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Mathlib/FieldTheory/ChevalleyWarning.lean | char_dvd_card_solutions | [
{
"state_after": "K : Type u_2\nσ : Type u_1\nι : Type ?u.638341\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf : MvPolynomial σ K\nh : totalDegree f < Fintype.card σ\nF : Unit → MvPolynomial σ K := fun x => f\n⊢ p ∣ Fintype.card { x // ↑(eval x) f = 0 }",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.638341\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf : MvPolynomial σ K\nh : totalDegree f < Fintype.card σ\n⊢ p ∣ Fintype.card { x // ↑(eval x) f = 0 }",
"tactic": "let F : Unit → MvPolynomial σ K := fun _ => f"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\nι : Type ?u.638341\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf : MvPolynomial σ K\nh : totalDegree f < Fintype.card σ\nF : Unit → MvPolynomial σ K := fun x => f\nthis : ∑ i : Unit, totalDegree (F i) < Fintype.card σ\n⊢ p ∣ Fintype.card { x // ↑(eval x) f = 0 }",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.638341\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf : MvPolynomial σ K\nh : totalDegree f < Fintype.card σ\nF : Unit → MvPolynomial σ K := fun x => f\n⊢ p ∣ Fintype.card { x // ↑(eval x) f = 0 }",
"tactic": "have : (∑ i : Unit, (F i).totalDegree) < Fintype.card σ := h"
},
{
"state_after": "case h.e'_4.h.h.e'_2.h.a\nK : Type u_2\nσ : Type u_1\nι : Type ?u.638341\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf : MvPolynomial σ K\nh : totalDegree f < Fintype.card σ\nF : Unit → MvPolynomial σ K := fun x => f\nthis : ∑ i : Unit, totalDegree (F i) < Fintype.card σ\nx✝ : σ → K\n⊢ ↑(eval x✝) f = 0 ↔ ∀ (i : Unit), i ∈ univ → ↑(eval x✝) (F i) = 0",
"state_before": "K : Type u_2\nσ : Type u_1\nι : Type ?u.638341\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf : MvPolynomial σ K\nh : totalDegree f < Fintype.card σ\nF : Unit → MvPolynomial σ K := fun x => f\nthis : ∑ i : Unit, totalDegree (F i) < Fintype.card σ\n⊢ p ∣ Fintype.card { x // ↑(eval x) f = 0 }",
"tactic": "convert char_dvd_card_solutions_of_sum_lt p this"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.h.e'_2.h.a\nK : Type u_2\nσ : Type u_1\nι : Type ?u.638341\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf : MvPolynomial σ K\nh : totalDegree f < Fintype.card σ\nF : Unit → MvPolynomial σ K := fun x => f\nthis : ∑ i : Unit, totalDegree (F i) < Fintype.card σ\nx✝ : σ → K\n⊢ ↑(eval x✝) f = 0 ↔ ∀ (i : Unit), i ∈ univ → ↑(eval x✝) (F i) = 0",
"tactic": "aesop"
}
]
| [
193,
8
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
1
]
|
Mathlib/Data/Nat/Choose/Basic.lean | Nat.multichoose_zero_right | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ multichoose n 0 = 1",
"tactic": "cases n <;> simp [multichoose]"
}
]
| [
364,
98
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
364,
1
]
|
Mathlib/GroupTheory/Submonoid/Operations.lean | MonoidHom.restrict_mker | []
| [
1196,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1195,
1
]
|
Mathlib/Analysis/Calculus/MeanValue.lean | Convex.is_const_of_fderivWithin_eq_zero | [
{
"state_after": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.244965\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_2\nG : Type u_3\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\nC : ℝ\ns : Set E\nx y : E\nf' g' : E → E →L[𝕜] G\nφ : E →L[𝕜] G\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nhf' : ∀ (x : E), x ∈ s → fderivWithin 𝕜 f s x = 0\nhx : x ∈ s\nhy : y ∈ s\nbound : ∀ (x : E), x ∈ s → ‖fderivWithin 𝕜 f s x‖ ≤ 0\n⊢ f x = f y",
"state_before": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.244965\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_2\nG : Type u_3\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\nC : ℝ\ns : Set E\nx y : E\nf' g' : E → E →L[𝕜] G\nφ : E →L[𝕜] G\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nhf' : ∀ (x : E), x ∈ s → fderivWithin 𝕜 f s x = 0\nhx : x ∈ s\nhy : y ∈ s\n⊢ f x = f y",
"tactic": "have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by\n simp only [hf' x hx, norm_zero, le_rfl]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.244965\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_2\nG : Type u_3\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\nC : ℝ\ns : Set E\nx y : E\nf' g' : E → E →L[𝕜] G\nφ : E →L[𝕜] G\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nhf' : ∀ (x : E), x ∈ s → fderivWithin 𝕜 f s x = 0\nhx : x ∈ s\nhy : y ∈ s\nbound : ∀ (x : E), x ∈ s → ‖fderivWithin 𝕜 f s x‖ ≤ 0\n⊢ f x = f y",
"tactic": "simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using\n hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.244965\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_2\nG : Type u_3\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\nC : ℝ\ns : Set E\nx✝ y : E\nf' g' : E → E →L[𝕜] G\nφ : E →L[𝕜] G\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nhf' : ∀ (x : E), x ∈ s → fderivWithin 𝕜 f s x = 0\nhx✝ : x✝ ∈ s\nhy : y ∈ s\nx : E\nhx : x ∈ s\n⊢ ‖fderivWithin 𝕜 f s x‖ ≤ 0",
"tactic": "simp only [hf' x hx, norm_zero, le_rfl]"
}
]
| [
596,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
591,
1
]
|
Mathlib/Data/Real/ENNReal.lean | ENNReal.mul_inv | [
{
"state_after": "case top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nha✝ : a ≠ 0 ∨ b ≠ ⊤\nhb✝ : a ≠ ⊤ ∨ b ≠ 0\nha : a ≠ 0 ∨ ⊤ ≠ ⊤\nhb : a ≠ ⊤ ∨ ⊤ ≠ 0\n⊢ (a * ⊤)⁻¹ = a⁻¹ * ⊤⁻¹\n\ncase coe\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na b✝ : ℝ≥0∞\nha✝ : a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : a ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha : a ≠ 0 ∨ ↑b ≠ ⊤\nhb : a ≠ ⊤ ∨ ↑b ≠ 0\n⊢ (a * ↑b)⁻¹ = a⁻¹ * (↑b)⁻¹",
"state_before": "α : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nha : a ≠ 0 ∨ b ≠ ⊤\nhb : a ≠ ⊤ ∨ b ≠ 0\n⊢ (a * b)⁻¹ = a⁻¹ * b⁻¹",
"tactic": "induction' b using recTopCoe with b"
},
{
"state_after": "case coe.top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na b✝ : ℝ≥0∞\nha✝² : a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a ≠ ⊤ ∨ ↑b ≠ 0\nha✝ : ⊤ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ⊤ ≠ ⊤ ∨ b✝ ≠ 0\nha : ⊤ ≠ 0 ∨ ↑b ≠ ⊤\nhb : ⊤ ≠ ⊤ ∨ ↑b ≠ 0\n⊢ (⊤ * ↑b)⁻¹ = ⊤⁻¹ * (↑b)⁻¹\n\ncase coe.coe\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"state_before": "case coe\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na b✝ : ℝ≥0∞\nha✝ : a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : a ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha : a ≠ 0 ∨ ↑b ≠ ⊤\nhb : a ≠ ⊤ ∨ ↑b ≠ 0\n⊢ (a * ↑b)⁻¹ = a⁻¹ * (↑b)⁻¹",
"tactic": "induction' a using recTopCoe with a"
},
{
"state_after": "case pos\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : a = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹\n\ncase neg\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"state_before": "case coe.coe\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"tactic": "by_cases h'a : a = 0"
},
{
"state_after": "case pos\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\nh'b : b = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹\n\ncase neg\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\nh'b : ¬b = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"state_before": "case neg\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"tactic": "by_cases h'b : b = 0"
},
{
"state_after": "case neg\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\nh'b : ¬b = 0\n⊢ a * b ≠ 0",
"state_before": "case neg\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\nh'b : ¬b = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"tactic": "rw [← ENNReal.coe_mul, ← ENNReal.coe_inv, ← ENNReal.coe_inv h'a, ← ENNReal.coe_inv h'b, ←\n ENNReal.coe_mul, mul_inv_rev, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\nh'b : ¬b = 0\n⊢ a * b ≠ 0",
"tactic": "simp [h'a, h'b]"
},
{
"state_after": "case top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nha✝ : a ≠ 0 ∨ b ≠ ⊤\nhb✝ : a ≠ ⊤ ∨ b ≠ 0\nhb : a ≠ ⊤ ∨ ⊤ ≠ 0\nha : a ≠ 0\n⊢ (a * ⊤)⁻¹ = a⁻¹ * ⊤⁻¹",
"state_before": "case top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nha✝ : a ≠ 0 ∨ b ≠ ⊤\nhb✝ : a ≠ ⊤ ∨ b ≠ 0\nha : a ≠ 0 ∨ ⊤ ≠ ⊤\nhb : a ≠ ⊤ ∨ ⊤ ≠ 0\n⊢ (a * ⊤)⁻¹ = a⁻¹ * ⊤⁻¹",
"tactic": "replace ha : a ≠ 0 := ha.neg_resolve_right rfl"
},
{
"state_after": "no goals",
