : List α\nh : ¬succ n✝ < length (hd :: tl)\n⊢ splitAt.go l (hd :: tl) (succ n✝) acc = (l, [])",
"tactic": "rw [length, succ_eq_add_one] at h"
},
{
"state_after": "case inr.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih : ∀ (xs : List α) (acc : Array α), ¬n✝ < length xs → splitAt.go l xs n✝ acc = (l, [])\nacc : Array α\nhd : α\ntl : List α\nh : ¬n✝ + 1 < length tl + 1\n⊢ splitAt.go l tl n✝ (Array.push acc hd) = (l, [])",
"state_before": "case inr.succ.cons\nι : Type ?u.283895\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nn : ℕ\nl✝ l : List α\nn✝ : ℕ\nih : ∀ (xs : List α) (acc : Array α), ¬n✝ < length xs → splitAt.go l xs n✝ acc = (l, [])\nacc : Array α\nhd : α\ntl : List α\nh : ¬n✝ + 1 < length tl + 1\n⊢ splitAt.go l (hd :: tl) (succ n✝) acc = (l, [])",
"tactic": "rw [splitAt.go]"
}
] |
[
2897,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2866,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithTop.coe_bit1
|
[] |
[
139,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
|
AlgebraicTopology.DoldKan.Γ₀.splitting_iso_hom_eq_id
|
[] |
[
261,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Algebra/Lie/Normalizer.lean
|
LieSubalgebra.ideal_in_normalizer
|
[
{
"state_after": "R : Type u_2\nL : Type u_1\nM : Type ?u.56361\nM' : Type ?u.56364\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhx : x ∈ normalizer H\nhy : y ∈ H\n⊢ ⁅y, x⁆ ∈ H",
"state_before": "R : Type u_2\nL : Type u_1\nM : Type ?u.56361\nM' : Type ?u.56364\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhx : x ∈ normalizer H\nhy : y ∈ H\n⊢ ⁅x, y⁆ ∈ H",
"tactic": "rw [← lie_skew, neg_mem_iff (G := L)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nL : Type u_1\nM : Type ?u.56361\nM' : Type ?u.56364\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhx : x ∈ normalizer H\nhy : y ∈ H\n⊢ ⁅y, x⁆ ∈ H",
"tactic": "exact hx ⟨y, hy⟩"
}
] |
[
158,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.nat_cast_zmod_eq_zero_iff_dvd
|
[
{
"state_after": "no goals",
"state_before": "a b : ℕ\n⊢ ↑a = 0 ↔ b ∣ a",
"tactic": "rw [← Nat.cast_zero, ZMod.nat_cast_eq_nat_cast_iff, Nat.modEq_zero_iff_dvd]"
}
] |
[
483,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.div_mul_le_self
|
[
{
"state_after": "no goals",
"state_before": "m : Nat\n⊢ m / 0 * 0 ≤ m",
"tactic": "simp"
}
] |
[
562,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
560,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
IsCompact.div_closedBall_one
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\nhs : IsCompact s\nhδ : 0 ≤ δ\n⊢ s / closedBall 1 δ = cthickening δ s",
"tactic": "simp [div_eq_mul_inv, hs.mul_closedBall_one hδ]"
}
] |
[
266,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.setToFun_congr_measure_of_add_left
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1664515\nG : Type ?u.1664518\n𝕜 : Type ?u.1664521\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : DominatedFinMeasAdditive μ' T C\nf : α → E\nhf : Integrable f\n⊢ μ' ≤ 1 • (μ + μ')",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1664515\nG : Type ?u.1664518\n𝕜 : Type ?u.1664521\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : DominatedFinMeasAdditive μ' T C\nf : α → E\nhf : Integrable f\n⊢ setToFun (μ + μ') T hT_add f = setToFun μ' T hT f",
"tactic": "refine' setToFun_congr_measure_of_integrable 1 one_ne_top _ hT_add hT f hf"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1664515\nG : Type ?u.1664518\n𝕜 : Type ?u.1664521\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : DominatedFinMeasAdditive μ' T C\nf : α → E\nhf : Integrable f\n⊢ μ' ≤ μ + μ'",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1664515\nG : Type ?u.1664518\n𝕜 : Type ?u.1664521\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : DominatedFinMeasAdditive μ' T C\nf : α → E\nhf : Integrable f\n⊢ μ' ≤ 1 • (μ + μ')",
"tactic": "rw [one_smul]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1664515\nG : Type ?u.1664518\n𝕜 : Type ?u.1664521\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : DominatedFinMeasAdditive μ' T C\nf : α → E\nhf : Integrable f\n⊢ 0 + μ' ≤ μ + μ'",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1664515\nG : Type ?u.1664518\n𝕜 : Type ?u.1664521\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : DominatedFinMeasAdditive μ' T C\nf : α → E\nhf : Integrable f\n⊢ μ' ≤ μ + μ'",
