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|---|---|---|---|---|---|---|
Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.VectorMeasure.measurable_of_not_zero_le_restrict | []
| [
1018,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1017,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/Slope.lean | slope_comm | [
{
"state_after": "no goals",
"state_before": "k : Type u_2\nE : Type u_1\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b : k\n⊢ slope f a b = slope f b a",
"tactic": "rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub]"
}
]
| [
96,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
]
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | HasFDerivAt.ccosh | []
| [
447,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
445,
1
]
|
Mathlib/Data/Finset/Basic.lean | Finset.sdiff_self | []
| [
2087,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2086,
1
]
|
Mathlib/Data/Finset/Pairwise.lean | List.pairwise_iff_coe_toFinset_pairwise | [
{
"state_after": "α : Type u_1\nι : Type ?u.7414\nι' : Type ?u.7417\nβ : Type ?u.7420\ninst✝ : DecidableEq α\nr : α → α → Prop\nl : List α\nhn : Nodup l\nhs : Symmetric r\nthis : IsSymm α r := { symm := hs }\n⊢ Set.Pairwise (↑(toFinset l)) r ↔ Pairwise r l",
"state_before": "α : Type u_1\nι : Type ?u.7414\nι' : Type ?u.7417\nβ : Type ?u.7420\ninst✝ : DecidableEq α\nr : α → α → Prop\nl : List α\nhn : Nodup l\nhs : Symmetric r\n⊢ Set.Pairwise (↑(toFinset l)) r ↔ Pairwise r l",
"tactic": "letI : IsSymm α r := ⟨hs⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.7414\nι' : Type ?u.7417\nβ : Type ?u.7420\ninst✝ : DecidableEq α\nr : α → α → Prop\nl : List α\nhn : Nodup l\nhs : Symmetric r\nthis : IsSymm α r := { symm := hs }\n⊢ Set.Pairwise (↑(toFinset l)) r ↔ Pairwise r l",
"tactic": "rw [coe_toFinset, hn.pairwise_coe]"
}
]
| [
81,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
]
|
Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_mono | []
| [
330,
92
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
1
]
|
Mathlib/Topology/Algebra/Order/Floor.lean | continuousOn_ceil | []
| [
63,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
]
|
Mathlib/Analysis/Calculus/MeanValue.lean | strictConvexOn_of_deriv2_pos' | []
| [
1238,
77
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1236,
1
]
|
Mathlib/Algebra/Periodic.lean | Function.Periodic.sub_antiperiod | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.215561\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh1 : Periodic f c₁\nh2 : Antiperiodic f c₂\n⊢ Antiperiodic f (c₁ - c₂)",
"tactic": "simpa only [sub_eq_add_neg] using h1.add_antiperiod h2.neg"
}
]
| [
523,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
]
|
Mathlib/Combinatorics/Derangements/Basic.lean | derangements.Equiv.RemoveNone.fiber_some | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ fiber (some a) ↔ f ∈ {f | fixedPoints ↑f ⊆ {a}}",
"state_before": "α : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\n⊢ fiber (some a) = {f | fixedPoints ↑f ⊆ {a}}",
"tactic": "ext f"
},
{
"state_after": "case h.mp\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ fiber (some a) → f ∈ {f | fixedPoints ↑f ⊆ {a}}\n\ncase h.mpr\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ {f | fixedPoints ↑f ⊆ {a}} → f ∈ fiber (some a)",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ fiber (some a) ↔ f ∈ {f | fixedPoints ↑f ⊆ {a}}",
"tactic": "constructor"
},
{
"state_after": "case h.mp\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ (∃ F, F ∈ derangements (Option α) ∧ ↑F none = some a ∧ removeNone F = f) → f ∈ {f | fixedPoints ↑f ⊆ {a}}",
"state_before": "case h.mp\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ fiber (some a) → f ∈ {f | fixedPoints ↑f ⊆ {a}}",
"tactic": "rw [RemoveNone.mem_fiber]"
},
{
"state_after": "case h.mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : x ∈ fixedPoints ↑(removeNone F)\n⊢ x ∈ {a}",
"state_before": "case h.mp\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ (∃ F, F ∈ derangements (Option α) ∧ ↑F none = some a ∧ removeNone F = f) → f ∈ {f | fixedPoints ↑f ⊆ {a}}",
"tactic": "rintro ⟨F, F_derangement, F_none, rfl⟩ x x_fixed"
},
{
"state_after": "case h.mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : ↑(removeNone F) x = x\n⊢ x ∈ {a}",
"state_before": "case h.mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : x ∈ fixedPoints ↑(removeNone F)\n⊢ x ∈ {a}",
"tactic": "rw [mem_fixedPoints_iff] at x_fixed"
},
{
"state_after": "case h.mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\n⊢ x ∈ {a}",
"state_before": "case h.mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : ↑(removeNone F) x = x\n⊢ x ∈ {a}",
"tactic": "apply_fun some at x_fixed"
},
{
"state_after": "case h.mp.intro.intro.intro.none\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\nFx : ↑F (some x) = none\n⊢ x ∈ {a}\n\ncase h.mp.intro.intro.intro.some\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\ny : α\nFx : ↑F (some x) = some y\n⊢ x ∈ {a}",
"state_before": "case h.mp.intro.intro.intro\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\n⊢ x ∈ {a}",