"state_before": "case top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nha✝ : a ≠ 0 ∨ b ≠ ⊤\nhb✝ : a ≠ ⊤ ∨ b ≠ 0\nhb : a ≠ ⊤ ∨ ⊤ ≠ 0\nha : a ≠ 0\n⊢ (a * ⊤)⁻¹ = a⁻¹ * ⊤⁻¹",
"tactic": "simp [ha]"
},
{
"state_after": "case coe.top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na b✝ : ℝ≥0∞\nha✝² : a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a ≠ ⊤ ∨ ↑b ≠ 0\nha✝ : ⊤ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ⊤ ≠ ⊤ ∨ b✝ ≠ 0\nha : ⊤ ≠ 0 ∨ ↑b ≠ ⊤\nhb : b ≠ 0\n⊢ (⊤ * ↑b)⁻¹ = ⊤⁻¹ * (↑b)⁻¹",
"state_before": "case coe.top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na b✝ : ℝ≥0∞\nha✝² : a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a ≠ ⊤ ∨ ↑b ≠ 0\nha✝ : ⊤ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ⊤ ≠ ⊤ ∨ b✝ ≠ 0\nha : ⊤ ≠ 0 ∨ ↑b ≠ ⊤\nhb : ⊤ ≠ ⊤ ∨ ↑b ≠ 0\n⊢ (⊤ * ↑b)⁻¹ = ⊤⁻¹ * (↑b)⁻¹",
"tactic": "replace hb : b ≠ 0 := coe_ne_zero.1 (hb.neg_resolve_left rfl)"
},
{
"state_after": "no goals",
"state_before": "case coe.top\nα : Type ?u.257111\nβ : Type ?u.257114\na✝ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na b✝ : ℝ≥0∞\nha✝² : a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a ≠ ⊤ ∨ ↑b ≠ 0\nha✝ : ⊤ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ⊤ ≠ ⊤ ∨ b✝ ≠ 0\nha : ⊤ ≠ 0 ∨ ↑b ≠ ⊤\nhb : b ≠ 0\n⊢ (⊤ * ↑b)⁻¹ = ⊤⁻¹ * (↑b)⁻¹",
"tactic": "simp [hb]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : a = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"tactic": "simp only [h'a, top_mul, ENNReal.inv_zero, ENNReal.coe_ne_top, zero_mul, Ne.def,\n not_false_iff, ENNReal.coe_zero, ENNReal.inv_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.257111\nβ : Type ?u.257114\na✝¹ b✝¹ c d : ℝ≥0∞\nr p q : ℝ≥0\na✝ b✝ : ℝ≥0∞\nha✝² : a✝ ≠ 0 ∨ b✝ ≠ ⊤\nhb✝² : a✝ ≠ ⊤ ∨ b✝ ≠ 0\nb : ℝ≥0\nha✝¹ : a✝ ≠ 0 ∨ ↑b ≠ ⊤\nhb✝¹ : a✝ ≠ ⊤ ∨ ↑b ≠ 0\na : ℝ≥0\nha✝ : ↑a ≠ 0 ∨ b✝ ≠ ⊤\nhb✝ : ↑a ≠ ⊤ ∨ b✝ ≠ 0\nha : ↑a ≠ 0 ∨ ↑b ≠ ⊤\nhb : ↑a ≠ ⊤ ∨ ↑b ≠ 0\nh'a : ¬a = 0\nh'b : b = 0\n⊢ (↑a * ↑b)⁻¹ = (↑a)⁻¹ * (↑b)⁻¹",
"tactic": "simp only [h'b, ENNReal.inv_zero, ENNReal.coe_ne_top, mul_top, Ne.def, not_false_iff,\n mul_zero, ENNReal.coe_zero, ENNReal.inv_eq_zero]"
}
]
| [
1442,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1426,
11
]
|
Mathlib/Analysis/Convex/Strict.lean | DirectedOn.strictConvex_sUnion | [
{
"state_after": "𝕜 : Type u_2\n𝕝 : Type ?u.6609\nE : Type u_1\nF : Type ?u.6615\nβ : Type ?u.6618\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx y : E\na b : 𝕜\nS : Set (Set E)\nhdir : DirectedOn (fun x x_1 => x ⊆ x_1) S\nhS : ∀ (s : Set E), s ∈ S → StrictConvex 𝕜 s\n⊢ StrictConvex 𝕜 (⋃ (i : ↑S), ↑i)",
"state_before": "𝕜 : Type u_2\n𝕝 : Type ?u.6609\nE : Type u_1\nF : Type ?u.6615\nβ : Type ?u.6618\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx y : E\na b : 𝕜\nS : Set (Set E)\nhdir : DirectedOn (fun x x_1 => x ⊆ x_1) S\nhS : ∀ (s : Set E), s ∈ S → StrictConvex 𝕜 s\n⊢ StrictConvex 𝕜 (⋃₀ S)",
"tactic": "rw [sUnion_eq_iUnion]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\n𝕝 : Type ?u.6609\nE : Type u_1\nF : Type ?u.6615\nβ : Type ?u.6618\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set E\nx y : E\na b : 𝕜\nS : Set (Set E)\nhdir : DirectedOn (fun x x_1 => x ⊆ x_1) S\nhS : ∀ (s : Set E), s ∈ S → StrictConvex 𝕜 s\n⊢ StrictConvex 𝕜 (⋃ (i : ↑S), ↑i)",
"tactic": "exact (directedOn_iff_directed.1 hdir).strictConvex_iUnion fun s => hS _ s.2"
}
]
| [
100,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
]
|
Mathlib/RingTheory/OreLocalization/Basic.lean | OreLocalization.mul'_char | [
{
"state_after": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\n⊢ r₁ * oreNum r₂ s₁ /ₒ (s₂ * oreDenom r₂ s₁) = r₁ * v /ₒ (s₂ * u)",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\n⊢ OreLocalization.mul' r₁ s₁ r₂ s₂ = r₁ * v /ₒ (s₂ * u)",
"tactic": "simp only [mul']"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nh₀ : r₂ * ↑(oreDenom r₂ s₁) = ↑s₁ * oreNum r₂ s₁\n⊢ r₁ * oreNum r₂ s₁ /ₒ (s₂ * oreDenom r₂ s₁) = r₁ * v /ₒ (s₂ * u)",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\n⊢ r₁ * oreNum r₂ s₁ /ₒ (s₂ * oreDenom r₂ s₁) = r₁ * v /ₒ (s₂ * u)",
"tactic": "have h₀ := ore_eq r₂ s₁"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nh₀ : r₂ * ↑(oreDenom r₂ s₁) = ↑s₁ * v₀\n⊢ r₁ * v₀ /ₒ (s₂ * oreDenom r₂ s₁) = r₁ * v /ₒ (s₂ * u)",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nh₀ : r₂ * ↑(oreDenom r₂ s₁) = ↑s₁ * oreNum r₂ s₁\n⊢ r₁ * oreNum r₂ s₁ /ₒ (s₂ * oreDenom r₂ s₁) = r₁ * v /ₒ (s₂ * u)",
"tactic": "set v₀ := oreNum r₂ s₁"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nh₀ : r₂ * ↑(oreDenom r₂ s₁) = ↑s₁ * v₀\n⊢ r₁ * v₀ /ₒ (s₂ * oreDenom r₂ s₁) = r₁ * v /ₒ (s₂ * u)",
"tactic": "set u₀ := oreDenom r₂ s₁"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"tactic": "rcases oreCondition (u₀ : R) u with ⟨r₃, s₃, h₃⟩"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"state_before": "case mk.mk\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"tactic": "have :=\n calc\n (s₁ : R) * (v * r₃) = r₂ * u * r₃ := by rw [← mul_assoc, ← huv]\n _ = r₂ * u₀ * s₃ := by rw [mul_assoc, mul_assoc, h₃]\n _ = s₁ * (v₀ * s₃) := by rw [← mul_assoc, h₀]"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"state_before": "case mk.mk\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"tactic": "rcases ore_left_cancel _ _ _ this with ⟨s₄, hs₄⟩"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * v /ₒ (s₂ * u) = r₁ * v₀ /ₒ (s₂ * u₀)",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * v₀ /ₒ (s₂ * u₀) = r₁ * v /ₒ (s₂ * u)",
"tactic": "symm"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ∃ u_1 v_1, r₁ * v₀ * ↑u_1 = r₁ * v * v_1 ∧ ↑(s₂ * u₀) * ↑u_1 = ↑(s₂ * u) * v_1",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * v /ₒ (s₂ * u) = r₁ * v₀ /ₒ (s₂ * u₀)",
"tactic": "rw [oreDiv_eq_iff]"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ∃ v_1, r₁ * v₀ * ↑(s₃ * s₄) = r₁ * v * v_1 ∧ ↑(s₂ * u₀) * ↑(s₃ * s₄) = ↑(s₂ * u) * v_1",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ∃ u_1 v_1, r₁ * v₀ * ↑u_1 = r₁ * v * v_1 ∧ ↑(s₂ * u₀) * ↑u_1 = ↑(s₂ * u) * v_1",
"tactic": "use s₃ * s₄"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * v₀ * ↑(s₃ * s₄) = r₁ * v * (r₃ * ↑s₄) ∧ ↑(s₂ * u₀) * ↑(s₃ * s₄) = ↑(s₂ * u) * (r₃ * ↑s₄)",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ∃ v_1, r₁ * v₀ * ↑(s₃ * s₄) = r₁ * v * v_1 ∧ ↑(s₂ * u₀) * ↑(s₃ * s₄) = ↑(s₂ * u) * v_1",
"tactic": "use r₃ * s₄"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * oreNum r₂ s₁ * (↑s₃ * ↑s₄) = r₁ * v * (r₃ * ↑s₄) ∧ ↑s₂ * ↑(oreDenom r₂ s₁) * (↑s₃ * ↑s₄) = ↑s₂ * ↑u * (r₃ * ↑s₄)",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * v₀ * ↑(s₃ * s₄) = r₁ * v * (r₃ * ↑s₄) ∧ ↑(s₂ * u₀) * ↑(s₃ * s₄) = ↑(s₂ * u) * (r₃ * ↑s₄)",
"tactic": "simp only [Submonoid.coe_mul]"
},
{
"state_after": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * oreNum r₂ s₁ * (↑s₃ * ↑s₄) = r₁ * v * (r₃ * ↑s₄)\n\ncase mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ↑s₂ * ↑(oreDenom r₂ s₁) * (↑s₃ * ↑s₄) = ↑s₂ * ↑u * (r₃ * ↑s₄)",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * oreNum r₂ s₁ * (↑s₃ * ↑s₄) = r₁ * v * (r₃ * ↑s₄) ∧ ↑s₂ * ↑(oreDenom r₂ s₁) * (↑s₃ * ↑s₄) = ↑s₂ * ↑u * (r₃ * ↑s₄)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\n⊢ ↑s₁ * (v * r₃) = r₂ * ↑u * r₃",
"tactic": "rw [← mul_assoc, ← huv]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\n⊢ r₂ * ↑u * r₃ = r₂ * ↑u₀ * ↑s₃",
"tactic": "rw [mul_assoc, mul_assoc, h₃]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\n⊢ r₂ * ↑u₀ * ↑s₃ = ↑s₁ * (v₀ * ↑s₃)",
"tactic": "rw [← mul_assoc, h₀]"
},
{
"state_after": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * (v * r₃ * ↑s₄) = r₁ * v * (r₃ * ↑s₄)",
"state_before": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * oreNum r₂ s₁ * (↑s₃ * ↑s₄) = r₁ * v * (r₃ * ↑s₄)",
"tactic": "rw [mul_assoc (b := v₀), ← mul_assoc (a := v₀), ← hs₄]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ r₁ * (v * r₃ * ↑s₄) = r₁ * v * (r₃ * ↑s₄)",
"tactic": "simp only [mul_assoc]"
},
{
"state_after": "case mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ↑s₂ * (↑u * r₃ * ↑s₄) = ↑s₂ * ↑u * (r₃ * ↑s₄)",
"state_before": "case mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ↑s₂ * ↑(oreDenom r₂ s₁) * (↑s₃ * ↑s₄) = ↑s₂ * ↑u * (r₃ * ↑s₄)",