"tactic": "nth_rw 1 [← zero_add μ']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1664515\nG : Type ?u.1664518\n𝕜 : Type ?u.1664521\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : DominatedFinMeasAdditive μ' T C\nf : α → E\nhf : Integrable f\n⊢ 0 + μ' ≤ μ + μ'",
"tactic": "exact add_le_add bot_le le_rfl"
}
] |
[
1669,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1662,
1
] |
Mathlib/GroupTheory/SemidirectProduct.lean
|
SemidirectProduct.map_comp_inl
|
[
{
"state_after": "no goals",
"state_before": "N : Type u_1\nG : Type u_4\nH : Type ?u.251009\ninst✝⁴ : Group N\ninst✝³ : Group G\ninst✝² : Group H\nφ : G →* MulAut N\nN₁ : Type u_2\nG₁ : Type u_3\ninst✝¹ : Group N₁\ninst✝ : Group G₁\nφ₁ : G₁ →* MulAut N₁\nf₁ : N →* N₁\nf₂ : G →* G₁\nh : ∀ (g : G), MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑φ₁ (↑f₂ g))) f₁\n⊢ MonoidHom.comp (map f₁ f₂ h) inl = MonoidHom.comp inl f₁",
"tactic": "ext <;> simp"
}
] |
[
300,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.range_subtype_map
|
[
{
"state_after": "case h.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ { val := x, property := hx } ∈ range (Subtype.map f h) ↔\n { val := x, property := hx } ∈ Subtype.val ⁻¹' (f '' {x | p x})",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\n⊢ range (Subtype.map f h) = Subtype.val ⁻¹' (f '' {x | p x})",
"tactic": "ext ⟨x, hx⟩"
},
{
"state_after": "case h.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }) ↔\n ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x",
"state_before": "case h.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ { val := x, property := hx } ∈ range (Subtype.map f h) ↔\n { val := x, property := hx } ∈ Subtype.val ⁻¹' (f '' {x | p x})",
"tactic": "rw [mem_preimage, mem_range, mem_image, Subtype.exists, Subtype.coe_mk]"
},
{
"state_after": "case h.mk.mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }) →\n ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x\n\ncase h.mk.mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x) →\n ∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"state_before": "case h.mk\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }) ↔\n ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x",
"tactic": "apply Iff.intro"
},
{
"state_after": "case h.mk.mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x) →\n ∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"state_before": "case h.mk.mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }) →\n ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x\n\ncase h.mk.mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x) →\n ∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"tactic": ". rintro ⟨a, b, hab⟩\n rw [Subtype.map, Subtype.mk.injEq] at hab\n use a\n trivial"
},
{
"state_after": "no goals",
"state_before": "case h.mk.mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x) →\n ∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"tactic": ". rintro ⟨a, b, hab⟩\n use a\n use b\n rw [Subtype.map, Subtype.mk.injEq]\n exact hab"
},
{
"state_after": "case h.mk.mp.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : p a\nhab : Subtype.map f h { val := a, property := b } = { val := x, property := hx }\n⊢ ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x",
"state_before": "case h.mk.mp\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }) →\n ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x",
"tactic": "rintro ⟨a, b, hab⟩"
},
{
"state_after": "case h.mk.mp.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : p a\nhab : f ↑{ val := a, property := b } = x\n⊢ ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x",
"state_before": "case h.mk.mp.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : p a\nhab : Subtype.map f h { val := a, property := b } = { val := x, property := hx }\n⊢ ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x",
"tactic": "rw [Subtype.map, Subtype.mk.injEq] at hab"
},
{
"state_after": "case h.mk.mp.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : p a\nhab : f ↑{ val := a, property := b } = x\n⊢ a ∈ {x | p x} ∧ f a = x",
"state_before": "case h.mk.mp.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : p a\nhab : f ↑{ val := a, property := b } = x\n⊢ ∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x",
"tactic": "use a"
},
{
"state_after": "no goals",
"state_before": "case h.mk.mp.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : p a\nhab : f ↑{ val := a, property := b } = x\n⊢ a ∈ {x | p x} ∧ f a = x",
"tactic": "trivial"
},
{
