file_path
stringlengths 11
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Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_eq_descFactorial_div_factorial | [
{
"state_after": "n k : ℕ\n⊢ k ! * choose n k = k ! * (descFactorial n k / k !)",
"state_before": "n k : ℕ\n⊢ choose n k = descFactorial n k / k !",
"tactic": "apply mul_left_cancel₀ (factorial_ne_zero k)"
},
{
"state_after": "n k : ℕ\n⊢ descFactorial n k = k ! * (descFactorial n k / k !)",
"state_before": "n k : ℕ\n⊢ k ! * choose n k = k ! * (descFactorial n k / k !)",
"tactic": "rw [← descFactorial_eq_factorial_mul_choose]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ descFactorial n k = k ! * (descFactorial n k / k !)",
"tactic": "exact (Nat.mul_div_cancel' <| factorial_dvd_descFactorial _ _).symm"
}
]
| [
273,
70
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
270,
1
]
|
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | intervalIntegral.integral_hasFDerivAt | []
| [
723,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
718,
1
]
|
Mathlib/Topology/PathConnected.lean | JoinedIn.somePath_mem | []
| [
841,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
840,
1
]
|
Mathlib/Data/Nat/Interval.lean | Nat.Ico_succ_succ | [
{
"state_after": "case a\na b c x : ℕ\n⊢ x ∈ Ico (succ a) (succ b) ↔ x ∈ Ioc a b",
"state_before": "a b c : ℕ\n⊢ Ico (succ a) (succ b) = Ioc a b",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case a\na b c x : ℕ\n⊢ x ∈ Ico (succ a) (succ b) ↔ x ∈ Ioc a b",
"tactic": "rw [mem_Ico, mem_Ioc, succ_le_iff, lt_succ_iff]"
}
]
| [
182,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
180,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | HasFDerivAt.hasFDerivWithinAt | []
| [
373,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
372,
1
]
|
Mathlib/Analysis/Calculus/ContDiff.lean | contDiff_add | []
| [
1191,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1190,
1
]
|
Mathlib/LinearAlgebra/Basic.lean | LinearEquiv.symm_neg | []
| [
2284,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2283,
1
]
|
Mathlib/Combinatorics/Young/YoungDiagram.lean | YoungDiagram.rowLens_ofRowLens_eq_self | []
| [
527,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
523,
1
]
|
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.toNat_eq_one_iff_unique | []
| [
1773,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1772,
1
]
|
Mathlib/RingTheory/WittVector/MulP.lean | WittVector.mulN_coeff | [
{
"state_after": "case zero\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ k : ℕ\n⊢ coeff (x * ↑Nat.zero) k = ↑(aeval x.coeff) (wittMulN p Nat.zero k)\n\ncase succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ coeff (x * ↑(Nat.succ n)) k = ↑(aeval x.coeff) (wittMulN p (Nat.succ n) k)",
"state_before": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn : ℕ\nx : 𝕎 R\nk : ℕ\n⊢ coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)",
"tactic": "induction' n with n ih generalizing k"
},
{
"state_after": "no goals",
"state_before": "case zero\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ k : ℕ\n⊢ coeff (x * ↑Nat.zero) k = ↑(aeval x.coeff) (wittMulN p Nat.zero k)",
"tactic": "simp only [Nat.zero_eq, Nat.cast_zero, MulZeroClass.mul_zero, zero_coeff, wittMulN,\n AlgHom.map_zero, Pi.zero_apply]"
},
{
"state_after": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ peval (wittAdd p k) ![(x * ↑n).coeff, x.coeff] =\n ↑(aeval fun i => ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)) (wittAdd p k)",
"state_before": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ coeff (x * ↑(Nat.succ n)) k = ↑(aeval x.coeff) (wittMulN p (Nat.succ n) k)",
"tactic": "rw [wittMulN, Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, mul_add, mul_one, aeval_bind₁,\n add_coeff]"
},
{
"state_after": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] = fun i => ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)",
"state_before": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ peval (wittAdd p k) ![(x * ↑n).coeff, x.coeff] =\n ↑(aeval fun i => ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)) (wittAdd p k)",
"tactic": "apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl"
},
{
"state_after": "case h.mk\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\nb : Fin 2\ni : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] (b, i) = ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] (b, i))",
"state_before": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] = fun i => ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)",
"tactic": "ext1 ⟨b, i⟩"
},
{
"state_after": "case h.mk.head\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk i : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] ({ val := 0, isLt := (_ : 0 < 2) }, i) =\n ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] ({ val := 0, isLt := (_ : 0 < 2) }, i))\n\ncase h.mk.tail.head\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk i : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] ({ val := 1, isLt := (_ : (fun a => a < 2) 1) }, i) =\n ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] ({ val := 1, isLt := (_ : (fun a => a < 2) 1) }, i))",
"state_before": "case h.mk\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk : ℕ\nb : Fin 2\ni : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] (b, i) = ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] (b, i))",
"tactic": "fin_cases b"
},
{
"state_after": "no goals",
"state_before": "case h.mk.head\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk i : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] ({ val := 0, isLt := (_ : 0 < 2) }, i) =\n ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] ({ val := 0, isLt := (_ : 0 < 2) }, i))",
"tactic": "simp [Function.uncurry, Matrix.cons_val_zero, ih]"
},
{
"state_after": "no goals",
"state_before": "case h.mk.tail.head\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nk✝ n : ℕ\nih : ∀ (k : ℕ), coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)\nk i : ℕ\n⊢ Function.uncurry ![(x * ↑n).coeff, x.coeff] ({ val := 1, isLt := (_ : (fun a => a < 2) 1) }, i) =\n ↑(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] ({ val := 1, isLt := (_ : (fun a => a < 2) 1) }, i))",
"tactic": "simp [Function.uncurry, Matrix.cons_val_one, Matrix.head_cons, aeval_X]"
}
]
| [
64,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
]
|
Mathlib/Topology/GDelta.lean | mem_residual_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.16113\nγ : Type ?u.16116\nι : Type ?u.16119\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ (∃ S, S ⊆ {t | IsOpen t ∧ Dense t} ∧ Set.Countable S ∧ ⋂₀ S ⊆ s) ↔\n ∃ S, (∀ (t : Set α), t ∈ S → IsOpen t) ∧ (∀ (t : Set α), t ∈ S → Dense t) ∧ Set.Countable S ∧ ⋂₀ S ⊆ s",
"tactic": "simp_rw [subset_def, mem_setOf, forall_and, and_assoc]"
}
]
| [
225,
95
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
]
|
Mathlib/GroupTheory/Subgroup/Pointwise.lean | Subgroup.sup_eq_closure | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.19854\nG : Type u_1\nA : Type ?u.19860\nS : Type ?u.19863\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns : Set G\nH K : Subgroup G\n⊢ closure ↑H ⊔ closure ↑K ≤ H ⊔ K",
"tactic": "rw [closure_eq, closure_eq]"
}
]
| [
155,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
]
|
Mathlib/Algebra/Star/Order.lean | StarOrderedRing.nonneg_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : NonUnitalSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\nx : R\n⊢ 0 ≤ x ↔ x ∈ AddSubmonoid.closure (Set.range fun s => star s * s)",
"tactic": "simp only [le_iff, zero_add, exists_eq_right']"
}
]
| [
147,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
]
|
Mathlib/GroupTheory/FreeGroup.lean | FreeGroup.toWord_eq_nil_iff | []
| [
1327,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1326,
1
]
|
Mathlib/Logic/Equiv/Basic.lean | Equiv.swap_swap | []
| [
1578,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1577,
1
]
|
Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | thickenedIndicatorAux_zero | [
{
"state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ thickenedIndicatorAux δ E x = 0",
"state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ¬x ∈ thickening δ E\n⊢ thickenedIndicatorAux δ E x = 0",
"tactic": "rw [thickening, mem_setOf_eq, not_lt] at x_out"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ = 0",