"tactic": "cases' Fx : F (some x) with y"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro.intro.none\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\nFx : ↑F (some x) = none\n⊢ x ∈ {a}",
"tactic": "rwa [removeNone_none F Fx, F_none, Option.some_inj, eq_comm] at x_fixed"
},
{
"state_after": "case h.mp.intro.intro.intro.some.h\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\ny : α\nFx : ↑F (some x) = some y\n⊢ False",
"state_before": "case h.mp.intro.intro.intro.some\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\ny : α\nFx : ↑F (some x) = some y\n⊢ x ∈ {a}",
"tactic": "exfalso"
},
{
"state_after": "case h.mp.intro.intro.intro.some.h\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : ↑F (some x) = some x\ny : α\nFx : ↑F (some x) = some y\n⊢ False",
"state_before": "case h.mp.intro.intro.intro.some.h\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : some (↑(removeNone F) x) = some x\ny : α\nFx : ↑F (some x) = some y\n⊢ False",
"tactic": "rw [removeNone_some F ⟨y, Fx⟩] at x_fixed"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro.intro.some.h\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nF : Perm (Option α)\nF_derangement : F ∈ derangements (Option α)\nF_none : ↑F none = some a\nx : α\nx_fixed : ↑F (some x) = some x\ny : α\nFx : ↑F (some x) = some y\n⊢ False",
"tactic": "exact F_derangement _ x_fixed"
},
{
"state_after": "case h.mpr\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ f ∈ fiber (some a)",
"state_before": "case h.mpr\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\n⊢ f ∈ {f | fixedPoints ↑f ⊆ {a}} → f ∈ fiber (some a)",
"tactic": "intro h_opfp"
},
{
"state_after": "case h.mpr\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ ↑Perm.decomposeOption.symm (some a, f) ∈ derangements (Option α) ∧\n ↑Perm.decomposeOption (↑Perm.decomposeOption.symm (some a, f)) = (some a, f)",
"state_before": "case h.mpr\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ f ∈ fiber (some a)",
"tactic": "use Equiv.Perm.decomposeOption.symm (some a, f)"
},
{
"state_after": "case h.mpr.left\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ ↑Perm.decomposeOption.symm (some a, f) ∈ derangements (Option α)\n\ncase h.mpr.right\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ ↑Perm.decomposeOption (↑Perm.decomposeOption.symm (some a, f)) = (some a, f)",
"state_before": "case h.mpr\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ ↑Perm.decomposeOption.symm (some a, f) ∈ derangements (Option α) ∧\n ↑Perm.decomposeOption (↑Perm.decomposeOption.symm (some a, f)) = (some a, f)",
"tactic": "constructor"
},
{
"state_after": "case h.mpr.left\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : Option α\n⊢ ↑(↑Perm.decomposeOption.symm (some a, f)) x ≠ x",
"state_before": "case h.mpr.left\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ ↑Perm.decomposeOption.symm (some a, f) ∈ derangements (Option α)",
"tactic": "intro x"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : Option α\n⊢ (fun x => ↑(Equiv.swap none (some a)) x) (↑(↑Perm.decomposeOption.symm (some a, f)) x) ≠\n (fun x => ↑(Equiv.swap none (some a)) x) x",
"state_before": "case h.mpr.left\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : Option α\n⊢ ↑(↑Perm.decomposeOption.symm (some a, f)) x ≠ x",
"tactic": "apply_fun fun x => Equiv.swap none (some a) x"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : Option α\n⊢ ↑(Equiv.swap none (some a)) ((↑(Equiv.swap none (some a)) ∘ ↑(optionCongr f)) x) ≠ ↑(Equiv.swap none (some a)) x",
"state_before": "α : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : Option α\n⊢ (fun x => ↑(Equiv.swap none (some a)) x) (↑(↑Perm.decomposeOption.symm (some a, f)) x) ≠\n (fun x => ↑(Equiv.swap none (some a)) x) x",
"tactic": "simp only [Perm.decomposeOption_symm_apply, swap_apply_self, Perm.coe_mul]"
},
{
"state_after": "case none\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\n⊢ ↑(Equiv.swap none (some a)) ((↑(Equiv.swap none (some a)) ∘ ↑(optionCongr f)) none) ≠ ↑(Equiv.swap none (some a)) none\n\ncase some\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\n⊢ ↑(Equiv.swap none (some a)) ((↑(Equiv.swap none (some a)) ∘ ↑(optionCongr f)) (some x)) ≠\n ↑(Equiv.swap none (some a)) (some x)",
"state_before": "α : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : Option α\n⊢ ↑(Equiv.swap none (some a)) ((↑(Equiv.swap none (some a)) ∘ ↑(optionCongr f)) x) ≠ ↑(Equiv.swap none (some a)) x",
"tactic": "cases' x with x"
},
{
"state_after": "case some\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"state_before": "case some\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\n⊢ ↑(Equiv.swap none (some a)) ((↑(Equiv.swap none (some a)) ∘ ↑(optionCongr f)) (some x)) ≠\n ↑(Equiv.swap none (some a)) (some x)",