"state_after": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ ∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"state_before": "case h.mk.mpr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\n⊢ (∃ x_1, x_1 ∈ {x | p x} ∧ f x_1 = x) →\n ∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"tactic": "rintro ⟨a, b, hab⟩"
},
{
"state_after": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ ∃ b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"state_before": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ ∃ a b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"tactic": "use a"
},
{
"state_after": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"state_before": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ ∃ b, Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"tactic": "use b"
},
{
"state_after": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ f ↑{ val := a, property := b } = x",
"state_before": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ Subtype.map f h { val := a, property := b } = { val := x, property := hx }",
"tactic": "rw [Subtype.map, Subtype.mk.injEq]"
},
{
"state_after": "no goals",
"state_before": "case h.mk.mpr.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.88408\nι : Sort ?u.88411\nι' : Sort ?u.88414\nf✝ : ι → α\ns t : Set α\np : α → Prop\nq : β → Prop\nf : α → β\nh : ∀ (x : α), p x → q (f x)\nx : β\nhx : q x\na : α\nb : a ∈ {x | p x}\nhab : f a = x\n⊢ f ↑{ val := a, property := b } = x",
"tactic": "exact hab"
}
] |
[
1018,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1005,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.disjoint_iUnion_left
|
[] |
[
2076,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2074,
1
] |
Mathlib/Data/Fin/VecNotation.lean
|
Matrix.empty_vecAlt1
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nm n o : ℕ\nm' : Type ?u.54086\nn' : Type ?u.54089\no' : Type ?u.54092\nα : Type u_1\nh : 0 = 0 + 0\n⊢ vecAlt1 h ![] = ![]",
"tactic": "simp"
}
] |
[
425,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasDerivAtFilter.const_add
|
[] |
[
127,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.algebraMap_fst
|
[] |
[
410,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
|
TensorAlgebra.algebraMap_eq_zero_iff
|
[] |
[
213,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.isCauSeq_abs
|
[] |
[
1262,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1258,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.neighborSet_bot
|
[] |
[
511,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Computability/Partrec.lean
|
Nat.rfind_min'
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ → Bool\nm : ℕ\npm : p m = true\nthis : true ∈ ↑p m\nn : ℕ\nhn : n ∈ rfind ↑p\nh : m < n\n⊢ False",
"tactic": "injection mem_unique this (rfind_min hn h)"
}
] |
[
118,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Topology/Algebra/WithZeroTopology.lean
|
WithZeroTopology.Iio_mem_nhds_zero
|
[] |
[
72,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Std/Data/List/Lemmas.lean
|
List.ne_and_not_mem_of_not_mem_cons
|
[] |
[
104,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
103,
1
] |
Mathlib/Algebra/Quandle.lean
|
Rack.assoc_iff_id
|
[
{
"state_after": "R✝ : Type ?u.30897\ninst✝¹ : Rack R✝\nR : Type u_1\ninst✝ : Rack R\nx y z : R\n⊢ (x ◃ y) ◃ x ◃ z = (x ◃ y) ◃ z ↔ x ◃ z = z",
"state_before": "R✝ : Type ?u.30897\ninst✝¹ : Rack R✝\nR : Type u_1\ninst✝ : Rack R\nx y z : R\n⊢ x ◃ y ◃ z = (x ◃ y) ◃ z ↔ x ◃ z = z",
"tactic": "rw [self_distrib]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.30897\ninst✝¹ : Rack R✝\nR : Type u_1\ninst✝ : Rack R\nx y z : R\n⊢ (x ◃ y) ◃ x ◃ z = (x ◃ y) ◃ z ↔ x ◃ z = z",
"tactic": "rw [left_cancel]"
}
] |
[
355,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
convex_iInter₂
|
[] |
[
112,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoDiv_zero_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\np : α\nhp : 0 < p\nb : α\n⊢ toIcoDiv (_ : 0 < 1) 0 b = ⌊b⌋",
"tactic": "simp [toIcoDiv_eq_floor]"
}
] |
[
1015,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.HasIntegral.mcShane_of_forall_isLittleO
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g✝ : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nB : ι →ᵇᵃ[↑I] ℝ\nhB0 : ∀ (J : Box ι), 0 ≤ ↑B J\ng : ι →ᵇᵃ[↑I] F\nH :\n ℝ≥0 →\n ∀ (x : ι → ℝ),\n x ∈ ↑Box.Icc I →\n ∀ (ε : ℝ),\n ε > 0 →\n ∃ δ, δ > 0 ∧ ∀ (J : Box ι), J ≤ I → ↑Box.Icc J ⊆ closedBall x δ → dist (↑(↑vol J) (f x)) (↑g J) ≤ ε * ↑B J\n⊢ ∀ (c : ℝ≥0) (x : ι → ℝ),\n x ∈ ↑Box.Icc I \\ ∅ →\n ∀ (ε : ℝ),\n ε > 0 →\n ∃ δ,\n δ > 0 ∧\n ∀ (J : Box ι),\n J ≤ I →\n ↑Box.Icc J ⊆ closedBall x δ →\n (McShane.bHenstock = true → x ∈ ↑Box.Icc J) →\n (McShane.bDistortion = true → Box.distortion J ≤ c) → dist (↑(↑vol J) (f x)) (↑g J) ≤ ε * ↑B J",