"state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ thickenedIndicatorAux δ E x = 0",
"tactic": "unfold thickenedIndicatorAux"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥",
"state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ = 0",
"tactic": "apply le_antisymm _ bot_le"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - ENNReal.ofReal δ / ENNReal.ofReal δ\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥",
"state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥",
"tactic": "have key := tsub_le_tsub\n (@rfl _ (1 : ℝ≥0∞)).le (ENNReal.div_le_div x_out (@rfl _ (ENNReal.ofReal δ : ℝ≥0∞)).le)"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - 1\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥",
"state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - ENNReal.ofReal δ / ENNReal.ofReal δ\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥",
"tactic": "rw [ENNReal.div_self (ne_of_gt (ENNReal.ofReal_pos.mpr δ_pos)) ofReal_ne_top] at key"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEdist x E\nkey : 1 - infEdist x E / ENNReal.ofReal δ ≤ 1 - 1\n⊢ 1 - infEdist x E / ENNReal.ofReal δ ≤ ⊥",
"tactic": "simpa using key"
}
]
| [
105,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
]
|
Mathlib/Algebra/BigOperators/Finprod.lean | finprod_mem_eq_prod_filter | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.196353\nι : Type ?u.196356\nG : Type ?u.196359\nM : Type u_2\nN : Type ?u.196365\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf : α → M\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhf : Set.Finite (mulSupport f)\nx : α\n⊢ (x ∈ s ∩ mulSupport fun i => f i) ↔\n x ∈ ↑(Finset.filter (fun x => x ∈ s) (Finite.toFinset hf)) ∩ mulSupport fun i => f i",
"state_before": "α : Type u_1\nβ : Type ?u.196353\nι : Type ?u.196356\nG : Type ?u.196359\nM : Type u_2\nN : Type ?u.196365\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf : α → M\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhf : Set.Finite (mulSupport f)\n⊢ (s ∩ mulSupport fun i => f i) = ↑(Finset.filter (fun x => x ∈ s) (Finite.toFinset hf)) ∩ mulSupport fun i => f i",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.196353\nι : Type ?u.196356\nG : Type ?u.196359\nM : Type u_2\nN : Type ?u.196365\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\nf : α → M\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nhf : Set.Finite (mulSupport f)\nx : α\n⊢ (x ∈ s ∩ mulSupport fun i => f i) ↔\n x ∈ ↑(Finset.filter (fun x => x ∈ s) (Finite.toFinset hf)) ∩ mulSupport fun i => f i",
"tactic": "simp [and_comm]"
}
]
| [
502,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
497,
1
]
|
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | LiouvilleWith.sub_int_iff | [
{
"state_after": "no goals",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\n⊢ LiouvilleWith p (x - ↑m) ↔ LiouvilleWith p x",
"tactic": "rw [← Rat.cast_coe_int, sub_rat_iff]"
}
]
| [
274,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
]
|
Mathlib/Topology/Constructions.lean | CofiniteTopology.mem_nhds_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.18624\nδ : Type ?u.18627\nε : Type ?u.18630\nζ : Type ?u.18633\na : CofiniteTopology α\ns : Set (CofiniteTopology α)\n⊢ s ∈ 𝓝 a ↔ a ∈ s ∧ Set.Finite (sᶜ)",
"tactic": "simp [nhds_eq]"
}
]
| [
305,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
]
|
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean | tendsto_atBot_iInf | []
| [
174,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
]
|
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.coe_ofBijective | []
| [
447,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
445,
1
]
|
Mathlib/Order/LocallyFinite.lean | Finset.coe_Icc | []
| [
348,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
347,
1
]
|
Mathlib/LinearAlgebra/Pi.lean | LinearMap.coe_single | []
| [
136,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
]
|
Mathlib/Algebra/Associated.lean | Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd | [
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a b : α\nn : ℕ\nhp : Prime p\nhpow : p ^ Nat.succ n ∣ a ^ Nat.succ n * b ^ n\nhb : ¬p ^ 2 ∣ b\nH : p ∣ a ^ Nat.succ n\n⊢ p ∣ a\n\ncase inr\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a b : α\nn : ℕ\nhp : Prime p\nhpow : p ^ Nat.succ n ∣ a ^ Nat.succ n * b ^ n\nhb : ¬p ^ 2 ∣ b\nhbdiv : p ∣ b ^ n\n⊢ p ∣ a",
"state_before": "α : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a b : α\nn : ℕ\nhp : Prime p\nhpow : p ^ Nat.succ n ∣ a ^ Nat.succ n * b ^ n\nhb : ¬p ^ 2 ∣ b\n⊢ p ∣ a",
"tactic": "cases' hp.dvd_or_dvd ((dvd_pow_self p (Nat.succ_ne_zero n)).trans hpow) with H hbdiv"
},
{
"state_after": "case inr.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhpow : p ^ Nat.succ n ∣ a ^ Nat.succ n * (p * x) ^ n\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\n⊢ p ∣ a",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a b : α\nn : ℕ\nhp : Prime p\nhpow : p ^ Nat.succ n ∣ a ^ Nat.succ n * b ^ n\nhb : ¬p ^ 2 ∣ b\nhbdiv : p ∣ b ^ n\n⊢ p ∣ a",
"tactic": "obtain ⟨x, rfl⟩ := hp.dvd_of_dvd_pow hbdiv"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\n⊢ p ∣ a",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhpow : p ^ Nat.succ n ∣ a ^ Nat.succ n * (p * x) ^ n\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\n⊢ p ∣ a",
"tactic": "obtain ⟨y, hy⟩ := hpow"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * x ^ n = p * y\n⊢ p ∣ a",
"state_before": "case inr.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\n⊢ p ∣ a",
"tactic": "have : a ^ n.succ * x ^ n = p * y := by\n refine' mul_left_cancel₀ (pow_ne_zero n hp.ne_zero) _\n rw [← mul_assoc _ p, ← pow_succ', ← hy, mul_pow, ← mul_assoc (a ^ n.succ), mul_comm _ (p ^ n),\n mul_assoc]"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * x ^ n = p * y\nhdvdx : p ∣ x ^ n\n⊢ p ^ 2 ∣ p * x",
"state_before": "case inr.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * x ^ n = p * y\n⊢ p ∣ a",
"tactic": "refine' hp.dvd_of_dvd_pow ((hp.dvd_or_dvd ⟨_, this⟩).resolve_right fun hdvdx => hb _)"
},
{
"state_after": "case inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\ny z : α\nhb : ¬p ^ 2 ∣ p * (p * z)\nhbdiv : p ∣ (p * (p * z)) ^ n\nhy : a ^ Nat.succ n * (p * (p * z)) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * (p * z) ^ n = p * y\nhdvdx : p ∣ (p * z) ^ n\n⊢ p ^ 2 ∣ p * (p * z)",
"state_before": "case inr.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * x ^ n = p * y\nhdvdx : p ∣ x ^ n\n⊢ p ^ 2 ∣ p * x",
"tactic": "obtain ⟨z, rfl⟩ := hp.dvd_of_dvd_pow hdvdx"
},
{
"state_after": "case inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\ny z : α\nhb : ¬p ^ 2 ∣ p * (p * z)\nhbdiv : p ∣ (p * (p * z)) ^ n\nhy : a ^ Nat.succ n * (p * (p * z)) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * (p * z) ^ n = p * y\nhdvdx : p ∣ (p * z) ^ n\n⊢ p * p ∣ p * p * z",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\ny z : α\nhb : ¬p ^ 2 ∣ p * (p * z)\nhbdiv : p ∣ (p * (p * z)) ^ n\nhy : a ^ Nat.succ n * (p * (p * z)) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * (p * z) ^ n = p * y\nhdvdx : p ∣ (p * z) ^ n\n⊢ p ^ 2 ∣ p * (p * z)",
"tactic": "rw [pow_two, ← mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\ny z : α\nhb : ¬p ^ 2 ∣ p * (p * z)\nhbdiv : p ∣ (p * (p * z)) ^ n\nhy : a ^ Nat.succ n * (p * (p * z)) ^ n = p ^ Nat.succ n * y\nthis : a ^ Nat.succ n * (p * z) ^ n = p * y\nhdvdx : p ∣ (p * z) ^ n\n⊢ p * p ∣ p * p * z",
"tactic": "exact dvd_mul_right _ _"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a b : α\nn : ℕ\nhp : Prime p\nhpow : p ^ Nat.succ n ∣ a ^ Nat.succ n * b ^ n\nhb : ¬p ^ 2 ∣ b\nH : p ∣ a ^ Nat.succ n\n⊢ p ∣ a",
"tactic": "exact hp.dvd_of_dvd_pow H"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\n⊢ p ^ n * (a ^ Nat.succ n * x ^ n) = p ^ n * (p * y)",
"state_before": "α : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\n⊢ a ^ Nat.succ n * x ^ n = p * y",
"tactic": "refine' mul_left_cancel₀ (pow_ne_zero n hp.ne_zero) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.44629\nγ : Type ?u.44632\nδ : Type ?u.44635\ninst✝ : CancelCommMonoidWithZero α\np a : α\nn : ℕ\nhp : Prime p\nx : α\nhb : ¬p ^ 2 ∣ p * x\nhbdiv : p ∣ (p * x) ^ n\ny : α\nhy : a ^ Nat.succ n * (p * x) ^ n = p ^ Nat.succ n * y\n⊢ p ^ n * (a ^ Nat.succ n * x ^ n) = p ^ n * (p * y)",
