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Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.canonicalEquiv_canonicalEquiv | [
{
"state_after": "case a\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_4\ninst✝⁶ : CommRing P'\ninst✝⁵ : Algebra R P'\nloc' : IsLocalization S P'\nP''✝ : Type ?u.928438\ninst✝⁴ : CommRing P''✝\ninst✝³ : Algebra R P''✝\nloc'' : IsLocalization S P''✝\nI✝ J : FractionalIdeal S P\ng : P →ₐ[R] P'\nP'' : Type u_1\ninst✝² : CommRing P''\ninst✝¹ : Algebra R P''\ninst✝ : IsLocalization S P''\nI : FractionalIdeal S P\nx✝ : P''\n⊢ x✝ ∈ ↑(canonicalEquiv S P' P'') (↑(canonicalEquiv S P P') I) ↔ x✝ ∈ ↑(canonicalEquiv S P P'') I",
"state_before": "R : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_4\ninst✝⁶ : CommRing P'\ninst✝⁵ : Algebra R P'\nloc' : IsLocalization S P'\nP''✝ : Type ?u.928438\ninst✝⁴ : CommRing P''✝\ninst✝³ : Algebra R P''✝\nloc'' : IsLocalization S P''✝\nI✝ J : FractionalIdeal S P\ng : P →ₐ[R] P'\nP'' : Type u_1\ninst✝² : CommRing P''\ninst✝¹ : Algebra R P''\ninst✝ : IsLocalization S P''\nI : FractionalIdeal S P\n⊢ ↑(canonicalEquiv S P' P'') (↑(canonicalEquiv S P P') I) = ↑(canonicalEquiv S P P'') I",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_4\ninst✝⁶ : CommRing P'\ninst✝⁵ : Algebra R P'\nloc' : IsLocalization S P'\nP''✝ : Type ?u.928438\ninst✝⁴ : CommRing P''✝\ninst✝³ : Algebra R P''✝\nloc'' : IsLocalization S P''✝\nI✝ J : FractionalIdeal S P\ng : P →ₐ[R] P'\nP'' : Type u_1\ninst✝² : CommRing P''\ninst✝¹ : Algebra R P''\ninst✝ : IsLocalization S P''\nI : FractionalIdeal S P\nx✝ : P''\n⊢ x✝ ∈ ↑(canonicalEquiv S P' P'') (↑(canonicalEquiv S P P') I) ↔ x✝ ∈ ↑(canonicalEquiv S P P'') I",
"tactic": "simp only [IsLocalization.map_map, RingHomInvPair.comp_eq₂, mem_canonicalEquiv_apply,\n exists_prop, exists_exists_and_eq_and]"
}
] | [
935,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
930,
1
] |
Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousWithinAt.zpow₀ | [] | [
343,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
341,
8
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean | Real.logb_eq_zero | [
{
"state_after": "b x y : ℝ\n⊢ (x = 0 ∨ x = 1 ∨ x = -1) ∨ b = 0 ∨ b = 1 ∨ b = -1 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1",
"state_before": "b x y : ℝ\n⊢ logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1",
"tactic": "simp_rw [logb, div_eq_zero_iff, log_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "b x y : ℝ\n⊢ (x = 0 ∨ x = 1 ∨ x = -1) ∨ b = 0 ∨ b = 1 ∨ b = -1 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1",
"tactic": "tauto"
}
] | [
385,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
383,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | Complex.arg_neg_eq_arg_sub_pi_of_im_pos | [
{
"state_after": "x : ℂ\nhi : 0 < x.im\n⊢ -arccos ((-x).re / ↑abs (-x)) = arccos (x.re / ↑abs x) - π",
"state_before": "x : ℂ\nhi : 0 < x.im\n⊢ arg (-x) = arg x - π",
"tactic": "rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)]"
},
{
"state_after": "no goals",
"state_before": "x : ℂ\nhi : 0 < x.im\n⊢ -arccos ((-x).re / ↑abs (-x)) = arccos (x.re / ↑abs x) - π",
"tactic": "simp [neg_div, Real.arccos_neg]"
}
] | [
377,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
375,
1
] |
Mathlib/Algebra/FreeMonoid/Basic.lean | FreeMonoid.recOn_of_mul | [] | [
177,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
] |
Mathlib/Data/Part.lean | Part.toOption_eq_none_iff | [] | [
310,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/Data/Polynomial/Derivative.lean | Polynomial.iterate_derivative_int_cast_mul | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Ring R\nn : ℤ\nk : ℕ\nf : R[X]\n⊢ (↑derivative^[k]) (↑n * f) = ↑n * (↑derivative^[k]) f",
"tactic": "induction' k with k ih generalizing f <;> simp [*]"
}
] | [
631,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
629,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.order_add_of_order_eq.aux | [
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ order (φ + ψ) = order φ",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ order (φ + ψ) ≤ order φ ⊓ order ψ",
"tactic": "suffices order (φ + ψ) = order φ by\n rw [le_inf_iff, this]\n exact ⟨le_rfl, le_of_lt H⟩"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\nthis : order (φ + ψ) = order φ\n⊢ order φ ≤ order φ ∧ order φ ≤ order ψ",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\nthis : order (φ + ψ) = order φ\n⊢ order (φ + ψ) ≤ order φ ⊓ order ψ",
"tactic": "rw [le_inf_iff, this]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\nthis : order (φ + ψ) = order φ\n⊢ order φ ≤ order φ ∧ order φ ≤ order ψ",
"tactic": "exact ⟨le_rfl, le_of_lt H⟩"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ (∀ (i : ℕ), ↑i = order φ → ↑(coeff R i) (φ + ψ) ≠ 0) ∧ ∀ (i : ℕ), ↑i < order φ → ↑(coeff R i) (φ + ψ) = 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ order (φ + ψ) = order φ",
"tactic": "rw [order_eq]"
},
{
"state_after": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ ∀ (i : ℕ), ↑i = order φ → ↑(coeff R i) (φ + ψ) ≠ 0\n\ncase right\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ ∀ (i : ℕ), ↑i < order φ → ↑(coeff R i) (φ + ψ) = 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ (∀ (i : ℕ), ↑i = order φ → ↑(coeff R i) (φ + ψ) ≠ 0) ∧ ∀ (i : ℕ), ↑i < order φ → ↑(coeff R i) (φ + ψ) = 0",
"tactic": "constructor"
},
{
"state_after": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\ni : ℕ\nhi : ↑i = order φ\n⊢ ↑(coeff R i) (φ + ψ) ≠ 0",
"state_before": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ ∀ (i : ℕ), ↑i = order φ → ↑(coeff R i) (φ + ψ) ≠ 0",
"tactic": "intro i hi"
},
{
"state_after": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\ni : ℕ\nH : ↑i < order ψ\nhi : ↑i = order φ\n⊢ ↑(coeff R i) (φ + ψ) ≠ 0",
"state_before": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\ni : ℕ\nhi : ↑i = order φ\n⊢ ↑(coeff R i) (φ + ψ) ≠ 0",
"tactic": "rw [← hi] at H"
},
{
"state_after": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\ni : ℕ\nH : ↑i < order ψ\nhi : ↑i = order φ\n⊢ ↑(coeff R i) φ ≠ 0",
"state_before": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\ni : ℕ\nH : ↑i < order ψ\nhi : ↑i = order φ\n⊢ ↑(coeff R i) (φ + ψ) ≠ 0",
"tactic": "rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero]"
},
{
"state_after": "no goals",
"state_before": "case left\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\ni : ℕ\nH : ↑i < order ψ\nhi : ↑i = order φ\n⊢ ↑(coeff R i) φ ≠ 0",
"tactic": "exact (order_eq_nat.1 hi.symm).1"
},
{
"state_after": "case right\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\ni : ℕ\nhi : ↑i < order φ\n⊢ ↑(coeff R i) (φ + ψ) = 0",
"state_before": "case right\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\n⊢ ∀ (i : ℕ), ↑i < order φ → ↑(coeff R i) (φ + ψ) = 0",
"tactic": "intro i hi"
},
{
"state_after": "no goals",
"state_before": "case right\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n_h : order φ ≠ order ψ\nH : order φ < order ψ\ni : ℕ\nhi : ↑i < order φ\n⊢ ↑(coeff R i) (φ + ψ) = 0",
"tactic": "rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H),\n zero_add]"
}
] | [
2359,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2346,
9
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean | Multiset.prod_zero | [] | [
81,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
] |
Mathlib/Algebra/Order/Field/Power.lean | zpow_le_iff_le | [] | [
87,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean | hasStrictFDerivAt_exp_of_mem_ball | [] | [
120,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean | le_tsub_of_add_le_left | [] | [
237,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
236,
1
] |
Mathlib/Algebra/Order/Floor.lean | Int.fract_floor | [] | [
917,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
916,
1
] |
Mathlib/Order/Max.lean | not_isTop | [] | [
222,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
220,
1
] |
Mathlib/Computability/PartrecCode.lean | Nat.Partrec.Code.encode_ofNatCode | [
{
"state_after": "no goals",
"state_before": "⊢ encodeCode (ofNatCode 0) = 0",
"tactic": "simp [ofNatCode, encodeCode]"
},
{
"state_after": "no goals",
"state_before": "⊢ encodeCode (ofNatCode 1) = 1",
"tactic": "simp [ofNatCode, encodeCode]"
},
{
"state_after": "no goals",
"state_before": "⊢ encodeCode (ofNatCode 2) = 2",
"tactic": "simp [ofNatCode, encodeCode]"
},
{
"state_after": "no goals",
"state_before": "⊢ encodeCode (ofNatCode 3) = 3",
"tactic": "simp [ofNatCode, encodeCode]"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"state_before": "n : ℕ\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "let m := n.div2.div2"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "have hm : m < n + 4 := by\n simp [Nat.div2_val]\n exact\n lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n (Nat.succ_le_succ (Nat.le_add_right _ _))"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "have _m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "have _m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "have IH := encode_ofNatCode m"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "have IH1 := encode_ofNatCode m.unpair.1"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "have IH2 := encode_ofNatCode m.unpair.2"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n⊢ encodeCode (ofNatCode (n + 4)) = bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n⊢ encodeCode (ofNatCode (n + 4)) = n + 4",
"tactic": "conv_rhs => rw [← Nat.bit_decomp n, ← Nat.bit_decomp n.div2]"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n⊢ encodeCode\n (match bodd n, bodd (div2 n) with\n | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n⊢ encodeCode (ofNatCode (n + 4)) = bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4",
"tactic": "simp [encodeCode, ofNatCode]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n⊢ encodeCode\n (match bodd n, bodd (div2 n) with\n | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4",
"tactic": "cases n.bodd <;> cases n.div2.bodd <;>\n simp [encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val]"
},
{
"state_after": "n : ℕ\nm : ℕ := div2 (div2 n)\n⊢ n / 2 / 2 < n + 4",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\n⊢ m < n + 4",
"tactic": "simp [Nat.div2_val]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nm : ℕ := div2 (div2 n)\n⊢ n / 2 / 2 < n + 4",
"tactic": "exact\n lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n (Nat.succ_le_succ (Nat.le_add_right _ _))"
}
] | [
188,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
9
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.degree_cubic | [
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.1382528\nha : a ≠ 0\n⊢ ↑3 = 3",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.1382528\nha : a ≠ 0\n⊢ degree (↑C a * X ^ 3 + ↑C b * X ^ 2 + ↑C c * X + ↑C d) = 3",
"tactic": "rw [add_assoc, add_assoc, ← add_assoc (C b * X ^ 2),\n degree_add_eq_left_of_degree_lt <| degree_quadratic_lt_degree_C_mul_X_cb ha,\n degree_C_mul_X_pow 3 ha]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.1382528\nha : a ≠ 0\n⊢ ↑3 = 3",
"tactic": "rfl"
}
] | [
1230,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1226,
1
] |
Mathlib/GroupTheory/DoubleCoset.lean | Doset.mem_doset_self | [] | [
50,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
49,
1
] |
Mathlib/Algebra/Order/Monoid/WithZero/Defs.lean | WithZero.covariantClass_add_le | [
{
"state_after": "α : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na b c : WithZero α\nhbc : b ≤ c\n⊢ a + b ≤ a + c",
"state_before": "α : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\n⊢ CovariantClass (WithZero α) (WithZero α) (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1",
"tactic": "refine ⟨fun a b c hbc => ?_⟩"
},
{
"state_after": "case h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nb c : WithZero α\nhbc : b ≤ c\n⊢ 0 + b ≤ 0 + c\n\ncase h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nb c : WithZero α\nhbc : b ≤ c\na✝ : α\n⊢ ↑a✝ + b ≤ ↑a✝ + c",
"state_before": "α : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na b c : WithZero α\nhbc : b ≤ c\n⊢ a + b ≤ a + c",