"tactic": "simp only [comp, optionCongr_apply, Option.map_some', swap_apply_self]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : x = a\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)\n\ncase neg\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : ¬x = a\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"state_before": "case some\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"tactic": "by_cases x_vs_a : x = a"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : ¬x = a\nne_1 : some x ≠ none\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : ¬x = a\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"tactic": "have ne_1 : some x ≠ none := Option.some_ne_none _"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : ¬x = a\nne_1 : some x ≠ none\nne_2 : some x ≠ some a\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : ¬x = a\nne_1 : some x ≠ none\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"tactic": "have ne_2 : some x ≠ some a := (Option.some_injective α).ne_iff.mpr x_vs_a"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : ¬x = a\nne_1 : some x ≠ none\nne_2 : some x ≠ some a\n⊢ ↑f x ≠ x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.81438\ninst✝ : DecidableEq α\na : α\nf : Perm α\nh_opfp : f ∈ {f | fixedPoints ↑f ⊆ {a}}\nx : α\nx_vs_a : ¬x = a\nne_1 : some x ≠ none\nne_2 : some x ≠ some a\n⊢ some (↑f x) ≠ ↑(Equiv.swap none (some a)) (some x)",
"tactic": "rw [swap_apply_of_ne_of_ne ne_1 ne_2, (Option.some_injective α).ne_iff]"
},
{
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{
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172,
28
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
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Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigO.mul_isLittleO | [
{
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"tactic": "simp only [IsLittleO_def] at *"
},
{
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{
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"tactic": "exact (hc'.mul (h₂ (div_pos cpos c'pos))).congr_const (mul_div_cancel' _ (ne_of_gt c'pos))"
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1596,
93
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/RingTheory/AlgebraicIndependent.lean | algebraicIndependent_iUnion_of_directed | [
{
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"tactic": "refine' algebraicIndependent_of_finite (⋃ i, s i) fun t ht ft => _"
},
{
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"tactic": "rcases finite_subset_iUnion ft ht with ⟨I, fi, hI⟩"
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"tactic": "rcases hs.finset_le fi.toFinset with ⟨i, hi⟩"
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"tactic": "exact (h i).mono (Subset.trans hI <| iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))"
}
]
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334,
95
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.finset_sum_apply | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.122815\nγ : Type ?u.122818\nδ : Type ?u.122821\nι : Type u_2\nR : Type ?u.122827\nR' : Type ?u.122830\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\nm : MeasurableSpace α\nI : Finset ι\nμ : ι → Measure α\ns : Set α\n⊢ ↑↑(∑ i in I, μ i) s = ∑ i in I, ↑↑(μ i) s",
"tactic": "rw [coe_finset_sum, Finset.sum_apply]"
}
]
| [
881,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
880,
1
]
|
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.mem_iInf | [
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\nN : Type ?u.19040\nP : Type ?u.19043\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ : Substructure L M\nι : Sort u_1\nS : ι → Substructure L M\nx : M\n⊢ (x ∈ ⨅ (i : ι), S i) ↔ ∀ (i : ι), x ∈ S i",
"tactic": "simp only [iInf, mem_sInf, Set.forall_range_iff]"
}
]
| [
227,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
226,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | AffineMap.snd_lineMap | []
| [
619,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
618,
1
]
|
Mathlib/Topology/Algebra/FilterBasis.lean | GroupFilterBasis.one | []
| [
100,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
]
|
Mathlib/RingTheory/Algebraic.lean | isAlgebraic_algebraMap_of_isAlgebraic | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type u_1\nA : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : Algebra R S\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : S\nx✝ : IsAlgebraic R a\nf : R[X]\nhf₁ : f ≠ 0\nhf₂ : ↑(aeval a) f = 0\n⊢ ↑(aeval (↑(algebraMap S A) a)) f = 0",
"tactic": "rw [aeval_algebraMap_apply, hf₂, map_zero]"
}
]
| [
144,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
]
|
Mathlib/Order/CompleteLattice.lean | Equiv.iSup_comp | []
| [
645,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
644,
1
]
|
Mathlib/Data/Matrix/Notation.lean | Matrix.row_cons | [
{
"state_after": "case a.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nx : α\nu : Fin m → α\ni✝ : Unit\nx✝ : Fin (Nat.succ m)\n⊢ row (vecCons x u) i✝ x✝ = vecCons x u x✝",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nx : α\nu : Fin m → α\n⊢ row (vecCons x u) = fun x_1 => vecCons x u",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nx : α\nu : Fin m → α\ni✝ : Unit\nx✝ : Fin (Nat.succ m)\n⊢ row (vecCons x u) i✝ x✝ = vecCons x u x✝",
"tactic": "rfl"
}
]
| [
187,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
1
]
|
Mathlib/Logic/Basic.lean | forall_eq_apply_imp_iff' | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.19831\nα : Sort ?u.19836\nκ : ι → Sort ?u.19833\np✝ q : α → Prop\nβ : Sort u_1\nf : α → β\np : β → Prop\n⊢ (∀ (b : β) (a : α), b = f a → p b) ↔ ∀ (a : α), p (f a)",
"tactic": "simp [forall_swap]"
}
]
| [