"tactic": "rw [mul_assoc (b := (u₀ : R)), ← mul_assoc (a := (u₀ : R)), h₃]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Monoid R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ u : { x // x ∈ S }\nv : R\nhuv : r₂ * ↑u = ↑s₁ * v\nv₀ : R := oreNum r₂ s₁\nu₀ : { x // x ∈ S } := oreDenom r₂ s₁\nh₀ : r₂ * ↑u₀ = ↑s₁ * v₀\nr₃ : R\ns₃ : { x // x ∈ S }\nh₃ : ↑u₀ * ↑s₃ = ↑u * r₃\nthis : ↑s₁ * (v * r₃) = ↑s₁ * (v₀ * ↑s₃)\ns₄ : { x // x ∈ S }\nhs₄ : v * r₃ * ↑s₄ = v₀ * ↑s₃ * ↑s₄\n⊢ ↑s₂ * (↑u * r₃ * ↑s₄) = ↑s₂ * ↑u * (r₃ * ↑s₄)",
"tactic": "simp only [mul_assoc]"
}
]
| [
218,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
9
]
|
Mathlib/Logic/Equiv/Basic.lean | Function.update_comp_equiv | [
{
"state_after": "no goals",
"state_before": "α' : Sort u_1\nα : Sort u_2\nβ : Sort u_3\ninst✝¹ : DecidableEq α'\ninst✝ : DecidableEq α\nf : α → β\ng : α' ≃ α\na : α\nv : β\n⊢ update f a v ∘ ↑g = update (f ∘ ↑g) (↑g.symm a) v",
"tactic": "rw [← update_comp_eq_of_injective _ g.injective, g.apply_symm_apply]"
}
]
| [
1921,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1918,
1
]
|
Mathlib/Data/Nat/Factors.lean | Nat.factors_chain | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ∀ {a : ℕ}, (∀ (p : ℕ), Prime p → p ∣ n → a ≤ p) → List.Chain (fun x x_1 => x ≤ x_1) a (factors n)",
"tactic": "match n with\n| 0 => simp\n| 1 => simp\n| k + 2 =>\n intro a h\n let m := minFac (k + 2)\n have : (k + 2) / m < (k + 2) := factors_lemma\n rw [factors]\n refine' List.Chain.cons ((le_minFac.2 h).resolve_left (by simp)) (factors_chain _)\n exact fun p pp d => minFac_le_of_dvd pp.two_le (d.trans <| div_dvd_of_dvd <| minFac_dvd _)"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ∀ {a : ℕ}, (∀ (p : ℕ), Prime p → p ∣ 0 → a ≤ p) → List.Chain (fun x x_1 => x ≤ x_1) a (factors 0)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ∀ {a : ℕ}, (∀ (p : ℕ), Prime p → p ∣ 1 → a ≤ p) → List.Chain (fun x x_1 => x ≤ x_1) a (factors 1)",
"tactic": "simp"
},
{
"state_after": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a (factors (k + 2))",
"state_before": "n k : ℕ\n⊢ ∀ {a : ℕ}, (∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p) → List.Chain (fun x x_1 => x ≤ x_1) a (factors (k + 2))",
"tactic": "intro a h"
},
{
"state_after": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a (factors (k + 2))",
"state_before": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a (factors (k + 2))",
"tactic": "let m := minFac (k + 2)"
},
{
"state_after": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a (factors (k + 2))",
"state_before": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a (factors (k + 2))",
"tactic": "have : (k + 2) / m < (k + 2) := factors_lemma"
},
{
"state_after": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a\n (let m := minFac (k + 2);\n let_fun this := (_ : (k + 2) / minFac (k + 2) < k + 2);\n m :: factors ((k + 2) / m))",
"state_before": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a (factors (k + 2))",
"tactic": "rw [factors]"
},
{
"state_after": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ ∀ (p : ℕ), Prime p → p ∣ (k + 2) / minFac (k + 2) → minFac (k + 2) ≤ p",
"state_before": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ List.Chain (fun x x_1 => x ≤ x_1) a\n (let m := minFac (k + 2);\n let_fun this := (_ : (k + 2) / minFac (k + 2) < k + 2);\n m :: factors ((k + 2) / m))",
"tactic": "refine' List.Chain.cons ((le_minFac.2 h).resolve_left (by simp)) (factors_chain _)"
},
{
"state_after": "no goals",
"state_before": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ ∀ (p : ℕ), Prime p → p ∣ (k + 2) / minFac (k + 2) → minFac (k + 2) ≤ p",
"tactic": "exact fun p pp d => minFac_le_of_dvd pp.two_le (d.trans <| div_dvd_of_dvd <| minFac_dvd _)"
},
{
"state_after": "no goals",
"state_before": "n k a : ℕ\nh : ∀ (p : ℕ), Prime p → p ∣ k + 2 → a ≤ p\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ ¬k + 2 = 1",
"tactic": "simp"
}
]
| [
104,
97
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
]
|
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | extChartAt_source_mem_nhdsWithin' | []
| [
1078,
65
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1076,
1
]
|
Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousWithinAt.zpow | []
| [
536,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
534,
1
]
|
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.L1.setToL1_const | []
| [
1158,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1156,
1
]
|
Mathlib/Topology/LocallyConstant/Basic.lean | IsLocallyConstant.one | []
| [
193,
81
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
]
|
Mathlib/Data/PFunctor/Univariate/M.lean | PFunctor.M.iselect_cons | [
{
"state_after": "no goals",
"state_before": "F : PFunctor\nX : Type ?u.28846\nf✝ : X → Obj F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited (M F)\nps : Path F\na : F.A\nf : B F a → M F\ni : B F a\n⊢ iselect ({ fst := a, snd := i } :: ps) (M.mk { fst := a, snd := f }) = iselect ps (f i)",
"tactic": "simp only [iselect, isubtree_cons]"
}
]
| [
574,
101
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
573,
1
]
|
Mathlib/Data/List/BigOperators/Basic.lean | MonoidHom.map_list_prod | []
| [
704,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
703,
11
]
|
Mathlib/SetTheory/Game/PGame.lean | PGame.leftMoves_mk | []
| [
142,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
]
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.closure_cycle_coprime_swap | [
{
"state_after": "ι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"state_before": "ι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (Fintype.card α)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"tactic": "rw [← Finset.card_univ, ← h2, ← h1.orderOf] at h0"
},
{
"state_after": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"state_before": "ι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"tactic": "cases' exists_pow_eq_self_of_coprime h0 with m hm"
},
{
"state_after": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"state_before": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"tactic": "have h2' : (σ ^ n).support = ⊤ := Eq.trans (support_pow_coprime h0) h2"
},
{
"state_after": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\nh1' : IsCycle ((σ ^ n) ^ ↑m)\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"state_before": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"tactic": "have h1' : IsCycle ((σ ^ n) ^ (m : ℤ)) := by rwa [← hm] at h1"
},
{
"state_after": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\nh1' : IsCycle (σ ^ n)\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"state_before": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\nh1' : IsCycle ((σ ^ n) ^ ↑m)\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"tactic": "replace h1' : IsCycle (σ ^ n) :=\n h1'.of_pow (le_trans (support_pow_le σ n) (ge_of_eq (congr_arg support hm)))"
},
{
"state_after": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\nh1' : IsCycle (σ ^ n)\n⊢ σ ^ n ∈ ↑(closure {σ, swap x (↑(σ ^ n) x)}) ∧ {swap x (↑(σ ^ n) x)} ⊆ ↑(closure {σ, swap x (↑(σ ^ n) x)})",
"state_before": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\nh1' : IsCycle (σ ^ n)\n⊢ closure {σ, swap x (↑(σ ^ n) x)} = ⊤",
"tactic": "rw [eq_top_iff, ← closure_cycle_adjacent_swap h1' h2' x, closure_le, Set.insert_subset]"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\nh1' : IsCycle (σ ^ n)\n⊢ σ ^ n ∈ ↑(closure {σ, swap x (↑(σ ^ n) x)}) ∧ {swap x (↑(σ ^ n) x)} ⊆ ↑(closure {σ, swap x (↑(σ ^ n) x)})",
"tactic": "exact\n ⟨Subgroup.pow_mem (closure _) (subset_closure (Set.mem_insert σ _)) n,\n Set.singleton_subset_iff.mpr (subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _)))⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2958039\nα : Type u_1\nβ : Type ?u.2958045\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : Nat.coprime n (orderOf σ)\nh1 : IsCycle σ\nh2 : support σ = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : support (σ ^ n) = ⊤\n⊢ IsCycle ((σ ^ n) ^ ↑m)",
"tactic": "rwa [← hm] at h1"
}
]
| [
1680,
101
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1668,
1
]
|
Mathlib/Analysis/Complex/AbsMax.lean | Complex.norm_eq_norm_of_isMaxOn_of_ball_subset | []
| [
208,
95
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