"tactic": "rw [← mul_assoc _ p, ← pow_succ', ← hy, mul_pow, ← mul_assoc (a ^ n.succ), mul_comm _ (p ^ n),\n mul_assoc]"
}
]
| [
143,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
]
|
Mathlib/Analysis/Asymptotics/Theta.lean | Asymptotics.isTheta_zero_right | []
| [
278,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
]
|
Mathlib/Data/MvPolynomial/Supported.lean | MvPolynomial.mem_supported | [
{
"state_after": "σ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ (∃ x, ↑(rename Subtype.val) x = p) ↔ ↑(vars p) ⊆ s",
"state_before": "σ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ p ∈ supported R s ↔ ↑(vars p) ⊆ s",
"tactic": "rw [supported_eq_range_rename, AlgHom.mem_range]"
},
{
"state_after": "case mp\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ (∃ x, ↑(rename Subtype.val) x = p) → ↑(vars p) ⊆ s\n\ncase mpr\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ ↑(vars p) ⊆ s → ∃ x, ↑(rename Subtype.val) x = p",
"state_before": "σ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ (∃ x, ↑(rename Subtype.val) x = p) ↔ ↑(vars p) ⊆ s",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\nq : MvPolynomial σ R\ns t : Set σ\np : MvPolynomial { x // x ∈ s } R\n⊢ ↑(vars (↑(rename Subtype.val) p)) ⊆ s",
"state_before": "case mp\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ (∃ x, ↑(rename Subtype.val) x = p) → ↑(vars p) ⊆ s",
"tactic": "rintro ⟨p, rfl⟩"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\nq : MvPolynomial σ R\ns t : Set σ\np : MvPolynomial { x // x ∈ s } R\n⊢ ↑(Finset.image Subtype.val (vars p)) ⊆ s",
"state_before": "case mp.intro\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\nq : MvPolynomial σ R\ns t : Set σ\np : MvPolynomial { x // x ∈ s } R\n⊢ ↑(vars (↑(rename Subtype.val) p)) ⊆ s",
"tactic": "refine' _root_.trans (Finset.coe_subset.2 (vars_rename _ _)) _"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\nq : MvPolynomial σ R\ns t : Set σ\np : MvPolynomial { x // x ∈ s } R\n⊢ ↑(Finset.image Subtype.val (vars p)) ⊆ s",
"tactic": "simp"
},
{
"state_after": "case mpr\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\nhs : ↑(vars p) ⊆ s\n⊢ ∃ x, ↑(rename Subtype.val) x = p",
"state_before": "case mpr\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ ↑(vars p) ⊆ s → ∃ x, ↑(rename Subtype.val) x = p",
"tactic": "intro hs"
},
{
"state_after": "no goals",
"state_before": "case mpr\nσ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\nhs : ↑(vars p) ⊆ s\n⊢ ∃ x, ↑(rename Subtype.val) x = p",
"tactic": "exact exists_rename_eq_of_vars_subset_range p ((↑) : s → σ) Subtype.val_injective (by simpa)"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nτ : Type ?u.120933\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\nhs : ↑(vars p) ⊆ s\n⊢ ↑(vars p) ⊆ Set.range Subtype.val",
"tactic": "simpa"
}
]
| [
86,
97
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
]
|
Mathlib/Analysis/Convex/Star.lean | StarConvex.preimage_add_left | [
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.72281\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y z : E\ns : Set E\nhs : StarConvex 𝕜 (z + x) s\n⊢ StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.72281\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y z : E\ns : Set E\nhs : StarConvex 𝕜 (x + z) s\n⊢ StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s)",
"tactic": "rw [add_comm] at hs"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.72281\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y z : E\ns : Set E\nhs : StarConvex 𝕜 (z + x) s\n⊢ StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s)",
"tactic": "simpa only [add_comm] using hs.preimage_add_right"
}
]
| [
264,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
]
|
Mathlib/Analysis/SpecificLimits/Basic.lean | tsum_geometric_of_lt_1 | []
| [
197,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
196,
1
]
|
Mathlib/Order/Filter/NAry.lean | Filter.map₂_eq_bot_iff | [
{
"state_after": "α : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\n⊢ (∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)) ↔ ∅ ∈ f ∨ ∅ ∈ g",
"state_before": "α : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\n⊢ map₂ m f g = ⊥ ↔ f = ⊥ ∨ g = ⊥",
"tactic": "simp only [← empty_mem_iff_bot, mem_map₂_iff, subset_empty_iff, image2_eq_empty_iff]"
},
{
"state_after": "case mp\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\n⊢ (∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)) → ∅ ∈ f ∨ ∅ ∈ g\n\ncase mpr\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\n⊢ ∅ ∈ f ∨ ∅ ∈ g → ∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)",
"state_before": "α : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\n⊢ (∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)) ↔ ∅ ∈ f ∨ ∅ ∈ g",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro.inl\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt✝ t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nt : Set β\nht : t ∈ g\nhs : ∅ ∈ f\n⊢ ∅ ∈ f ∨ ∅ ∈ g\n\ncase mp.intro.intro.intro.intro.inr\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\ns : Set α\nhs : s ∈ f\nht : ∅ ∈ g\n⊢ ∅ ∈ f ∨ ∅ ∈ g",
"state_before": "case mp\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\n⊢ (∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)) → ∅ ∈ f ∨ ∅ ∈ g",
"tactic": "rintro ⟨s, t, hs, ht, rfl | rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.inl\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt✝ t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nt : Set β\nht : t ∈ g\nhs : ∅ ∈ f\n⊢ ∅ ∈ f ∨ ∅ ∈ g",
"tactic": "exact Or.inl hs"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.inr\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\ns : Set α\nhs : s ∈ f\nht : ∅ ∈ g\n⊢ ∅ ∈ f ∨ ∅ ∈ g",
"tactic": "exact Or.inr ht"
},
{
"state_after": "case mpr.inl\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh✝ h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nh : ∅ ∈ f\n⊢ ∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)\n\ncase mpr.inr\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh✝ h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nh : ∅ ∈ g\n⊢ ∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)",
"state_before": "case mpr\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\n⊢ ∅ ∈ f ∨ ∅ ∈ g → ∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)",
"tactic": "rintro (h | h)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh✝ h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nh : ∅ ∈ f\n⊢ ∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)",
"tactic": "exact ⟨_, _, h, univ_mem, Or.inl rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr\nα : Type u_2\nα' : Type ?u.22051\nβ : Type u_3\nβ' : Type ?u.22057\nγ : Type u_1\nγ' : Type ?u.22063\nδ : Type ?u.22066\nδ' : Type ?u.22069\nε : Type ?u.22072\nε' : Type ?u.22075\nm : α → β → γ\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nh✝ h₁ h₂ : Filter γ\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nu : Set γ\nv : Set δ\na : α\nb : β\nc : γ\nh : ∅ ∈ g\n⊢ ∃ s t, s ∈ f ∧ t ∈ g ∧ (s = ∅ ∨ t = ∅)",
"tactic": "exact ⟨_, _, univ_mem, h, Or.inr rfl⟩"
}
]
| [
124,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
]
|
Std/Data/String/Lemmas.lean | Substring.ValidFor.foldl | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl m r : List Char\nf : α → Char → α\ninit : α\n⊢ Substring.foldl f init\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } } =\n List.foldl f init m",
"tactic": "simp [-List.append_assoc, Substring.foldl, foldlAux_of_valid]"
}
]
| [
923,
78
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
922,
1
]
|
Mathlib/GroupTheory/Submonoid/Operations.lean | Submonoid.coe_mul | []
| [
679,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
678,
1
]
|
Mathlib/Algebra/GroupPower/Identities.lean | sum_four_sq_mul_sum_four_sq | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\na b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R\n⊢ (x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2) =\n (x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄) ^ 2 + (x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃) ^ 2 +\n (x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂) ^ 2 +\n (x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁) ^ 2",
"tactic": "ring"
}
]
| [
63,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
]
|
Mathlib/Algebra/Ring/Basic.lean | AddMonoidHom.mulRight_apply | []