"tactic": "simpa only [McShane, Bool.coe_sort_false, false_imp_iff, true_imp_iff, diff_empty] using H"
}
] |
[
849,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
843,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderHom.fst_prod_snd
|
[
{
"state_after": "case h.h.mk\nF : Type ?u.52049\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.52058\nδ : Type ?u.52061\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nx : α\ny : β\n⊢ ↑(OrderHom.prod fst snd) (x, y) = ↑id (x, y)",
"state_before": "F : Type ?u.52049\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.52058\nδ : Type ?u.52061\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\n⊢ OrderHom.prod fst snd = id",
"tactic": "ext ⟨x, y⟩ : 2"
},
{
"state_after": "no goals",
"state_before": "case h.h.mk\nF : Type ?u.52049\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.52058\nδ : Type ?u.52061\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nx : α\ny : β\n⊢ ↑(OrderHom.prod fst snd) (x, y) = ↑id (x, y)",
"tactic": "rfl"
}
] |
[
460,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.degree_interpolate_erase_lt
|
[
{
"state_after": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ degree (↑(interpolate (Finset.erase s i) v) r) < ↑(card (Finset.erase s i))",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ degree (↑(interpolate (Finset.erase s i) v) r) < ↑(card s - 1)",
"tactic": "rw [← Finset.card_erase_of_mem hi]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ degree (↑(interpolate (Finset.erase s i) v) r) < ↑(card (Finset.erase s i))",
"tactic": "exact degree_interpolate_lt _ (Set.InjOn.mono (coe_subset.mpr (erase_subset _ _)) hvs)"
}
] |
[
356,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.forall₂_zip
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33435\nδ : Type ?u.33438\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₁ : R a✝ b✝\nh₂ : Forall₂ R l₁✝ l₂✝\nx : α\ny : β\nhx : (x, y) = (a✝, b✝) ∨ (x, y) ∈ zipWith Prod.mk l₁✝ l₂✝\n⊢ R x y",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33435\nδ : Type ?u.33438\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₁ : R a✝ b✝\nh₂ : Forall₂ R l₁✝ l₂✝\nx : α\ny : β\nhx : (x, y) ∈ zip (a✝ :: l₁✝) (b✝ :: l₂✝)\n⊢ R x y",
"tactic": "rw [zip, zipWith, mem_cons] at hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33435\nδ : Type ?u.33438\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₁ : R a✝ b✝\nh₂ : Forall₂ R l₁✝ l₂✝\nx : α\ny : β\nhx : (x, y) = (a✝, b✝) ∨ (x, y) ∈ zipWith Prod.mk l₁✝ l₂✝\n⊢ R x y",
"tactic": "match hx with\n| Or.inl rfl => exact h₁\n| Or.inr h₃ => exact forall₂_zip h₂ h₃"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33435\nδ : Type ?u.33438\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₂ : Forall₂ R l₁✝ l₂✝\nx : α\ny : β\nhx : (x, y) = (a✝, b✝) ∨ (x, y) ∈ zipWith Prod.mk l₁✝ l₂✝\nh₁ : R x y\n⊢ R x y",
"tactic": "exact h₁"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33435\nδ : Type ?u.33438\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na✝ : α\nb✝ : β\nl₁✝ : List α\nl₂✝ : List β\nh₂ : Forall₂ R l₁✝ l₂✝\nx : α\ny : β\nhx : (x, y) = (a✝, b✝) ∨ (x, y) ∈ zipWith Prod.mk l₁✝ l₂✝\nx✝¹ : β\nx✝ : α\nh₃ : (x, y) ∈ zipWith Prod.mk l₁✝ l₂✝\nh₁ : R x✝ x✝¹\n⊢ R x y",
"tactic": "exact forall₂_zip h₂ h₃"
}
] |
[
194,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Control/Traversable/Equiv.lean
|
Equiv.id_map
|
[
{
"state_after": "no goals",
"state_before": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝¹ : Functor t\ninst✝ : LawfulFunctor t\nα : Type u\nx : t' α\n⊢ Equiv.map eqv id x = x",
"tactic": "simp [Equiv.map, id_map]"
}
] |
[
61,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
11
] |
Mathlib/GroupTheory/NielsenSchreier.lean
|
IsFreeGroupoid.SpanningTree.treeHom_eq
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Groupoid G\ninst✝¹ : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\na : G\np : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a\n⊢ treeHom T a = homOfPath T p",
"tactic": "rw [treeHom, Unique.default_eq]"
}
] |
[
182,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
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