"tactic": "induction a using WithZero.recZeroCoe"
},
{
"state_after": "case h₂.h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nc : WithZero α\na✝ : α\nhbc : 0 ≤ c\n⊢ ↑a✝ + 0 ≤ ↑a✝ + c\n\ncase h₂.h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nc : WithZero α\na✝¹ a✝ : α\nhbc : ↑a✝ ≤ c\n⊢ ↑a✝¹ + ↑a✝ ≤ ↑a✝¹ + c",
"state_before": "case h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nb c : WithZero α\nhbc : b ≤ c\na✝ : α\n⊢ ↑a✝ + b ≤ ↑a✝ + c",
"tactic": "induction b using WithZero.recZeroCoe"
},
{
"state_after": "no goals",
"state_before": "case h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nb c : WithZero α\nhbc : b ≤ c\n⊢ 0 + b ≤ 0 + c",
"tactic": "rwa [zero_add, zero_add]"
},
{
"state_after": "case h₂.h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nc : WithZero α\na✝ : α\nhbc : 0 ≤ c\n⊢ ↑a✝ ≤ ↑a✝ + c",
"state_before": "case h₂.h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nc : WithZero α\na✝ : α\nhbc : 0 ≤ c\n⊢ ↑a✝ + 0 ≤ ↑a✝ + c",
"tactic": "rw [add_zero]"
},
{
"state_after": "case h₂.h₁.h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝ : α\nhbc : 0 ≤ 0\n⊢ ↑a✝ ≤ ↑a✝ + 0\n\ncase h₂.h₁.h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ : α\nhbc : 0 ≤ ↑a✝\n⊢ ↑a✝¹ ≤ ↑a✝¹ + ↑a✝",
"state_before": "case h₂.h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nc : WithZero α\na✝ : α\nhbc : 0 ≤ c\n⊢ ↑a✝ ≤ ↑a✝ + c",
"tactic": "induction c using WithZero.recZeroCoe"
},
{
"state_after": "no goals",
"state_before": "case h₂.h₁.h₁\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝ : α\nhbc : 0 ≤ 0\n⊢ ↑a✝ ≤ ↑a✝ + 0",
"tactic": "rw [add_zero]"
},
{
"state_after": "case h₂.h₁.h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ : α\nhbc : 0 ≤ ↑a✝\n⊢ a✝¹ ≤ a✝¹ + a✝",
"state_before": "case h₂.h₁.h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ : α\nhbc : 0 ≤ ↑a✝\n⊢ ↑a✝¹ ≤ ↑a✝¹ + ↑a✝",
"tactic": "rw [← coe_add, coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "case h₂.h₁.h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ : α\nhbc : 0 ≤ ↑a✝\n⊢ a✝¹ ≤ a✝¹ + a✝",
"tactic": "exact le_add_of_nonneg_right (h _)"
},
{
"state_after": "case h₂.h₂.intro.intro\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ c : α\nhbc' : a✝ ≤ c\nhbc : ↑a✝ ≤ ↑c\n⊢ ↑a✝¹ + ↑a✝ ≤ ↑a✝¹ + ↑c",
"state_before": "case h₂.h₂\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\nc : WithZero α\na✝¹ a✝ : α\nhbc : ↑a✝ ≤ c\n⊢ ↑a✝¹ + ↑a✝ ≤ ↑a✝¹ + c",
"tactic": "rcases WithBot.coe_le_iff.1 hbc with ⟨c, rfl, hbc'⟩"
},
{
"state_after": "case h₂.h₂.intro.intro\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ c : α\nhbc' : a✝ ≤ c\nhbc : ↑a✝ ≤ ↑c\n⊢ ↑a✝¹ + ↑a✝ ≤ ↑(a✝¹ + c)",
"state_before": "case h₂.h₂.intro.intro\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ c : α\nhbc' : a✝ ≤ c\nhbc : ↑a✝ ≤ ↑c\n⊢ ↑a✝¹ + ↑a✝ ≤ ↑a✝¹ + ↑c",
"tactic": "refine le_trans ?_ (le_of_eq <| coe_add _ _)"
},
{
"state_after": "case h₂.h₂.intro.intro\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ c : α\nhbc' : a✝ ≤ c\nhbc : ↑a✝ ≤ ↑c\n⊢ a✝¹ + a✝ ≤ a✝¹ + c",
"state_before": "case h₂.h₂.intro.intro\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ c : α\nhbc' : a✝ ≤ c\nhbc : ↑a✝ ≤ ↑c\n⊢ ↑a✝¹ + ↑a✝ ≤ ↑(a✝¹ + c)",
"tactic": "rw [← coe_add, coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "case h₂.h₂.intro.intro\nα : Type u\ninst✝² : AddZeroClass α\ninst✝¹ : Preorder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : ∀ (a : α), 0 ≤ a\na✝¹ a✝ c : α\nhbc' : a✝ ≤ c\nhbc : ↑a✝ ≤ ↑c\n⊢ a✝¹ + a✝ ≤ a✝¹ + c",
"tactic": "exact add_le_add_left hbc' _"
}
] | [
121,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
11
] |
Mathlib/Data/Nat/Choose/Central.lean | Nat.two_le_centralBinom | [] | [
69,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/Algebra/Parity.lean | IsSquare_sq | [] | [
129,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean | Subsemigroup.not_mem_of_not_mem_closure | [] | [
319,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Mathlib/Topology/SubsetProperties.lean | CompactExhaustion.mem_diff_shiftr_find | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.169884\nπ : ι → Type ?u.169889\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nK : CompactExhaustion α\nx : α\n⊢ ¬CompactExhaustion.find (shiftr K) x ≤ CompactExhaustion.find K x",
"tactic": "simp only [find_shiftr, not_le, Nat.lt_succ_self]"
}
] | [
1503,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1501,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.SimpleFunc.setToSimpleFunc_zero | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.125421\nF : Type u_2\nF' : Type u_3\nG : Type ?u.125430\n𝕜 : Type ?u.125433\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm✝ : MeasurableSpace α\nμ : Measure α\nm : MeasurableSpace α\nf : α →ₛ F\n⊢ setToSimpleFunc 0 f = 0",
"tactic": "simp [setToSimpleFunc]"
}
] | [
290,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.filter_add | [] | [
1098,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1097,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean | MeasureTheory.Measure.map_div_left_eq_self | [] | [
468,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
466,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean | Pell.asq_pos | [
{
"state_after": "a : ℕ\na1 : 1 < a\nthis : a * 1 ≤ a * a\n⊢ a ≤ a * a",
"state_before": "a : ℕ\na1 : 1 < a\n⊢ a ≤ a * a",
"tactic": "have := @Nat.mul_le_mul_left 1 a a (le_of_lt a1)"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nthis : a * 1 ≤ a * a\n⊢ a ≤ a * a",
"tactic": "rwa [mul_one] at this"
}
] | [
180,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
] |
Mathlib/Order/SuccPred/Basic.lean | Order.pred_eq_iff_covby | [
{
"state_after": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\nb : α\ninst✝ : NoMinOrder α\n⊢ pred b ⋖ b",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\na b : α\ninst✝ : NoMinOrder α\n⊢ pred b = a → a ⋖ b",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\nb : α\ninst✝ : NoMinOrder α\n⊢ pred b ⋖ b",
"tactic": "exact pred_covby _"
}
] | [
847,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
844,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization | [
{
"state_after": "R✝ : Type ?u.108992\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhR : HasUnitMulPowIrreducibleFactorization R\nthis : UniqueFactorizationMonoid R := HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid hR\n⊢ DiscreteValuationRing R",
"state_before": "R✝ : Type ?u.108992\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhR : HasUnitMulPowIrreducibleFactorization R\n⊢ DiscreteValuationRing R",
"tactic": "letI : UniqueFactorizationMonoid R := hR.toUniqueFactorizationMonoid"
},
{
"state_after": "R✝ : Type ?u.108992\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhR : HasUnitMulPowIrreducibleFactorization R\nthis : UniqueFactorizationMonoid R := HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid hR\n⊢ ∃ p, Irreducible p",
"state_before": "R✝ : Type ?u.108992\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhR : HasUnitMulPowIrreducibleFactorization R\nthis : UniqueFactorizationMonoid R := HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid hR\n⊢ DiscreteValuationRing R",
"tactic": "apply of_ufd_of_unique_irreducible _ hR.unique_irreducible"
},
{
"state_after": "case intro.intro\nR✝ : Type ?u.108992\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R\nhp : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nthis : UniqueFactorizationMonoid R :=\n HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid (Exists.intro p { left := hp, right := H })\n⊢ ∃ p, Irreducible p",
"state_before": "R✝ : Type ?u.108992\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhR : HasUnitMulPowIrreducibleFactorization R\nthis : UniqueFactorizationMonoid R := HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid hR\n⊢ ∃ p, Irreducible p",
"tactic": "obtain ⟨p, hp, H⟩ := hR"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR✝ : Type ?u.108992\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R\nhp : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nthis : UniqueFactorizationMonoid R :=\n HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid (Exists.intro p { left := hp, right := H })\n⊢ ∃ p, Irreducible p",
"tactic": "exact ⟨p, hp⟩"
}
] | [
319,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.tendsto_atTop_finset_of_monotone | [
{
"state_after": "ι : Type ?u.269296\nι' : Type ?u.269299\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.269308\ninst✝ : Preorder β\nf : β → Finset α\nh : Monotone f\nh' : ∀ (x : α), ∃ n, x ∈ f n\n⊢ ∀ (i : α), ∀ᶠ (a : β) in atTop, f a ∈ Ici {i}",
"state_before": "ι : Type ?u.269296\nι' : Type ?u.269299\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.269308\ninst✝ : Preorder β\nf : β → Finset α\nh : Monotone f\nh' : ∀ (x : α), ∃ n, x ∈ f n\n⊢ Tendsto f atTop atTop",
"tactic": "simp only [atTop_finset_eq_iInf, tendsto_iInf, tendsto_principal]"
},
{
"state_after": "ι : Type ?u.269296\nι' : Type ?u.269299\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.269308\ninst✝ : Preorder β\nf : β → Finset α\nh : Monotone f\nh' : ∀ (x : α), ∃ n, x ∈ f n\na : α\n⊢ ∀ᶠ (a_1 : β) in atTop, f a_1 ∈ Ici {a}",
"state_before": "ι : Type ?u.269296\nι' : Type ?u.269299\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.269308\ninst✝ : Preorder β\nf : β → Finset α\nh : Monotone f\nh' : ∀ (x : α), ∃ n, x ∈ f n\n⊢ ∀ (i : α), ∀ᶠ (a : β) in atTop, f a ∈ Ici {i}",
"tactic": "intro a"
},
{
"state_after": "case intro\nι : Type ?u.269296\nι' : Type ?u.269299\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.269308\ninst✝ : Preorder β\nf : β → Finset α\nh : Monotone f\nh' : ∀ (x : α), ∃ n, x ∈ f n\na : α\nb : β\nhb : a ∈ f b\n⊢ ∀ᶠ (a_1 : β) in atTop, f a_1 ∈ Ici {a}",
"state_before": "ι : Type ?u.269296\nι' : Type ?u.269299\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.269308\ninst✝ : Preorder β\nf : β → Finset α\nh : Monotone f\nh' : ∀ (x : α), ∃ n, x ∈ f n\na : α\n⊢ ∀ᶠ (a_1 : β) in atTop, f a_1 ∈ Ici {a}",
"tactic": "rcases h' a with ⟨b, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.269296\nι' : Type ?u.269299\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.269308\ninst✝ : Preorder β\nf : β → Finset α\nh : Monotone f\nh' : ∀ (x : α), ∃ n, x ∈ f n\na : α\nb : β\nhb : a ∈ f b\n⊢ ∀ᶠ (a_1 : β) in atTop, f a_1 ∈ Ici {a}",
"tactic": "exact (eventually_ge_atTop b).mono fun b' hb' => (Finset.singleton_subset_iff.2 hb).trans (h hb')"
}
] | [
1379,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1374,
1
] |
Mathlib/Algebra/Group/Basic.lean | exists_npow_eq_one_of_zpow_eq_one | [
{
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"state_before": "α : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d : G\nn : ℤ\nhn : n ≠ 0\nx : G\nh : x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"tactic": "cases' n with n n"
},
{
"state_after": "case ofNat\nα : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d x : G\nn : ℕ\nhn : Int.ofNat n ≠ 0\nh : x ^ ↑n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"state_before": "case ofNat\nα : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d x : G\nn : ℕ\nhn : Int.ofNat n ≠ 0\nh : x ^ Int.ofNat n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"tactic": "simp only [Int.ofNat_eq_coe] at h"
},
{
"state_after": "case ofNat\nα : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d x : G\nn : ℕ\nhn : Int.ofNat n ≠ 0\nh : x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"state_before": "case ofNat\nα : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d x : G\nn : ℕ\nhn : Int.ofNat n ≠ 0\nh : x ^ ↑n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"tactic": "rw [zpow_ofNat] at h"
},
{
"state_after": "case ofNat\nα : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d x : G\nn : ℕ\nhn : Int.ofNat n ≠ 0\nh : x ^ n = 1\nn0 : n = 0\n⊢ Int.ofNat n = 0",
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"tactic": "refine' ⟨n, Nat.pos_of_ne_zero fun n0 ↦ hn ?_, h⟩"
},
{
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"tactic": "rw [n0]"
},
{
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"tactic": "rfl"
},
{