815,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
814,
9
]
|
Mathlib/Algebra/Order/Group/Abs.lean | eq_of_abs_sub_eq_zero | []
| [
352,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
]
|
Mathlib/Data/Sym/Sym2.lean | Sym2.other_mem | [
{
"state_after": "case h.e'_5\nα : Type u_1\nβ : Type ?u.35146\nγ : Type ?u.35149\na : α\nz : Sym2 α\nh : a ∈ z\n⊢ z = Quotient.mk (Rel.setoid α) (a, Mem.other h)",
"state_before": "α : Type u_1\nβ : Type ?u.35146\nγ : Type ?u.35149\na : α\nz : Sym2 α\nh : a ∈ z\n⊢ Mem.other h ∈ z",
"tactic": "convert mem_mk''_right a <| Mem.other h"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nα : Type u_1\nβ : Type ?u.35146\nγ : Type ?u.35149\na : α\nz : Sym2 α\nh : a ∈ z\n⊢ z = Quotient.mk (Rel.setoid α) (a, Mem.other h)",
"tactic": "rw [other_spec h]"
}
]
| [
371,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
]
|
Mathlib/LinearAlgebra/Dual.lean | Subspace.dualLift_of_mem | [
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nW : Subspace K V\nφ : Module.Dual K { x // x ∈ W }\nw : V\nhw : w ∈ W\n⊢ ↑(↑(dualLift W) φ) w = ↑φ { val := w, property := hw }",
"tactic": "convert dualLift_of_subtype ⟨w, hw⟩"
}
]
| [
1026,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1025,
1
]
|
Mathlib/Topology/Algebra/Order/Field.lean | TopologicalRing.of_norm | [
{
"state_after": "case h0\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\n⊢ ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ TopologicalRing R",
"state_before": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\n⊢ TopologicalRing R",
"tactic": "have h0 : ∀ f : R → R, ∀ c ≥ (0 : 𝕜), (∀ x, norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)"
},
{
"state_after": "case hmul\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)\n\ncase hmul_left\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)\n\ncase hmul_right\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)",
"state_before": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ TopologicalRing R",
"tactic": "apply TopologicalRing.of_addGroup_of_nhds_zero"
},
{
"state_after": "case hmul_left\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)\n\ncase hmul_right\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)",
"state_before": "case hmul\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)\n\ncase hmul_left\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)\n\ncase hmul_right\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)",
"tactic": "case hmul =>\n refine ((nhds_basis.prod nhds_basis).tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_\n refine ⟨(1, ε), ⟨one_pos, ε0⟩, fun (x, y) ⟨hx, hy⟩ => ?_⟩\n simp only [sub_zero] at *\n calc norm (x * y) ≤ norm x * norm y := norm_mul_le _ _\n _ < ε := mul_lt_of_le_one_of_lt_of_nonneg hx.le hy (norm_nonneg _)"
},
{
"state_after": "case hmul_right\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)",
"state_before": "case hmul_left\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)\n\ncase hmul_right\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)",
"tactic": "case hmul_left => exact fun x => h0 _ (norm x) (norm_nonneg _) (norm_mul_le x)"
},
{
"state_after": "case h0\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nf : R → R\nc : 𝕜\nc0 : c ≥ 0\nhf : ∀ (x : R), norm (f x) ≤ c * norm x\nε : 𝕜\nε0 : 0 < ε\n⊢ ∃ ia, 0 < ia ∧ ∀ (x : R), x ∈ {x | norm x < ia} → f x ∈ {x | norm x < ε}",
"state_before": "case h0\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\n⊢ ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)",
"tactic": "refine fun f c c0 hf ↦ (nhds_basis.tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_"
},
{
"state_after": "case h0.intro.intro\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nf : R → R\nc : 𝕜\nc0 : c ≥ 0\nhf : ∀ (x : R), norm (f x) ≤ c * norm x\nε : 𝕜\nε0 : 0 < ε\nδ : 𝕜\nδ0 : 0 < δ\nhδ : c * δ < ε\n⊢ ∃ ia, 0 < ia ∧ ∀ (x : R), x ∈ {x | norm x < ia} → f x ∈ {x | norm x < ε}",
"state_before": "case h0\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nf : R → R\nc : 𝕜\nc0 : c ≥ 0\nhf : ∀ (x : R), norm (f x) ≤ c * norm x\nε : 𝕜\nε0 : 0 < ε\n⊢ ∃ ia, 0 < ia ∧ ∀ (x : R), x ∈ {x | norm x < ia} → f x ∈ {x | norm x < ε}",
"tactic": "rcases exists_pos_mul_lt ε0 c with ⟨δ, δ0, hδ⟩"
},
{
"state_after": "case h0.intro.intro\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nf : R → R\nc : 𝕜\nc0 : c ≥ 0\nhf : ∀ (x : R), norm (f x) ≤ c * norm x\nε : 𝕜\nε0 : 0 < ε\nδ : 𝕜\nδ0 : 0 < δ\nhδ : c * δ < ε\nx : R\nhx : x ∈ {x | norm x < δ}\n⊢ c * norm x < ε",
"state_before": "case h0.intro.intro\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nf : R → R\nc : 𝕜\nc0 : c ≥ 0\nhf : ∀ (x : R), norm (f x) ≤ c * norm x\nε : 𝕜\nε0 : 0 < ε\nδ : 𝕜\nδ0 : 0 < δ\nhδ : c * δ < ε\n⊢ ∃ ia, 0 < ia ∧ ∀ (x : R), x ∈ {x | norm x < ia} → f x ∈ {x | norm x < ε}",
"tactic": "refine ⟨δ, δ0, fun x hx ↦ (hf _).trans_lt ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case h0.intro.intro\nR : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nf : R → R\nc : 𝕜\nc0 : c ≥ 0\nhf : ∀ (x : R), norm (f x) ≤ c * norm x\nε : 𝕜\nε0 : 0 < ε\nδ : 𝕜\nδ0 : 0 < δ\nhδ : c * δ < ε\nx : R\nhx : x ∈ {x | norm x < δ}\n⊢ c * norm x < ε",
"tactic": "exact (mul_le_mul_of_nonneg_left (le_of_lt hx) c0).trans_lt hδ"