205,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isLittleO_one_left_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.297892\nE : Type u_3\nF : Type u_2\nG : Type ?u.297901\nE' : Type ?u.297904\nF' : Type ?u.297907\nG' : Type ?u.297910\nE'' : Type ?u.297913\nF'' : Type ?u.297916\nG'' : Type ?u.297919\nR : Type ?u.297922\nR' : Type ?u.297925\n𝕜 : Type ?u.297928\n𝕜' : Type ?u.297931\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝¹ : One F\ninst✝ : NormOneClass F\n_x : α\n⊢ 0 ≤ ‖1‖",
"tactic": "simp only [norm_one, zero_le_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.297892\nE : Type u_3\nF : Type u_2\nG : Type ?u.297901\nE' : Type ?u.297904\nF' : Type ?u.297907\nG' : Type ?u.297910\nE'' : Type ?u.297913\nF'' : Type ?u.297916\nG'' : Type ?u.297919\nR : Type ?u.297922\nR' : Type ?u.297925\n𝕜 : Type ?u.297928\n𝕜' : Type ?u.297931\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝¹ : One F\ninst✝ : NormOneClass F\n⊢ (∀ (n : ℕ), ∀ᶠ (x : α) in l, ↑n * ‖1‖ ≤ ‖f x‖) ↔ ∀ (n : ℕ), True → ∀ᶠ (x : α) in l, ‖f x‖ ∈ Ici ↑n",
"tactic": "simp only [norm_one, mul_one, true_imp_iff, mem_Ici]"
}
]
| [
1346,
70
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1339,
1
]
|
Mathlib/Data/Set/Intervals/Basic.lean | Set.Iic_inter_Ioc_of_le | []
| [
715,
73
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
714,
1
]
|
Mathlib/Order/Hom/Bounded.lean | BotHom.coe_copy | []
| [
414,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
]
|
Mathlib/LinearAlgebra/BilinearForm.lean | BilinForm.flip_apply | []
| [
336,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
335,
1
]
|
Mathlib/Data/Polynomial/AlgebraMap.lean | Polynomial.aeval_mul | []
| [
254,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
253,
1
]
|
Mathlib/MeasureTheory/Group/Arithmetic.lean | Finset.measurable_prod' | []
| [
934,
87
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
932,
1
]
|
Mathlib/Algebra/Lie/Normalizer.lean | LieSubmodule.mem_normalizer | []
| [
64,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
]
|
Mathlib/Data/Set/Basic.lean | Set.setOf_or | []
| [
306,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
]
|
Mathlib/Algebra/Order/Group/Abs.lean | sub_lt_of_abs_sub_lt_left | []
| [
301,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
]
|
Mathlib/GroupTheory/Perm/List.lean | List.formPerm_eq_of_isRotated | [
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.774035\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhd : Nodup l\nn : ℕ\n⊢ formPerm l = formPerm (rotate l n)",
"state_before": "α : Type u_1\nβ : Type ?u.774035\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl l' : List α\nhd : Nodup l\nh : l ~r l'\n⊢ formPerm l = formPerm l'",
"tactic": "obtain ⟨n, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.774035\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nhd : Nodup l\nn : ℕ\n⊢ formPerm l = formPerm (rotate l n)",
"tactic": "exact (formPerm_rotate l hd n).symm"
}
]
| [
296,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
]
|
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.findMin_dual | []
| [
591,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
589,
1
]
|
Mathlib/Order/BoundedOrder.lean | bot_unique | []
| [
369,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
]
|
Mathlib/Data/Fin/VecNotation.lean | Matrix.vecHead_vecAlt0 | []
| [
362,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
360,
1
]
|
Mathlib/Data/List/Func.lean | List.Func.get_map | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\nx✝ : ℕ\nh : x✝ < length []\n⊢ get x✝ (map f []) = f (get x✝ [])",
"tactic": "cases h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\nn : ℕ\nhead✝ : α\nas : List α\nh1 : n + 1 < length (head✝ :: as)\nh2 : n < length as\n⊢ get (n + 1) (map f (head✝ :: as)) = f (get (n + 1) (head✝ :: as))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\nn : ℕ\nhead✝ : α\nas : List α\nh1 : n + 1 < length (head✝ :: as)\n⊢ get (n + 1) (map f (head✝ :: as)) = f (get (n + 1) (head✝ :: as))",
"tactic": "have h2 : n < length as := by\n rw [← Nat.succ_le_iff, ← Nat.lt_succ_iff]\n apply h1"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\nn : ℕ\nhead✝ : α\nas : List α\nh1 : n + 1 < length (head✝ :: as)\nh2 : n < length as\n⊢ get (n + 1) (map f (head✝ :: as)) = f (get (n + 1) (head✝ :: as))",
"tactic": "apply get_map h2"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\nn : ℕ\nhead✝ : α\nas : List α\nh1 : n + 1 < length (head✝ :: as)\n⊢ Nat.succ n < Nat.succ (length as)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\nn : ℕ\nhead✝ : α\nas : List α\nh1 : n + 1 < length (head✝ :: as)\n⊢ n < length as",
"tactic": "rw [← Nat.succ_le_iff, ← Nat.lt_succ_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\nf : α → β\nn : ℕ\nhead✝ : α\nas : List α\nh1 : n + 1 < length (head✝ :: as)\n⊢ Nat.succ n < Nat.succ (length as)",
"tactic": "apply h1"
}
]
| [
205,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
]
|
Mathlib/Analysis/Convex/Function.lean | concaveOn_iff_pairwise_pos | []
| [
356,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
]
|
Mathlib/MeasureTheory/Group/Measure.lean | MeasureTheory.Measure.measurePreserving_inv | []
| [
390,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
389,
1
]
|
Mathlib/Data/Complex/Exponential.lean | Real.add_one_le_exp | [
{
"state_after": "case inl\nx : ℝ\nh : 0 ≤ x\n⊢ x + 1 ≤ exp x\n\ncase inr\nx : ℝ\nh : x < 0\n⊢ x + 1 ≤ exp x",
"state_before": "x : ℝ\n⊢ x + 1 ≤ exp x",
"tactic": "cases' le_or_lt 0 x with h h"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nh : x < 0\n⊢ x + 1 ≤ exp x",
"tactic": "exact (add_one_lt_exp_of_neg h).le"
},
{
"state_after": "no goals",
"state_before": "case inl\nx : ℝ\nh : 0 ≤ x\n⊢ x + 1 ≤ exp x",
"tactic": "exact Real.add_one_le_exp_of_nonneg h"
}
]
| [
1990,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1987,
1
]
|
Mathlib/Algebra/Squarefree.lean | Squarefree.squarefree_of_dvd | []
| [
88,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
]
|
Mathlib/Analysis/InnerProductSpace/Projection.lean | orthogonalProjection_mem_subspace_eq_self | [
{
"state_after": "case a\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.512356\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv : { x // x ∈ K }\n⊢ ↑(↑(orthogonalProjection K) ↑v) = ↑v",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.512356\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv : { x // x ∈ K }\n⊢ ↑(orthogonalProjection K) ↑v = v",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.512356\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv : { x // x ∈ K }\n⊢ ↑(↑(orthogonalProjection K) ↑v) = ↑v",
"tactic": "apply eq_orthogonalProjection_of_mem_of_inner_eq_zero <;> simp"
}
]
| [
519,
65
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
517,
1
]
|
Mathlib/Data/Finset/Basic.lean | Finset.pair_eq_singleton | []
| [
1113,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1112,
1
]
|
Mathlib/SetTheory/ZFC/Ordinal.lean | ZFSet.isTransitive_iff_sUnion_subset | [
{
"state_after": "case intro.intro\nx y✝ z✝ : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y ∈ ⋃₀ x\nz : ZFSet\nhz : z ∈ x\nhz' : y ∈ z\n⊢ y ∈ x",
"state_before": "x y✝ z : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y ∈ ⋃₀ x\n⊢ y ∈ x",
"tactic": "rcases mem_sUnion.1 hy with ⟨z, hz, hz'⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nx y✝ z✝ : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y ∈ ⋃₀ x\nz : ZFSet\nhz : z ∈ x\nhz' : y ∈ z\n⊢ y ∈ x",
"tactic": "exact h.mem_trans hz' hz"
}
]
| [
99,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
]
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.MeasurePreserving.set_lintegral_comp_emb | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1688080\nδ : Type ?u.1688083\nm : MeasurableSpace α\nμ ν✝ : Measure α\nmb : MeasurableSpace β\nν : Measure β\ng : α → β\nhg : MeasurePreserving g\nhge : MeasurableEmbedding g\nf : β → ℝ≥0∞\ns : Set α\n⊢ (∫⁻ (a : α) in s, f (g a) ∂μ) = ∫⁻ (b : β) in g '' s, f b ∂ν",
"tactic": "rw [← hg.set_lintegral_comp_preimage_emb hge, preimage_image_eq _ hge.injective]"
}
]
| [
1357,
83
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1354,
1
]
|
Mathlib/Topology/Algebra/Module/StrongTopology.lean | ContinuousLinearMap.strongUniformity.uniformEmbedding_coeFn | []