| [
90,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
]
|
Mathlib/Data/Fin/Basic.lean | Fin.cast_trans | [
{
"state_after": "case h\nn m k : ℕ\nh : n = m\nh' : m = k\ni : Fin n\n⊢ ↑(↑(cast h') (↑(cast h) i)) = ↑(↑(cast (_ : n = k)) i)",
"state_before": "n m k : ℕ\nh : n = m\nh' : m = k\ni : Fin n\n⊢ ↑(cast h') (↑(cast h) i) = ↑(cast (_ : n = k)) i",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nn m k : ℕ\nh : n = m\nh' : m = k\ni : Fin n\n⊢ ↑(↑(cast h') (↑(cast h) i)) = ↑(↑(cast (_ : n = k)) i)",
"tactic": "simp"
}
]
| [
1095,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1092,
1
]
|
Mathlib/Analysis/Normed/MulAction.lean | nnnorm_smul | []
| [
103,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
]
|
Mathlib/Data/PFunctor/Univariate/M.lean | PFunctor.Approx.agree_trival | [
{
"state_after": "no goals",
"state_before": "F : PFunctor\nx : CofixA F 0\ny : CofixA F 1\n⊢ Agree x y",
"tactic": "constructor"
}
]
| [
89,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
]
|
Mathlib/LinearAlgebra/Matrix/ToLin.lean | LinearMap.toMatrix_reindexRange | [
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝⁹ : CommSemiring R\nl : Type ?u.1797707\nm : Type u_4\nn : Type u_5\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_1\nM₂ : Type u_2\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\ninst✝¹ : DecidableEq M₁\ninst✝ : DecidableEq M₂\nf : M₁ →ₗ[R] M₂\nk : m\ni : n\n⊢ ↑(toMatrix (Basis.reindexRange v₁) (Basis.reindexRange v₂)) f\n { val := ↑v₂ k, property := (_ : ↑v₂ k ∈ Set.range ↑v₂) }\n { val := ↑v₁ i, property := (_ : ↑v₁ i ∈ Set.range ↑v₁) } =\n ↑(toMatrix v₁ v₂) f k i",
"tactic": "simp_rw [LinearMap.toMatrix_apply, Basis.reindexRange_self, Basis.reindexRange_repr]"
}
]
| [
641,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
636,
1
]
|
Mathlib/Topology/SubsetProperties.lean | IsIrreducible.isPreirreducible | []
| [
1699,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1698,
1
]
|
Mathlib/Data/Polynomial/Derivative.lean | Polynomial.iterate_derivative_nat_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✝ : Semiring R\nn k : ℕ\nf : R[X]\n⊢ (↑derivative^[k]) (↑n * f) = ↑n * (↑derivative^[k]) f",
"tactic": "induction' k with k ih generalizing f <;> simp [*]"
}
]
| [
363,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
]
|
Mathlib/CategoryTheory/Sites/Grothendieck.lean | CategoryTheory.GrothendieckTopology.arrow_max | [
{
"state_after": "C : Type u\ninst✝ : Category C\nX Y : C\nS✝ R : Sieve X\nJ : GrothendieckTopology C\nf : Y ⟶ X\nS : Sieve X\nhf : S.arrows f\n⊢ ⊤ ∈ sieves J Y",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nS✝ R : Sieve X\nJ : GrothendieckTopology C\nf : Y ⟶ X\nS : Sieve X\nhf : S.arrows f\n⊢ Covers J S f",
"tactic": "rw [Covers, (Sieve.pullback_eq_top_iff_mem f).1 hf]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nS✝ R : Sieve X\nJ : GrothendieckTopology C\nf : Y ⟶ X\nS : Sieve X\nhf : S.arrows f\n⊢ ⊤ ∈ sieves J Y",
"tactic": "apply J.top_mem"
}
]
| [
195,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
]
|
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.interpolate_one | [
{
"state_after": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhs : Finset.Nonempty s\n⊢ ∑ x in s, Lagrange.basis s v x = 1",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhs : Finset.Nonempty s\n⊢ ↑(interpolate s v) 1 = 1",
"tactic": "simp_rw [interpolate_apply, Pi.one_apply, map_one, one_mul]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhs : Finset.Nonempty s\n⊢ ∑ x in s, Lagrange.basis s v x = 1",
"tactic": "exact sum_basis hvs hs"
}
]
| [
320,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
]
|
Mathlib/Logic/Encodable/Basic.lean | Encodable.decode_sum_val | []
| [
300,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
299,
1
]
|
Mathlib/Order/UpperLower/Basic.lean | Set.monotone_mem | []
| [
267,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
]
|
Mathlib/FieldTheory/Minpoly/Basic.lean | minpoly.aeval | [
{
"state_after": "A : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\n⊢ ↑(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\n⊢ ↑(Polynomial.aeval x) (minpoly A x) = 0",
"tactic": "delta minpoly"
},
{
"state_after": "case inl\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ ↑(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0\n\ncase inr\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : ¬IsIntegral A x\n⊢ ↑(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\n⊢ ↑(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0",
"tactic": "split_ifs with hx"
},
{
"state_after": "case inl\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ ↑(Polynomial.aeval x)\n (WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx) =\n 0",
"state_before": "case inl\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ ↑(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0",
"tactic": "rw [dif_pos hx]"
},
{
"state_after": "no goals",
"state_before": "case inl\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : IsIntegral A x\n⊢ ↑(Polynomial.aeval x)\n (WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx) =\n 0",
"tactic": "exact (degree_lt_wf.min_mem _ hx).2"
},
{
"state_after": "case inr\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : ¬IsIntegral A x\n⊢ ↑(Polynomial.aeval x) 0 = 0",
"state_before": "case inr\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : ¬IsIntegral A x\n⊢ ↑(Polynomial.aeval x)\n (if hx : IsIntegral A x then\n WellFounded.min (_ : WellFounded fun p q => degree p < degree q)\n (fun x_1 => Monic x_1 ∧ eval₂ (algebraMap A B) x x_1 = 0) hx\n else 0) =\n 0",
"tactic": "rw [dif_neg hx]"
},
{
"state_after": "no goals",
"state_before": "case inr\nA : Type u_2\nB : Type u_1\nB' : Type ?u.17400\ninst✝⁴ : CommRing A\ninst✝³ : Ring B\ninst✝² : Ring B'\ninst✝¹ : Algebra A B\ninst✝ : Algebra A B'\nx : B\nhx : ¬IsIntegral A x\n⊢ ↑(Polynomial.aeval x) 0 = 0",
"tactic": "exact aeval_zero _"
}
]
| [
91,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
]
|
Mathlib/Analysis/Normed/Order/Lattice.lean | norm_abs_sub_abs | []
| [
187,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
1
]
|
Mathlib/Topology/UniformSpace/Completion.lean | UniformSpace.Completion.uniformContinuous_extension₂ | []
| [
698,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
697,
1
]
|
Mathlib/Data/Matrix/Basic.lean | Matrix.map_id | [
{
"state_after": "case a.h\nl : Type ?u.6402\nm : Type u_1\nn : Type u_2\no : Type ?u.6411\nm' : o → Type ?u.6416\nn' : o → Type ?u.6421\nR : Type ?u.6424\nS : Type ?u.6427\nα : Type v\nβ : Type w\nγ : Type ?u.6434\nM : Matrix m n α\ni✝ : m\nx✝ : n\n⊢ map M id i✝ x✝ = M i✝ x✝",
"state_before": "l : Type ?u.6402\nm : Type u_1\nn : Type u_2\no : Type ?u.6411\nm' : o → Type ?u.6416\nn' : o → Type ?u.6421\nR : Type ?u.6424\nS : Type ?u.6427\nα : Type v\nβ : Type w\nγ : Type ?u.6434\nM : Matrix m n α\n⊢ map M id = M",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.6402\nm : Type u_1\nn : Type u_2\no : Type ?u.6411\nm' : o → Type ?u.6416\nn' : o → Type ?u.6421\nR : Type ?u.6424\nS : Type ?u.6427\nα : Type v\nβ : Type w\nγ : Type ?u.6434\nM : Matrix m n α\ni✝ : m\nx✝ : n\n⊢ map M id i✝ x✝ = M i✝ x✝",
"tactic": "rfl"
}
]
| [
157,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
]
|
Mathlib/Algebra/GroupPower/Order.lean | abs_le_of_sq_le_sq | [
{
"state_after": "no goals",
"state_before": "β : Type ?u.265331\nA : Type ?u.265334\nG : Type ?u.265337\nM : Type ?u.265340\nR : Type u_1\ninst✝ : LinearOrderedRing R\nx y : R\nh : x ^ 2 ≤ y ^ 2\nhy : 0 ≤ y\n⊢ abs x ≤ y",
"tactic": "rwa [← abs_of_nonneg hy, ← sq_le_sq]"
}
]
| [
718,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
717,
1
]
|
Mathlib/Data/Matrix/Basic.lean | Matrix.diagonal_pow | []
| [
1222,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1220,
1
]
|
Mathlib/RingTheory/AlgebraicIndependent.lean | AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.1169857\nR : Type u_2\nK : Type ?u.1169863\nA : Type u_3\nA' : Type ?u.1169869\nA'' : Type ?u.1169872\nV : Type u\nV' : Type ?u.1169877\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing A'\ninst✝³ : CommRing A''\ninst✝² : Algebra R A\ninst✝¹ : Algebra R A'\ninst✝ : Algebra R A''\na b : R\nhx : AlgebraicIndependent R x\ni : ι\n⊢ ↑(mvPolynomialOptionEquivPolynomialAdjoin hx) (X (some i)) = ↑Polynomial.C (↑(aevalEquiv hx) (X i))",
