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Mathlib/Combinatorics/Young/YoungDiagram.lean | YoungDiagram.rowLen_eq_card | [
{
"state_after": "no goals",
"state_before": "μ : YoungDiagram\ni : ℕ\n⊢ rowLen μ i = Finset.card (row μ i)",
"tactic": "simp [row_eq_prod]"
}
] | [
326,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
325,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | NormedCommGroup.nhds_one_basis_norm_lt | [
{
"state_after": "case h.e'_5.h.h.e'_2.h.h.e'_3.h.e'_3\n𝓕 : Type ?u.129849\n𝕜 : Type ?u.129852\nα : Type ?u.129855\nι : Type ?u.129858\nκ : Type ?u.129861\nE : Type u_1\nF : Type ?u.129867\nG : Type ?u.129870\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ x✝¹ : ℝ\nx✝ : E\n⊢ x✝ = x✝ / 1",
"state_before": "𝓕 : Type ?u.129849\n𝕜 : Type ?u.129852\nα : Type ?u.129855\nι : Type ?u.129858\nκ : Type ?u.129861\nE : Type u_1\nF : Type ?u.129867\nG : Type ?u.129870\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\n⊢ HasBasis (𝓝 1) (fun ε => 0 < ε) fun ε => {y | ‖y‖ < ε}",
"tactic": "convert NormedCommGroup.nhds_basis_norm_lt (1 : E)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.h.e'_2.h.h.e'_3.h.e'_3\n𝓕 : Type ?u.129849\n𝕜 : Type ?u.129852\nα : Type ?u.129855\nι : Type ?u.129858\nκ : Type ?u.129861\nE : Type u_1\nF : Type ?u.129867\nG : Type ?u.129870\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ x✝¹ : ℝ\nx✝ : E\n⊢ x✝ = x✝ / 1",
"tactic": "simp"
}
] | [
774,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
771,
1
] |
Mathlib/Algebra/Star/Pointwise.lean | Set.star_mem_star | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns t : Set α\na : α\ninst✝ : InvolutiveStar α\n⊢ a⋆ ∈ s⋆ ↔ a ∈ s",
"tactic": "simp only [mem_star, star_star]"
}
] | [
65,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/RingTheory/Localization/Integral.lean | IsFractionRing.isAlgebraic_iff' | [
{
"state_after": "R : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\n⊢ (∀ (x : S), IsAlgebraic R x) ↔ ∀ (x : K), IsAlgebraic R x",
"state_before": "R : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\n⊢ Algebra.IsAlgebraic R S ↔ Algebra.IsAlgebraic R K",
"tactic": "simp only [Algebra.IsAlgebraic]"
},
{
"state_after": "case mp\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\n⊢ (∀ (x : S), IsAlgebraic R x) → ∀ (x : K), IsAlgebraic R x\n\ncase mpr\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\n⊢ (∀ (x : K), IsAlgebraic R x) → ∀ (x : S), IsAlgebraic R x",
"state_before": "R : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\n⊢ (∀ (x : S), IsAlgebraic R x) ↔ ∀ (x : K), IsAlgebraic R x",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\nx : K\n⊢ IsAlgebraic R x",
"state_before": "case mp\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\n⊢ (∀ (x : S), IsAlgebraic R x) → ∀ (x : K), IsAlgebraic R x",
"tactic": "intro h x"
},
{
"state_after": "case mp\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\nx : K\n⊢ IsIntegral (FractionRing R) x",
"state_before": "case mp\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\nx : K\n⊢ IsAlgebraic R x",
"tactic": "rw [IsFractionRing.isAlgebraic_iff R (FractionRing R) K, isAlgebraic_iff_isIntegral]"
},
{
"state_after": "case mp.intro.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a / ↑(algebraMap S K) b)",
"state_before": "case mp\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\nx : K\n⊢ IsIntegral (FractionRing R) x",
"tactic": "obtain ⟨a : S, b, ha, rfl⟩ := @div_surjective S _ _ _ _ _ _ x"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a / ↑(algebraMap S K) b)",
"state_before": "case mp.intro.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a / ↑(algebraMap S K) b)",
"tactic": "obtain ⟨f, hf₁, hf₂⟩ := h b"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a * (↑(algebraMap S K) b)⁻¹)",
"state_before": "case mp.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a / ↑(algebraMap S K) b)",
"tactic": "rw [div_eq_mul_inv]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a)\n\ncase mp.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) b)⁻¹",
"state_before": "case mp.intro.intro.intro.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a * (↑(algebraMap S K) b)⁻¹)",
"tactic": "refine' isIntegral_mul _ _"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsAlgebraic (FractionRing R) (↑(algebraMap S K) a)",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) a)",
"tactic": "rw [← isAlgebraic_iff_isIntegral]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsAlgebraic R (↑(algebraMap S K) a)",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsAlgebraic (FractionRing R) (↑(algebraMap S K) a)",
"tactic": "refine'\n _root_.isAlgebraic_of_larger_base_of_injective\n (NoZeroSMulDivisors.algebraMap_injective R (FractionRing R)) _"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsAlgebraic R (↑(algebraMap S K) a)",
"tactic": "exact isAlgebraic_algebraMap_of_isAlgebraic (h a)"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsAlgebraic (FractionRing R) (↑(algebraMap S K) b)⁻¹",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsIntegral (FractionRing R) (↑(algebraMap S K) b)⁻¹",
"tactic": "rw [← isAlgebraic_iff_isIntegral]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ reverse (Polynomial.map (algebraMap R (FractionRing R)) f) ≠ 0 ∧\n ↑(aeval (↑(algebraMap S K) b)⁻¹) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ IsAlgebraic (FractionRing R) (↑(algebraMap S K) b)⁻¹",
"tactic": "use (f.map (algebraMap R (FractionRing R))).reverse"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2.left\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ reverse (Polynomial.map (algebraMap R (FractionRing R)) f) ≠ 0\n\ncase mp.intro.intro.intro.intro.intro.refine'_2.right\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ ↑(aeval (↑(algebraMap S K) b)⁻¹) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ reverse (Polynomial.map (algebraMap R (FractionRing R)) f) ≠ 0 ∧\n ↑(aeval (↑(algebraMap S K) b)⁻¹) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2.left\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ reverse (Polynomial.map (algebraMap R (FractionRing R)) f) ≠ 0",
"tactic": "rwa [Ne.def, Polynomial.reverse_eq_zero, ← Polynomial.degree_eq_bot,\n Polynomial.degree_map_eq_of_injective\n (NoZeroSMulDivisors.algebraMap_injective R (FractionRing R)),\n Polynomial.degree_eq_bot]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2.right\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\nthis : Invertible (↑(algebraMap S K) b)\n⊢ ↑(aeval (↑(algebraMap S K) b)⁻¹) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2.right\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\n⊢ ↑(aeval (↑(algebraMap S K) b)⁻¹) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0",
"tactic": "have : Invertible (algebraMap S K b) :=\n IsUnit.invertible\n (isUnit_of_mem_nonZeroDivisors\n (mem_nonZeroDivisors_iff_ne_zero.2 fun h =>\n nonZeroDivisors.ne_zero ha\n ((injective_iff_map_eq_zero (algebraMap S K)).1\n (NoZeroSMulDivisors.algebraMap_injective _ _) b h)))"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2.right\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : S), IsAlgebraic R x\na b : S\nha : b ∈ nonZeroDivisors S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval b) f = 0\nthis : Invertible (↑(algebraMap S K) b)\n⊢ ↑(aeval (↑(algebraMap S K) b)⁻¹) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0",
"tactic": "rw [Polynomial.aeval_def, ← invOf_eq_inv, Polynomial.eval₂_reverse_eq_zero_iff,\n Polynomial.eval₂_map, ← IsScalarTower.algebraMap_eq, ← Polynomial.aeval_def,\n Polynomial.aeval_algebraMap_apply, hf₂, RingHom.map_zero]"
},
{
"state_after": "case mpr\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\n⊢ IsAlgebraic R x",
"state_before": "case mpr\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\n⊢ (∀ (x : K), IsAlgebraic R x) → ∀ (x : S), IsAlgebraic R x",
"tactic": "intro h x"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval (↑(algebraMap S K) x)) f = 0\n⊢ IsAlgebraic R x",
"state_before": "case mpr\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\n⊢ IsAlgebraic R x",
"tactic": "obtain ⟨f, hf₁, hf₂⟩ := h (algebraMap S K x)"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval (↑(algebraMap S K) x)) f = 0\n⊢ ↑(aeval x) f = 0",
"state_before": "case mpr.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval (↑(algebraMap S K) x)) f = 0\n⊢ IsAlgebraic R x",
"tactic": "use f, hf₁"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(algebraMap S K) (↑(aeval x) f) = 0\n⊢ ↑(aeval x) f = 0",