"state_after": "case negSucc\nα : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d x : G\nn : ℕ\nhn : Int.negSucc n ≠ 0\nh : x ^ (n + 1) = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"state_before": "case negSucc\nα : Type ?u.63936\nβ : Type ?u.63939\nG : Type u_1\ninst✝ : Group G\na b c d x : G\nn : ℕ\nhn : Int.negSucc n ≠ 0\nh : x ^ Int.negSucc n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1",
"tactic": "rw [zpow_negSucc, inv_eq_one] at h"
},
{
"state_after": "no goals",
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"tactic": "refine' ⟨n + 1, n.succ_pos, h⟩"
}
] | [
889,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
880,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inv.lean | differentiableWithinAt_inv | [] | [
89,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/Data/Matrix/Notation.lean | Matrix.smul_vec3 | [
{
"state_after": "no goals",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nR : Type u_1\ninst✝ : SMul R α\nx : R\na₀ a₁ a₂ : α\n⊢ x • ![a₀, a₁, a₂] = ![x • a₀, x • a₁, x • a₂]",
"tactic": "rw [smul_cons, smul_cons, smul_cons, smul_empty]"
}
] | [
485,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
483,
1
] |
Mathlib/Order/SymmDiff.lean | bihimp_fst | [] | [
836,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
834,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Basic.lean | Units.mk0_mul | [
{
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"state_before": "α : Type ?u.18576\nM₀ : Type ?u.18579\nG₀ : Type u_1\nM₀' : Type ?u.18585\nG₀' : Type ?u.18588\nF : Type ?u.18591\nF' : Type ?u.18594\ninst✝¹ : MonoidWithZero M₀\ninst✝ : GroupWithZero G₀\na b c x y : G₀\nhxy : x * y ≠ 0\n⊢ mk0 (x * y) hxy = mk0 x (_ : x ≠ 0) * mk0 y (_ : y ≠ 0)",
"tactic": "ext"
},
{
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"state_before": "case a\nα : Type ?u.18576\nM₀ : Type ?u.18579\nG₀ : Type u_1\nM₀' : Type ?u.18585\nG₀' : Type ?u.18588\nF : Type ?u.18591\nF' : Type ?u.18594\ninst✝¹ : MonoidWithZero M₀\ninst✝ : GroupWithZero G₀\na b c x y : G₀\nhxy : x * y ≠ 0\n⊢ ↑(mk0 (x * y) hxy) = ↑(mk0 x (_ : x ≠ 0) * mk0 y (_ : y ≠ 0))",
"tactic": "rfl"
}
] | [
281,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
278,
1
] |
Mathlib/Topology/Constructions.lean | CofiniteTopology.isClosed_iff | [
{
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"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.17786\nδ : Type ?u.17789\nε : Type ?u.17792\nζ : Type ?u.17795\ns : Set (CofiniteTopology α)\n⊢ IsClosed s ↔ s = univ ∨ Set.Finite s",
"tactic": "simp only [← isOpen_compl_iff, isOpen_iff', compl_compl, compl_empty_iff]"
}
] | [
291,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.smul_toProd | [] | [
158,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
157,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | lt_mul_of_le_of_one_lt_of_pos | [] | [
770,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
768,
1
] |
Mathlib/Data/Finset/Sups.lean | Finset.infs_inter_subset_left | [] | [
346,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
345,
1
] |
Mathlib/Analysis/Convex/Gauge.lean | gauge_zero' | [
{
"state_after": "case h\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ gauge 0 x = OfNat.ofNat 0 x",
"state_before": "𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\n⊢ gauge 0 = 0",
"tactic": "ext x"
},
{
"state_after": "case h\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = OfNat.ofNat 0 x",
"state_before": "case h\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ gauge 0 x = OfNat.ofNat 0 x",
"tactic": "rw [gauge_def']"
},
{
"state_after": "case h.inl\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ 0} = OfNat.ofNat 0 0\n\ncase h.inr\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nhx : x ≠ 0\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = OfNat.ofNat 0 x",
"state_before": "case h\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = OfNat.ofNat 0 x",
"tactic": "obtain rfl | hx := eq_or_ne x 0"
},
{
"state_after": "no goals",
"state_before": "case h.inl\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ 0} = OfNat.ofNat 0 0",
"tactic": "simp only [csInf_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]"
},
{
"state_after": "case h.inr\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nhx : x ≠ 0\n⊢ sInf {r | r ∈ Ioi 0 ∧ (r = 0 ∨ x = 0)} = 0",
"state_before": "case h.inr\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nhx : x ≠ 0\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = OfNat.ofNat 0 x",
"tactic": "simp only [mem_zero, Pi.zero_apply, inv_eq_zero, smul_eq_zero]"
},
{
"state_after": "case h.e'_2.h.e'_3\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nhx : x ≠ 0\n⊢ {r | r ∈ Ioi 0 ∧ (r = 0 ∨ x = 0)} = ∅",
"state_before": "case h.inr\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nhx : x ≠ 0\n⊢ sInf {r | r ∈ Ioi 0 ∧ (r = 0 ∨ x = 0)} = 0",
"tactic": "convert Real.sInf_empty"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_3\n𝕜 : Type ?u.24181\nE : Type u_1\nF : Type ?u.24187\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\na : ℝ\nx : E\nhx : x ≠ 0\n⊢ {r | r ∈ Ioi 0 ∧ (r = 0 ∨ x = 0)} = ∅",
"tactic": "exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr.1) hx"
}
] | [
117,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean | Set.image_mul_prod | [] | [
351,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
350,
1
] |
Std/Data/Int/DivMod.lean | Int.dvd_sub | [
{
"state_after": "no goals",
"state_before": "a✝ d e : Int\n⊢ a✝ * d - a✝ * e = a✝ * (d - e)",
"tactic": "rw [Int.mul_sub]"
}
] | [
619,
64
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
618,
11
] |
Mathlib/Analysis/Normed/Group/Basic.lean | norm_le_norm_add_norm_div | [
{
"state_after": "𝓕 : Type ?u.86263\n𝕜 : Type ?u.86266\nα : Type ?u.86269\nι : Type ?u.86272\nκ : Type ?u.86275\nE : Type u_1\nF : Type ?u.86281\nG : Type ?u.86284\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : E\n⊢ ‖v‖ ≤ ‖u‖ + ‖v / u‖",
"state_before": "𝓕 : Type ?u.86263\n𝕜 : Type ?u.86266\nα : Type ?u.86269\nι : Type ?u.86272\nκ : Type ?u.86275\nE : Type u_1\nF : Type ?u.86281\nG : Type ?u.86284\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : E\n⊢ ‖v‖ ≤ ‖u‖ + ‖u / v‖",
"tactic": "rw [norm_div_rev]"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.86263\n𝕜 : Type ?u.86266\nα : Type ?u.86269\nι : Type ?u.86272\nκ : Type ?u.86275\nE : Type u_1\nF : Type ?u.86281\nG : Type ?u.86284\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : E\n⊢ ‖v‖ ≤ ‖u‖ + ‖v / u‖",
"tactic": "exact norm_le_norm_add_norm_div' v u"
}
] | [
587,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
585,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.Adj.ne | [] | [
211,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
11
] |
Mathlib/Analysis/Convex/Segment.lean | Prod.image_mk_segment_right | [
{
"state_after": "case h.mk\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.327345\nι : Type ?u.327348\nπ : ι → Type ?u.327353\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ny₁ y₂ : F\nx' : E\ny' : F\n⊢ (x', y') ∈ (fun y => (x, y)) '' [y₁-[𝕜]y₂] ↔ (x', y') ∈ [(x, y₁)-[𝕜](x, y₂)]",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.327345\nι : Type ?u.327348\nπ : ι → Type ?u.327353\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ny₁ y₂ : F\n⊢ (fun y => (x, y)) '' [y₁-[𝕜]y₂] = [(x, y₁)-[𝕜](x, y₂)]",
"tactic": "ext ⟨x', y'⟩"
},
{
"state_after": "case h.mk\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.327345\nι : Type ?u.327348\nπ : ι → Type ?u.327353\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ny₁ y₂ : F\nx' : E\ny' : F\n⊢ (∃ b b_1 h h h, x = x' ∧ b • y₁ + b_1 • y₂ = y') ↔ ∃ a b h h h, a • x + b • x = x' ∧ a • y₁ + b • y₂ = y'",
"state_before": "case h.mk\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.327345\nι : Type ?u.327348\nπ : ι → Type ?u.327353\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ny₁ y₂ : F\nx' : E\ny' : F\n⊢ (x', y') ∈ (fun y => (x, y)) '' [y₁-[𝕜]y₂] ↔ (x', y') ∈ [(x, y₁)-[𝕜](x, y₂)]",
"tactic": "simp_rw [Set.mem_image, segment, Set.mem_setOf, Prod.smul_mk, Prod.mk_add_mk, Prod.mk.inj_iff, ←\n exists_and_right, @exists_comm F, exists_eq_left']"
},
{
"state_after": "case h.mk\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.327345\nι : Type ?u.327348\nπ : ι → Type ?u.327353\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ny₁ y₂ : F\nx' : E\ny' : F\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ x = x' ∧ a • y₁ + b • y₂ = y' ↔ a • x + b • x = x' ∧ a • y₁ + b • y₂ = y'",
"state_before": "case h.mk\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.327345\nι : Type ?u.327348\nπ : ι → Type ?u.327353\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ny₁ y₂ : F\nx' : E\ny' : F\n⊢ (∃ b b_1 h h h, x = x' ∧ b • y₁ + b_1 • y₂ = y') ↔ ∃ a b h h h, a • x + b • x = x' ∧ a • y₁ + b • y₂ = y'",
"tactic": "refine' exists₅_congr fun a b ha hb hab => _"
},
{
"state_after": "no goals",
"state_before": "case h.mk\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.327345\nι : Type ?u.327348\nπ : ι → Type ?u.327353\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ny₁ y₂ : F\nx' : E\ny' : F\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ x = x' ∧ a • y₁ + b • y₂ = y' ↔ a • x + b • x = x' ∧ a • y₁ + b • y₂ = y'",
"tactic": "rw [Convex.combo_self hab]"
}
] | [
618,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Real.sqrtTwoAddSeries_zero | [
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ sqrtTwoAddSeries x 0 = x",
"tactic": "simp"
}
] | [
682,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
682,
1
] |
Mathlib/Logic/Equiv/Option.lean | Equiv.optionSubtype_apply_symm_apply | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.18695\ninst✝ : DecidableEq β\nx : β\ne : { e // ↑e none = x }\nb : { y // y ≠ x }\n⊢ some\n (↑(↑{\n toFun := fun e =>\n { toFun := fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) },\n invFun := fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true),\n left_inv :=\n (_ :\n ∀ (a : α),\n (fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n ((fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) }) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : { y // y ≠ x }),\n (fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) })\n ((fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true)) b) =\n b) },\n invFun := fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b => if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b => if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b => if h : b = x then none else some (↑e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none) },\n left_inv :=\n (_ :\n ∀ (e : { e // ↑e none = x }),\n (fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n ((fun e =>\n { toFun := fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) },\n invFun := fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true),\n left_inv :=\n (_ :\n ∀ (a : α),\n (fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n ((fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) }) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : { y // y ≠ x }),\n (fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) })\n ((fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n b) =\n b) })\n e) =\n e),\n right_inv :=\n (_ :\n ∀ (e : α ≃ { y // y ≠ x }),\n (fun e =>\n { toFun := fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) },\n invFun := fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true),\n left_inv :=\n (_ :\n ∀ (a : α),\n (fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n ((fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) }) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : { y // y ≠ x }),\n (fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) })\n ((fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n b) =\n b) })\n ((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n e) =\n e) }\n e).symm\n b) =\n ↑(↑e).symm ↑b",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.18695\ninst✝ : DecidableEq β\nx : β\ne : { e // ↑e none = x }\nb : { y // y ≠ x }\n⊢ some (↑(↑(optionSubtype x) e).symm b) = ↑(↑e).symm ↑b",