"tactic": "rw [AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply, aeval_X, Option.elim,\n Polynomial.map_C, RingHom.coe_coe]"
}
]
| [
464,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
459,
1
]
|
Mathlib/Algebra/Order/Kleene.lean | kstar_zero | []
| [
257,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
]
|
Mathlib/Data/Set/Pointwise/SMul.lean | Set.smul_set_range | []
| [
460,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
458,
1
]
|
Mathlib/Data/Fin/VecNotation.lean | Matrix.neg_cons | [
{
"state_after": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.86190\nn' : Type ?u.86193\no' : Type ?u.86196\ninst✝ : Neg α\nx : α\nv : Fin n → α\ni : Fin (Nat.succ n)\n⊢ (-vecCons x v) i = vecCons (-x) (-v) i",
"state_before": "α : Type u\nm n o : ℕ\nm' : Type ?u.86190\nn' : Type ?u.86193\no' : Type ?u.86196\ninst✝ : Neg α\nx : α\nv : Fin n → α\n⊢ -vecCons x v = vecCons (-x) (-v)",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nm n o : ℕ\nm' : Type ?u.86190\nn' : Type ?u.86193\no' : Type ?u.86196\ninst✝ : Neg α\nx : α\nv : Fin n → α\ni : Fin (Nat.succ n)\n⊢ (-vecCons x v) i = vecCons (-x) (-v) i",
"tactic": "refine' Fin.cases _ _ i <;> simp"
}
]
| [
583,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
581,
1
]
|
Mathlib/Order/Filter/Bases.lean | Filter.HasBasis.coprod | []
| [
956,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
952,
1
]
|
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | Ordinal.CNF_foldr | [
{
"state_after": "b o : Ordinal\n⊢ foldr (fun p r => b ^ p.fst * p.snd + r) 0 [] = 0",
"state_before": "b o : Ordinal\n⊢ foldr (fun p r => b ^ p.fst * p.snd + r) 0 (CNF b 0) = 0",
"tactic": "rw [CNF_zero]"
},
{
"state_after": "no goals",
"state_before": "b o : Ordinal\n⊢ foldr (fun p r => b ^ p.fst * p.snd + r) 0 [] = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "b o✝ o : Ordinal\nho : o ≠ 0\nIH : foldr (fun p r => b ^ p.fst * p.snd + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o\n⊢ foldr (fun p r => b ^ p.fst * p.snd + r) 0 (CNF b o) = o",
"tactic": "rw [CNF_ne_zero ho, foldr_cons, IH, div_add_mod]"
}
]
| [
119,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
]
|
Mathlib/Analysis/Convex/Function.lean | ConcaveOn.comp | []
| [
141,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
]
|
Mathlib/RingTheory/Ideal/LocalRing.lean | LocalRing.of_isUnit_or_isUnit_of_isUnit_add | []
| [
59,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
57,
1
]
|
Mathlib/CategoryTheory/Generator.lean | CategoryTheory.StructuredArrow.isCoseparating_proj_preimage | [
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nS : D\nT : C ⥤ D\n𝒢 : Set C\nh𝒢 : IsCoseparating 𝒢\nX Y : StructuredArrow S T\nf g : X ⟶ Y\nhfg : ∀ (G : StructuredArrow S T), G ∈ (proj S T).toPrefunctor.obj ⁻¹' 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nG : C\nhG : G ∈ 𝒢\nh : Y.right ⟶ G\n⊢ f.right ≫ h = g.right ≫ h",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nS : D\nT : C ⥤ D\n𝒢 : Set C\nh𝒢 : IsCoseparating 𝒢\n⊢ IsCoseparating ((proj S T).toPrefunctor.obj ⁻¹' 𝒢)",
"tactic": "refine' fun X Y f g hfg => ext _ _ (h𝒢 _ _ fun G hG h => _)"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nS : D\nT : C ⥤ D\n𝒢 : Set C\nh𝒢 : IsCoseparating 𝒢\nX Y : StructuredArrow S T\nf g : X ⟶ Y\nhfg : ∀ (G : StructuredArrow S T), G ∈ (proj S T).toPrefunctor.obj ⁻¹' 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nG : C\nhG : G ∈ 𝒢\nh : Y.right ⟶ G\n⊢ f.right ≫ h = g.right ≫ h",
"tactic": "exact congr_arg CommaMorphism.right (hfg (mk (Y.hom ≫ T.map h)) hG (homMk h rfl))"
}
]
| [
367,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
364,
1
]
|
Std/Data/Array/Lemmas.lean | Array.set_set | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.19284\na : Array α\ni : Fin (size a)\nv v' : α\n⊢ i.val < size (set a i v)",
"tactic": "simp [i.2]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : Array α\ni : Fin (size a)\nv v' : α\n⊢ set (set a i v) { val := i.val, isLt := (_ : i.val < size (set a i v)) } v' = set a i v'",
"tactic": "simp [set, List.set_set]"
}
]
| [
111,
86
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
110,
1
]
|
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.Integrable.edist_toL1_toL1 | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1322600\nδ : Type ?u.1322603\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : α → β\nhf : Integrable f\nhg : Integrable g\n⊢ (∫⁻ (a : α), ↑‖f a - g a‖₊ ∂μ) = ∫⁻ (a : α), edist (f a) (g a) ∂μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1322600\nδ : Type ?u.1322603\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : α → β\nhf : Integrable f\nhg : Integrable g\n⊢ edist (toL1 f hf) (toL1 g hg) = ∫⁻ (a : α), edist (f a) (g a) ∂μ",
"tactic": "simp [Integrable.toL1, snorm, snorm']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1322600\nδ : Type ?u.1322603\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : α → β\nhf : Integrable f\nhg : Integrable g\n⊢ (∫⁻ (a : α), ↑‖f a - g a‖₊ ∂μ) = ∫⁻ (a : α), edist (f a) (g a) ∂μ",
"tactic": "simp [edist_eq_coe_nnnorm_sub]"
}
]
| [
1408,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1405,
1
]
|
Mathlib/Topology/Constructions.lean | isOpen_range_inl | []
| [
926,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
925,
1
]
|
Mathlib/NumberTheory/LucasLehmer.lean | LucasLehmer.mersenne_int_pos | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : 0 < p\n⊢ 1 < 2 ^ p",
"tactic": "exact_mod_cast Nat.one_lt_two_pow p hp"
}
]
| [
101,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
]
|
Mathlib/Data/Set/Intervals/Infinite.lean | Set.Ico_infinite | []
| [
52,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
51,
1
]
|
Mathlib/Data/Set/Basic.lean | Set.inter_compl_nonempty_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u s t : Set α\nx : α\n⊢ x ∈ s ∧ ¬x ∈ t ↔ x ∈ s ∩ tᶜ",
"tactic": "simp [mem_compl]"
}
]
| [
1784,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1783,
1
]
|
Mathlib/RingTheory/Ideal/Basic.lean | Ring.ne_bot_of_isMaximal_of_not_isField | [
{
"state_after": "α : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal M\nnot_field : ¬IsField R\nh : M = ⊥\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal M\nnot_field : ¬IsField R\n⊢ M ≠ ⊥",
"tactic": "rintro h"
},
{
"state_after": "α : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal ⊥\nnot_field : ¬IsField R\nh : M = ⊥\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal M\nnot_field : ¬IsField R\nh : M = ⊥\n⊢ False",
"tactic": "rw [h] at max"
},
{
"state_after": "case mk.intro\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nnot_field : ¬IsField R\nh : M = ⊥\n_h1 : ⊥ ≠ ⊤\nh2 : ∀ (b : Ideal R), ⊥ < b → b = ⊤\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nmax : Ideal.IsMaximal ⊥\nnot_field : ¬IsField R\nh : M = ⊥\n⊢ False",
"tactic": "rcases max with ⟨⟨_h1, h2⟩⟩"
},
{
"state_after": "case mk.intro.intro.intro\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nnot_field : ¬IsField R\nh : M = ⊥\n_h1 : ⊥ ≠ ⊤\nh2 : ∀ (b : Ideal R), ⊥ < b → b = ⊤\nI : Ideal R\nhIbot : ⊥ < I\nhItop : I < ⊤\n⊢ False",
"state_before": "case mk.intro\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nnot_field : ¬IsField R\nh : M = ⊥\n_h1 : ⊥ ≠ ⊤\nh2 : ∀ (b : Ideal R), ⊥ < b → b = ⊤\n⊢ False",
"tactic": "obtain ⟨I, hIbot, hItop⟩ := not_isField_iff_exists_ideal_bot_lt_and_lt_top.mp not_field"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.intro\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nM : Ideal R\nnot_field : ¬IsField R\nh : M = ⊥\n_h1 : ⊥ ≠ ⊤\nh2 : ∀ (b : Ideal R), ⊥ < b → b = ⊤\nI : Ideal R\nhIbot : ⊥ < I\nhItop : I < ⊤\n⊢ False",
"tactic": "exact ne_of_lt hItop (h2 I hIbot)"
}
]
| [
800,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
794,
1
]
|
Mathlib/Analysis/Calculus/Deriv/Add.lean | hasDerivAt_neg | []
| [
251,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
]
|
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.tail_congr | [
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ Computation.LiftRel Equiv ((fun o => Option.recOn o nil Prod.snd) <$> destruct s)\n ((fun o => Option.recOn o nil Prod.snd) <$> destruct t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ tail s ~ʷ tail t",