"state_before": "case mpr.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval (↑(algebraMap S K) x)) f = 0\n⊢ ↑(aeval x) f = 0",
"tactic": "rw [Polynomial.aeval_algebraMap_apply] at hf₂"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nR : Type u_2\ninst✝¹³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Type ?u.599719\ninst✝¹⁰ : CommRing P\nA : Type ?u.599725\nK : Type u_1\ninst✝⁹ : CommRing A\ninst✝⁸ : IsDomain A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : NoZeroSMulDivisors R K\ninst✝¹ : IsFractionRing S K\ninst✝ : IsScalarTower R S K\nh : ∀ (x : K), IsAlgebraic R x\nx : S\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(algebraMap S K) (↑(aeval x) f) = 0\n⊢ ↑(aeval x) f = 0",
"tactic": "exact\n (injective_iff_map_eq_zero (algebraMap S K)).1 (NoZeroSMulDivisors.algebraMap_injective _ _) _\n hf₂"
}
] | [
428,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
389,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | LocalHomeomorph.extend_symm_continuousWithinAt_comp_right_iff | [
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_2\nH : Type u_5\nE' : Type ?u.146554\nM' : Type ?u.146557\nH' : Type ?u.146560\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t : Set M\nX : Type u_1\ninst✝ : TopologicalSpace X\ng : M → X\ns : Set M\nx : M\n⊢ ContinuousWithinAt (g ∘ ↑(LocalEquiv.symm (extend f I))) (↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I)\n (↑(extend f I) x) ↔\n ContinuousWithinAt ((g ∘ ↑(LocalHomeomorph.symm f)) ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' (↑(LocalHomeomorph.symm f) ⁻¹' s) ∩ range ↑I) (↑I (↑f x))",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_2\nH : Type u_5\nE' : Type ?u.146554\nM' : Type ?u.146557\nH' : Type ?u.146560\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t : Set M\nX : Type u_1\ninst✝ : TopologicalSpace X\ng : M → X\ns : Set M\nx : M\n⊢ ContinuousWithinAt (g ∘ ↑(LocalEquiv.symm (extend f I))) (↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I)\n (↑(extend f I) x) ↔\n ContinuousWithinAt (g ∘ ↑(LocalHomeomorph.symm f)) (↑(LocalHomeomorph.symm f) ⁻¹' s) (↑f x)",
"tactic": "rw [← I.symm_continuousWithinAt_comp_right_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_2\nH : Type u_5\nE' : Type ?u.146554\nM' : Type ?u.146557\nH' : Type ?u.146560\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t : Set M\nX : Type u_1\ninst✝ : TopologicalSpace X\ng : M → X\ns : Set M\nx : M\n⊢ ContinuousWithinAt (g ∘ ↑(LocalEquiv.symm (extend f I))) (↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I)\n (↑(extend f I) x) ↔\n ContinuousWithinAt ((g ∘ ↑(LocalHomeomorph.symm f)) ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' (↑(LocalHomeomorph.symm f) ⁻¹' s) ∩ range ↑I) (↑I (↑f x))",
"tactic": "rfl"
}
] | [
874,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
870,
1
] |
Mathlib/Algebra/Homology/Single.lean | HomologicalComplex.single_map_f_self | [
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝³ : Category V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA B : V\nf : A ⟶ B\n⊢ Hom.f ((single V c j).map f) j = (singleObjXSelf V c j A).hom ≫ f ≫ (singleObjXSelf V c j B).inv",
"tactic": "simp"
}
] | [
88,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/Topology/Sheaves/Presheaf.lean | TopCat.Presheaf.pushforward_eq'_hom_app | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (Functor.op (Opens.map f)).obj U = (Functor.op (Opens.map g)).obj U",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : TopCat\nf g : X ⟶ Y\nh : f = g\nℱ : Presheaf C X\nU : (Opens ↑Y)ᵒᵖ\n⊢ (eqToHom (_ : f _* ℱ = g _* ℱ)).app U =\n ℱ.map (eqToHom (_ : (Functor.op (Opens.map f)).obj U = (Functor.op (Opens.map g)).obj U))",
"tactic": "rw [eqToHom_app, eqToHom_map]"
}
] | [
181,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Icc_prod_eq | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b : α × β\n⊢ Icc a b = Icc a.fst b.fst ×ˢ Icc a.snd b.snd",
"tactic": "simp"
}
] | [
1905,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1905,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.adj_and_reachable_delete_edges_iff_exists_cycle | [
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ (Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) ↔\n ∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ Adj G v w ∧ Reachable (G \\ fromEdgeSet {Quotient.mk (Sym2.Rel.setoid V) (v, w)}) v w ↔\n ∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "rw [reachable_delete_edges_iff_exists_walk]"
},
{
"state_after": "case mp\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ (Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) →\n ∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n\ncase mpr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ (∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) →\n Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ (Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) ↔\n ∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ ∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "case mp\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ (Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) →\n ∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "rintro ⟨h, p, hp⟩"
},
{
"state_after": "case mp.intro.intro.refine_1\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ Walk.IsCycle (Walk.cons (_ : Adj G w v) ↑(Walk.toPath p))\n\ncase mp.intro.intro.refine_2\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.cons (_ : Adj G w v) ↑(Walk.toPath p))",
"state_before": "case mp.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ ∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "refine ⟨w, Walk.cons h.symm p.toPath, ?_, ?_⟩"
},
{
"state_after": "case mp.intro.intro.refine_1.he\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ ¬Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges ↑(Walk.toPath p)",
"state_before": "case mp.intro.intro.refine_1\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ Walk.IsCycle (Walk.cons (_ : Adj G w v) ↑(Walk.toPath p))",
"tactic": "apply Path.cons_isCycle"
},
{
"state_after": "case mp.intro.intro.refine_1.he\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges ↑(Walk.toPath p)",
"state_before": "case mp.intro.intro.refine_1.he\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ ¬Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges ↑(Walk.toPath p)",
"tactic": "rw [Sym2.eq_swap]"
},
{
"state_after": "case mp.intro.intro.refine_1.he\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh✝ : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges ↑(Walk.toPath p)\n⊢ False",
"state_before": "case mp.intro.intro.refine_1.he\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges ↑(Walk.toPath p)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refine_1.he\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh✝ : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nh : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges ↑(Walk.toPath p)\n⊢ False",
"tactic": "cases hp (Walk.edges_toPath_subset p h)"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.refine_2\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : Adj G v w\np : Walk G v w\nhp : ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.cons (_ : Adj G w v) ↑(Walk.toPath p))",
"tactic": "simp only [Sym2.eq_swap, Walk.edges_cons, List.mem_cons, eq_self_iff_true, true_or_iff]"
},
{
"state_after": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\n⊢ Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "case mpr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ (∃ u p, Walk.IsCycle p ∧ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) →\n Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "rintro ⟨u, c, hc, he⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\n⊢ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\n⊢ Adj G v w ∧ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "refine ⟨c.adj_of_mem_edges he, ?_⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ False",
"state_before": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\n⊢ ∃ p, ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "by_contra' hb"
},
{
"state_after": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nhb' : ∀ (p : Walk G w v), Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p\n⊢ False",
"state_before": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ False",
"tactic": "have hb' : ∀ p : G.Walk w v, ⟦(w, v)⟧ ∈ p.edges := by\n intro p\n simpa [Sym2.eq_swap] using hb p.reverse"
},
{