"tactic": "dsimp only [optionSubtype]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.18695\ninst✝ : DecidableEq β\nx : β\ne : { e // ↑e none = x }\nb : { y // y ≠ x }\n⊢ some\n (↑(↑{\n toFun := fun e =>\n { toFun := fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) },\n invFun := fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true),\n left_inv :=\n (_ :\n ∀ (a : α),\n (fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n ((fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) }) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : { y // y ≠ x }),\n (fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) })\n ((fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true)) b) =\n b) },\n invFun := fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b => if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b => if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b => if h : b = x then none else some (↑e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none) },\n left_inv :=\n (_ :\n ∀ (e : { e // ↑e none = x }),\n (fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n ((fun e =>\n { toFun := fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) },\n invFun := fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true),\n left_inv :=\n (_ :\n ∀ (a : α),\n (fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n ((fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) }) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : { y // y ≠ x }),\n (fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) })\n ((fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n b) =\n b) })\n e) =\n e),\n right_inv :=\n (_ :\n ∀ (e : α ≃ { y // y ≠ x }),\n (fun e =>\n { toFun := fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) },\n invFun := fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true),\n left_inv :=\n (_ :\n ∀ (a : α),\n (fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n ((fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) }) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : { y // y ≠ x }),\n (fun a => { val := ↑↑e (some a), property := (_ : ↑↑e (some a) ≠ x) })\n ((fun b => Option.get (↑(↑e).symm ↑b) (_ : isSome (↑(↑e).symm ↑b) = true))\n b) =\n b) })\n ((fun e =>\n {\n val :=\n { toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) },\n property :=\n (_ :\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none =\n ↑{ toFun := fun a => casesOn' a x (Subtype.val ∘ ↑e),\n invFun := fun b =>\n if h : b = x then none else some (↑e.symm { val := b, property := h }),\n left_inv :=\n (_ :\n ∀ (a : Option α),\n (fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n ((fun a => casesOn' a x (Subtype.val ∘ ↑e)) a) =\n a),\n right_inv :=\n (_ :\n ∀ (b : β),\n (fun a => casesOn' a x (Subtype.val ∘ ↑e))\n ((fun b =>\n if h : b = x then none\n else some (↑e.symm { val := b, property := h }))\n b) =\n b) }\n none) })\n e) =\n e) }\n e).symm\n b) =\n ↑(↑e).symm ↑b",
"tactic": "simp"
}
] | [
237,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | Class.coe_inter | [] | [
1659,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1658,
1
] |
Mathlib/Order/UpperLower/Basic.lean | isLowerSet_Iic | [] | [
214,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean | OnePoint.nhdsWithin_compl_infty_eq | [
{
"state_after": "case refine'_1\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ ∀ (i : Set (OnePoint X)), ∞ ∈ i ∧ IsOpen i → ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ i ∩ {∞}ᶜ\n\ncase refine'_2\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ ∀ (i' : Set X), IsClosed i' ∧ IsCompact i' → ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' i'ᶜ",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ 𝓝[{∞}ᶜ] ∞ = map some (coclosedCompact X)",
"tactic": "refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _"
},
{
"state_after": "case refine'_1.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt : Set X\ns : Set (OnePoint X)\nhs : ∞ ∈ s\nhso : IsOpen s\n⊢ ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ s ∩ {∞}ᶜ",
"state_before": "case refine'_1\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ ∀ (i : Set (OnePoint X)), ∞ ∈ i ∧ IsOpen i → ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ i ∩ {∞}ᶜ",
"tactic": "rintro s ⟨hs, hso⟩"
},
{
"state_after": "case refine'_1.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt : Set X\ns : Set (OnePoint X)\nhs : ∞ ∈ s\nhso : IsOpen s\n⊢ some '' (some ⁻¹' s)ᶜᶜ ⊆ s ∩ {∞}ᶜ",
"state_before": "case refine'_1.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt : Set X\ns : Set (OnePoint X)\nhs : ∞ ∈ s\nhso : IsOpen s\n⊢ ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ s ∩ {∞}ᶜ",
"tactic": "refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt : Set X\ns : Set (OnePoint X)\nhs : ∞ ∈ s\nhso : IsOpen s\n⊢ some '' (some ⁻¹' s)ᶜᶜ ⊆ s ∩ {∞}ᶜ",
"tactic": "simp [Subset.rfl]"
},
{
"state_after": "case refine'_2.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt s : Set X\nh₁ : IsClosed s\nh₂ : IsCompact s\n⊢ ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' sᶜ",
"state_before": "case refine'_2\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ ∀ (i' : Set X), IsClosed i' ∧ IsCompact i' → ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' i'ᶜ",
"tactic": "rintro s ⟨h₁, h₂⟩"
},
{
"state_after": "case refine'_2.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt s : Set X\nh₁ : IsClosed s\nh₂ : IsCompact s\n⊢ (some '' s)ᶜ ∩ {∞}ᶜ ⊆ some '' sᶜ",
"state_before": "case refine'_2.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt s : Set X\nh₁ : IsClosed s\nh₂ : IsCompact s\n⊢ ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' sᶜ",
"tactic": "refine' ⟨_, ⟨mem_compl infty_not_mem_image_coe, isOpen_compl_image_coe.2 ⟨h₁, h₂⟩⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\ns✝ : Set (OnePoint X)\nt s : Set X\nh₁ : IsClosed s\nh₂ : IsCompact s\n⊢ (some '' s)ᶜ ∩ {∞}ᶜ ⊆ some '' sᶜ",
"tactic": "simp [compl_image_coe, ← diff_eq, subset_preimage_image]"
}
] | [
318,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
311,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.IsPrime.isPrimary | [] | [
1852,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1851,
1
] |
Mathlib/Analysis/ODE/PicardLindelof.lean | PicardLindelof.FunSpace.dist_iterate_next_le | [
{
"state_after": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nv : PicardLindelof E\nf : FunSpace v\ninst✝ : CompleteSpace E\nf₁ f₂ : FunSpace v\nn : ℕ\nt : ↑(Icc v.tMin v.tMax)\n⊢ (↑v.L * abs (↑t - ↑v.t₀)) ^ n / ↑n ! * dist f₁ f₂ ≤ (↑v.L * tDist v) ^ n / ↑n ! * dist f₁ f₂",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nv : PicardLindelof E\nf : FunSpace v\ninst✝ : CompleteSpace E\nf₁ f₂ : FunSpace v\nn : ℕ\n⊢ dist ((next^[n]) f₁) ((next^[n]) f₂) ≤ (↑v.L * tDist v) ^ n / ↑n ! * dist f₁ f₂",
"tactic": "refine' dist_le_of_forall fun t => (dist_iterate_next_apply_le _ _ _ _).trans _"
},
{
"state_after": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nv : PicardLindelof E\nf : FunSpace v\ninst✝ : CompleteSpace E\nf₁ f₂ : FunSpace v\nn : ℕ\nt : ↑(Icc v.tMin v.tMax)\nthis : abs (↑t - ↑v.t₀) ≤ tDist v\n⊢ (↑v.L * abs (↑t - ↑v.t₀)) ^ n / ↑n ! * dist f₁ f₂ ≤ (↑v.L * tDist v) ^ n / ↑n ! * dist f₁ f₂",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nv : PicardLindelof E\nf : FunSpace v\ninst✝ : CompleteSpace E\nf₁ f₂ : FunSpace v\nn : ℕ\nt : ↑(Icc v.tMin v.tMax)\n⊢ (↑v.L * abs (↑t - ↑v.t₀)) ^ n / ↑n ! * dist f₁ f₂ ≤ (↑v.L * tDist v) ^ n / ↑n ! * dist f₁ f₂",
"tactic": "have : |(t - v.t₀ : ℝ)| ≤ v.tDist := v.dist_t₀_le t"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nv : PicardLindelof E\nf : FunSpace v\ninst✝ : CompleteSpace E\nf₁ f₂ : FunSpace v\nn : ℕ\nt : ↑(Icc v.tMin v.tMax)\nthis : abs (↑t - ↑v.t₀) ≤ tDist v\n⊢ (↑v.L * abs (↑t - ↑v.t₀)) ^ n / ↑n ! * dist f₁ f₂ ≤ (↑v.L * tDist v) ^ n / ↑n ! * dist f₁ f₂",
"tactic": "gcongr"
}
] | [
338,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
334,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean | EsakiaHom.copy_eq | [] | [
277,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
276,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.IsCycleOn.pow_apply_eq | [
{
"state_after": "case inl\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns t : Set α\na b x y : α\nn : ℕ\nhf : IsCycleOn f ↑{a}\nha : a ∈ {a}\n⊢ ↑(f ^ n) a = a ↔ card {a} ∣ n\n\ncase inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"state_before": "ι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"tactic": "obtain rfl | hs := Finset.eq_singleton_or_nontrivial ha"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"tactic": "classical\n have h : ∀ x ∈ s.attach, ¬f ↑x = ↑x := fun x _ => hf.apply_ne hs x.2\n have := (hf.isCycle_subtypePerm hs).orderOf\n simp only [coe_sort_coe, support_subtype_perm, ne_eq, decide_not, Bool.not_eq_true',\n decide_eq_false_iff_not, mem_attach, forall_true_left, Subtype.forall, filter_true_of_mem h,\n card_attach] at this\n rw [← this, orderOf_dvd_iff_pow_eq_one,\n (hf.isCycle_subtypePerm hs).pow_eq_one_iff'\n (ne_of_apply_ne ((↑) : s → α) <| hf.apply_ne hs (⟨a, ha⟩ : s).2)]\n simp"
},
{
"state_after": "case inl\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns t : Set α\na b x y : α\nn : ℕ\nhf : ↑f a = a\nha : a ∈ {a}\n⊢ ↑(f ^ n) a = a ↔ card {a} ∣ n",
"state_before": "case inl\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns t : Set α\na b x y : α\nn : ℕ\nhf : IsCycleOn f ↑{a}\nha : a ∈ {a}\n⊢ ↑(f ^ n) a = a ↔ card {a} ∣ n",
"tactic": "rw [coe_singleton, isCycleOn_singleton] at hf"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns t : Set α\na b x y : α\nn : ℕ\nhf : ↑f a = a\nha : a ∈ {a}\n⊢ ↑(f ^ n) a = a ↔ card {a} ∣ n",
"tactic": "simpa using IsFixedPt.iterate hf n"
},
{
"state_after": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"state_before": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"tactic": "have h : ∀ x ∈ s.attach, ¬f ↑x = ↑x := fun x _ => hf.apply_ne hs x.2"
},
{
"state_after": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\nthis :\n orderOf (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)) =\n card (support (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)))\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"state_before": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"tactic": "have := (hf.isCycle_subtypePerm hs).orderOf"
},
{
"state_after": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\nthis : orderOf (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)) = card s\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"state_before": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\nthis :\n orderOf (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)) =\n card (support (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)))\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"tactic": "simp only [coe_sort_coe, support_subtype_perm, ne_eq, decide_not, Bool.not_eq_true',\n decide_eq_false_iff_not, mem_attach, forall_true_left, Subtype.forall, filter_true_of_mem h,\n card_attach] at this"
},
{
"state_after": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\nthis : orderOf (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)) = card s\n⊢ ↑(f ^ n) a = a ↔\n ↑(subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s) ^ n) { val := a, property := ha } = { val := a, property := ha }",
"state_before": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\nthis : orderOf (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)) = card s\n⊢ ↑(f ^ n) a = a ↔ card s ∣ n",
"tactic": "rw [← this, orderOf_dvd_iff_pow_eq_one,\n (hf.isCycle_subtypePerm hs).pow_eq_one_iff'\n (ne_of_apply_ne ((↑) : s → α) <| hf.apply_ne hs (⟨a, ha⟩ : s).2)]"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.1658946\nα : Type u_1\nβ : Type ?u.1658952\nf g : Perm α\ns✝ t : Set α\na b x y : α\ns : Finset α\nhf : IsCycleOn f ↑s\nha : a ∈ s\nn : ℕ\nhs : Set.Nontrivial ↑s\nh : ∀ (x : { x // x ∈ s }), x ∈ attach s → ¬↑f ↑x = ↑x\nthis : orderOf (subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s)) = card s\n⊢ ↑(f ^ n) a = a ↔\n ↑(subtypePerm f (_ : ∀ (x : α), x ∈ ↑s ↔ ↑f x ∈ ↑s) ^ n) { val := a, property := ha } = { val := a, property := ha }",