"tactic": "apply flatten_congr"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ Computation.LiftRel Equiv (Computation.bind (destruct s) (Computation.pure ∘ fun o => Option.rec nil Prod.snd o))\n (Computation.bind (destruct t) (Computation.pure ∘ fun o => Option.rec nil Prod.snd o))",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ Computation.LiftRel Equiv (Computation.map (fun o => Option.rec nil Prod.snd o) (destruct s))\n (Computation.map (fun o => Option.rec nil Prod.snd o) (destruct t))",
"tactic": "rw [← Computation.bind_pure, ← Computation.bind_pure]"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ ∀ {a b : Option (α × WSeq α)},\n BisimO (fun x x_1 => x ~ʷ x_1) a b →\n Computation.LiftRel Equiv ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) a)\n ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) b)",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ Computation.LiftRel Equiv (Computation.bind (destruct s) (Computation.pure ∘ fun o => Option.rec nil Prod.snd o))\n (Computation.bind (destruct t) (Computation.pure ∘ fun o => Option.rec nil Prod.snd o))",
"tactic": "apply liftRel_bind _ _ (destruct_congr h)"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na b : Option (α × WSeq α)\nh : BisimO (fun x x_1 => x ~ʷ x_1) a b\n⊢ Computation.LiftRel Equiv ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) a)\n ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) b)",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ ∀ {a b : Option (α × WSeq α)},\n BisimO (fun x x_1 => x ~ʷ x_1) a b →\n Computation.LiftRel Equiv ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) a)\n ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) b)",
"tactic": "intro a b h"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na b : Option (α × WSeq α)\nh : BisimO (fun x x_1 => x ~ʷ x_1) a b\n⊢ Option.rec nil Prod.snd a ~ʷ Option.rec nil Prod.snd b",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na b : Option (α × WSeq α)\nh : BisimO (fun x x_1 => x ~ʷ x_1) a b\n⊢ Computation.LiftRel Equiv ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) a)\n ((Computation.pure ∘ fun o => Option.rec nil Prod.snd o) b)",
"tactic": "simp [-liftRel_pure_left, -liftRel_pure_right]"
},
{
"state_after": "case a.none.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nh : BisimO (fun x x_1 => x ~ʷ x_1) none none\n⊢ Option.rec nil Prod.snd none ~ʷ Option.rec nil Prod.snd none\n\ncase a.none.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nb : α × WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) none (some b)\n⊢ Option.rec nil Prod.snd none ~ʷ Option.rec nil Prod.snd (some b)\n\ncase a.some.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na : α × WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some a) none\n⊢ Option.rec nil Prod.snd (some a) ~ʷ Option.rec nil Prod.snd none\n\ncase a.some.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na b : α × WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some a) (some b)\n⊢ Option.rec nil Prod.snd (some a) ~ʷ Option.rec nil Prod.snd (some b)",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na b : Option (α × WSeq α)\nh : BisimO (fun x x_1 => x ~ʷ x_1) a b\n⊢ Option.rec nil Prod.snd a ~ʷ Option.rec nil Prod.snd b",
"tactic": "cases' a with a <;> cases' b with b"
},
{
"state_after": "no goals",
"state_before": "case a.none.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nh : BisimO (fun x x_1 => x ~ʷ x_1) none none\n⊢ Option.rec nil Prod.snd none ~ʷ Option.rec nil Prod.snd none",
"tactic": "trivial"
},
{
"state_after": "no goals",
"state_before": "case a.none.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nb : α × WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) none (some b)\n⊢ Option.rec nil Prod.snd none ~ʷ Option.rec nil Prod.snd (some b)",
"tactic": "cases h"
},
{
"state_after": "case a.some.none.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nfst✝ : α\nsnd✝ : WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some (fst✝, snd✝)) none\n⊢ Option.rec nil Prod.snd (some (fst✝, snd✝)) ~ʷ Option.rec nil Prod.snd none",
"state_before": "case a.some.none\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na : α × WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some a) none\n⊢ Option.rec nil Prod.snd (some a) ~ʷ Option.rec nil Prod.snd none",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case a.some.none.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nfst✝ : α\nsnd✝ : WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some (fst✝, snd✝)) none\n⊢ Option.rec nil Prod.snd (some (fst✝, snd✝)) ~ʷ Option.rec nil Prod.snd none",
"tactic": "cases h"
},
{
"state_after": "case a.some.some.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nb : α × WSeq α\na : α\ns' : WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some b)\n⊢ Option.rec nil Prod.snd (some (a, s')) ~ʷ Option.rec nil Prod.snd (some b)",
"state_before": "case a.some.some\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na b : α × WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some a) (some b)\n⊢ Option.rec nil Prod.snd (some a) ~ʷ Option.rec nil Prod.snd (some b)",
"tactic": "cases' a with a s'"
},
{
"state_after": "case a.some.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na : α\ns' : WSeq α\nb : α\nt' : WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some (b, t'))\n⊢ Option.rec nil Prod.snd (some (a, s')) ~ʷ Option.rec nil Prod.snd (some (b, t'))",
"state_before": "case a.some.some.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\nb : α × WSeq α\na : α\ns' : WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some b)\n⊢ Option.rec nil Prod.snd (some (a, s')) ~ʷ Option.rec nil Prod.snd (some b)",
"tactic": "cases' b with b t'"
},
{
"state_after": "no goals",
"state_before": "case a.some.some.mk.mk\nα : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh✝ : s ~ʷ t\na : α\ns' : WSeq α\nb : α\nt' : WSeq α\nh : BisimO (fun x x_1 => x ~ʷ x_1) (some (a, s')) (some (b, t'))\n⊢ Option.rec nil Prod.snd (some (a, s')) ~ʷ Option.rec nil Prod.snd (some (b, t'))",
"tactic": "exact h.right"
}
]
| [
1189,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1176,
1
]
|
Mathlib/Data/Int/Cast/Field.lean | Int.cast_neg_natCast | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.5\nR : Type u_1\ninst✝ : DivisionRing R\nn : ℕ\n⊢ ↑(-↑n) = -↑n",
"tactic": "simp"
}
]
| [
38,
87
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
38,
1
]
|
Mathlib/Algebra/MonoidAlgebra/Basic.lean | MonoidAlgebra.mul_single_apply_aux | [
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.494305\ninst✝¹ : Semiring k\ninst✝ : Mul G\nf : MonoidAlgebra k G\nr : k\nx y z : G\nH : ∀ (a : G), a * x = z ↔ a = y\n⊢ ↑(f * single x r) z = ↑f y * r",
"tactic": "classical exact\n have A :\n ∀ a₁ b₁,\n ((single x r).sum fun a₂ b₂ => ite (a₁ * a₂ = z) (b₁ * b₂) 0) =\n ite (a₁ * x = z) (b₁ * r) 0 :=\n fun a₁ b₁ => sum_single_index <| by simp\n calc\n (HMul.hMul (β := MonoidAlgebra k G) f (single x r)) z =\n sum f fun a b => if a = y then b * r else 0 := by simp only [mul_apply, A, H]\n _ = if y ∈ f.support then f y * r else 0 := (f.support.sum_ite_eq' _ _)\n _ = f y * r := by split_ifs with h <;> simp at h <;> simp [h]"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.494305\ninst✝¹ : Semiring k\ninst✝ : Mul G\nf : MonoidAlgebra k G\nr : k\nx y z : G\nH : ∀ (a : G), a * x = z ↔ a = y\n⊢ ↑(f * single x r) z = ↑f y * r",
"tactic": "exact\nhave A :\n∀ a₁ b₁,\n((single x r).sum fun a₂ b₂ => ite (a₁ * a₂ = z) (b₁ * b₂) 0) =\nite (a₁ * x = z) (b₁ * r) 0 :=\nfun a₁ b₁ => sum_single_index <| by simp\ncalc\n(HMul.hMul (β := MonoidAlgebra k G) f (single x r)) z =\nsum f fun a b => if a = y then b * r else 0 := by simp only [mul_apply, A, H]\n_ = if y ∈ f.support then f y * r else 0 := (f.support.sum_ite_eq' _ _)\n_ = f y * r := by split_ifs with h <;> simp at h <;> simp [h]"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.494305\ninst✝¹ : Semiring k\ninst✝ : Mul G\nf : MonoidAlgebra k G\nr : k\nx y z : G\nH : ∀ (a : G), a * x = z ↔ a = y\na₁ : G\nb₁ : k\n⊢ (if a₁ * x = z then b₁ * 0 else 0) = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.494305\ninst✝¹ : Semiring k\ninst✝ : Mul G\nf : MonoidAlgebra k G\nr : k\nx y z : G\nH : ∀ (a : G), a * x = z ↔ a = y\nA :\n ∀ (a₁ : G) (b₁ : k),\n (sum (single x r) fun a₂ b₂ => if a₁ * a₂ = z then b₁ * b₂ else 0) = if a₁ * x = z then b₁ * r else 0\n⊢ ↑(f * single x r) z = sum f fun a b => if a = y then b * r else 0",