"state_after": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nhb' : ∀ (p : Walk G w v), Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p\nhvc : v ∈ Walk.support c\n⊢ False",
"state_before": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nhb' : ∀ (p : Walk G w v), Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p\n⊢ False",
"tactic": "have hvc : v ∈ c.support := Walk.fst_mem_support_of_mem_edges c he"
},
{
"state_after": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nhb' : ∀ (p : Walk G w v), Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p\nhvc : v ∈ Walk.support c\n⊢ Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges (Walk.rotate c hvc)",
"state_before": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nhb' : ∀ (p : Walk G w v), Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p\nhvc : v ∈ Walk.support c\n⊢ False",
"tactic": "refine reachable_deleteEdges_iff_exists_cycle.aux hb' (c.rotate hvc) (hc.isTrail.rotate hvc)\n ?_ (Walk.start_mem_support _)"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nhb' : ∀ (p : Walk G w v), Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p\nhvc : v ∈ Walk.support c\n⊢ Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges (Walk.rotate c hvc)",
"tactic": "rwa [(Walk.rotate_edges c hvc).mem_iff, Sym2.eq_swap]"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\np : Walk G w v\n⊢ Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n⊢ ∀ (p : Walk G w v), Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p",
"tactic": "intro p"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w u : V\nc : Walk G u u\nhc : Walk.IsCycle c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\np : Walk G w v\n⊢ Quotient.mk (Sym2.Rel.setoid V) (w, v) ∈ Walk.edges p",
"tactic": "simpa [Sym2.eq_swap] using hb p.reverse"
}
] | [
2521,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2499,
1
] |
Mathlib/Data/Sym/Sym2.lean | Sym2.rel_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1901\nγ : Type ?u.1904\nx y z w : α\n⊢ Rel α (x, y) (z, w) ↔ x = z ∧ y = w ∨ x = w ∧ y = z",
"tactic": "aesop (rule_sets [Sym2])"
}
] | [
94,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Data/Nat/Digits.lean | Nat.modEq_digits_sum | [
{
"state_after": "n✝ b b' : ℕ\nh : b' % b = 1\nn : ℕ\n⊢ n ≡ ofDigits 1 (digits b' n) [MOD b]",
"state_before": "n✝ b b' : ℕ\nh : b' % b = 1\nn : ℕ\n⊢ n ≡ List.sum (digits b' n) [MOD b]",
"tactic": "rw [← ofDigits_one]"
},
{
"state_after": "case h.e'_3.h.e'_3\nn✝ b b' : ℕ\nh : b' % b = 1\nn : ℕ\n⊢ 1 = b' % b",
"state_before": "n✝ b b' : ℕ\nh : b' % b = 1\nn : ℕ\n⊢ ofDigits b' (digits b' n) ≡ ofDigits 1 (digits b' n) [MOD b]",
"tactic": "convert ofDigits_modEq b' b (digits b' n)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_3\nn✝ b b' : ℕ\nh : b' % b = 1\nn : ℕ\n⊢ 1 = b' % b",
"tactic": "exact h.symm"
}
] | [
548,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/RingTheory/Adjoin/FG.lean | is_noetherian_subring_closure | [] | [
217,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
] |
Mathlib/Order/CompactlyGenerated.lean | CompleteLattice.wellFounded_iff_isSupFiniteCompact | [] | [
281,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
279,
1
] |
Mathlib/Data/Polynomial/Mirror.lean | Polynomial.coeff_mirror | [
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\n⊢ coeff (mirror p) n = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)\n\ncase neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : ¬natDegree p < n\n⊢ coeff (mirror p) n = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ coeff (mirror p) n = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"tactic": "by_cases h2 : p.natDegree < n"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\n⊢ coeff (mirror p) n = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : ¬natDegree p < n\n⊢ coeff (mirror p) n = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"tactic": "rw [not_lt] at h2"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\n⊢ coeff (mirror p) n = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"tactic": "rw [revAt_le (h2.trans (Nat.le_add_right _ _))]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : natTrailingDegree p ≤ n\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)\n\ncase neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : ¬natTrailingDegree p ≤ n\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)",
"tactic": "by_cases h3 : p.natTrailingDegree ≤ n"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : n < natTrailingDegree p\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : ¬natTrailingDegree p ≤ n\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)",
"tactic": "rw [not_le] at h3"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : n < natTrailingDegree p\n⊢ coeff (mirror p) n = 0",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : n < natTrailingDegree p\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)",
"tactic": "rw [coeff_eq_zero_of_natDegree_lt (lt_tsub_iff_right.mpr (Nat.add_lt_add_left h3 _))]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : n < natTrailingDegree p\n⊢ coeff (mirror p) n = 0",
"tactic": "exact coeff_eq_zero_of_lt_natTrailingDegree (by rwa [mirror_natTrailingDegree])"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\n⊢ 0 = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\n⊢ coeff (mirror p) n = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"tactic": "rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\nh1 : n ≤ natDegree p + natTrailingDegree p\n⊢ 0 = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)\n\ncase neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\nh1 : ¬n ≤ natDegree p + natTrailingDegree p\n⊢ 0 = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\n⊢ 0 = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"tactic": "by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\n⊢ natDegree (mirror p) < n",
"tactic": "rwa [mirror_natDegree]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\nh1 : n ≤ natDegree p + natTrailingDegree p\n⊢ natDegree p + natTrailingDegree p - n < natTrailingDegree p",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\nh1 : n ≤ natDegree p + natTrailingDegree p\n⊢ 0 = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"tactic": "rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\nh1 : n ≤ natDegree p + natTrailingDegree p\n⊢ natDegree p + natTrailingDegree p - n < natTrailingDegree p",
"tactic": "exact (tsub_lt_iff_left h1).mpr (Nat.add_lt_add_right h2 _)"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : natDegree p < n\nh1 : ¬n ≤ natDegree p + natTrailingDegree p\n⊢ 0 = coeff p (↑(revAt (natDegree p + natTrailingDegree p)) n)",
"tactic": "rw [← revAtFun_eq, revAtFun, if_neg h1, coeff_eq_zero_of_natDegree_lt h2]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : natTrailingDegree p ≤ n\n⊢ coeff (mirror p) n = coeff p (natDegree p + natTrailingDegree p - n)",
"tactic": "rw [← tsub_add_eq_add_tsub h2, ← tsub_tsub_assoc h2 h3, mirror, coeff_mul_X_pow', if_pos h3,\n coeff_reverse, revAt_le (tsub_le_self.trans h2)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nh2 : n ≤ natDegree p\nh3 : n < natTrailingDegree p\n⊢ n < natTrailingDegree (mirror p)",
"tactic": "rwa [mirror_natTrailingDegree]"
}
] | [
100,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Std/Data/RBMap/WF.lean | Std.RBNode.Ordered.setRed | [
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\nt : RBNode α\n⊢ Ordered cmp\n (match t with\n | node c a v b => node red a v b\n | nil => nil) ↔\n Ordered cmp t",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\nt : RBNode α\n⊢ Ordered cmp (setRed t) ↔ Ordered cmp t",
"tactic": "unfold setRed"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\nt : RBNode α\n⊢ Ordered cmp\n (match t with\n | node c a v b => node red a v b\n | nil => nil) ↔\n Ordered cmp t",
"tactic": "split <;> simp [Ordered]"
}
] | [
233,
42
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
232,
11
] |
Mathlib/SetTheory/Ordinal/Exponential.lean | Ordinal.add_log_le_log_mul | [
{
"state_after": "case pos\nx y b : Ordinal\nhx : x ≠ 0\nhy : y ≠ 0\nhb : 1 < b\n⊢ log b x + log b y ≤ log b (x * y)\n\ncase neg\nx y b : Ordinal\nhx : x ≠ 0\nhy : y ≠ 0\nhb : ¬1 < b\n⊢ log b x + log b y ≤ log b (x * y)",
"state_before": "x y b : Ordinal\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ log b x + log b y ≤ log b (x * y)",
"tactic": "by_cases hb : 1 < b"
},
{
"state_after": "no goals",
"state_before": "case neg\nx y b : Ordinal\nhx : x ≠ 0\nhy : y ≠ 0\nhb : ¬1 < b\n⊢ log b x + log b y ≤ log b (x * y)",
"tactic": "simp only [log_of_not_one_lt_left hb, zero_add, le_refl]"