"tactic": "simp"
}
] | [
869,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
855,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | Complex.tendsto_euler_sin_prod | [
{
"state_after": "z : ℂ\nA :\n Tendsto\n (fun n =>\n ((↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 (Complex.sin (↑π * z)))\n⊢ Tendsto (fun n => ↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))",
"state_before": "z : ℂ\n⊢ Tendsto (fun n => ↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))",
"tactic": "have A :\n Tendsto\n (fun n : ℕ =>\n ((π * z * ∏ j in Finset.range n, ((1 : ℂ) - z ^ 2 / ((j : ℂ) + 1) ^ 2)) *\n ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n)) /\n (∫ x in (0 : ℝ)..π / 2, cos x ^ (2 * n) : ℝ))\n atTop (𝓝 <| _) :=\n Tendsto.congr (fun n => sin_pi_mul_eq z n) tendsto_const_nhds"
},
{
"state_after": "z : ℂ\nA :\n Tendsto\n (fun n =>\n ((↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 (Complex.sin (↑π * z)))\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\n⊢ Tendsto (fun n => ↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))",
"state_before": "z : ℂ\nA :\n Tendsto\n (fun n =>\n ((↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 (Complex.sin (↑π * z)))\n⊢ Tendsto (fun n => ↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))",
"tactic": "have : 𝓝 (Complex.sin (π * z)) = 𝓝 (Complex.sin (π * z) * 1) := by rw [mul_one]"
},
{
"state_after": "z : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto (fun n => ↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))",
"state_before": "z : ℂ\nA :\n Tendsto\n (fun n =>\n ((↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 (Complex.sin (↑π * z)))\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\n⊢ Tendsto (fun n => ↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))",
"tactic": "simp_rw [this, mul_div_assoc] at A"
},
{
"state_after": "z : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto\n (fun n =>\n (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 1)",
"state_before": "z : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto (fun n => ↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))",
"tactic": "convert (tendsto_mul_iff_of_ne_zero _ one_ne_zero).mp A"
},
{
"state_after": "z : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis :\n Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)\n⊢ Tendsto\n (fun n =>\n (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 1)\n\ncase this\nz : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)",
"state_before": "z : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto\n (fun n =>\n (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 1)",
"tactic": "suffices :\n Tendsto\n (fun n : ℕ =>\n (∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) /\n (∫ x in (0 : ℝ)..π / 2, cos x ^ n : ℝ))\n atTop (𝓝 1)"
},
{
"state_after": "case this\nz : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)",
"state_before": "z : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis :\n Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)\n⊢ Tendsto\n (fun n =>\n (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 1)\n\ncase this\nz : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)",
"tactic": "exact this.comp (tendsto_id.const_mul_atTop' zero_lt_two)"
},
{
"state_after": "case this\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\n⊢ Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)",
"state_before": "case this\nz : ℂ\nthis : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\n⊢ Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)",
"tactic": "have : ContinuousOn (fun x : ℝ => Complex.cos (2 * z * x)) (Icc 0 (π / 2)) :=\n (Complex.continuous_cos.comp (continuous_const.mul Complex.continuous_ofReal)).continuousOn"
},
{
"state_after": "case h.e'_3\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\n⊢ (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) =\n fun n => (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)\n\ncase h.e'_5\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\n⊢ 𝓝 1 = 𝓝 (Complex.cos (2 * z * ↑0))",
"state_before": "case this\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\n⊢ Tendsto\n (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n))\n atTop (𝓝 1)",
"tactic": "convert tendsto_integral_cos_pow_mul_div this using 1"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\nA :\n Tendsto\n (fun n =>\n ((↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))\n atTop (𝓝 (Complex.sin (↑π * z)))\n⊢ 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)",
"tactic": "rw [mul_one]"
},
{
"state_after": "case h.e'_3.h\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\nn : ℕ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) =\n (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)",
"state_before": "case h.e'_3\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\n⊢ (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) =\n fun n => (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)",
"tactic": "ext1 n"
},
{
"state_after": "case h.e'_3.h.e_a.e_f.h\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\nn : ℕ\nx : ℝ\n⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)",
"state_before": "case h.e'_3.h\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\nn : ℕ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) =\n (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)",
"tactic": "congr 2 with x : 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e_a.e_f.h\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\nn : ℕ\nx : ℝ\n⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nz : ℂ\nthis✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)\nA :\n Tendsto\n (fun n =>\n (↑π * z * ∏ j in Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *\n ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /\n ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))\n atTop (𝓝 (Complex.sin (↑π * z) * 1))\nthis : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))\n⊢ 𝓝 1 = 𝓝 (Complex.cos (2 * z * ↑0))",
"tactic": "rw [Complex.ofReal_zero, MulZeroClass.mul_zero, Complex.cos_zero]"
}
] | [
331,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
306,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | charmatrix_apply_natDegree_le | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝³ : CommRing R\nn G : Type v\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nα β : Type v\ninst✝ : DecidableEq α\nM : Matrix n n R\ni j : n\n⊢ natDegree (charmatrix M i j) ≤ if i = j then 1 else 0",
"tactic": "split_ifs with h <;> simp [h, natDegree_X_sub_C_le]"
}
] | [
57,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/Topology/ContinuousOn.lean | DenseRange.nhdsWithin_neBot | [] | [
410,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
] |
Mathlib/Topology/Order.lean | nhds_iInf | [] | [
661,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
659,
1
] |
Mathlib/NumberTheory/Divisors.lean | Nat.one_mem_divisors | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 1 ∈ divisors n ↔ n ≠ 0",
"tactic": "simp"
}
] | [
104,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.map_atBot_eq_of_gc | [] | [
1517,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1514,
1
] |
Mathlib/Data/List/Basic.lean | List.eq_cons_of_mem_head? | [
{
"state_after": "ι : Type ?u.41922\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx a : α\nl : List α\nh : a = x\n⊢ a :: l = x :: tail (a :: l)",
"state_before": "ι : Type ?u.41922\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx a : α\nl : List α\nh : x ∈ head? (a :: l)\n⊢ a :: l = x :: tail (a :: l)",
"tactic": "simp only [head?, Option.mem_def, Option.some_inj] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.41922\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nx a : α\nl : List α\nh : a = x\n⊢ a :: l = x :: tail (a :: l)",
"tactic": "exact h ▸ rfl"
}
] | [
879,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
875,
1
] |
Mathlib/Data/W/Cardinal.lean | WType.cardinal_mk_le_max_aleph0_of_finite | [
{
"state_after": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nh : IsEmpty α\n⊢ (#WType β) ≤ max (#α) ℵ₀",
"state_before": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\n⊢ IsEmpty α → (#WType β) ≤ max (#α) ℵ₀",
"tactic": "intro h"
},
{
"state_after": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nh : IsEmpty α\n⊢ 0 ≤ max (#α) ℵ₀",
"state_before": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nh : IsEmpty α\n⊢ (#WType β) ≤ max (#α) ℵ₀",
"tactic": "rw [Cardinal.mk_eq_zero (WType β)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nh : IsEmpty α\n⊢ 0 ≤ max (#α) ℵ₀",
"tactic": "exact zero_le _"
},
{
"state_after": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nhn : Nonempty α\nm : Cardinal := max (#α) ℵ₀\n⊢ 1 ≤ ⨆ (a : α), m ^ (#β a)",
"state_before": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nhn : Nonempty α\nm : Cardinal := max (#α) ℵ₀\n⊢ Order.succ 0 ≤ ⨆ (a : α), m ^ (#β a)",
"tactic": "rw [succ_zero]"
},
{
"state_after": "case intro\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ 1 ≤ ⨆ (a : α), m ^ (#β a)",
"state_before": "α : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nhn : Nonempty α\nm : Cardinal := max (#α) ℵ₀\n⊢ 1 ≤ ⨆ (a : α), m ^ (#β a)",
"tactic": "obtain ⟨a⟩ : Nonempty α := hn"
},
{
"state_after": "case intro\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ 1 ≤ m ^ (#β a)",
"state_before": "case intro\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ 1 ≤ ⨆ (a : α), m ^ (#β a)",
"tactic": "refine' le_trans _ (le_ciSup (bddAbove_range.{u, u} _) a)"
},
{
"state_after": "case intro\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ ?m.2512 ^ 0 ≤ m ^ (#β a)\n\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ Cardinal",
"state_before": "case intro\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ 1 ≤ m ^ (#β a)",
"tactic": "rw [← power_zero]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ ?m.2512 ^ 0 ≤ m ^ (#β a)\n\nα : Type u\nβ : α → Type u\ninst✝ : ∀ (a : α), Finite (β a)\nm : Cardinal := max (#α) ℵ₀\na : α\n⊢ Cardinal",
"tactic": "exact\n power_le_power_left\n (pos_iff_ne_zero.1 (aleph0_pos.trans_le (le_max_right _ _))) (zero_le _)"
}
] | [
84,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean | LipschitzWith.dist_left | [] | [
407,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
406,
11
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.mem_cycleFactorsFinset_iff | [
{
"state_after": "case intro.intro\nι : Type ?u.2735472\nα : Type u_1\nβ : Type ?u.2735478\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f p : Perm α\nl : List (Perm α)\nhl : List.Nodup l\nhl' : List.toFinset l = cycleFactorsFinset f\n⊢ p ∈ cycleFactorsFinset f ↔ IsCycle p ∧ ∀ (a : α), a ∈ support p → ↑p a = ↑f a",
"state_before": "ι : Type ?u.2735472\nα : Type u_1\nβ : Type ?u.2735478\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f p : Perm α\n⊢ p ∈ cycleFactorsFinset f ↔ IsCycle p ∧ ∀ (a : α), a ∈ support p → ↑p a = ↑f a",
"tactic": "obtain ⟨l, hl, hl'⟩ := f.cycleFactorsFinset.exists_list_nodup_eq"
},
{
"state_after": "case intro.intro\nι : Type ?u.2735472\nα : Type u_1\nβ : Type ?u.2735478\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f p : Perm α\nl : List (Perm α)\nhl : List.Nodup l\nhl' : List.toFinset l = cycleFactorsFinset f\n⊢ p ∈ List.toFinset l ↔ IsCycle p ∧ ∀ (a : α), a ∈ support p → ↑p a = ↑f a",
"state_before": "case intro.intro\nι : Type ?u.2735472\nα : Type u_1\nβ : Type ?u.2735478\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f p : Perm α\nl : List (Perm α)\nhl : List.Nodup l\nhl' : List.toFinset l = cycleFactorsFinset f\n⊢ p ∈ cycleFactorsFinset f ↔ IsCycle p ∧ ∀ (a : α), a ∈ support p → ↑p a = ↑f a",
"tactic": "rw [← hl']"
},
{
"state_after": "case intro.intro\nι : Type ?u.2735472\nα : Type u_1\nβ : Type ?u.2735478\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f p : Perm α\nl : List (Perm α)\nhl : List.Nodup l\nhl' : (∀ (f : Perm α), f ∈ l → IsCycle f) ∧ List.Pairwise Disjoint l ∧ List.prod l = f\n⊢ p ∈ List.toFinset l ↔ IsCycle p ∧ ∀ (a : α), a ∈ support p → ↑p a = ↑f a",