"tactic": "simp only [mul_apply, A, H]"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.494305\ninst✝¹ : Semiring k\ninst✝ : Mul G\nf : MonoidAlgebra k G\nr : k\nx y z : G\nH : ∀ (a : G), a * x = z ↔ a = y\nA :\n ∀ (a₁ : G) (b₁ : k),\n (sum (single x r) fun a₂ b₂ => if a₁ * a₂ = z then b₁ * b₂ else 0) = if a₁ * x = z then b₁ * r else 0\n⊢ (if y ∈ f.support then ↑f y * r else 0) = ↑f y * r",
"tactic": "split_ifs with h <;> simp at h <;> simp [h]"
}
]
| [
553,
70
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
]
|
Mathlib/Analysis/SpecificLimits/Normed.lean | tendsto_pow_atTop_nhds_0_of_norm_lt_1 | [
{
"state_after": "α : Type ?u.130810\nβ : Type ?u.130813\nι : Type ?u.130816\nR : Type u_1\ninst✝ : NormedRing R\nx : R\nh : ‖x‖ < 1\n⊢ Tendsto (fun n => ‖x‖ ^ n) atTop (𝓝 0)",
"state_before": "α : Type ?u.130810\nβ : Type ?u.130813\nι : Type ?u.130816\nR : Type u_1\ninst✝ : NormedRing R\nx : R\nh : ‖x‖ < 1\n⊢ Tendsto (fun n => x ^ n) atTop (𝓝 0)",
"tactic": "apply squeeze_zero_norm' (eventually_norm_pow_le x)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.130810\nβ : Type ?u.130813\nι : Type ?u.130816\nR : Type u_1\ninst✝ : NormedRing R\nx : R\nh : ‖x‖ < 1\n⊢ Tendsto (fun n => ‖x‖ ^ n) atTop (𝓝 0)",
"tactic": "exact tendsto_pow_atTop_nhds_0_of_lt_1 (norm_nonneg _) h"
}
]
| [
271,
59
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
]
|
Mathlib/Data/Finset/Sups.lean | Finset.sups_assoc | []
| [
200,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
]
|
Mathlib/SetTheory/ZFC/Basic.lean | Class.coe_insert | []
| [
1649,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1648,
1
]
|
Mathlib/CategoryTheory/Equivalence.lean | CategoryTheory.Equivalence.funInvIdAssoc_inv_app | [
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\ne : C ≌ D\nF : C ⥤ E\nX : C\n⊢ (𝟙 (F.obj X) ≫ F.map (e.unitIso.hom.app X)) ≫ 𝟙 (F.obj (e.inverse.obj (e.functor.obj X))) = F.map ((unit e).app X)",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\ne : C ≌ D\nF : C ⥤ E\nX : C\n⊢ (funInvIdAssoc e F).inv.app X = F.map ((unit e).app X)",
"tactic": "dsimp [funInvIdAssoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\ne : C ≌ D\nF : C ⥤ E\nX : C\n⊢ (𝟙 (F.obj X) ≫ F.map (e.unitIso.hom.app X)) ≫ 𝟙 (F.obj (e.inverse.obj (e.functor.obj X))) = F.map ((unit e).app X)",
"tactic": "aesop_cat"
}
]
| [
335,
12
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
332,
1
]
|
Mathlib/LinearAlgebra/Matrix/Basis.lean | mul_basis_toMatrix | [
{
"state_after": "ι : Type u_1\nι' : Type u_2\nκ : Type u_5\nκ' : Type ?u.502195\nR : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type ?u.502388\nM₂ : Type ?u.502391\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_6\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Fintype κ\ninst✝³ : Fintype κ'\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nb₁ : Basis ι R M\nb₂ : Basis ι' R M\nb₃ : Basis κ R N\nA : Matrix κ ι R\nthis : ↑(toMatrix b₁ b₃) (↑(toLin b₁ b₃) A) ⬝ Basis.toMatrix b₁ ↑b₂ = ↑(toMatrix b₂ b₃) (↑(toLin b₁ b₃) A)\n⊢ A ⬝ Basis.toMatrix b₁ ↑b₂ = ↑(toMatrix b₂ b₃) (↑(toLin b₁ b₃) A)",
"state_before": "ι : Type u_1\nι' : Type u_2\nκ : Type u_5\nκ' : Type ?u.502195\nR : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type ?u.502388\nM₂ : Type ?u.502391\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_6\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Fintype κ\ninst✝³ : Fintype κ'\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nb₁ : Basis ι R M\nb₂ : Basis ι' R M\nb₃ : Basis κ R N\nA : Matrix κ ι R\n⊢ A ⬝ Basis.toMatrix b₁ ↑b₂ = ↑(toMatrix b₂ b₃) (↑(toLin b₁ b₃) A)",
"tactic": "have := linearMap_toMatrix_mul_basis_toMatrix b₂ b₁ b₃ (Matrix.toLin b₁ b₃ A)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type u_2\nκ : Type u_5\nκ' : Type ?u.502195\nR : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type ?u.502388\nM₂ : Type ?u.502391\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_6\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Fintype κ\ninst✝³ : Fintype κ'\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nb₁ : Basis ι R M\nb₂ : Basis ι' R M\nb₃ : Basis κ R N\nA : Matrix κ ι R\nthis : ↑(toMatrix b₁ b₃) (↑(toLin b₁ b₃) A) ⬝ Basis.toMatrix b₁ ↑b₂ = ↑(toMatrix b₂ b₃) (↑(toLin b₁ b₃) A)\n⊢ A ⬝ Basis.toMatrix b₁ ↑b₂ = ↑(toMatrix b₂ b₃) (↑(toLin b₁ b₃) A)",
"tactic": "rwa [LinearMap.toMatrix_toLin] at this"
}
]
| [
213,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
]
|
Mathlib/GroupTheory/Perm/Basic.lean | Equiv.Perm.ofSubtype_apply_of_mem | []
| [
450,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
449,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_inter_mul_prod_diff | [
{
"state_after": "case h.e'_3.a.h.e\nι : Type ?u.772557\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns t : Finset α\nf : α → β\nx✝ : α\na✝ : x✝ ∈ s\n⊢ f = piecewise t f f",
"state_before": "ι : Type ?u.772557\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns t : Finset α\nf : α → β\n⊢ (∏ x in s ∩ t, f x) * ∏ x in s \\ t, f x = ∏ x in s, f x",
"tactic": "convert (s.prod_piecewise t f f).symm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.a.h.e\nι : Type ?u.772557\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns t : Finset α\nf : α → β\nx✝ : α\na✝ : x✝ ∈ s\n⊢ f = piecewise t f f",
"tactic": "simp [Finset.piecewise]"
}
]
| [
1547,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1544,
1
]
|
Mathlib/Order/LiminfLimsup.lean | Filter.limsup_nat_add | []
| [
814,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
813,
1
]
|
Mathlib/Computability/Primrec.lean | Nat.Primrec.casesOn' | [
{
"state_after": "no goals",
"state_before": "f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nn : ℕ\n⊢ unpaired\n (fun z n =>\n Nat.rec (f z)\n (fun y IH =>\n g\n (Nat.pair (unpair (Nat.pair z (Nat.pair y IH))).fst\n (unpair (unpair (Nat.pair z (Nat.pair y IH))).snd).fst))\n n)\n n =\n unpaired (fun z n => Nat.casesOn n (f z) fun y => g (Nat.pair z y)) n",
"tactic": "simp"
}
]
| [
118,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
]
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | Real.rpow_left_injOn | [
{
"state_after": "x✝ y✝ z✝ x : ℝ\nhx : x ≠ 0\ny : ℝ\nhy : y ∈ {y | 0 ≤ y}\nz : ℝ\nhz : z ∈ {y | 0 ≤ y}\nhyz : y ^ x = z ^ x\n⊢ y = z",
"state_before": "x✝ y z x : ℝ\nhx : x ≠ 0\n⊢ InjOn (fun y => y ^ x) {y | 0 ≤ y}",
"tactic": "rintro y hy z hz (hyz : y ^ x = z ^ x)"
},
{
"state_after": "no goals",
"state_before": "x✝ y✝ z✝ x : ℝ\nhx : x ≠ 0\ny : ℝ\nhy : y ∈ {y | 0 ≤ y}\nz : ℝ\nhz : z ∈ {y | 0 ≤ y}\nhyz : y ^ x = z ^ x\n⊢ y = z",
"tactic": "rw [← rpow_one y, ← rpow_one z, ← _root_.mul_inv_cancel hx, rpow_mul hy, rpow_mul hz, hyz]"
}
]
| [
588,
93
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
586,
1
]
|
Mathlib/Data/Finset/Prod.lean | Finset.product_eq_biUnion | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.18842\ns✝ s' : Finset α\nt✝ t' : Finset β\na : α\nb : β\ninst✝ : DecidableEq (α × β)\ns : Finset α\nt : Finset β\nx✝ : α × β\nx : α\ny : β\n⊢ (x, y) ∈ s ×ˢ t ↔ (x, y) ∈ Finset.biUnion s fun a => image (fun b => (a, b)) t",
"tactic": "simp only [mem_product, mem_biUnion, mem_image, exists_prop, Prod.mk.inj_iff, and_left_comm,\n exists_and_left, exists_eq_right, exists_eq_left]"
}
]
| [
124,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
]
|
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | DiscreteValuationRing.addVal_def' | []
| [
417,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
415,
1
]
|
Mathlib/Data/Real/ENNReal.lean | ENNReal.le_of_forall_pos_le_add | [
{
"state_after": "α : Type ?u.129734\nβ : Type ?u.129737\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : b < a\n⊢ ∃ ε, 0 < ε ∧ b < ⊤ ∧ b + ↑ε < a",
"state_before": "α : Type ?u.129734\nβ : Type ?u.129737\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : ∀ (ε : ℝ≥0), 0 < ε → b < ⊤ → a ≤ b + ↑ε\n⊢ a ≤ b",
"tactic": "contrapose! h"
},
{
"state_after": "case intro.intro\nα : Type ?u.129734\nβ : Type ?u.129737\na b c d : ℝ≥0∞\nr✝ p q : ℝ≥0\nh : b < a\nr : ℝ≥0\nhr0 : 0 < r\nhr : b + ↑r < a\n⊢ ∃ ε, 0 < ε ∧ b < ⊤ ∧ b + ↑ε < a",
"state_before": "α : Type ?u.129734\nβ : Type ?u.129737\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : b < a\n⊢ ∃ ε, 0 < ε ∧ b < ⊤ ∧ b + ↑ε < a",