},
{
"state_after": "case pos\nx y b : Ordinal\nhx : x ≠ 0\nhy : y ≠ 0\nhb : 1 < b\n⊢ b ^ log b x * b ^ log b y ≤ x * y",
"state_before": "case pos\nx y b : Ordinal\nhx : x ≠ 0\nhy : y ≠ 0\nhb : 1 < b\n⊢ log b x + log b y ≤ log b (x * y)",
"tactic": "rw [← opow_le_iff_le_log hb (mul_ne_zero hx hy), opow_add]"
},
{
"state_after": "no goals",
"state_before": "case pos\nx y b : Ordinal\nhx : x ≠ 0\nhy : y ≠ 0\nhb : 1 < b\n⊢ b ^ log b x * b ^ log b y ≤ x * y",
"tactic": "exact mul_le_mul' (opow_log_le_self b hx) (opow_log_le_self b hy)"
}
] | [
449,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
443,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean | MulHom.eq_of_eqOn_dense | [] | [
499,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
497,
1
] |
Mathlib/Order/OrdContinuous.lean | OrderIso.leftOrdContinuous | [] | [
278,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
275,
11
] |
Mathlib/LinearAlgebra/BilinearForm.lean | BilinForm.mem_selfAdjointSubmodule | [] | [
1149,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1147,
1
] |
Mathlib/LinearAlgebra/ProjectiveSpace/Basic.lean | Projectivization.mk''_submodule | [] | [
188,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean | fderiv_add | [] | [
174,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
172,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.trans_assoc | [] | [
859,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
857,
1
] |
Mathlib/Algebra/Associated.lean | Associated.of_mul_right | [
{
"state_after": "α : Type u_1\nβ : Type ?u.231684\nγ : Type ?u.231687\nδ : Type ?u.231690\ninst✝ : CancelCommMonoidWithZero α\na b c d : α\n⊢ b * a ~ᵤ d * c → b ~ᵤ d → b ≠ 0 → a ~ᵤ c",
"state_before": "α : Type u_1\nβ : Type ?u.231684\nγ : Type ?u.231687\nδ : Type ?u.231690\ninst✝ : CancelCommMonoidWithZero α\na b c d : α\n⊢ a * b ~ᵤ c * d → b ~ᵤ d → b ≠ 0 → a ~ᵤ c",
"tactic": "rw [mul_comm a, mul_comm c]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.231684\nγ : Type ?u.231687\nδ : Type ?u.231690\ninst✝ : CancelCommMonoidWithZero α\na b c d : α\n⊢ b * a ~ᵤ d * c → b ~ᵤ d → b ≠ 0 → a ~ᵤ c",
"tactic": "exact Associated.of_mul_left"
}
] | [
667,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
665,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.coe_adj_sub | [] | [
144,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
] |
Std/Data/Rat/Lemmas.lean | Rat.mkRat_mul_left | [
{
"state_after": "no goals",
"state_before": "n : Int\nd a : Nat\na0 : a ≠ 0\n⊢ mkRat (↑a * n) (a * d) = mkRat n d",
"tactic": "if d0 : d = 0 then simp [d0] else\nrw [← normalize_eq_mkRat d0, ← normalize_mul_left d0 a0, normalize_eq_mkRat]"
},
{
"state_after": "no goals",
"state_before": "n : Int\nd a : Nat\na0 : a ≠ 0\nd0 : d = 0\n⊢ mkRat (↑a * n) (a * d) = mkRat n d",
"tactic": "simp [d0]"
},
{
"state_after": "no goals",
"state_before": "n : Int\nd a : Nat\na0 : a ≠ 0\nd0 : ¬d = 0\n⊢ mkRat (↑a * n) (a * d) = mkRat n d",
"tactic": "rw [← normalize_eq_mkRat d0, ← normalize_mul_left d0 a0, normalize_eq_mkRat]"
}
] | [
107,
79
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
105,
1
] |
src/lean/Init/Control/Lawful.lean | ExceptT.seqRight_eq | [
{
"state_after": "m : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (do\n let _ ← x\n y) =\n do\n let f ← const α id <$> x\n f <$> y",
"state_before": "m : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y",
"tactic": "show (x >>= fun _ => y) = (const α id <$> x) >>= fun f => f <$> y"
},
{
"state_after": "m : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (do\n let _ ← x\n y) =\n do\n let f ← x >>= pure ∘ const α id\n f <$> y",
"state_before": "m : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (do\n let _ ← x\n y) =\n do\n let f ← const α id <$> x\n f <$> y",
"tactic": "rw [← ExceptT.bind_pure_comp]"
},
{
"state_after": "case h\nm : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (run do\n let _ ← x\n y) =\n run do\n let f ← x >>= pure ∘ const α id\n f <$> y",
"state_before": "m : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (do\n let _ ← x\n y) =\n do\n let f ← x >>= pure ∘ const α id\n f <$> y",
"tactic": "apply ext"
},
{
"state_after": "case h\nm : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (do\n let x ← run x\n match x with\n | Except.ok x => run y\n | Except.error e => pure (Except.error e)) =\n do\n let x ← run x\n let x ←\n match x with\n | Except.ok x => pure (Except.ok id)\n | Except.error e => pure (Except.error e)\n match x with\n | Except.ok x => Except.map x <$> run y\n | Except.error e => pure (Except.error e)",
"state_before": "case h\nm : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (run do\n let _ ← x\n y) =\n run do\n let f ← x >>= pure ∘ const α id\n f <$> y",
"tactic": "simp [run_bind]"
},
{
"state_after": "case h.h\nm : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ ∀ (a : Except ε α),\n (match a with\n | Except.ok x => run y\n | Except.error e => pure (Except.error e)) =\n do\n let x ←\n match a with\n | Except.ok x => pure (Except.ok id)\n | Except.error e => pure (Except.error e)\n match x with\n | Except.ok x => Except.map x <$> run y\n | Except.error e => pure (Except.error e)",
"state_before": "case h\nm : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ (do\n let x ← run x\n match x with\n | Except.ok x => run y\n | Except.error e => pure (Except.error e)) =\n do\n let x ← run x\n let x ←\n match x with\n | Except.ok x => pure (Except.ok id)\n | Except.error e => pure (Except.error e)\n match x with\n | Except.ok x => Except.map x <$> run y\n | Except.error e => pure (Except.error e)",
"tactic": "apply bind_congr"
},
{
"state_after": "case h.h\nm : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\na : Except ε α\n⊢ (match a with\n | Except.ok x => run y\n | Except.error e => pure (Except.error e)) =\n do\n let x ←\n match a with\n | Except.ok x => pure (Except.ok id)\n | Except.error e => pure (Except.error e)\n match x with\n | Except.ok x => Except.map x <$> run y\n | Except.error e => pure (Except.error e)",
"state_before": "case h.h\nm : Type u_1 → Type u_2\nε α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : ExceptT ε m α\ny : ExceptT ε m β\n⊢ ∀ (a : Except ε α),\n (match a with\n | Except.ok x => run y\n | Except.error e => pure (Except.error e)) =\n do\n let x ←\n match a with\n | Except.ok x => pure (Except.ok id)\n | Except.error e => pure (Except.error e)\n match x with\n | Except.ok x => Except.map x <$> run y\n | Except.error e => pure (Except.error e)",
"tactic": "intro a"
},
{
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"tactic": "cases a <;> simp"
}
] | [
161,
28
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
155,
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Mathlib/MeasureTheory/Integral/SetIntegral.lean | MeasureTheory.set_integral_le_nonneg | [
{
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"tactic": "rw [← integral_indicator hs, ←\n integral_indicator (stronglyMeasurable_const.measurableSet_le hf)]"
},
{
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"tactic": "exact\n integral_mono (hfi.indicator hs)\n (hfi.indicator (stronglyMeasurable_const.measurableSet_le hf))\n (indicator_le_indicator_nonneg s f)"
}
] | [
751,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
744,
1
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Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigOWith_top | [
{
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"tactic": "rw [IsBigOWith_def]"
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{
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"tactic": "rfl"
}
] | [
1274,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1274,
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Mathlib/Topology/PartitionOfUnity.lean | BumpCovering.toPartitionOfUnity_eq_mul_prod | [] | [
464,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/List/Indexes.lean | List.mapIdxM_eq_mmap_enum | [
{
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"tactic": "simp only [mapIdxM, mapIdxMGo_eq_mapIdxMAuxSpec, Array.toList_eq, Array.data_toArray,\n nil_append, mapIdxMAuxSpec, Array.size_toArray, length_nil, id_map', enum]"
}
] | [
336,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.cof_ne_zero | [] | [
497,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
496,
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Mathlib/Computability/Primrec.lean | Primrec.vector_get | [
{
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"tactic": "rw [Vector.get_eq_get, ← List.get?_eq_get]"
},
{