"state_before": "case intro.intro\nι : Type ?u.2735472\nα : Type u_1\nβ : Type ?u.2735478\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f p : Perm α\nl : List (Perm α)\nhl : List.Nodup l\nhl' : List.toFinset l = cycleFactorsFinset f\n⊢ p ∈ List.toFinset l ↔ IsCycle p ∧ ∀ (a : α), a ∈ support p → ↑p a = ↑f a",
"tactic": "rw [eq_comm, cycleFactorsFinset_eq_list_toFinset hl] at hl'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type ?u.2735472\nα : Type u_1\nβ : Type ?u.2735478\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf✝ f p : Perm α\nl : List (Perm α)\nhl : List.Nodup l\nhl' : (∀ (f : Perm α), f ∈ l → IsCycle f) ∧ List.Pairwise Disjoint l ∧ List.prod l = f\n⊢ p ∈ List.toFinset l ↔ IsCycle p ∧ ∀ (a : α), a ∈ support p → ↑p a = ↑f a",
"tactic": "simpa [List.mem_toFinset, Ne.def, ← hl'.right.right] using\n mem_list_cycles_iff hl'.left hl'.right.left"
}
] | [
1413,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1407,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.add_eq_right | [
{
"state_after": "no goals",
"state_before": "a b : Cardinal\nhb : ℵ₀ ≤ b\nha : a ≤ b\n⊢ a + b = b",
"tactic": "rw [add_comm, add_eq_left hb ha]"
}
] | [
784,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
783,
1
] |
Mathlib/Data/Sym/Sym2.lean | Sym2.congr_left | [
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.4752\nγ : Type ?u.4755\na b c : α\nh : Quotient.mk (Rel.setoid α) (b, a) = Quotient.mk (Rel.setoid α) (c, a)\n⊢ b = c\n\ncase mpr\nα : Type u_1\nβ : Type ?u.4752\nγ : Type ?u.4755\na b c : α\nh : b = c\n⊢ Quotient.mk (Rel.setoid α) (b, a) = Quotient.mk (Rel.setoid α) (c, a)",
"state_before": "α : Type u_1\nβ : Type ?u.4752\nγ : Type ?u.4755\na b c : α\n⊢ Quotient.mk (Rel.setoid α) (b, a) = Quotient.mk (Rel.setoid α) (c, a) ↔ b = c",
"tactic": "constructor <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.4752\nγ : Type ?u.4755\na b c : α\nh : b = c\n⊢ Quotient.mk (Rel.setoid α) (b, a) = Quotient.mk (Rel.setoid α) (c, a)",
"tactic": "rw [h]"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.4752\nγ : Type ?u.4755\na b c : α\nh : (b, a) ≈ (c, a)\n⊢ b = c",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.4752\nγ : Type ?u.4755\na b c : α\nh : Quotient.mk (Rel.setoid α) (b, a) = Quotient.mk (Rel.setoid α) (c, a)\n⊢ b = c",
"tactic": "rw [Quotient.eq] at h"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.4752\nγ : Type ?u.4755\na b c : α\nh : (b, a) ≈ (c, a)\n⊢ b = c",
"tactic": "cases h <;> rfl"
}
] | [
165,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Algebra/Group/WithOne/Basic.lean | MulEquiv.withOneCongr_symm | [] | [
158,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/Data/List/Nodup.lean | List.disjoint_of_nodup_append | [] | [
223,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Std/Data/List/Lemmas.lean | List.cons_ne_nil | [] | [
20,
63
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
20,
1
] |
src/lean/Init/Core.lean | Iff.trans | [] | [
669,
43
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
666,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean | RightDerivMeasurableAux.a_mono | [
{
"state_after": "case intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nL : F\nr ε δ : ℝ\nh : ε ≤ δ\nx r' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : ℝ), y ∈ Icc x (x + r') → ∀ (z : ℝ), z ∈ Icc x (x + r') → ‖f z - f y - (z - y) • L‖ ≤ ε * r\n⊢ x ∈ A f L r δ",
"state_before": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nL : F\nr ε δ : ℝ\nh : ε ≤ δ\n⊢ A f L r ε ⊆ A f L r δ",
"tactic": "rintro x ⟨r', r'r, hr'⟩"
},
{
"state_after": "case intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nL : F\nr ε δ : ℝ\nh : ε ≤ δ\nx r' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : ℝ), y ∈ Icc x (x + r') → ∀ (z : ℝ), z ∈ Icc x (x + r') → ‖f z - f y - (z - y) • L‖ ≤ ε * r\ny : ℝ\nhy : y ∈ Icc x (x + r')\nz : ℝ\nhz : z ∈ Icc x (x + r')\n⊢ 0 ≤ r",
"state_before": "case intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nL : F\nr ε δ : ℝ\nh : ε ≤ δ\nx r' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : ℝ), y ∈ Icc x (x + r') → ∀ (z : ℝ), z ∈ Icc x (x + r') → ‖f z - f y - (z - y) • L‖ ≤ ε * r\n⊢ x ∈ A f L r δ",
"tactic": "refine' ⟨r', r'r, fun y hy z hz => (hr' y hy z hz).trans (mul_le_mul_of_nonneg_right h _)⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nL : F\nr ε δ : ℝ\nh : ε ≤ δ\nx r' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : ℝ), y ∈ Icc x (x + r') → ∀ (z : ℝ), z ∈ Icc x (x + r') → ‖f z - f y - (z - y) • L‖ ≤ ε * r\ny : ℝ\nhy : y ∈ Icc x (x + r')\nz : ℝ\nhz : z ∈ Icc x (x + r')\n⊢ 0 ≤ r",
"tactic": "linarith [hy.1, hy.2, r'r.2]"
}
] | [
516,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
513,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.CommSq.cone_snd | [] | [
88,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.leadingCoeff_mul_X | [] | [
1026,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1025,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean | TensorProduct.rTensorHomToHomRTensor_apply | [] | [
852,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
850,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean | Matrix.Commute.mul_zpow | [
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA B : M\nh : Commute A B\nn : ℕ\n⊢ (A * B) ^ ↑n = A ^ ↑n * B ^ ↑n",
"tactic": "simp [h.mul_pow n, -mul_eq_mul]"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA B : M\nh : Commute A B\nn : ℕ\n⊢ (A * B) ^ -[n+1] = A ^ -[n+1] * B ^ -[n+1]",
"tactic": "rw [zpow_negSucc, zpow_negSucc, zpow_negSucc, mul_eq_mul _⁻¹, ← mul_inv_rev, ← mul_eq_mul,\n h.mul_pow n.succ, (h.pow_pow _ _).eq]"
}
] | [
314,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
310,
1
] |
Mathlib/Order/Filter/Lift.lean | Filter.map_lift'_eq2 | [] | [
307,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
] |
Std/Data/String/Lemmas.lean | String.data_take | [
{
"state_after": "no goals",
"state_before": "s : String\nn : Nat\n⊢ (take s n).data = List.take n s.data",
"tactic": "rw [take_eq]"
}
] | [
1099,
96
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1099,
9
] |
Mathlib/RingTheory/Multiplicity.lean | multiplicity.multiplicity_add_eq_min | [
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p a < multiplicity p b\n⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)\n\ncase inr.inl\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p a = multiplicity p b\n⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)\n\ncase inr.inr\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p b < multiplicity p a\n⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)",
"state_before": "α : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\n⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)",
"tactic": "rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p a < multiplicity p b\n⊢ multiplicity p a ≤ multiplicity p b",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p a < multiplicity p b\n⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)",
"tactic": "rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p a < multiplicity p b\n⊢ multiplicity p a ≤ multiplicity p b",
"tactic": "exact le_of_lt hab"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p a = multiplicity p b\n⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)",
"tactic": "contradiction"
},
{
"state_after": "case inr.inr\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p b < multiplicity p a\n⊢ multiplicity p b ≤ multiplicity p a",
"state_before": "case inr.inr\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p b < multiplicity p a\n⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)",
"tactic": "rw [multiplicity_add_of_gt hab, min_eq_right]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\ninst✝¹ : Ring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≠ multiplicity p b\nhab : multiplicity p b < multiplicity p a\n⊢ multiplicity p b ≤ multiplicity p a",
"tactic": "exact le_of_lt hab"
}
] | [
467,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
460,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean | TensorProduct.map_tmul | [] | [
726,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
725,
1
] |
Mathlib/Data/Nat/Choose/Sum.lean | Commute.add_pow | [
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\n⊢ (x + y) ^ n = ∑ m in range (n + 1), x ^ m * y ^ (n - m) * ↑(Nat.choose n m)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\n⊢ (x + y) ^ n = ∑ m in range (n + 1), x ^ m * y ^ (n - m) * ↑(Nat.choose n m)",
"tactic": "let t : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * choose n m"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\n⊢ (x + y) ^ n = ∑ m in range (n + 1), t n m",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\n⊢ (x + y) ^ n = ∑ m in range (n + 1), x ^ m * y ^ (n - m) * ↑(Nat.choose n m)",
"tactic": "change (x + y) ^ n = ∑ m in range (n + 1), t n m"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\n⊢ (x + y) ^ n = ∑ m in range (n + 1), t n m",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\n⊢ (x + y) ^ n = ∑ m in range (n + 1), t n m",
"tactic": "have h_first : ∀ n, t n 0 = y ^ n := fun n ↦ by\n simp only [choose_zero_right, _root_.pow_zero, Nat.cast_one, mul_one, one_mul, tsub_zero]"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\n⊢ (x + y) ^ n = ∑ m in range (n + 1), t n m",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\n⊢ (x + y) ^ n = ∑ m in range (n + 1), t n m",
"tactic": "have h_last : ∀ n, t n n.succ = 0 := fun n ↦ by\n simp only [ge_iff_le, choose_succ_self, cast_zero, mul_zero]"
},
{
"state_after": "case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\n⊢ (x + y) ^ zero = ∑ m in range (zero + 1), t zero m\n\ncase succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ (x + y) ^ succ n = ∑ m in range (succ n + 1), t (succ n) m",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\n⊢ (x + y) ^ n = ∑ m in range (n + 1), t n m",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nn : ℕ\n⊢ t n 0 = y ^ n",
"tactic": "simp only [choose_zero_right, _root_.pow_zero, Nat.cast_one, mul_one, one_mul, tsub_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nn : ℕ\n⊢ t n (succ n) = 0",
"tactic": "simp only [ge_iff_le, choose_succ_self, cast_zero, mul_zero]"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\n⊢ (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\n⊢ ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)",
"tactic": "intro n i h_mem"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\n⊢ (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)",
"tactic": "have h_le : i ≤ n := Nat.le_of_lt_succ (mem_range.mp h_mem)"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ ((fun m => x ^ m * y ^ (succ n - m) * ↑(Nat.choose (succ n) m)) ∘ succ) i =\n x * (x ^ i * y ^ (n - i) * ↑(Nat.choose n i)) + y * (x ^ succ i * y ^ (n - succ i) * ↑(Nat.choose n (succ i)))",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)",
"tactic": "dsimp only"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n i) +\n x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x * (x ^ i * y ^ (n - i) * ↑(Nat.choose n i)) + y * (x ^ succ i * y ^ (n - succ i) * ↑(Nat.choose n (succ i)))",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ ((fun m => x ^ m * y ^ (succ n - m) * ↑(Nat.choose (succ n) m)) ∘ succ) i =\n x * (x ^ i * y ^ (n - i) * ↑(Nat.choose n i)) + y * (x ^ succ i * y ^ (n - succ i) * ↑(Nat.choose n (succ i)))",
"tactic": "rw [Function.comp_apply, choose_succ_succ, Nat.cast_add, mul_add]"
},
{
"state_after": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n i) = x * (x ^ i * y ^ (n - i) * ↑(Nat.choose n i))\n\ncase e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n y * (x ^ succ i * y ^ (n - succ i) * ↑(Nat.choose n (succ i)))",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n i) +\n x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x * (x ^ i * y ^ (n - i) * ↑(Nat.choose n i)) + y * (x ^ succ i * y ^ (n - succ i) * ↑(Nat.choose n (succ i)))",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n i) = x * (x ^ i * y ^ (n - i) * ↑(Nat.choose n i))",