| [
104,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
]
|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | IsLeast.csInf_eq | []
| [
569,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
568,
1
]
|
Mathlib/Deprecated/Group.lean | RingHom.to_isAddGroupHom | []
| [
391,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
390,
1
]
|
Mathlib/Analysis/Convex/Star.lean | StarConvex.smul_mem | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.110353\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ t • x ∈ s",
"tactic": "simpa using hs.add_smul_mem (by simpa using hx) ht₀ ht₁"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.110353\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\ns t✝ : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ 0 + x ∈ s",
"tactic": "simpa using hx"
}
]
| [
344,
92
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
]
|
Std/Data/List/Lemmas.lean | List.sublist_of_cons_sublist | []
| [
342,
28
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
341,
1
]
|
Mathlib/Data/Polynomial/UnitTrinomial.lean | Polynomial.trinomial_natDegree | [
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nh : i ∈ support (trinomial k m n u v w)\n⊢ ↑i ≤ ↑n",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\n⊢ natDegree (trinomial k m n u v w) = n",
"tactic": "refine'\n natDegree_eq_of_degree_eq_some\n ((Finset.sup_le fun i h => _).antisymm <|\n le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nh : i ∈ {k, m, n}\n⊢ ↑i ≤ ↑n",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nh : i ∈ support (trinomial k m n u v w)\n⊢ ↑i ≤ ↑n",
"tactic": "replace h := support_trinomial' k m n u v w h"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nh : i = k ∨ i = m ∨ i = n\n⊢ ↑i ≤ ↑n",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nh : i ∈ {k, m, n}\n⊢ ↑i ≤ ↑n",
"tactic": "rw [mem_insert, mem_insert, mem_singleton] at h"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nu v w : R\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nhkm : i < m\n⊢ ↑i ≤ ↑n\n\ncase inr.inl\nR : Type u_1\ninst✝ : Semiring R\nk n : ℕ\nu v w : R\nhw : w ≠ 0\ni : ℕ\nhkm : k < i\nhmn : i < n\n⊢ ↑i ≤ ↑n\n\ncase inr.inr\nR : Type u_1\ninst✝ : Semiring R\nk m : ℕ\nu v w : R\nhkm : k < m\nhw : w ≠ 0\ni : ℕ\nhmn : m < i\n⊢ ↑i ≤ ↑i",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nh : i = k ∨ i = m ∨ i = n\n⊢ ↑i ≤ ↑n",
"tactic": "rcases h with (rfl | rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk m n : ℕ\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w ≠ 0\n⊢ coeff (trinomial k m n u v w) n ≠ 0",
"tactic": "rwa [trinomial_leading_coeff' hkm hmn]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nu v w : R\nhmn : m < n\nhw : w ≠ 0\ni : ℕ\nhkm : i < m\n⊢ ↑i ≤ ↑n",
"tactic": "exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nR : Type u_1\ninst✝ : Semiring R\nk n : ℕ\nu v w : R\nhw : w ≠ 0\ni : ℕ\nhkm : k < i\nhmn : i < n\n⊢ ↑i ≤ ↑n",
"tactic": "exact WithBot.coe_le_coe.mpr hmn.le"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nR : Type u_1\ninst✝ : Semiring R\nk m : ℕ\nu v w : R\nhkm : k < m\nhw : w ≠ 0\ni : ℕ\nhmn : m < i\n⊢ ↑i ≤ ↑i",
"tactic": "exact le_rfl"
}
]
| [
81,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
]
|
Mathlib/Data/Num/Lemmas.lean | ZNum.zneg_zneg | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.658241\nn : ZNum\n⊢ - -n = n",
"tactic": "cases n <;> rfl"
}
]
| [
1079,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1079,
1
]
|
Mathlib/AlgebraicTopology/SimplexCategory.lean | SimplexCategory.isIso_of_bijective | []
| [
564,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
561,
1
]
|
Mathlib/Data/Polynomial/Coeff.lean | Polynomial.coeff_monomial_zero_mul | []
| [
292,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
]
|
Mathlib/Data/Nat/Factors.lean | Nat.dvd_of_factors_subperm | [
{
"state_after": "case inl\na : ℕ\nha : a ≠ 0\nh : factors a <+~ factors 0\n⊢ a ∣ 0\n\ncase inr\na b : ℕ\nha : a ≠ 0\nh : factors a <+~ factors b\nhb : b > 0\n⊢ a ∣ b",
"state_before": "a b : ℕ\nha : a ≠ 0\nh : factors a <+~ factors b\n⊢ a ∣ b",
"tactic": "rcases b.eq_zero_or_pos with (rfl | hb)"
},
{
"state_after": "case inr.zero\nb : ℕ\nhb : b > 0\nha : zero ≠ 0\nh : factors zero <+~ factors b\n⊢ zero ∣ b\n\ncase inr.succ.zero\nb : ℕ\nhb : b > 0\nha : succ zero ≠ 0\nh : factors (succ zero) <+~ factors b\n⊢ succ zero ∣ b\n\ncase inr.succ.succ\nb : ℕ\nhb : b > 0\na : ℕ\nha : succ (succ a) ≠ 0\nh : factors (succ (succ a)) <+~ factors b\n⊢ succ (succ a) ∣ b",
"state_before": "case inr\na b : ℕ\nha : a ≠ 0\nh : factors a <+~ factors b\nhb : b > 0\n⊢ a ∣ b",
"tactic": "rcases a with (_ | _ | a)"
},
{
"state_after": "case inr.succ.succ\nb : ℕ\nhb : b > 0\na : ℕ\nha : succ (succ a) ≠ 0\nh : factors (succ (succ a)) <+~ factors b\n⊢ b = succ (succ a) * prod (List.diff (factors b) (factors (succ (succ a))))",
"state_before": "case inr.succ.succ\nb : ℕ\nhb : b > 0\na : ℕ\nha : succ (succ a) ≠ 0\nh : factors (succ (succ a)) <+~ factors b\n⊢ succ (succ a) ∣ b",
"tactic": "use (@List.diff _ instBEq b.factors a.succ.succ.factors).prod"
},
{
"state_after": "case inr.succ.succ\nb : ℕ\nhb : b > 0\na : ℕ\nha : succ (succ a) ≠ 0\nh : factors (succ (succ a)) <+~ factors b\n⊢ b = prod (factors (succ (succ a))) * prod (List.diff (factors b) (factors (succ (succ a))))",
"state_before": "case inr.succ.succ\nb : ℕ\nhb : b > 0\na : ℕ\nha : succ (succ a) ≠ 0\nh : factors (succ (succ a)) <+~ factors b\n⊢ b = succ (succ a) * prod (List.diff (factors b) (factors (succ (succ a))))",
"tactic": "nth_rw 1 [← Nat.prod_factors ha]"
},
{
"state_after": "no goals",
"state_before": "case inr.succ.succ\nb : ℕ\nhb : b > 0\na : ℕ\nha : succ (succ a) ≠ 0\nh : factors (succ (succ a)) <+~ factors b\n⊢ b = prod (factors (succ (succ a))) * prod (List.diff (factors b) (factors (succ (succ a))))",
"tactic": "rw [← List.prod_append,\n List.Perm.prod_eq <| List.subperm_append_diff_self_of_count_le <| List.subperm_ext_iff.mp h,\n Nat.prod_factors hb.ne']"
},
{
"state_after": "no goals",
"state_before": "case inl\na : ℕ\nha : a ≠ 0\nh : factors a <+~ factors 0\n⊢ a ∣ 0",
"tactic": "exact dvd_zero _"
},
{
"state_after": "no goals",
"state_before": "case inr.zero\nb : ℕ\nhb : b > 0\nha : zero ≠ 0\nh : factors zero <+~ factors b\n⊢ zero ∣ b",
"tactic": "exact (ha rfl).elim"
},
{
"state_after": "no goals",
"state_before": "case inr.succ.zero\nb : ℕ\nhb : b > 0\nha : succ zero ≠ 0\nh : factors (succ zero) <+~ factors b\n⊢ succ zero ∣ b",
"tactic": "exact one_dvd _"
}
]
| [
253,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
241,
1
]
|
Mathlib/Data/Finset/Basic.lean | Finset.toList_cons | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.481863\nγ : Type ?u.481866\na : α\ns : Finset α\nh : ¬a ∈ s\n⊢ List.Nodup (a :: toList s)",
"tactic": "simp [h, nodup_toList s]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.481863\nγ : Type ?u.481866\na : α\ns : Finset α\nh : ¬a ∈ s\nx : α\n⊢ x ∈ toList (cons a s h) ↔ x ∈ a :: toList s",
"tactic": "simp only [List.mem_cons, Finset.mem_toList, Finset.mem_cons]"
}
]
| [
3409,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3407,
1
]
|
Mathlib/Analysis/Normed/Group/Basic.lean | Pi.sum_nnnorm_apply_le_nnnorm' | []
| [
2557,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2556,
1
]
|
Mathlib/Order/Filter/Basic.lean | Filter.comap_comap | []
| [
2087,
77
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2086,
1
]
|
Mathlib/Order/Bounds/Basic.lean | le_of_isLUB_le_isGLB | []
| [
1063,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1058,
1
]
|
Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | Path.Homotopy.trans_refl_reparam | [
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\np : Path x₀ x₁\n⊢ Continuous fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t ∈ I) }",
"tactic": "continuity"
},
{
"state_after": "case a.h\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\np : Path x₀ x₁\nx✝ : ↑I\n⊢ ↑(Path.trans p (Path.refl x₁)) x✝ =\n ↑(Path.reparam p (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t ∈ I) })\n (_ : Continuous fun x => { val := transReflReparamAux x, property := (_ : transReflReparamAux x ∈ I) })\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t ∈ I) }) 0 = 0)\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t ∈ I) }) 1 = 1))\n x✝",