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"tactic": "rfl"
}
] | [
1312,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/List/Basic.lean | List.indexOf_cons_eq | [
{
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"tactic": "rw [e]"
},
{
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"tactic": "exact indexOf_cons_self b l"
}
] | [
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48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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] |
Mathlib/Analysis/Convex/Between.lean | Wbtw.map | [
{
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"tactic": "rw [Wbtw, ← affineSegment_image]"
},
{
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"tactic": "exact Set.mem_image_of_mem _ h"
}
] | [
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Topology/ContinuousOn.lean | continuousOn_congr | [] | [
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63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Submodule.toSubalgebra_mk | [
{
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"tactic": "intro r"
},
{
"state_after": "R : Type ?u.480281\nA : Type ?u.480284\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈\n { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },\n add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),\n zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier",
"state_before": "R : Type ?u.480281\nA : Type ?u.480284\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ ↑(algebraMap R A) r ∈\n { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },\n add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),\n zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier",
"tactic": "rw [Algebra.algebraMap_eq_smul_one]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.480281\nA : Type ?u.480284\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\np s : Submodule R A\nh1 : 1 ∈ s\nhmul : ∀ (x y : A), x ∈ s → y ∈ s → x * y ∈ s\nr : R\n⊢ r • 1 ∈\n { toSubmonoid := { toSubsemigroup := { carrier := ↑s, mul_mem' := hmul }, one_mem' := h1 },\n add_mem' := (_ : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s),\n zero_mem' := (_ : 0 ∈ s) }.toSubmonoid.toSubsemigroup.carrier",
"tactic": "apply s.smul_mem _ h1"
}
] | [
577,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
573,
1
] |
Mathlib/Data/Num/Lemmas.lean | ZNum.cast_bitm1 | [
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\n⊢ ↑(ZNum.bitm1 (- -n)) = _root_.bit0 ↑n - 1",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\n⊢ ↑(ZNum.bitm1 n) = _root_.bit0 ↑n - 1",
"tactic": "conv =>\n lhs\n rw [← zneg_zneg n]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\n⊢ -_root_.bit1 ↑(-n) = _root_.bit0 ↑n - 1",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\n⊢ ↑(ZNum.bitm1 (- -n)) = _root_.bit0 ↑n - 1",
"tactic": "rw [← zneg_bit1, cast_zneg, cast_bit1]"
},
{
"state_after": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\nthis : ↑(-1 + ↑n + ↑n) = ↑(↑n + ↑n + -1)\n⊢ -_root_.bit1 ↑(-n) = _root_.bit0 ↑n - 1",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\n⊢ -_root_.bit1 ↑(-n) = _root_.bit0 ↑n - 1",
"tactic": "have : ((-1 + n + n : ℤ) : α) = (n + n + -1 : ℤ) := by simp [add_comm, add_left_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\nthis : ↑(-1 + ↑n + ↑n) = ↑(↑n + ↑n + -1)\n⊢ -_root_.bit1 ↑(-n) = _root_.bit0 ↑n - 1",
"tactic": "simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg] using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\nn : ZNum\n⊢ ↑(-1 + ↑n + ↑n) = ↑(↑n + ↑n + -1)",
"tactic": "simp [add_comm, add_left_comm]"
}
] | [
1161,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1155,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean | Antitone.continuousAt_iff_leftLim_eq_rightLim | [] | [
370,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.filter_True | [
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.320628\nγ : Type ?u.320631\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ns : Finset α\na✝ : α\n⊢ a✝ ∈ filter (fun x => True) s ↔ a✝ ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.320628\nγ : Type ?u.320631\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ns : Finset α\n⊢ filter (fun x => True) s = s",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.320628\nγ : Type ?u.320631\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ns : Finset α\na✝ : α\n⊢ a✝ ∈ filter (fun x => True) s ↔ a✝ ∈ s",
"tactic": "simp"
}
] | [
2682,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2681,
1
] |
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.map_congr | [] | [
1513,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1512,
1
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | MeasureTheory.ProbabilityMeasure.coe_toWeakDualBCNN | [] | [
239,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
237,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Cardinal.le_range_of_union_finset_eq_top | [
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\n⊢ (#β) ≤ (#↑(range f))",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ (#β) ≤ (#↑(range f))",
"tactic": "have k : _root_.Infinite (range f) := by\n rw [infinite_coe_iff]\n apply mt (union_finset_finite_of_range_finite f)\n rw [w]\n exact infinite_univ"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : ¬(#β) ≤ (#↑(range f))\n⊢ False",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\n⊢ (#β) ≤ (#↑(range f))",
"tactic": "by_contra h"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\n⊢ False",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : ¬(#β) ≤ (#↑(range f))\n⊢ False",
"tactic": "simp only [not_le] at h"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\n⊢ False",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\n⊢ False",
"tactic": "let u : ∀ b, ∃ a, b ∈ f a := fun b => by simpa using (w.ge : _) (Set.mem_univ b)"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\n⊢ False",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\n⊢ False",
"tactic": "let u' : β → range f := fun b => ⟨f (u b).choose, by simp⟩"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\nv' : ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a)\n⊢ False",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\n⊢ False",
"tactic": "have v' : ∀ a, u' ⁻¹' {⟨f a, by simp⟩} ≤ f a := by\n rintro a p m\n simp at m\n rw [← m]\n apply fun b => (u b).choose_spec"
},
{
"state_after": "case intro.mk.intro\nα✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\nv' : ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a)\na : α\np : Infinite ↑(u' ⁻¹' {{ val := f a, property := (_ : ∃ y, f y = f a) }})\n⊢ False",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\nv' : ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a)\n⊢ False",
"tactic": "obtain ⟨⟨-, ⟨a, rfl⟩⟩, p⟩ := exists_infinite_fiber u' h k"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro\nα✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\nv' : ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a)\na : α\np : Infinite ↑(u' ⁻¹' {{ val := f a, property := (_ : ∃ y, f y = f a) }})\n⊢ False",
"tactic": "exact (@Infinite.of_injective _ _ p (inclusion (v' a)) (inclusion_injective _)).false"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ Set.Infinite (range f)",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ Infinite ↑(range f)",
"tactic": "rw [infinite_coe_iff]"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ ¬Set.Finite (⋃ (a : α), ↑(f a))",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ Set.Infinite (range f)",
"tactic": "apply mt (union_finset_finite_of_range_finite f)"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ ¬Set.Finite ⊤",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ ¬Set.Finite (⋃ (a : α), ↑(f a))",
"tactic": "rw [w]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\n⊢ ¬Set.Finite ⊤",
"tactic": "exact infinite_univ"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nb : β\n⊢ ∃ a, b ∈ f a",
"tactic": "simpa using (w.ge : _) (Set.mem_univ b)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nb : β\n⊢ f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\na : α\n⊢ f a ∈ range f",
"tactic": "simp"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\na : α\np : β\nm : p ∈ u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }}\n⊢ p ∈ ↑(f a)",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\n⊢ ∀ (a : α), u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }} ≤ ↑(f a)",