"tactic": "rw [pow_succ x, succ_sub_succ, mul_assoc, mul_assoc, mul_assoc]"
},
{
"state_after": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))",
"state_before": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n y * (x ^ succ i * y ^ (n - succ i) * ↑(Nat.choose n (succ i)))",
"tactic": "rw [← mul_assoc y, ← mul_assoc y, (h.symm.pow_right i.succ).eq]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\nh_eq : i = n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))\n\ncase neg\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\nh_eq : ¬i = n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))",
"state_before": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))",
"tactic": "by_cases h_eq : i = n"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\nh_eq : i = n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))",
"tactic": "rw [h_eq, choose_succ_self, Nat.cast_zero, mul_zero, mul_zero]"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\nh_eq : ¬i = n\n⊢ x ^ succ i * y ^ succ (n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\nh_eq : ¬i = n\n⊢ x ^ succ i * y ^ (succ n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))",
"tactic": "rw [succ_sub (lt_of_le_of_ne h_le h_eq)]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nn i : ℕ\nh_mem : i ∈ range (succ n)\nh_le : i ≤ n\nh_eq : ¬i = n\n⊢ x ^ succ i * y ^ succ (n - succ i) * ↑(Nat.choose n (succ i)) =\n x ^ succ i * y * y ^ (n - succ i) * ↑(Nat.choose n (succ i))",
"tactic": "rw [pow_succ y, mul_assoc, mul_assoc, mul_assoc, mul_assoc]"
},
{
"state_after": "case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\n⊢ 1 = t zero zero",
"state_before": "case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\n⊢ (x + y) ^ zero = ∑ m in range (zero + 1), t zero m",
"tactic": "rw [_root_.pow_zero, sum_range_succ, range_zero, sum_empty, zero_add]"
},
{
"state_after": "case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\n⊢ 1 = x ^ zero * y ^ (zero - zero) * ↑(Nat.choose zero zero)",
"state_before": "case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\n⊢ 1 = t zero zero",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\n⊢ 1 = x ^ zero * y ^ (zero - zero) * ↑(Nat.choose zero zero)",
"tactic": "rw [_root_.pow_zero, _root_.pow_zero, choose_self, Nat.cast_one, mul_one, mul_one]"
},
{
"state_after": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ (x + y) ^ succ n = ∑ k in range (n + 1), t (succ n) (k + 1) + y ^ succ n",
"state_before": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ (x + y) ^ succ n = ∑ m in range (succ n + 1), t (succ n) m",
"tactic": "rw [sum_range_succ', h_first]"
},
{
"state_after": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ (x + y) ^ succ n = ∑ x_1 in range (succ n), x * t n x_1 + (∑ x in range (succ n), y * t n (succ x) + y ^ succ n)",
"state_before": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ (x + y) ^ succ n = ∑ k in range (n + 1), t (succ n) (k + 1) + y ^ succ n",
"tactic": "erw [sum_congr rfl (h_middle n), sum_add_distrib, add_assoc]"
},
{
"state_after": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ ∑ x_1 in range (n + 1), x * t n x_1 + ∑ x in range (n + 1), y * t n x =\n ∑ x_1 in range (succ n), x * t n x_1 + (∑ x in range (succ n), y * t n (succ x) + y ^ succ n)",
"state_before": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ (x + y) ^ succ n = ∑ x_1 in range (succ n), x * t n x_1 + (∑ x in range (succ n), y * t n (succ x) + y ^ succ n)",
"tactic": "rw [pow_succ (x + y), ih, add_mul, mul_sum, mul_sum]"
},
{
"state_after": "case succ.e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ ∑ x in range (n + 1), y * t n x = ∑ x in range (succ n), y * t n (succ x) + y ^ succ n",
"state_before": "case succ\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ ∑ x_1 in range (n + 1), x * t n x_1 + ∑ x in range (n + 1), y * t n x =\n ∑ x_1 in range (succ n), x * t n x_1 + (∑ x in range (succ n), y * t n (succ x) + y ^ succ n)",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case succ.e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(Nat.choose n m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n (succ n) = 0\nh_middle : ∀ (n i : ℕ), i ∈ range (succ n) → (t (succ n) ∘ succ) i = x * t n i + y * t n (succ i)\nn : ℕ\nih : (x + y) ^ n = ∑ m in range (n + 1), t n m\n⊢ ∑ x in range (n + 1), y * t n x = ∑ x in range (succ n), y * t n (succ x) + y ^ succ n",
"tactic": "rw [sum_range_succ', sum_range_succ, h_first, h_last, mul_zero, add_zero, _root_.pow_succ]"
}
] | [
71,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | extChartAt_preimage_mem_nhdsWithin' | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.219932\nM' : Type ?u.219935\nH' : Type ?u.219938\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\nx' : M\nh : x' ∈ (extChartAt I x).source\nht : t ∈ 𝓝[s] x'\n⊢ ↑(LocalEquiv.symm (extChartAt I x)) ⁻¹' t ∈\n 𝓝[↑(LocalEquiv.symm (extChartAt I x)) ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x'",
"tactic": "rwa [← map_extChartAt_symm_nhdsWithin' I x h, mem_map] at ht"
}
] | [
1208,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1205,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.prod_mk_left | [] | [
1630,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1628,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_X_pow | [] | [
1263,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1262,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | mem_nhds_left | [] | [
797,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
796,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.StronglyMeasurable.smul_const | [] | [
478,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
476,
11
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.unique_ext_iff | [] | [
279,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
278,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.restrict_congr_meas | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.307127\nγ : Type ?u.307130\nδ : Type ?u.307133\nι : Type ?u.307136\nR : Type ?u.307139\nR' : Type ?u.307142\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nhs : MeasurableSet s\nH : restrict μ s = restrict ν s\nt : Set α\nhts : t ⊆ s\nht : MeasurableSet t\n⊢ ↑↑μ t = ↑↑ν t",
"tactic": "rw [← inter_eq_self_of_subset_left hts, ← restrict_apply ht, H, restrict_apply ht]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.307127\nγ : Type ?u.307130\nδ : Type ?u.307133\nι : Type ?u.307136\nR : Type ?u.307139\nR' : Type ?u.307142\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nhs : MeasurableSet s\nH : ∀ (t : Set α), t ⊆ s → MeasurableSet t → ↑↑μ t = ↑↑ν t\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(restrict μ s) t = ↑↑(restrict ν s) t",
"tactic": "rw [restrict_apply ht, restrict_apply ht, H _ (inter_subset_right _ _) (ht.inter hs)]"
}
] | [
1803,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1798,
1
] |
Std/Data/Int/Lemmas.lean | Int.add_le_add_iff_left | [] | [
780,
57
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
779,
11
] |
Mathlib/Data/Rat/Cast.lean | Rat.cast_min | [] | [
353,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
] |
Mathlib/Order/Bounds/Basic.lean | IsGreatest.isLUB | [] | [
283,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.aevalTower_comp_C | [] | [
1607,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1606,
1
] |
Mathlib/Algebra/EuclideanDomain/Defs.lean | EuclideanDomain.mul_right_not_lt | [
{
"state_after": "R : Type u\ninst✝ : EuclideanDomain R\na b : R\nh : a ≠ 0\n⊢ ¬b * a ≺ b",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\na b : R\nh : a ≠ 0\n⊢ ¬a * b ≺ b",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\na b : R\nh : a ≠ 0\n⊢ ¬b * a ≺ b",
"tactic": "exact mul_left_not_lt b h"
}
] | [
154,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean | TrivSqZeroExt.fst_nat_cast | [] | [
523,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
522,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean | Measurable.indicator | [] | [
342,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
340,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.leadingCoeff_ne_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.514573\n⊢ leadingCoeff p ≠ 0 ↔ p ≠ 0",
"tactic": "rw [Ne.def, leadingCoeff_eq_zero]"
}
] | [
668,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
668,
1
] |
Mathlib/Data/Set/Image.lean | Subtype.preimage_val_eq_preimage_val_iff | [] | [
1453,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1451,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.filter_union_filter_of_codisjoint | [] | [
2956,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2954,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.leadingCoeff_linear | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.910503\nha : a ≠ 0\n⊢ leadingCoeff (↑C a * X + ↑C b) = a",
"tactic": "rw [add_comm, leadingCoeff_add_of_degree_lt (degree_C_lt_degree_C_mul_X ha),\n leadingCoeff_C_mul_X]"
}
] | [
1170,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1168,
1
] |
Mathlib/Computability/TuringMachine.lean | Turing.TM1to1.stepAux_read | [
{
"state_after": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL R : ListBlank Γ\n⊢ ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL R : ListBlank Γ\n⊢ stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)",
"tactic": "suffices ∀ f, stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc R.head)) v (trTape' enc0 (L.cons R.head) R.tail) by\n rw [read, this, stepAux_move, encdec, trTape'_move_left enc0]\n simp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons]"
},
{
"state_after": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\n⊢ ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v (trTape' enc0 L (ListBlank.cons a R)) =\n stepAux (f (enc (ListBlank.head (ListBlank.cons a R)))) v\n (trTape' enc0 (ListBlank.cons (ListBlank.head (ListBlank.cons a R)) L) (ListBlank.tail (ListBlank.cons a R)))",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL R : ListBlank Γ\n⊢ ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))",
"tactic": "obtain ⟨a, R, rfl⟩ := R.exists_cons"
},
{
"state_after": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\n⊢ ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))",
"state_before": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\n⊢ ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v (trTape' enc0 L (ListBlank.cons a R)) =\n stepAux (f (enc (ListBlank.head (ListBlank.cons a R)))) v\n (trTape' enc0 (ListBlank.cons (ListBlank.head (ListBlank.cons a R)) L) (ListBlank.tail (ListBlank.cons a R)))",
"tactic": "simp only [ListBlank.head_cons, ListBlank.tail_cons, trTape', ListBlank.cons_bind,\n ListBlank.append_assoc]"
},
{
"state_after": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\n⊢ ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ) (L' R' : ListBlank Bool) (l₁ l₂ : List Bool) (h : List.length l₂ = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"state_before": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\n⊢ ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))",
"tactic": "suffices ∀ i f L' R' l₁ l₂ h,\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f ⟨l₂, h⟩) v (Tape.mk' (ListBlank.append (l₂.reverseAux l₁) L') R') by\n intro f\n exact this n f (L.bind (fun x => (enc x).1.reverse) _)\n (R.bind (fun x => (enc x).1) _) [] _ (enc a).2"
},
{
"state_after": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\n⊢ ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ) (L' R' : ListBlank Bool) (l₁ l₂ : List Bool) (h : List.length l₂ = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"state_before": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\n⊢ ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ) (L' R' : ListBlank Bool) (l₁ l₂ : List Bool) (h : List.length l₂ = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"tactic": "clear f L a R"
},
{