},
{
"state_after": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\nε : 𝕜\nε0 : 0 < ε\n⊢ ∃ ia,\n (0 < ia.fst ∧ 0 < ia.snd) ∧\n ∀ (x : R × R),\n x ∈ {x | norm x < ia.fst} ×ˢ {x | norm x < ia.snd} → uncurry (fun x x_1 => x * x_1) x ∈ {x | norm x < ε}",
"state_before": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)",
"tactic": "refine ((nhds_basis.prod nhds_basis).tendsto_iff nhds_basis).2 fun ε ε0 ↦ ?_"
},
{
"state_after": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\nε : 𝕜\nε0 : 0 < ε\nx✝¹ : R × R\nx y : R\nx✝ : (x, y) ∈ {x | norm x < (1, ε).fst} ×ˢ {x | norm x < (1, ε).snd}\nhx : (x, y).fst ∈ {x | norm x < (1, ε).fst}\nhy : (x, y).snd ∈ {x | norm x < (1, ε).snd}\n⊢ uncurry (fun x x_1 => x * x_1) (x, y) ∈ {x | norm x < ε}",
"state_before": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\nε : 𝕜\nε0 : 0 < ε\n⊢ ∃ ia,\n (0 < ia.fst ∧ 0 < ia.snd) ∧\n ∀ (x : R × R),\n x ∈ {x | norm x < ia.fst} ×ˢ {x | norm x < ia.snd} → uncurry (fun x x_1 => x * x_1) x ∈ {x | norm x < ε}",
"tactic": "refine ⟨(1, ε), ⟨one_pos, ε0⟩, fun (x, y) ⟨hx, hy⟩ => ?_⟩"
},
{
"state_after": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\nε : 𝕜\nε0 : 0 < ε\nx✝¹ : R × R\nx y : R\nx✝ : (x, y) ∈ {x | norm x < 1} ×ˢ {x | norm x < ε}\nhx : x ∈ {x | norm x < 1}\nhy : y ∈ {x | norm x < ε}\n⊢ uncurry (fun x x_1 => x * x_1) (x, y) ∈ {x | norm x < ε}",
"state_before": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\nε : 𝕜\nε0 : 0 < ε\nx✝¹ : R × R\nx y : R\nx✝ : (x, y) ∈ {x | norm x < (1, ε).fst} ×ˢ {x | norm x < (1, ε).snd}\nhx : (x, y).fst ∈ {x | norm x < (1, ε).fst}\nhy : (x, y).snd ∈ {x | norm x < (1, ε).snd}\n⊢ uncurry (fun x x_1 => x * x_1) (x, y) ∈ {x | norm x < ε}",
"tactic": "simp only [sub_zero] at *"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\nε : 𝕜\nε0 : 0 < ε\nx✝¹ : R × R\nx y : R\nx✝ : (x, y) ∈ {x | norm x < 1} ×ˢ {x | norm x < ε}\nhx : x ∈ {x | norm x < 1}\nhy : y ∈ {x | norm x < ε}\n⊢ uncurry (fun x x_1 => x * x_1) (x, y) ∈ {x | norm x < ε}",
"tactic": "calc norm (x * y) ≤ norm x * norm y := norm_mul_le _ _\n_ < ε := mul_lt_of_le_one_of_lt_of_nonneg hx.le hy (norm_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\n𝕜 : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nnorm : R → 𝕜\nnorm_nonneg : ∀ (x : R), 0 ≤ norm x\nnorm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y\nnhds_basis : HasBasis (𝓝 0) (fun x => 0 < x) fun ε => {x | norm x < ε}\nh0 : ∀ (f : R → R) (c : 𝕜), c ≥ 0 → (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)\n⊢ ∀ (x₀ : R), Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)",
"tactic": "exact fun x => h0 _ (norm x) (norm_nonneg _) (norm_mul_le x)"
}
]
| [
52,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
32,
1
]
|
Mathlib/Algebra/Star/Order.lean | conjugate_nonneg' | [
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : NonUnitalSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : StarOrderedRing R\na : R\nha : 0 ≤ a\nc : R\n⊢ 0 ≤ c * a * star c",
"tactic": "simpa only [star_star] using conjugate_nonneg ha (star c)"
}
]
| [
178,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
]
|
Mathlib/GroupTheory/GroupAction/Defs.lean | smul_sub | [
{
"state_after": "no goals",
"state_before": "M : Type u_2\nN : Type ?u.62983\nG : Type ?u.62986\nA : Type u_1\nB : Type ?u.62992\nα : Type ?u.62995\nβ : Type ?u.62998\nγ : Type ?u.63001\nδ : Type ?u.63004\ninst✝² : Monoid M\ninst✝¹ : AddGroup A\ninst✝ : DistribMulAction M A\nr : M\nx y : A\n⊢ r • (x - y) = r • x - r • y",
"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg, smul_add, smul_neg]"
}
]
| [
956,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
955,
1
]
|
Mathlib/Init/Algebra/Order.lean | ne_of_lt | []
| [
105,
88
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
]
|
Mathlib/Data/PEquiv.lean | PEquiv.self_trans_symm | [
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ a✝ ∈ ↑(PEquiv.trans f (PEquiv.symm f)) x✝ ↔ a✝ ∈ ↑(ofSet {a | isSome (↑f a) = true}) x✝",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\n⊢ PEquiv.trans f (PEquiv.symm f) = ofSet {a | isSome (↑f a) = true}",
"tactic": "ext"
},
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ a✝ ∈ Option.bind (↑f x✝) ↑(PEquiv.symm f) ↔ a✝ ∈ ↑(ofSet {a | isSome (↑f a) = true}) x✝",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ a✝ ∈ ↑(PEquiv.trans f (PEquiv.symm f)) x✝ ↔ a✝ ∈ ↑(ofSet {a | isSome (↑f a) = true}) x✝",
"tactic": "dsimp [PEquiv.trans]"
},
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ (∃ a, ↑f x✝ = some a ∧ ↑f a✝ = some a) ↔ a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1}",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ a✝ ∈ Option.bind (↑f x✝) ↑(PEquiv.symm f) ↔ a✝ ∈ ↑(ofSet {a | isSome (↑f a) = true}) x✝",
"tactic": "simp only [eq_some_iff f, Option.isSome_iff_exists, Option.mem_def, bind_eq_some',\n ofSet_eq_some_iff]"
},
{
"state_after": "case h.a.mp\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ (∃ a, ↑f x✝ = some a ∧ ↑f a✝ = some a) → a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1}\n\ncase h.a.mpr\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1} → ∃ a, ↑f x✝ = some a ∧ ↑f a✝ = some a",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ (∃ a, ↑f x✝ = some a ∧ ↑f a✝ = some a) ↔ a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1}",