"tactic": "rintro a p m"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\na : α\np : β\nm : f (Exists.choose (_ : ∃ a, p ∈ f a)) = f a\n⊢ p ∈ ↑(f a)",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\na : α\np : β\nm : p ∈ u' ⁻¹' {{ val := f a, property := (_ : f a ∈ range f) }}\n⊢ p ∈ ↑(f a)",
"tactic": "simp at m"
},
{
"state_after": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\na : α\np : β\nm : f (Exists.choose (_ : ∃ a, p ∈ f a)) = f a\n⊢ p ∈ ↑(f (Exists.choose (_ : ∃ a, p ∈ f a)))",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\na : α\np : β\nm : f (Exists.choose (_ : ∃ a, p ∈ f a)) = f a\n⊢ p ∈ ↑(f a)",
"tactic": "rw [← m]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.149865\nr : α✝ → α✝ → Prop\nα : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : (⋃ (a : α), ↑(f a)) = ⊤\nk : Infinite ↑(range f)\nh : (#↑(range f)) < (#β)\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b =>\n Eq.mp\n (Eq.trans Mathlib.Data.Set.Lattice._auxLemma.3\n (congrArg Exists (funext fun i => Mathlib.Data.Finset.Basic._auxLemma.4)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) :=\n fun b =>\n { val := f (Exists.choose (_ : ∃ a, b ∈ f a)), property := (_ : f (Exists.choose (_ : ∃ a, b ∈ f a)) ∈ range f) }\na : α\np : β\nm : f (Exists.choose (_ : ∃ a, p ∈ f a)) = f a\n⊢ p ∈ ↑(f (Exists.choose (_ : ∃ a, p ∈ f a)))",
"tactic": "apply fun b => (u b).choose_spec"
}
] | [
1058,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1041,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean | ContinuousMultilinearMap.le_op_norm | [
{
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"tactic": "have A : 0 ≤ ∏ i, ‖m i‖ := prod_nonneg fun j _ => norm_nonneg _"
},
{
"state_after": "case inl\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\nA : 0 ≤ ∏ i : ι, ‖m i‖\nh : 0 = ∏ i : ι, ‖m i‖\n⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖\n\ncase inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\nA : 0 ≤ ∏ i : ι, ‖m i‖\nhlt : 0 < ∏ i : ι, ‖m i‖\n⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖",
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"tactic": "cases' A.eq_or_lt with h hlt"
},
{
"state_after": "case inl.intro.intro\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\nA : 0 ≤ ∏ i : ι, ‖m i‖\nh : 0 = ∏ i : ι, ‖m i‖\ni : ι\nleft✝ : i ∈ univ\nhi : ‖m i‖ = 0\n⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖",
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"tactic": "rcases prod_eq_zero_iff.1 h.symm with ⟨i, _, hi⟩"
},
{
"state_after": "case inl.intro.intro\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\nA : 0 ≤ ∏ i : ι, ‖m i‖\nh : 0 = ∏ i : ι, ‖m i‖\ni : ι\nleft✝ : i ∈ univ\nhi : m i = 0\n⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖",
"state_before": "case inl.intro.intro\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\nA : 0 ≤ ∏ i : ι, ‖m i‖\nh : 0 = ∏ i : ι, ‖m i‖\ni : ι\nleft✝ : i ∈ univ\nhi : ‖m i‖ = 0\n⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖",
"tactic": "rw [norm_eq_zero] at hi"
},
{
"state_after": "case inl.intro.intro\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\nA : 0 ≤ ∏ i : ι, ‖m i‖\nh : 0 = ∏ i : ι, ‖m i‖\ni : ι\nleft✝ : i ∈ univ\nhi : m i = 0\nthis : ↑f m = 0\n⊢ ‖↑f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖",
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"tactic": "have : f m = 0 := f.map_coord_zero i hi"
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Mathlib/Topology/Instances/Real.lean | Int.tendsto_zmultiplesHom_cofinite | [
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Mathlib/Algebra/Polynomial/BigOperators.lean | Polynomial.natDegree_prod_of_monic | [
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211,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | Asymptotics.IsBigO.rpow | [] | [
232,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
229,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean | integral_cos | [
{
"state_after": "case hderiv\na b : ℝ\nn : ℕ\n⊢ deriv sin = fun x => cos x\n\ncase hdiff\na b : ℝ\nn : ℕ\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ sin x\n\ncase hcont\na b : ℝ\nn : ℕ\n⊢ ContinuousOn (fun x => cos x) [[a, b]]",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, cos x) = sin b - sin a",
"tactic": "rw [integral_deriv_eq_sub']"
},
{
"state_after": "no goals",
"state_before": "case hderiv\na b : ℝ\nn : ℕ\n⊢ deriv sin = fun x => cos x",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case hdiff\na b : ℝ\nn : ℕ\n⊢ ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ sin x",
"tactic": "simp only [differentiableAt_sin, implies_true]"
},
{
"state_after": "no goals",
"state_before": "case hcont\na b : ℝ\nn : ℕ\n⊢ ContinuousOn (fun x => cos x) [[a, b]]",
"tactic": "exact continuousOn_cos"
}
] | [
514,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
510,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.toAddSubmonoid_injective | [] | [
252,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean | MeasurableSpace.generateFrom_measurableSet | [] | [
402,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
400,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean | add_le_mul | [] | [
878,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
876,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean | ContinuousLinearMap.toSpanSingleton_smul | [] | [
263,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
] |
Mathlib/CategoryTheory/Functor/Category.lean | CategoryTheory.NatTrans.app_naturality | [] | [
85,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/Deprecated/Subgroup.lean | isSubgroup_iUnion_of_directed | [] | [
131,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Std/Data/Array/Lemmas.lean | Array.getElem?_eq_data_get? | [
{
"state_after": "case pos\nα : Type u_1\na : Array α\ni : Nat\nh✝¹ h✝ : i < size a\n⊢ List.get a.data { val := i, isLt := (_ : i < size a) } = a[i]",
"state_before": "α : Type u_1\na : Array α\ni : Nat\n⊢ a[i]? = List.get? a.data i",
"tactic": "by_cases i < a.size <;> simp_all [getElem?_pos, getElem?_neg, List.get?_eq_get, eq_comm]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\na : Array α\ni : Nat\nh✝¹ h✝ : i < size a\n⊢ List.get a.data { val := i, isLt := (_ : i < size a) } = a[i]",
"tactic": "rfl"
}
] | [
57,
96
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
56,
1
] |
Mathlib/Algebra/AlgebraicCard.lean | Algebraic.cardinal_mk_lift_of_infinite | [] | [
71,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
67,
1
] |
Std/Data/String/Lemmas.lean | String.firstDiffPos_eq | [
{
"state_after": "no goals",
"state_before": "a b : String\n⊢ firstDiffPos a b = { byteIdx := utf8Len (List.takeWhile₂ (fun x x_1 => decide (x = x_1)) a.data b.data).fst }",
"tactic": "simpa [firstDiffPos] using\n firstDiffPos_loop_eq [] [] a.1 b.1 ((utf8Len a.1).min (utf8Len b.1)) 0 rfl rfl (by simp)"
},
{
"state_after": "no goals",
"state_before": "a b : String\n⊢ Nat.min (utf8Len a.data) (utf8Len b.data) = min (utf8Len [] + utf8Len a.data) (utf8Len [] + utf8Len b.data)",
"tactic": "simp"
}
] | [
385,
93
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
382,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.Subsequent.moveRight_mk_left | [
{
"state_after": "no goals",
"state_before": "xl : Type u_1\ni : xl\nxr : Type u_1\nxR : xr → PGame\nxL : xl → PGame\nj : RightMoves (xL i)\n⊢ Subsequent (PGame.moveRight (xL i) j) (mk xl xr xL xR)",
"tactic": "pgame_wf_tac"
}
] | [
309,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
9
] |
Mathlib/Order/OmegaCompletePartialOrder.lean | OmegaCompletePartialOrder.ContinuousHom.map_continuous' | [
{
"state_after": "α : Type u\nα' : Type ?u.60176\nβ✝ : Type v\nβ' : Type ?u.60181\nγ✝ : Type ?u.60184\nφ : Type ?u.60187\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : β → γ\ng : α → Part β\nhg : Continuous' g\n⊢ Continuous' fun x => g x >>= pure ∘ f",
"state_before": "α : Type u\nα' : Type ?u.60176\nβ✝ : Type v\nβ' : Type ?u.60181\nγ✝ : Type ?u.60184\nφ : Type ?u.60187\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : β → γ\ng : α → Part β\nhg : Continuous' g\n⊢ Continuous' fun x => f <$> g x",
"tactic": "simp only [map_eq_bind_pure_comp]"
},
{
"state_after": "α : Type u\nα' : Type ?u.60176\nβ✝ : Type v\nβ' : Type ?u.60181\nγ✝ : Type ?u.60184\nφ : Type ?u.60187\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : β → γ\ng : α → Part β\nhg : Continuous' g\n⊢ Continuous' fun x => pure ∘ f",
"state_before": "α : Type u\nα' : Type ?u.60176\nβ✝ : Type v\nβ' : Type ?u.60181\nγ✝ : Type ?u.60184\nφ : Type ?u.60187\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : β → γ\ng : α → Part β\nhg : Continuous' g\n⊢ Continuous' fun x => g x >>= pure ∘ f",