"state_before": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\np : Path x₀ x₁\n⊢ Path.trans p (Path.refl x₁) =\n Path.reparam p (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t ∈ I) })\n (_ : Continuous fun x => { val := transReflReparamAux x, property := (_ : transReflReparamAux x ∈ I) })\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t ∈ I) }) 0 = 0)\n (_ : (fun t => { val := transReflReparamAux t, property := (_ : transReflReparamAux t ∈ I) }) 1 = 1)",
"tactic": "ext"
},
{
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{
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{
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{
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175,
9
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
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|
Mathlib/Data/Set/Function.lean | Set.restrict_extend_range | [
{
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"tactic": "classical\nexact restrict_dite _ _"
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{
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123,
26
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
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|
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | Pmf.normalize_apply | []
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251,
86
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
251,
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|
Mathlib/Topology/Instances/ENNReal.lean | Continuous.ennreal_mul | []
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373,
70
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
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Mathlib/Algebra/Ring/AddAut.lean | AddAut.mulRight_symm_apply | []
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49,
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
48,
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|
Mathlib/Data/Polynomial/Eval.lean | Polynomial.comp_eq_sum_left | [
{
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"tactic": "rw [comp, eval₂_eq_sum]"
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531,
97
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
531,
1
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|
Mathlib/Data/Matrix/Kronecker.lean | Matrix.kroneckerMap_map_left | []
| [
79,
21
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
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|
Mathlib/RingTheory/Polynomial/Basic.lean | Polynomial.linearIndependent_powers_iff_aeval | [
{
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{
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"tactic": "simp only [Finsupp.total_apply, aeval_endomorphism, forall_iff_forall_finsupp, Sum, support,\n coeff, ofFinsupp_eq_zero]"
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"tactic": "exact Iff.rfl"
}
]
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991,
16
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
986,
1
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|
Mathlib/Data/Complex/Module.lean | Complex.coe_basisOneI_repr | []
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165,
6
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
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|
Mathlib/RingTheory/UniqueFactorizationDomain.lean | UniqueFactorizationMonoid.count_normalizedFactors_eq | [
{
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"tactic": "rw [← PartENat.natCast_inj]"
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{
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"tactic": "convert (multiplicity_eq_count_normalizedFactors hp hx0).symm"
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{
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"tactic": "exact (multiplicity.eq_coe_iff.mpr ⟨hle, hlt⟩).symm"
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{
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{
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"tactic": "exact hnorm.symm"
}
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1007,
54
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
998,
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|
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.EqOnSource.trans' | []
| [
974,
37
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
972,
1
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|
Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.closure_eq_iInter_thickening' | [
{
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1371,
58
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1368,
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Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.cos_coe | []
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327,
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326,
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Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval₂_sub | [
{
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}
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1257,
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1255,
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Mathlib/Algebra/GroupPower/Ring.lean | zero_pow_eq | [
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50,
41
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Mathlib/RingTheory/Finiteness.lean | RingHom.Finite.id | []
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681,
23
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680,
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Mathlib/Combinatorics/Additive/SalemSpencer.lean | mulRothNumber_union_le | [
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390,
86
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381,
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Mathlib/LinearAlgebra/Span.lean | Submodule.iSup_induction | [
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{
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634,
61
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
630,
1
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Mathlib/Order/MinMax.lean | min_lt_of_right_lt | []
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112,
32
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
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|
Mathlib/Data/Multiset/Basic.lean | Multiset.count_cons_self | []
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2354,
29
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2353,
1
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|
Mathlib/Analysis/Normed/Field/UnitBall.lean | coe_div_unitSphere | []
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142,
6
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140,
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Mathlib/Init/Algebra/Order.lean | not_lt | []
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373,
31
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372,
1
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Mathlib/Algebra/Order/Monoid/WithTop.lean | WithBot.coe_nat | []
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539,
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538,
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|
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | SeminormFamily.withSeminorms_of_nhds | [
{
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425,
10
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
420,
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|
Mathlib/Topology/Basic.lean | nhds_le_of_le | [
{
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881,
50
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
880,
1
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|
Mathlib/Data/ULift.lean | ULift.up_inj | []
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114,
22
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113,
1
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|
Mathlib/ModelTheory/LanguageMap.lean | FirstOrder.Language.LHom.funext | [