"state_after": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\ni : ℕ\nf : Vector Bool i → Stmt Bool Λ' σ\nL' R' : ListBlank Bool\nl₁ l₂ : List Bool\nh✝ : List.length l₂ = i\n⊢ stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h✝ }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"state_before": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\n⊢ ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ) (L' R' : ListBlank Bool) (l₁ l₂ : List Bool) (h : List.length l₂ = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"tactic": "intro i f L' R' l₁ l₂ _"
},
{
"state_after": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁ l₂ : List Bool\nf : Vector Bool (List.length l₂) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"state_before": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\ni : ℕ\nf : Vector Bool i → Stmt Bool Λ' σ\nL' R' : ListBlank Bool\nl₁ l₂ : List Bool\nh✝ : List.length l₂ = i\n⊢ stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h✝ }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"tactic": "subst i"
},
{
"state_after": "case intro.intro.nil\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂ : List Bool\nf✝ : Vector Bool (List.length l₂) → Stmt Bool Λ' σ\nl₁ : List Bool\nf : Vector Bool (List.length []) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length []) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append [] R')) =\n stepAux (f { val := [], property := (_ : List.length [] = List.length []) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux [] l₁) L') R')\n\ncase intro.intro.cons\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length (a :: l₂)) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (a :: l₂) R')) =\n stepAux (f { val := a :: l₂, property := (_ : List.length (a :: l₂) = List.length (a :: l₂)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R')",
"state_before": "case intro.intro\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁ l₂ : List Bool\nf : Vector Bool (List.length l₂) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')",
"tactic": "induction' l₂ with a l₂ IH generalizing l₁"
},
{
"state_after": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length (a :: l₂)) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (a :: l₂) R')) =\n stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))\n\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R')) =\n stepAux (f { val := a :: l₂, property := (_ : List.length (a :: l₂) = List.length (a :: l₂)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R')",
"state_before": "case intro.intro.cons\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length (a :: l₂)) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (a :: l₂) R')) =\n stepAux (f { val := a :: l₂, property := (_ : List.length (a :: l₂) = List.length (a :: l₂)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R')",
"tactic": "trans\n stepAux (readAux l₂.length fun v ↦ f (a ::ᵥ v)) v\n (Tape.mk' ((L'.append l₁).cons a) (R'.append l₂))"
},
{
"state_after": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (f (a ::ᵥ { val := l₂, property := (_ : List.length l₂ = List.length l₂) })) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ (a :: l₁)) L') R') =\n stepAux (f { val := a :: l₂, property := (_ : List.length (a :: l₂) = List.length (a :: l₂)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R')",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R')) =\n stepAux (f { val := a :: l₂, property := (_ : List.length (a :: l₂) = List.length (a :: l₂)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R')",
"tactic": "rw [← ListBlank.append, IH]"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (f (a ::ᵥ { val := l₂, property := (_ : List.length l₂ = List.length l₂) })) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ (a :: l₁)) L') R') =\n stepAux (f { val := a :: l₂, property := (_ : List.length (a :: l₂) = List.length (a :: l₂)) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R')",
"tactic": "rfl"
},
{
"state_after": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL R : ListBlank Γ\nthis :\n ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n⊢ stepAux (f (ListBlank.head R)) v\n (trTape' enc0 (ListBlank.tail (ListBlank.cons (ListBlank.head R) L))\n (ListBlank.cons (ListBlank.head (ListBlank.cons (ListBlank.head R) L)) (ListBlank.tail R))) =\n stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL R : ListBlank Γ\nthis :\n ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n⊢ stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)",
"tactic": "rw [read, this, stepAux_move, encdec, trTape'_move_left enc0]"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL R : ListBlank Γ\nthis :\n ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v (trTape' enc0 L R) =\n stepAux (f (enc (ListBlank.head R))) v (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))\n⊢ stepAux (f (ListBlank.head R)) v\n (trTape' enc0 (ListBlank.tail (ListBlank.cons (ListBlank.head R) L))\n (ListBlank.cons (ListBlank.head (ListBlank.cons (ListBlank.head R) L)) (ListBlank.tail R))) =\n stepAux (f (ListBlank.head R)) v (trTape' enc0 L R)",
"tactic": "simp only [ListBlank.head_cons, ListBlank.cons_head_tail, ListBlank.tail_cons]"
},
{
"state_after": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf✝ : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\nthis :\n ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ) (L' R' : ListBlank Bool) (l₁ l₂ : List Bool) (h : List.length l₂ = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nf : Vector Bool n → Stmt Bool Λ' σ\n⊢ stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\nthis :\n ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ) (L' R' : ListBlank Bool) (l₁ l₂ : List Bool) (h : List.length l₂ = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\n⊢ ∀ (f : Vector Bool n → Stmt Bool Λ' σ),\n stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))",
"tactic": "intro f"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nf✝ : Γ → Stmt Bool Λ' σ\nv : σ\nL : ListBlank Γ\na : Γ\nR : ListBlank Γ\nthis :\n ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ) (L' R' : ListBlank Bool) (l₁ l₂ : List Bool) (h : List.length l₂ = i),\n stepAux (readAux i f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := h }) v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nf : Vector Bool n → Stmt Bool Λ' σ\n⊢ stepAux (readAux n f) v\n (Tape.mk'\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))\n (ListBlank.append (Vector.toList (enc a))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) =\n stepAux (f (enc a)) v\n (Tape.mk'\n (ListBlank.append (List.reverse (Vector.toList (enc a)))\n (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x)))\n (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)))\n (ListBlank.bind R (fun x => Vector.toList (enc x))\n (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))",
"tactic": "exact this n f (L.bind (fun x => (enc x).1.reverse) _)\n (R.bind (fun x => (enc x).1) _) [] _ (enc a).2"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.nil\nΓ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂ : List Bool\nf✝ : Vector Bool (List.length l₂) → Stmt Bool Λ' σ\nl₁ : List Bool\nf : Vector Bool (List.length []) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length []) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append [] R')) =\n stepAux (f { val := [], property := (_ : List.length [] = List.length []) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux [] l₁) L') R')",
"tactic": "rfl"
},
{
"state_after": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ (bif ListBlank.head (ListBlank.cons a (ListBlank.append l₂ R')) then\n stepAux (readAux (List.length l₂) fun v => f (true ::ᵥ v)) v\n (Tape.move Dir.right (Tape.mk' (ListBlank.append l₁ L') (ListBlank.cons a (ListBlank.append l₂ R'))))\n else\n stepAux (readAux (List.length l₂) fun v => f (false ::ᵥ v)) v\n (Tape.move Dir.right (Tape.mk' (ListBlank.append l₁ L') (ListBlank.cons a (ListBlank.append l₂ R'))))) =\n stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ stepAux (readAux (List.length (a :: l₂)) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (a :: l₂) R')) =\n stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))",
"tactic": "dsimp [readAux, stepAux]"
},
{
"state_after": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ (bif a then\n stepAux (readAux (List.length l₂) fun v => f (true ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))\n else\n stepAux (readAux (List.length l₂) fun v => f (false ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))) =\n stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ (bif ListBlank.head (ListBlank.cons a (ListBlank.append l₂ R')) then\n stepAux (readAux (List.length l₂) fun v => f (true ::ᵥ v)) v\n (Tape.move Dir.right (Tape.mk' (ListBlank.append l₁ L') (ListBlank.cons a (ListBlank.append l₂ R'))))\n else\n stepAux (readAux (List.length l₂) fun v => f (false ::ᵥ v)) v\n (Tape.move Dir.right (Tape.mk' (ListBlank.append l₁ L') (ListBlank.cons a (ListBlank.append l₂ R'))))) =\n stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "Γ : Type u_3\ninst✝² : Inhabited Γ\nΛ : Type u_2\ninst✝¹ : Inhabited Λ\nσ : Type u_1\ninst✝ : Inhabited σ\nn : ℕ\nenc : Γ → Vector Bool n\ndec : Vector Bool n → Γ\nenc0 : enc default = Vector.replicate n false\nM : Λ → Stmt₁\nencdec : ∀ (a : Γ), dec (enc a) = a\nv : σ\nL' R' : ListBlank Bool\nl₁✝ l₂✝ : List Bool\nf✝ : Vector Bool (List.length l₂✝) → Stmt Bool Λ' σ\na : Bool\nl₂ : List Bool\nIH :\n ∀ (l₁ : List Bool) (f : Vector Bool (List.length l₂) → Stmt Bool Λ' σ),\n stepAux (readAux (List.length l₂) f) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) =\n stepAux (f { val := l₂, property := (_ : List.length l₂ = List.length l₂) }) v\n (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R')\nl₁ : List Bool\nf : Vector Bool (List.length (a :: l₂)) → Stmt Bool Λ' σ\n⊢ (bif a then\n stepAux (readAux (List.length l₂) fun v => f (true ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))\n else\n stepAux (readAux (List.length l₂) fun v => f (false ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))) =\n stepAux (readAux (List.length l₂) fun v => f (a ::ᵥ v)) v\n (Tape.mk' (ListBlank.cons a (ListBlank.append l₁ L')) (ListBlank.append l₂ R'))",
"tactic": "cases a <;> rfl"
}
] | [
1850,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1822,
1
] |
Mathlib/Topology/Inseparable.lean | SeparationQuotient.tendsto_lift₂_nhdsWithin | [
{
"state_after": "X : Type u_1\nY : Type u_2\nZ : Type ?u.107519\nα : Type u_3\nι : Type ?u.107525\nπ : ι → Type ?u.107530\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx✝ y✝ z : X\ns✝ : Set X\nf✝ : X → Y\nt : Set (SeparationQuotient X)\nf : X → Y → α\nhf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d\nx : X\ny : Y\ns : Set (SeparationQuotient X × SeparationQuotient Y)\nl : Filter α\n⊢ Tendsto (uncurry (lift₂ f hf)) (Filter.map (Prod.map mk mk) (𝓝 (x, y) ⊓ 𝓟 (Prod.map mk mk ⁻¹' s))) l ↔\n Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.107519\nα : Type u_3\nι : Type ?u.107525\nπ : ι → Type ?u.107530\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx✝ y✝ z : X\ns✝ : Set X\nf✝ : X → Y\nt : Set (SeparationQuotient X)\nf : X → Y → α\nhf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d\nx : X\ny : Y\ns : Set (SeparationQuotient X × SeparationQuotient Y)\nl : Filter α\n⊢ Tendsto (uncurry (lift₂ f hf)) (𝓝[s] (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l",
"tactic": "rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.107519\nα : Type u_3\nι : Type ?u.107525\nπ : ι → Type ?u.107530\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx✝ y✝ z : X\ns✝ : Set X\nf✝ : X → Y\nt : Set (SeparationQuotient X)\nf : X → Y → α\nhf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d\nx : X\ny : Y\ns : Set (SeparationQuotient X × SeparationQuotient Y)\nl : Filter α\n⊢ Tendsto (uncurry (lift₂ f hf)) (Filter.map (Prod.map mk mk) (𝓝 (x, y) ⊓ 𝓟 (Prod.map mk mk ⁻¹' s))) l ↔\n Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l",
"tactic": "rfl"
}
] | [
604,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
598,
9
] |
Mathlib/FieldTheory/Subfield.lean | Subfield.comap_top | [] | [
812,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
811,
1
] |
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