"tactic": "constructor"
},
{
"state_after": "case h.a.mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\nb : β\nhb₁ : ↑f x✝ = some b\nhb₂ : ↑f a✝ = some b\n⊢ a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1}",
"state_before": "case h.a.mp\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ (∃ a, ↑f x✝ = some a ∧ ↑f a✝ = some a) → a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1}",
"tactic": "rintro ⟨b, hb₁, hb₂⟩"
},
{
"state_after": "no goals",
"state_before": "case h.a.mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\nb : β\nhb₁ : ↑f x✝ = some b\nhb₂ : ↑f a✝ = some b\n⊢ a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1}",
"tactic": "exact ⟨PEquiv.inj _ hb₂ hb₁, b, hb₂⟩"
},
{
"state_after": "no goals",
"state_before": "case h.a.mpr\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\nx✝ a✝ : α\n⊢ a✝ = x✝ ∧ a✝ ∈ {a | ∃ a_1, ↑f a = some a_1} → ∃ a, ↑f x✝ = some a ∧ ↑f a✝ = some a",
"tactic": "simp (config := { contextual := true })"
}
]
| [
281,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
]
|
Mathlib/Algebra/Ring/BooleanRing.lean | ofBoolAlg_symmDiff | [
{
"state_after": "α : Type u_1\nβ : Type ?u.38685\nγ : Type ?u.38688\ninst✝² : BooleanRing α\ninst✝¹ : BooleanRing β\ninst✝ : BooleanRing γ\na b : AsBoolAlg α\n⊢ ↑ofBoolAlg ((a ⊔ b) \\ (a ⊓ b)) = ↑ofBoolAlg a + ↑ofBoolAlg b",
"state_before": "α : Type u_1\nβ : Type ?u.38685\nγ : Type ?u.38688\ninst✝² : BooleanRing α\ninst✝¹ : BooleanRing β\ninst✝ : BooleanRing γ\na b : AsBoolAlg α\n⊢ ↑ofBoolAlg (a ∆ b) = ↑ofBoolAlg a + ↑ofBoolAlg b",
"tactic": "rw [symmDiff_eq_sup_sdiff_inf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.38685\nγ : Type ?u.38688\ninst✝² : BooleanRing α\ninst✝¹ : BooleanRing β\ninst✝ : BooleanRing γ\na b : AsBoolAlg α\n⊢ ↑ofBoolAlg ((a ⊔ b) \\ (a ⊓ b)) = ↑ofBoolAlg a + ↑ofBoolAlg b",
"tactic": "exact of_boolalg_symm_diff_aux _ _"
}
]
| [
321,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
319,
1
]
|
Mathlib/Algebra/Order/ToIntervalMod.lean | iUnion_Icc_int_cast | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedRing α\ninst✝ : Archimedean α\na : α\n⊢ (⋃ (n : ℤ), Icc (↑n) (↑n + 1)) = univ",
"tactic": "simpa only [zero_add] using iUnion_Icc_add_int_cast (0 : α)"
}
]
| [
1118,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1117,
1
]
|
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | BilinForm.toQuadraticForm_multiset_sum | []
| [
702,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
700,
1
]
|
Mathlib/GroupTheory/Exponent.lean | Monoid.exponent_eq_zero_iff | [
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝ : Monoid G\n⊢ exponent G = 0 ↔ ¬ExponentExists G",
"tactic": "simp only [exponentExists_iff_ne_zero, Classical.not_not]"
}
]
| [
92,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
]
|
Mathlib/Algebra/CharP/Basic.lean | Int.cast_injOn_of_ringChar_ne_two | [
{
"state_after": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na : ℤ\nha : a ∈ {0, 1, -1}\nb : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\n⊢ a = b",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\n⊢ Set.InjOn Int.cast {0, 1, -1}",
"tactic": "intro a ha b hb h"
},
{
"state_after": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na : ℤ\nha : a ∈ {0, 1, -1}\nb : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\n⊢ a - b = 0",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na : ℤ\nha : a ∈ {0, 1, -1}\nb : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\n⊢ a = b",
"tactic": "apply eq_of_sub_eq_zero"
},
{
"state_after": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na : ℤ\nha : a ∈ {0, 1, -1}\nb : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\nhf : ¬a - b = 0\n⊢ False",
"state_before": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na : ℤ\nha : a ∈ {0, 1, -1}\nb : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\n⊢ a - b = 0",
"tactic": "by_contra hf"
},
{
"state_after": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\n⊢ False",
"state_before": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na : ℤ\nha : a ∈ {0, 1, -1}\nb : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\nhf : ¬a - b = 0\n⊢ False",
"tactic": "replace ha : a = 0 ∨ a = 1 ∨ a = -1 := ha"
},
{
"state_after": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\n⊢ False",
"state_before": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nhb : b ∈ {0, 1, -1}\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\n⊢ False",
"tactic": "replace hb : b = 0 ∨ b = 1 ∨ b = -1 := hb"
},
{
"state_after": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n⊢ False",
"state_before": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\n⊢ False",
"tactic": "have hh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2 := by\n rcases ha with (ha | ha | ha) <;> rcases hb with (hb | hb | hb)\n pick_goal 5\n pick_goal 9\n iterate 3 rw [ha, hb, sub_self] at hf; tauto\n all_goals rw [ha, hb]; norm_num"
},
{
"state_after": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\nh' : ↑(a - b) = 0\n⊢ False",
"state_before": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n⊢ False",