"tactic": "rcases lt_iff_exists_add_pos_lt.1 h with ⟨r, hr0, hr⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type ?u.129734\nβ : Type ?u.129737\na b c d : ℝ≥0∞\nr✝ p q : ℝ≥0\nh : b < a\nr : ℝ≥0\nhr0 : 0 < r\nhr : b + ↑r < a\n⊢ ∃ ε, 0 < ε ∧ b < ⊤ ∧ b + ↑ε < a",
"tactic": "exact ⟨r, hr0, h.trans_le le_top, hr⟩"
}
]
| [
842,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
839,
1
]
|
Mathlib/Topology/SubsetProperties.lean | isClopen_biInter | []
| [
1607,
86
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1605,
1
]
|
Mathlib/Data/List/Basic.lean | List.insert_neg | []
| [
255,
11
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
]
|
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ioi_subset_Ioo_union_Ici | []
| [
1258,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1257,
1
]
|
Mathlib/Data/Polynomial/FieldDivision.lean | Polynomial.modByMonic_eq_mod | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np✝ q p : R[X]\nhq : Monic q\n⊢ p %ₘ q = p %ₘ (q * ↑C (leadingCoeff q)⁻¹)",
"tactic": "simp only [Monic.def.1 hq, inv_one, mul_one, C_1]"
}
]
| [
202,
101
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
]
|
Mathlib/Probability/ProbabilityMassFunction/Monad.lean | Pmf.toOuterMeasure_bind_apply | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.66368\np : Pmf α\nf : α → Pmf β\ng : β → Pmf γ\ns : Set β\n⊢ ↑(toOuterMeasure (bind p f)) s = ∑' (b : β), if b ∈ s then ∑' (a : α), ↑p a * ↑(f a) b else 0",
"tactic": "simp [toOuterMeasure_apply, Set.indicator_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.66368\np : Pmf α\nf : α → Pmf β\ng : β → Pmf γ\ns : Set β\nb : β\n⊢ (if b ∈ s then ∑' (a : α), ↑p a * ↑(f a) b else 0) = ∑' (a : α), ↑p a * if b ∈ s then ↑(f a) b else 0",
"tactic": "split_ifs <;> simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.66368\np : Pmf α\nf : α → Pmf β\ng : β → Pmf γ\ns : Set β\na : α\nb : β\n⊢ (if b ∈ s then ↑(f a) b else 0) = if b ∈ s then ↑(f a) b else 0",
"tactic": "split_ifs <;> rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.66368\np : Pmf α\nf : α → Pmf β\ng : β → Pmf γ\ns : Set β\na : α\n⊢ (↑p a * ∑' (b : β), if b ∈ s then ↑(f a) b else 0) = ↑p a * ↑(toOuterMeasure (f a)) s",
"tactic": "simp only [toOuterMeasure_apply, Set.indicator_apply]"
}
]
| [
184,
83
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
172,
1
]
|
Mathlib/MeasureTheory/Constructions/Pi.lean | MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Icc | []
| [
511,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
509,
1
]
|
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | Real.Angle.expMapCircle_neg | [
{
"state_after": "case h\nx✝ : ℝ\n⊢ expMapCircle (-↑x✝) = (expMapCircle ↑x✝)⁻¹",
"state_before": "θ : Angle\n⊢ expMapCircle (-θ) = (expMapCircle θ)⁻¹",
"tactic": "induction θ using Real.Angle.induction_on"
},
{
"state_after": "no goals",
"state_before": "case h\nx✝ : ℝ\n⊢ expMapCircle (-↑x✝) = (expMapCircle ↑x✝)⁻¹",
"tactic": "simp_rw [← Real.Angle.coe_neg, Real.Angle.expMapCircle_coe, _root_.expMapCircle_neg]"
}
]
| [
138,
87
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
]
|
Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean | Filter.ZeroAtFilter.smul | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\n𝕜 : Type u_1\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Zero 𝕜\ninst✝² : Zero β\ninst✝¹ : SMulWithZero 𝕜 β\ninst✝ : ContinuousSMul 𝕜 β\nl : Filter α\nf : α → β\nc : 𝕜\nhf : ZeroAtFilter l f\n⊢ ZeroAtFilter l (c • f)",
"tactic": "simpa using hf.const_smul c"
}
]
| [
55,
87
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
]
|
Mathlib/Data/List/MinMax.lean | List.argmin_nil | []
| [
117,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
]
|
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | IsCyclic.exponent_eq_zero_of_infinite | []
| [
594,
86
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
591,
1
]
|
Mathlib/Topology/Sets/Closeds.lean | TopologicalSpace.Clopens.coe_sdiff | []
| [
338,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
9
]
|
Mathlib/LinearAlgebra/Finsupp.lean | mem_span_set | [
{
"state_after": "R : Type u_2\nM : Type u_1\nN : Type ?u.764012\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nm : M\ns : Set M\n⊢ m ∈ span R (_root_.id '' s) ↔ ∃ c, ↑c.support ⊆ s ∧ (Finsupp.sum c fun mi r => r • mi) = m",
"state_before": "R : Type u_2\nM : Type u_1\nN : Type ?u.764012\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nm : M\ns : Set M\n⊢ m ∈ span R s ↔ ∃ c, ↑c.support ⊆ s ∧ (Finsupp.sum c fun mi r => r • mi) = m",
"tactic": "conv_lhs => rw [← Set.image_id s]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\nN : Type ?u.764012\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nm : M\ns : Set M\n⊢ m ∈ span R (_root_.id '' s) ↔ ∃ c, ↑c.support ⊆ s ∧ (Finsupp.sum c fun mi r => r • mi) = m",
"tactic": "exact Finsupp.mem_span_image_iff_total R (v := _root_.id (α := M))"
}
]
| [
1192,
69
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1186,
1
]
|
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.AlgHom.fieldRange_eq_top | []
| [
286,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
284,
1
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.QuasiMeasurePreserving.smul_ae_eq_of_ae_eq | [
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.524934\nβ : Type ?u.524937\nγ : Type ?u.524940\nδ : Type ?u.524943\nι : Type ?u.524946\nR : Type ?u.524949\nR' : Type ?u.524952\nm0 : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t✝ : Set α✝\nμa μa' : Measure α✝\nμb μb' : Measure β\nμc : Measure γ\nf : α✝ → β\nG : Type u_1\nα : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ng : G\nh_qmp : QuasiMeasurePreserving ((fun x x_1 => x • x_1) g⁻¹)\nh_ae_eq : s =ᵐ[μ] t\n⊢ g • s =ᵐ[μ] g • t",
"tactic": "simpa only [← preimage_smul_inv] using h_qmp.ae_eq h_ae_eq"
}
]
| [
2591,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2587,
1
]
|
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | HomogeneousLocalization.ext_iff_val | [
{
"state_after": "case h\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : Submonoid A\ng : HomogeneousLocalization 𝒜 x\na✝ : NumDenSameDeg 𝒜 x\nh : val (Quotient.mk'' a✝) = val g\n⊢ Quotient.mk'' a✝ = g",
"state_before": "ι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : Submonoid A\nf g : HomogeneousLocalization 𝒜 x\nh : val f = val g\n⊢ f = g",
"tactic": "induction f using Quotient.inductionOn'"
},
{
"state_after": "case h.h\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : Submonoid A\na✝¹ a✝ : NumDenSameDeg 𝒜 x\nh : val (Quotient.mk'' a✝¹) = val (Quotient.mk'' a✝)\n⊢ Quotient.mk'' a✝¹ = Quotient.mk'' a✝",
"state_before": "case h\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : Submonoid A\ng : HomogeneousLocalization 𝒜 x\na✝ : NumDenSameDeg 𝒜 x\nh : val (Quotient.mk'' a✝) = val g\n⊢ Quotient.mk'' a✝ = g",
"tactic": "induction g using Quotient.inductionOn'"
},
{
"state_after": "case h.h\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : Submonoid A\na✝¹ a✝ : NumDenSameDeg 𝒜 x\nh : val (Quotient.mk'' a✝¹) = val (Quotient.mk'' a✝)\n⊢ Setoid.r a✝¹ a✝",
"state_before": "case h.h\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : Submonoid A\na✝¹ a✝ : NumDenSameDeg 𝒜 x\nh : val (Quotient.mk'' a✝¹) = val (Quotient.mk'' a✝)\n⊢ Quotient.mk'' a✝¹ = Quotient.mk'' a✝",
"tactic": "rw [Quotient.eq'']"
},
{
"state_after": "no goals",
"state_before": "case h.h\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : DecidableEq ι\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nx : Submonoid A\na✝¹ a✝ : NumDenSameDeg 𝒜 x\nh : val (Quotient.mk'' a✝¹) = val (Quotient.mk'' a✝)\n⊢ Setoid.r a✝¹ a✝",
"tactic": "simpa only [Quotient.liftOn'_mk] using h"
}
]
| [
541,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
535,
1
]
|
Mathlib/Data/Multiset/Bind.lean | Multiset.bind_congr | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.25404\nδ : Type ?u.25407\na : α\ns t : Multiset α\nf✝ g✝ f g : α → Multiset β\nm : Multiset α\n⊢ (∀ (a : α), a ∈ m → f a = g a) → bind m f = bind m g",
"tactic": "simp (config := { contextual := true }) [bind]"
}
]
| [
152,
100
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
]
|
Mathlib/Order/GaloisConnection.lean | GaloisCoinsertion.strictMono_l | []
| [
840,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
839,
1
]
|
Mathlib/FieldTheory/PerfectClosure.lean | injective_pow_p | []
| [
126,
46
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
]
|
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