"tactic": "apply bind_continuous' _ _ hg"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type ?u.60176\nβ✝ : Type v\nβ' : Type ?u.60181\nγ✝ : Type ?u.60184\nφ : Type ?u.60187\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β✝\ninst✝³ : OmegaCompletePartialOrder γ✝\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\nβ γ : Type v\nf : β → γ\ng : α → Part β\nhg : Continuous' g\n⊢ Continuous' fun x => pure ∘ f",
"tactic": "apply const_continuous'"
}
] | [
664,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
662,
1
] |
Mathlib/Algebra/Hom/Group.lean | map_div | [] | [
443,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
1
] |
Mathlib/Data/Set/Image.lean | Set.mem_image | [] | [
213,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
212,
1
] |
Mathlib/Data/Set/Sigma.lean | Set.sigma_inter_sigma | [
{
"state_after": "case h.mk\nι : Type u_1\nι' : Type ?u.8988\nα : ι → Type u_2\nβ : ι → Type ?u.8998\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx✝ : (i : ι) × α i\ni j : ι\na : α i\nx : ι\ny : α x\n⊢ { fst := x, snd := y } ∈ Set.Sigma s₁ t₁ ∩ Set.Sigma s₂ t₂ ↔\n { fst := x, snd := y } ∈ Set.Sigma (s₁ ∩ s₂) fun i => t₁ i ∩ t₂ i",
"state_before": "ι : Type u_1\nι' : Type ?u.8988\nα : ι → Type u_2\nβ : ι → Type ?u.8998\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx : (i : ι) × α i\ni j : ι\na : α i\n⊢ Set.Sigma s₁ t₁ ∩ Set.Sigma s₂ t₂ = Set.Sigma (s₁ ∩ s₂) fun i => t₁ i ∩ t₂ i",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nι : Type u_1\nι' : Type ?u.8988\nα : ι → Type u_2\nβ : ι → Type ?u.8998\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\nu : Set ((i : ι) × α i)\nx✝ : (i : ι) × α i\ni j : ι\na : α i\nx : ι\ny : α x\n⊢ { fst := x, snd := y } ∈ Set.Sigma s₁ t₁ ∩ Set.Sigma s₂ t₂ ↔\n { fst := x, snd := y } ∈ Set.Sigma (s₁ ∩ s₂) fun i => t₁ i ∩ t₂ i",
"tactic": "simp [and_assoc, and_left_comm]"
}
] | [
146,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean | Polynomial.eq_of_monic_of_associated | [
{
"state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : Monic p\nhq : Monic q\nu : R[X]ˣ\nhu : p * ↑u = q\n⊢ p = q",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : Monic p\nhq : Monic q\nhpq : Associated p q\n⊢ p = q",
"tactic": "obtain ⟨u, hu⟩ := hpq"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : leadingCoeff p = 1\nhq : leadingCoeff q = 1\nu : R[X]ˣ\nhu : p * ↑u = q\n⊢ p = q",
"state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : Monic p\nhq : Monic q\nu : R[X]ˣ\nhu : p * ↑u = q\n⊢ p = q",
"tactic": "unfold Monic at hp hq"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : leadingCoeff p = 1\nhq : leadingCoeff q = 1\nu : R[X]ˣ\nhu : p * ↑C (coeff (↑u) 0) = q\n⊢ p = q",
"state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : leadingCoeff p = 1\nhq : leadingCoeff q = 1\nu : R[X]ˣ\nhu : p * ↑u = q\n⊢ p = q",
"tactic": "rw [eq_C_of_degree_le_zero (degree_coe_units _).le] at hu"
},
{
"state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : leadingCoeff p = 1\nu : R[X]ˣ\nhq : coeff (↑u) 0 = 1\nhu : p * ↑C (coeff (↑u) 0) = q\n⊢ p = q",
"state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : leadingCoeff p = 1\nhq : leadingCoeff q = 1\nu : R[X]ˣ\nhu : p * ↑C (coeff (↑u) 0) = q\n⊢ p = q",
"tactic": "rw [← hu, leadingCoeff_mul, hp, one_mul, leadingCoeff_C] at hq"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : leadingCoeff p = 1\nu : R[X]ˣ\nhq : coeff (↑u) 0 = 1\nhu : p * ↑C (coeff (↑u) 0) = q\n⊢ p = q",
"tactic": "rwa [hq, C_1, mul_one] at hu"
}
] | [
441,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
436,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean | LocalEquiv.transEquiv_eq_trans | [] | [
795,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
794,
1
] |
Std/Data/List/Lemmas.lean | List.filter_append | [
{
"state_after": "α : Type u_1\np : α → Bool\na : α\nl₁ l₂ : List α\n⊢ (match p a with\n | true => a :: filter p (l₁ ++ l₂)\n | false => filter p (l₁ ++ l₂)) =\n (match p a with\n | true => a :: filter p l₁\n | false => filter p l₁) ++\n filter p l₂",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl₁ l₂ : List α\n⊢ filter p (a :: l₁ ++ l₂) = filter p (a :: l₁) ++ filter p l₂",
"tactic": "simp [filter]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\na : α\nl₁ l₂ : List α\n⊢ (match p a with\n | true => a :: filter p (l₁ ++ l₂)\n | false => filter p (l₁ ++ l₂)) =\n (match p a with\n | true => a :: filter p l₁\n | false => filter p l₁) ++\n filter p l₂",
"tactic": "split <;> simp [filter_append l₁]"
}
] | [
1117,
71
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1114,
9
] |
Mathlib/Topology/Constructions.lean | openEmbedding_sigma_map | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.553929\nδ : Type ?u.553932\nε : Type ?u.553935\nζ : Type ?u.553938\nι : Type u_1\nκ : Type u_2\nσ : ι → Type u_3\nτ : κ → Type u_4\ninst✝² : (i : ι) → TopologicalSpace (σ i)\ninst✝¹ : (k : κ) → TopologicalSpace (τ k)\ninst✝ : TopologicalSpace α\nf₁ : ι → κ\nf₂ : (i : ι) → σ i → τ (f₁ i)\nh : Injective f₁\n⊢ OpenEmbedding (Sigma.map f₁ f₂) ↔ ∀ (i : ι), OpenEmbedding (f₂ i)",
"tactic": "simp only [openEmbedding_iff_embedding_open, isOpenMap_sigma_map, embedding_sigma_map h,\n forall_and]"
}
] | [
1572,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1569,
1
] |
Mathlib/Data/Set/Basic.lean | Set.not_monotoneOn_not_antitoneOn_iff_exists_lt_lt | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\n⊢ ¬MonotoneOn f s ∧ ¬AntitoneOn f s ↔ ∃ a x b x c x, a < b ∧ b < c ∧ (f a < f b ∧ f c < f b ∨ f b < f a ∧ f b < f c)",
"tactic": "simp [monotoneOn_iff_monotone, antitoneOn_iff_antitone, and_assoc, exists_and_left,\n not_monotone_not_antitone_iff_exists_lt_lt, @and_left_comm (_ ∈ s)]"
}
] | [
2735,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2730,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | Cycle.formPerm_apply_mem_eq_next | [
{
"state_after": "case h\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nx : α\na✝ : List α\nh : Nodup (Quot.mk Setoid.r a✝)\nhx : x ∈ Quot.mk Setoid.r a✝\n⊢ ↑(formPerm (Quot.mk Setoid.r a✝) h) x = next (Quot.mk Setoid.r a✝) h x hx",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ s' s : Cycle α\nh : Nodup s\nx : α\nhx : x ∈ s\n⊢ ↑(formPerm s h) x = next s h x hx",
"tactic": "induction s using Quot.inductionOn"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nx : α\na✝ : List α\nh : Nodup (Quot.mk Setoid.r a✝)\nhx : x ∈ Quot.mk Setoid.r a✝\n⊢ ↑(formPerm (Quot.mk Setoid.r a✝) h) x = next (Quot.mk Setoid.r a✝) h x hx",
"tactic": "simpa using List.formPerm_apply_mem_eq_next h _ (by simp_all)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nx : α\na✝ : List α\nh : Nodup (Quot.mk Setoid.r a✝)\nhx : x ∈ Quot.mk Setoid.r a✝\n⊢ x ∈ a✝",
"tactic": "simp_all"
}
] | [
186,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
183,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | contDiffOn_of_locally_contDiffOn | [
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : ∀ (x : E), x ∈ s → ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)\nx : E\nxs : x ∈ s\n⊢ ContDiffWithinAt 𝕜 n f s x",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : ∀ (x : E), x ∈ s → ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "intro x xs"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : ∀ (x : E), x ∈ s → ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)\nx : E\nxs : x ∈ s\nu : Set E\nu_open : IsOpen u\nxu : x ∈ u\nhu : ContDiffOn 𝕜 n f (s ∩ u)\n⊢ ContDiffWithinAt 𝕜 n f s x",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : ∀ (x : E), x ∈ s → ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)\nx : E\nxs : x ∈ s\n⊢ ContDiffWithinAt 𝕜 n f s x",
"tactic": "rcases h x xs with ⟨u, u_open, xu, hu⟩"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : ∀ (x : E), x ∈ s → ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)\nx : E\nxs : x ∈ s\nu : Set E\nu_open : IsOpen u\nxu : x ∈ u\nhu : ContDiffOn 𝕜 n f (s ∩ u)\n⊢ u ∈ 𝓝 x",
"state_before": "case intro.intro.intro\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : ∀ (x : E), x ∈ s → ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)\nx : E\nxs : x ∈ s\nu : Set E\nu_open : IsOpen u\nxu : x ∈ u\nhu : ContDiffOn 𝕜 n f (s ∩ u)\n⊢ ContDiffWithinAt 𝕜 n f s x",
"tactic": "apply (contDiffWithinAt_inter _).1 (hu x ⟨xs, xu⟩)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : ∀ (x : E), x ∈ s → ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)\nx : E\nxs : x ∈ s\nu : Set E\nu_open : IsOpen u\nxu : x ∈ u\nhu : ContDiffOn 𝕜 n f (s ∩ u)\n⊢ u ∈ 𝓝 x",
"tactic": "exact IsOpen.mem_nhds u_open xu"