"tactic": "have h' : ((a - b : ℤ) : R) = 0 := by exact_mod_cast sub_eq_zero_of_eq h"
},
{
"state_after": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\n⊢ False",
"state_before": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\nh' : ↑(a - b) = 0\n⊢ False",
"tactic": "have h'' : ((b - a : ℤ) : R) = 0 := by exact_mod_cast sub_eq_zero_of_eq h.symm"
},
{
"state_after": "case h.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 1\n⊢ False\n\ncase h.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : b - a = 1\n⊢ False\n\ncase h.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 2\n⊢ False\n\ncase h.inr.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : b - a = 2\n⊢ False",
"state_before": "case h\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\n⊢ False",
"tactic": "rcases hh with (hh | hh | hh | hh)"
},
{
"state_after": "case inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2",
"tactic": "rcases ha with (ha | ha | ha) <;> rcases hb with (hb | hb | hb)"
},
{
"state_after": "case inr.inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2",
"state_before": "case inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2",
"tactic": "pick_goal 5"
},
{
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{
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"tactic": "all_goals rw [ha, hb]; norm_num"
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"tactic": "rw [ha, hb, sub_self] at hf"
},
{
"state_after": "case inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2",
"state_before": "case inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬0 = 0\nha : a = 0\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inl.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 1\nhb : b = -1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 0\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n\ncase inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2",
"tactic": "tauto"
},
{
"state_after": "case inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ -1 - 1 = 1 ∨ 1 - -1 = 1 ∨ -1 - 1 = 2 ∨ 1 - -1 = 2",
"state_before": "case inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2",
"tactic": "rw [ha, hb]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = -1\nhb : b = 1\n⊢ -1 - 1 = 1 ∨ 1 - -1 = 1 ∨ -1 - 1 = 2 ∨ 1 - -1 = 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\n⊢ ↑(a - b) = 0",
"tactic": "exact_mod_cast sub_eq_zero_of_eq h"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nhh : a - b = 1 ∨ b - a = 1 ∨ a - b = 2 ∨ b - a = 2\nh' : ↑(a - b) = 0\n⊢ ↑(b - a) = 0",
"tactic": "exact_mod_cast sub_eq_zero_of_eq h.symm"
},
{
"state_after": "case h.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : 1 = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 1\n⊢ False",
"state_before": "case h.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 1\n⊢ False",
"tactic": "rw [hh, (by norm_cast : ((1 : ℤ) : R) = 1)] at h'"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : 1 = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 1\n⊢ False",
"tactic": "exact one_ne_zero h'"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑1 = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 1\n⊢ ↑1 = 1",
"tactic": "norm_cast"
},
{
"state_after": "case h.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : 1 = 0\nhh : b - a = 1\n⊢ False",
"state_before": "case h.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : b - a = 1\n⊢ False",
"tactic": "rw [hh, (by norm_cast : ((1 : ℤ) : R) = 1)] at h''"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : 1 = 0\nhh : b - a = 1\n⊢ False",
"tactic": "exact one_ne_zero h''"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑1 = 0\nhh : b - a = 1\n⊢ ↑1 = 1",
"tactic": "norm_cast"
},
{
"state_after": "case h.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : 2 = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 2\n⊢ False",
"state_before": "case h.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 2\n⊢ False",
"tactic": "rw [hh, (by norm_cast : ((2 : ℤ) : R) = 2)] at h'"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inr.inl\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : 2 = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 2\n⊢ False",
"tactic": "exact Ring.two_ne_zero hR h'"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑2 = 0\nh'' : ↑(b - a) = 0\nhh : a - b = 2\n⊢ ↑2 = 2",
"tactic": "norm_cast"
},
{
"state_after": "case h.inr.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : 2 = 0\nhh : b - a = 2\n⊢ False",
"state_before": "case h.inr.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑(b - a) = 0\nhh : b - a = 2\n⊢ False",
"tactic": "rw [hh, (by norm_cast : ((2 : ℤ) : R) = 2)] at h''"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inr.inr\nR✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : 2 = 0\nhh : b - a = 2\n⊢ False",
"tactic": "exact Ring.two_ne_zero hR h''"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.964532\nR : Type u_1\ninst✝¹ : NonAssocRing R\ninst✝ : Nontrivial R\nhR : ringChar R ≠ 2\na b : ℤ\nh : ↑a = ↑b\nhf : ¬a - b = 0\nha : a = 0 ∨ a = 1 ∨ a = -1\nhb : b = 0 ∨ b = 1 ∨ b = -1\nh' : ↑(a - b) = 0\nh'' : ↑2 = 0\nhh : b - a = 2\n⊢ ↑2 = 2",
"tactic": "norm_cast"