{
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{
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"tactic": "cases' G with Gf Gr"
},
{
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},
{
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"tactic": "exact And.intro h_fun h_rel"
}
]
| [
122,
30
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
11
]
|
Mathlib/Order/Heyting/Basic.lean | PUnit.compl_eq | []
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1355,
6
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1354,
1
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|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.IsCycle.cycleFactorsFinset_eq_singleton | []
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1458,
50
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1456,
1
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|
Mathlib/Algebra/TrivSqZeroExt.lean | TrivSqZeroExt.inl_one | []
| [
464,
6
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
463,
1
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|
Mathlib/Analysis/Convex/StoneSeparation.lean | exists_convex_convex_compl_subset | [
{
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"tactic": "let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\n⊢ ∃ C, Convex 𝕜 C ∧ Convex 𝕜 (Cᶜ) ∧ s ⊆ C ∧ t ⊆ Cᶜ",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\n⊢ ∃ C, Convex 𝕜 C ∧ Convex 𝕜 (Cᶜ) ∧ s ⊆ C ∧ t ⊆ Cᶜ",
"tactic": "obtain ⟨C, hC, hsC, hCmax⟩ :=\n zorn_subset_nonempty S\n (fun c hcS hc ⟨_, _⟩ =>\n ⟨⋃₀ c,\n ⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,\n disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,\n fun s => subset_sUnion_of_mem⟩)\n s ⟨hs, hst⟩"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\n⊢ False",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\n⊢ ∃ C, Convex 𝕜 C ∧ Convex 𝕜 (Cᶜ) ∧ s ⊆ C ∧ t ⊆ Cᶜ",
"tactic": "refine'\n ⟨C, hC.1, convex_iff_segment_subset.2 fun x hx y hy z hz hzC => _, hsC, hC.2.subset_compl_left⟩"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nh : ∀ (c : E), c ∈ Cᶜ → ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ False\n\ncase h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\n⊢ ∀ (c : E), c ∈ Cᶜ → ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\n⊢ False",
"tactic": "suffices h : ∀ c ∈ Cᶜ, ∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\n⊢ ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\n⊢ ∀ (c : E), c ∈ Cᶜ → ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)",
"tactic": "rintro c hc"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ False",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\n⊢ ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)",
"tactic": "by_contra' h"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh✝ : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\nh : Disjoint (↑(convexHull 𝕜).toOrderHom (insert c C)) t\n⊢ False\n\ncase h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ Disjoint (↑(convexHull 𝕜).toOrderHom (insert c C)) t",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ False",
"tactic": "suffices h : Disjoint (convexHull 𝕜 (insert c C)) t"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ Disjoint (⋃ (y : E) (_ : y ∈ ↑(convexHull 𝕜).toOrderHom C), segment 𝕜 c y) t",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ Disjoint (↑(convexHull 𝕜).toOrderHom (insert c C)) t",
"tactic": "rw [convexHull_insert ⟨z, hzC⟩, convexJoin_singleton_left]"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\na : E\nha : a ∈ ↑(convexHull 𝕜).toOrderHom C\nb : E\nhb : b ∈ segment 𝕜 c a ⊓ t\n⊢ a ∈ C",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ Disjoint (⋃ (y : E) (_ : y ∈ ↑(convexHull 𝕜).toOrderHom C), segment 𝕜 c y) t",
"tactic": "refine' disjoint_iUnion₂_left.2 fun a ha => disjoint_iff_inf_le.mpr fun b hb => h a _ ⟨b, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nc : E\nhc : c ∈ Cᶜ\nh : ∀ (a : E), a ∈ C → ¬Set.Nonempty (segment 𝕜 c a ∩ t)\na : E\nha : a ∈ ↑(convexHull 𝕜).toOrderHom C\nb : E\nhb : b ∈ segment 𝕜 c a ⊓ t\n⊢ a ∈ C",
"tactic": "rwa [← hC.1.convexHull_eq]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nh : ∀ (c : E), c ∈ Cᶜ → ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)\np : E\nhp : p ∈ C\nu : E\nhu : u ∈ segment 𝕜 x p\nhut : u ∈ t\n⊢ False",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nh : ∀ (c : E), c ∈ Cᶜ → ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)\n⊢ False",
"tactic": "obtain ⟨p, hp, u, hu, hut⟩ := h x hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nh : ∀ (c : E), c ∈ Cᶜ → ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)\np : E\nhp : p ∈ C\nu : E\nhu : u ∈ segment 𝕜 x p\nhut : u ∈ t\nq : E\nhq : q ∈ C\nv : E\nhv : v ∈ segment 𝕜 y q\nhvt : v ∈ t\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nι : Type ?u.47333\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhC : C ∈ S\nhsC : s ⊆ C\nhCmax : ∀ (a : Set E), a ∈ S → C ⊆ a → a = C\nx : E\nhx : x ∈ Cᶜ\ny : E\nhy : y ∈ Cᶜ\nz : E\nhz : z ∈ segment 𝕜 x y\nhzC : z ∈ C\nh : ∀ (c : E), c ∈ Cᶜ → ∃ a, a ∈ C ∧ Set.Nonempty (segment 𝕜 c a ∩ t)\np : E\nhp : p ∈ C\nu : E\nhu : u ∈ segment 𝕜 x p\nhut : u ∈ t\n⊢ False",
"tactic": "obtain ⟨q, hq, v, hv, hvt⟩ := h y hy"
},
{
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"tactic": "refine'\n not_disjoint_segment_convexHull_triple hz hu hv\n (hC.2.symm.mono (ht.segment_subset hut hvt) <| convexHull_min _ hC.1)"
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{
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"tactic": "simpa [insert_subset, hp, hq, singleton_subset_iff.2 hzC]"
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{
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"tactic": "rw [←\n hCmax _ ⟨convex_convexHull _ _, h⟩ ((subset_insert _ _).trans <| subset_convexHull _ _)] at hc"
},
{
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"tactic": "exact hc (subset_convexHull _ _ <| mem_insert _ _)"
}
]
| [
114,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
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|
Mathlib/LinearAlgebra/Matrix/Circulant.lean | Matrix.Fin.circulant_isSymm_iff | [
{
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"tactic": "simp [IsSymm.ext_iff, IsEmpty.forall_iff]"
}
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| [
199,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
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|
Mathlib/Analysis/Normed/Group/HomCompletion.lean | NormedAddGroupHom.zero_completion | []
| [
136,
42
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
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|
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | BoxIntegral.TaggedPrepartition.mem_iUnion | [
{
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"tactic": "convert Set.mem_iUnion₂"
},
{
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"tactic": "rw [Box.mem_coe, mem_toPrepartition, exists_prop]"
}
]
| [
84,
52
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
1
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|
Mathlib/Logic/IsEmpty.lean | isEmpty_pprod | [
{
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"tactic": "simp only [← not_nonempty_iff, nonempty_pprod, not_and_or]"
}
]
| [
176,
61
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
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|
Mathlib/Algebra/Periodic.lean | Function.Antiperiodic.sub_eq | [
{
"state_after": "no goals",
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"tactic": "simp only [← neg_eq_iff_eq_neg, ← h (x - c), sub_add_cancel]"
}
]
| [
408,
88
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
1
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|
Mathlib/Algebra/Periodic.lean | Multiset.periodic_prod | []
| [
94,
89
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
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|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineEquiv.span_eq_top_iff | [
{
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},
{
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"tactic": "intro h"
},
{
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"tactic": "have : s = e.symm '' (e '' s) := by rw [← image_comp]; simp"