}
] | [
731,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
726,
1
] |
Mathlib/Data/MvPolynomial/Rename.lean | MvPolynomial.renameEquiv_refl | [] | [
168,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/ModelTheory/LanguageMap.lean | FirstOrder.Language.LHom.comp_id | [
{
"state_after": "case mk\nL : Language\nL' : Language\nM : Type w\ninst✝ : Structure L M\nϕ : L →ᴸ L'\nL'' : Language\nonFunction✝ : ⦃n : ℕ⦄ → Functions L n → Functions L' n\nonRelation✝ : ⦃n : ℕ⦄ → Relations L n → Relations L' n\n⊢ { onFunction := onFunction✝, onRelation := onRelation✝ } ∘' LHom.id L =\n { onFunction := onFunction✝, onRelation := onRelation✝ }",
"state_before": "L : Language\nL' : Language\nM : Type w\ninst✝ : Structure L M\nϕ : L →ᴸ L'\nL'' : Language\nF : L →ᴸ L'\n⊢ F ∘' LHom.id L = F",
"tactic": "cases F"
},
{
"state_after": "no goals",
"state_before": "case mk\nL : Language\nL' : Language\nM : Type w\ninst✝ : Structure L M\nϕ : L →ᴸ L'\nL'' : Language\nonFunction✝ : ⦃n : ℕ⦄ → Functions L n → Functions L' n\nonRelation✝ : ⦃n : ℕ⦄ → Relations L n → Relations L' n\n⊢ { onFunction := onFunction✝, onRelation := onRelation✝ } ∘' LHom.id L =\n { onFunction := onFunction✝, onRelation := onRelation✝ }",
"tactic": "rfl"
}
] | [
165,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | Right.neg_of_mul_neg_right | [] | [
596,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
595,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean | Finsupp.single_sum | [] | [
292,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
] |
Mathlib/Topology/Instances/EReal.lean | EReal.continuousAt_add_top_top | [
{
"state_after": "α : Type ?u.21891\ninst✝ : TopologicalSpace α\n⊢ ∀ (x : ℝ), ∀ᶠ (a : EReal × EReal) in 𝓝 (⊤, ⊤), ↑x < a.fst + a.snd",
"state_before": "α : Type ?u.21891\ninst✝ : TopologicalSpace α\n⊢ ContinuousAt (fun p => p.fst + p.snd) (⊤, ⊤)",
"tactic": "simp only [ContinuousAt, tendsto_nhds_top_iff_real, top_add_top]"
},
{
"state_after": "α : Type ?u.21891\ninst✝ : TopologicalSpace α\nr : ℝ\nx✝ : EReal × EReal\nh : ↑0 < x✝.fst ∧ ↑r < x✝.snd\n⊢ ↑r < x✝.fst + x✝.snd",
"state_before": "α : Type ?u.21891\ninst✝ : TopologicalSpace α\n⊢ ∀ (x : ℝ), ∀ᶠ (a : EReal × EReal) in 𝓝 (⊤, ⊤), ↑x < a.fst + a.snd",
"tactic": "refine fun r ↦ ((lt_mem_nhds (coe_lt_top 0)).prod_nhds\n (lt_mem_nhds <| coe_lt_top r)).mono fun _ h ↦ ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.21891\ninst✝ : TopologicalSpace α\nr : ℝ\nx✝ : EReal × EReal\nh : ↑0 < x✝.fst ∧ ↑r < x✝.snd\n⊢ ↑r < x✝.fst + x✝.snd",
"tactic": "simpa only [coe_zero, zero_add] using add_lt_add h.1 h.2"
}
] | [
202,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/GroupTheory/Sylow.lean | Sylow.coe_comapOfInjective | [] | [
127,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean | Submodule.adjoint_orthogonalProjection | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.878200\nG : Type ?u.878203\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : InnerProductSpace 𝕜 F\ninst✝⁴ : InnerProductSpace 𝕜 G\ninst✝³ : CompleteSpace E\ninst✝² : CompleteSpace G\ninst✝¹ : CompleteSpace F\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ U }\n⊢ ↑adjoint (orthogonalProjection U) = Submodule.subtypeL U",
"tactic": "rw [← U.adjoint_subtypeL, adjoint_adjoint]"
}
] | [
201,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | CategoryTheory.Limits.coprod.associator_naturality | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX Y : C\ninst✝¹ : Category C\ninst✝ : HasBinaryCoproducts C\nX₁ X₂ X₃ Y₁ Y₂ Y₃ : C\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\nf₃ : X₃ ⟶ Y₃\n⊢ map (map f₁ f₂) f₃ ≫ (associator Y₁ Y₂ Y₃).hom = (associator X₁ X₂ X₃).hom ≫ map f₁ (map f₂ f₃)",
"tactic": "simp"
}
] | [
1155,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1151,
1
] |
Mathlib/CategoryTheory/Limits/ConcreteCategory.lean | CategoryTheory.Limits.Concrete.colimit_rep_eq_of_exists | [] | [
238,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/Logic/Encodable/Basic.lean | Encodable.encode_nat | [] | [
129,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | Filter.Tendsto.bddAbove_range | [] | [
55,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
] |
Mathlib/Data/Num/Lemmas.lean | PosNum.lt_iff_cmp | [] | [
826,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
825,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.inf_le_iff | [] | [
724,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
723,
11
] |
Std/Data/String/Lemmas.lean | String.posOf_eq | [] | [
315,
60
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
315,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.mod_mod | [] | [
1100,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1099,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean | SimpleGraph.interedges_biUnion_left | [] | [
363,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/Order/WithBot.lean | WithBot.ne_bot_iff_exists | [] | [
169,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/LinearAlgebra/FreeModule/Rank.lean | rank_tensorProduct' | [
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nN✝ : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : StrongRankCondition R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : Module.Free R M\ninst✝⁵ : AddCommGroup N✝\ninst✝⁴ : Module R N✝\ninst✝³ : Module.Free R N✝\nN : Type v\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Free R N\n⊢ Module.rank R (M ⊗[R] N) = Module.rank R M * Module.rank R N",
"tactic": "simp"
}
] | [
125,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean | Matrix.isSymm_diagonal | [] | [
129,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.ne_insert_of_not_mem | [
{
"state_after": "α : Type u_1\nβ : Type ?u.112192\nγ : Type ?u.112195\ninst✝ : DecidableEq α\ns✝ t✝ u v : Finset α\na✝ b : α\ns t : Finset α\na : α\nh : s = insert a t\n⊢ a ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.112192\nγ : Type ?u.112195\ninst✝ : DecidableEq α\ns✝ t✝ u v : Finset α\na✝ b : α\ns t : Finset α\na : α\nh : ¬a ∈ s\n⊢ s ≠ insert a t",
"tactic": "contrapose! h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.112192\nγ : Type ?u.112195\ninst✝ : DecidableEq α\ns✝ t✝ u v : Finset α\na✝ b : α\ns t : Finset α\na : α\nh : s = insert a t\n⊢ a ∈ s",
"tactic": "simp [h]"
}
] | [
1157,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1155,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | RingEquiv.ofLeftInverse_symm_apply | [] | [
1333,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1331,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/Minpoly.lean | LinearMap.minpoly_toMatrix | [] | [
73,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean | MulEquiv.map_one | [] | [
538,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
537,
11
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | ContDiffAt.contDiffWithinAt | [] | [
1331,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1330,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean | LocalEquiv.symm_source | [] | [
321,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
320,
1
] |
Mathlib/Topology/Algebra/GroupWithZero.lean | Continuous.div_const | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5613\nG₀ : Type u_2\ninst✝³ : DivInvMonoid G₀\ninst✝² : TopologicalSpace G₀\ninst✝¹ : ContinuousMul G₀\nf : α → G₀\ns : Set α\nl : Filter α\ninst✝ : TopologicalSpace α\nhf : Continuous f\ny : G₀\n⊢ Continuous fun x => f x / y",
"tactic": "simpa only [div_eq_mul_inv] using hf.mul continuous_const"
}
] | [
79,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.lsub_lt_ord | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.29527\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal\nc : Ordinal\nhι : (#ι) < cof c\n⊢ Cardinal.lift (#ι) < cof c",
"tactic": "rwa [(#ι).lift_id]"
}
] | [
340,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
1
] |
Mathlib/Algebra/Hom/Group.lean | MulHom.coe_copy | [] | [
843,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
841,
1
] |
Mathlib/Control/Basic.lean | Functor.map_map | [] | [
28,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
27,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | mul_lt_of_mul_lt_of_nonneg_right | [] | [
263,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
] |
Std/Data/List/Lemmas.lean | List.isSuffix.eq_of_length | [] | [
1657,
25
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1656,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.toDual_inf' | [] | [
1064,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1062,
1
] |
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