},
{
"state_after": "k : Type u_2\nV₁ : Type u_4\nP₁ : Type u_1\nV₂ : Type u_5\nP₂ : Type u_3\nV₃ : Type ?u.582825\nP₃ : Type ?u.582828\ninst✝⁹ : Ring k\ninst✝⁸ : AddCommGroup V₁\ninst✝⁷ : Module k V₁\ninst✝⁶ : AffineSpace V₁ P₁\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module k V₂\ninst✝³ : AffineSpace V₂ P₂\ninst✝² : AddCommGroup V₃\ninst✝¹ : Module k V₃\ninst✝ : AffineSpace V₃ P₃\nf : P₁ →ᵃ[k] P₂\ns : Set P₁\ne : P₁ ≃ᵃ[k] P₂\nh : affineSpan k (↑e '' s) = ⊤\nthis : s = ↑(symm e) '' (↑e '' s)\n⊢ affineSpan k (↑(symm e) '' (↑e '' s)) = ⊤",
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},
{
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},
{
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"tactic": "simp"
}
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1599,
77
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1593,
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|
Mathlib/MeasureTheory/Measure/AEDisjoint.lean | MeasureTheory.AEDisjoint.iUnion_left_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\n⊢ AEDisjoint μ (⋃ (i : ι), s i) t ↔ ∀ (i : ι), AEDisjoint μ (s i) t",
"tactic": "simp only [AEDisjoint, iUnion_inter, measure_iUnion_null_iff]"
}
]
| [
98,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
]
|
Mathlib/Topology/MetricSpace/MetricSeparated.lean | IsMetricSeparated.mono | []
| [
70,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
]
|
Mathlib/Topology/Algebra/Module/Basic.lean | Module.punctured_nhds_neBot | [
{
"state_after": "case intro\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ NeBot (𝓝[{x}ᶜ] x)",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx : M\n⊢ NeBot (𝓝[{x}ᶜ] x)",
"tactic": "rcases exists_ne (0 : M) with ⟨y, hy⟩"
},
{
"state_after": "case intro\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\nthis : Tendsto (fun c => x + c • y) (𝓝[{0}ᶜ] 0) (𝓝[{x}ᶜ] x)\n⊢ NeBot (𝓝[{x}ᶜ] x)\n\ncase this\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ Tendsto (fun c => x + c • y) (𝓝[{0}ᶜ] 0) (𝓝[{x}ᶜ] x)",
"state_before": "case intro\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ NeBot (𝓝[{x}ᶜ] x)",
"tactic": "suffices : Tendsto (fun c : R => x + c • y) (𝓝[≠] 0) (𝓝[≠] x)"
},
{
"state_after": "case this\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ Tendsto (fun c => x + c • y) (𝓝[{0}ᶜ] 0) (𝓝[{x}ᶜ] x)",
"state_before": "case intro\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\nthis : Tendsto (fun c => x + c • y) (𝓝[{0}ᶜ] 0) (𝓝[{x}ᶜ] x)\n⊢ NeBot (𝓝[{x}ᶜ] x)\n\ncase this\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ Tendsto (fun c => x + c • y) (𝓝[{0}ᶜ] 0) (𝓝[{x}ᶜ] x)",
"tactic": "exact this.neBot"
},
{
"state_after": "case this.refine'_1\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ Tendsto (fun c => x + c • y) (𝓝 0) (𝓝 x)\n\ncase this.refine'_2\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ ∀ (a : R), a ∈ {0}ᶜ → x + a • y ∈ {x}ᶜ",
"state_before": "case this\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ Tendsto (fun c => x + c • y) (𝓝[{0}ᶜ] 0) (𝓝[{x}ᶜ] x)",
"tactic": "refine' Tendsto.inf _ (tendsto_principal_principal.2 <| _)"
},
{
"state_after": "case h.e'_5.h.e'_3\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ x = x + 0 • y",
"state_before": "case this.refine'_1\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ Tendsto (fun c => x + c • y) (𝓝 0) (𝓝 x)",
"tactic": "convert tendsto_const_nhds.add ((@tendsto_id R _).smul_const y)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e'_3\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ x = x + 0 • y",
"tactic": "rw [zero_smul, add_zero]"
},
{
"state_after": "case this.refine'_2\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\nc : R\nhc : c ∈ {0}ᶜ\n⊢ x + c • y ∈ {x}ᶜ",
"state_before": "case this.refine'_2\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\n⊢ ∀ (a : R), a ∈ {0}ᶜ → x + a • y ∈ {x}ᶜ",
"tactic": "intro c hc"
},
{
"state_after": "no goals",
"state_before": "case this.refine'_2\nR : Type u_2\nM : Type u_1\ninst✝⁹ : Ring R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : ContinuousAdd M\ninst✝⁴ : Module R M\ninst✝³ : ContinuousSMul R M\ninst✝² : Nontrivial M\ninst✝¹ : NeBot (𝓝[{0}ᶜ] 0)\ninst✝ : NoZeroSMulDivisors R M\nx y : M\nhy : y ≠ 0\nc : R\nhc : c ∈ {0}ᶜ\n⊢ x + c • y ∈ {x}ᶜ",
"tactic": "simpa [hy] using hc"
}
]
| [
96,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
]
|
Mathlib/Algebra/Order/Monoid/Lemmas.lean | StrictMonoOn.const_mul' | []
| [
1384,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1382,
1
]
|
Mathlib/Algebra/Hom/Freiman.lean | FreimanHom.cancel_right | []
| [
260,
73
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
1
]
|
Mathlib/SetTheory/Ordinal/Basic.lean | Ordinal.card_eq_zero | [
{
"state_after": "α✝ : Type ?u.102030\nβ : Type ?u.102033\nγ : Type ?u.102036\nr✝ : α✝ → α✝ → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\nh : card (type r) = 0\nthis : IsEmpty { α := α, r := r, wo := x✝ }.α\n⊢ type r = 0",
"state_before": "α✝ : Type ?u.102030\nβ : Type ?u.102033\nγ : Type ?u.102036\nr✝ : α✝ → α✝ → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\nh : card (type r) = 0\n⊢ type r = 0",
"tactic": "haveI := Cardinal.mk_eq_zero_iff.1 h"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.102030\nβ : Type ?u.102033\nγ : Type ?u.102036\nr✝ : α✝ → α✝ → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nα : Type u_1\nr : α → α → Prop\nx✝ : IsWellOrder α r\nh : card (type r) = 0\nthis : IsEmpty { α := α, r := r, wo := x✝ }.α\n⊢ type r = 0",
"tactic": "apply type_eq_zero_of_empty"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.102030\nβ : Type ?u.102033\nγ : Type ?u.102036\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\ne : o = 0\n⊢ card o = 0",
"tactic": "simp only [e, card_zero]"
}
]
| [
634,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
630,
1
]
|
Mathlib/Data/Nat/Hyperoperation.lean | hyperoperation_two | [
{
"state_after": "case h.h\nm k : ℕ\n⊢ hyperoperation 2 m k = m * k",
"state_before": "⊢ hyperoperation 2 = fun x x_1 => x * x_1",
"tactic": "ext (m k)"
},
{
"state_after": "case h.h.zero\nm : ℕ\n⊢ hyperoperation 2 m Nat.zero = m * Nat.zero\n\ncase h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ hyperoperation 2 m (Nat.succ bn) = m * Nat.succ bn",
"state_before": "case h.h\nm k : ℕ\n⊢ hyperoperation 2 m k = m * k",
"tactic": "induction' k with bn bih"
},
{
"state_after": "case h.h.zero\nm : ℕ\n⊢ 0 = m * Nat.zero",
"state_before": "case h.h.zero\nm : ℕ\n⊢ hyperoperation 2 m Nat.zero = m * Nat.zero",
"tactic": "rw [hyperoperation]"
},
{
"state_after": "no goals",
"state_before": "case h.h.zero\nm : ℕ\n⊢ 0 = m * Nat.zero",
"tactic": "exact (Nat.mul_zero m).symm"
},
{
"state_after": "case h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ (fun x x_1 => x + x_1) m (m * bn) = m * Nat.succ bn",
"state_before": "case h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ hyperoperation 2 m (Nat.succ bn) = m * Nat.succ bn",
"tactic": "rw [hyperoperation_recursion, hyperoperation_one, bih]"
},
{
"state_after": "case h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ m + m * bn = m * Nat.succ bn",
"state_before": "case h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ (fun x x_1 => x + x_1) m (m * bn) = m * Nat.succ bn",
"tactic": "dsimp only"
},
{
"state_after": "case h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ m * 1 + m * bn = m * Nat.succ bn",
"state_before": "case h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ m + m * bn = m * Nat.succ bn",
"tactic": "nth_rewrite 1 [← mul_one m]"
},
{
"state_after": "no goals",
"state_before": "case h.h.succ\nm bn : ℕ\nbih : hyperoperation 2 m bn = m * bn\n⊢ m * 1 + m * bn = m * Nat.succ bn",
"tactic": "rw [← mul_add, add_comm, Nat.succ_eq_add_one]"
}
]
| [
83,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
]
|
Mathlib/Order/Bounds/Basic.lean | IsLeast.union | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t✝ : Set α\na✝ b✝ : α\ninst✝ : LinearOrder γ\na b : γ\ns t : Set γ\nha : IsLeast s a\nhb : IsLeast t b\n⊢ min a b ∈ s ∪ t",
"tactic": "cases' le_total a b with h h <;> simp [h, ha.1, hb.1]"
}
]
| [
461,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
1
]
|
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