}
]
| [
735,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
710,
1
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|
Mathlib/Computability/Primrec.lean | Primrec.of_equiv_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ✝ : Type ?u.27775\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β✝\ninst✝ : Primcodable σ\nβ : Type u_1\ne : β ≃ α\nf : σ → β\nthis : Primcodable β := Primcodable.ofEquiv α e\nh : Primrec fun a => ↑e (f a)\na : σ\n⊢ ↑e.symm (↑e (f a)) = f a",
"tactic": "simp"
}
]
| [
310,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
306,
1
]
|
Mathlib/Data/Set/Intervals/OrderIso.lean | OrderIso.image_Iio | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na : α\n⊢ ↑e '' Iio a = Iio (↑e a)",
"tactic": "rw [e.image_eq_preimage, e.symm.preimage_Iio, e.symm_symm]"
}
]
| [
81,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
]
|
Mathlib/Topology/UniformSpace/Cauchy.lean | completeSpace_of_isComplete_univ | []
| [
401,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
400,
1
]
|
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.eq_X_mul_shift_add_const | [
{
"state_after": "case h.zero\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\n⊢ ↑(coeff R Nat.zero) φ = ↑(coeff R Nat.zero) ((X * mk fun p => ↑(coeff R (p + 1)) φ) + ↑(C R) (↑(constantCoeff R) φ))\n\ncase h.succ\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\n⊢ ↑(coeff R (Nat.succ n)) φ =\n ↑(coeff R (Nat.succ n)) ((X * mk fun p => ↑(coeff R (p + 1)) φ) + ↑(C R) (↑(constantCoeff R) φ))",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\n⊢ φ = (X * mk fun p => ↑(coeff R (p + 1)) φ) + ↑(C R) (↑(constantCoeff R) φ)",
"tactic": "ext (_ | n)"
},
{
"state_after": "no goals",
"state_before": "case h.zero\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\n⊢ ↑(coeff R Nat.zero) φ = ↑(coeff R Nat.zero) ((X * mk fun p => ↑(coeff R (p + 1)) φ) + ↑(C R) (↑(constantCoeff R) φ))",
"tactic": "simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X,\n zero_mul, coeff_zero_C, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case h.succ\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\n⊢ ↑(coeff R (Nat.succ n)) φ =\n ↑(coeff R (Nat.succ n)) ((X * mk fun p => ↑(coeff R (p + 1)) φ) + ↑(C R) (↑(constantCoeff R) φ))",
"tactic": "simp only [coeff_succ_X_mul, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero,\n if_false, add_zero]"
}
]
| [
1663,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1657,
1
]
|
Mathlib/Data/Set/Intervals/WithBotTop.lean | WithTop.preimage_coe_Ioc | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ some ⁻¹' Ioc ↑a ↑b = Ioc a b",
"tactic": "simp [← Ioi_inter_Iic]"
}
]
| [
68,
101
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
]
|
Mathlib/LinearAlgebra/Determinant.lean | LinearEquiv.det_trans | []
| [
396,
16
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
394,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Images.lean | CategoryTheory.Limits.ImageMap.factor_map | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nm : ImageMap sq\n⊢ (factorThruImage f.hom ≫ m.map) ≫ image.ι g.hom = (sq.left ≫ factorThruImage g.hom) ≫ image.ι g.hom",
"tactic": "simp"
}
]
| [
699,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
697,
1
]
|
Mathlib/Data/Nat/Basic.lean | Nat.pred_eq_sub_one | []
| [
308,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
]
|
Mathlib/Data/List/Perm.lean | List.Perm.bagInter_left | [
{
"state_after": "case nil\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\nt₁ t₂ : List α\np : t₁ ~ t₂\n⊢ List.bagInter [] t₁ = List.bagInter [] t₂\n\ncase cons\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\na : α\nl : List α\nIH : ∀ {t₁ t₂ : List α}, t₁ ~ t₂ → List.bagInter l t₁ = List.bagInter l t₂\nt₁ t₂ : List α\np : t₁ ~ t₂\n⊢ List.bagInter (a :: l) t₁ = List.bagInter (a :: l) t₂",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl t₁ t₂ : List α\np : t₁ ~ t₂\n⊢ List.bagInter l t₁ = List.bagInter l t₂",
"tactic": "induction' l with a l IH generalizing t₁ t₂ p"
},
{
"state_after": "case pos\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\na : α\nl : List α\nIH : ∀ {t₁ t₂ : List α}, t₁ ~ t₂ → List.bagInter l t₁ = List.bagInter l t₂\nt₁ t₂ : List α\np : t₁ ~ t₂\nh : a ∈ t₁\n⊢ List.bagInter (a :: l) t₁ = List.bagInter (a :: l) t₂\n\ncase neg\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\na : α\nl : List α\nIH : ∀ {t₁ t₂ : List α}, t₁ ~ t₂ → List.bagInter l t₁ = List.bagInter l t₂\nt₁ t₂ : List α\np : t₁ ~ t₂\nh : ¬a ∈ t₁\n⊢ List.bagInter (a :: l) t₁ = List.bagInter (a :: l) t₂",
"state_before": "case cons\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\na : α\nl : List α\nIH : ∀ {t₁ t₂ : List α}, t₁ ~ t₂ → List.bagInter l t₁ = List.bagInter l t₂\nt₁ t₂ : List α\np : t₁ ~ t₂\n⊢ List.bagInter (a :: l) t₁ = List.bagInter (a :: l) t₂",
"tactic": "by_cases h : a ∈ t₁"
},
{
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"state_before": "case nil\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\nt₁ t₂ : List α\np : t₁ ~ t₂\n⊢ List.bagInter [] t₁ = List.bagInter [] t₂",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\na : α\nl : List α\nIH : ∀ {t₁ t₂ : List α}, t₁ ~ t₂ → List.bagInter l t₁ = List.bagInter l t₂\nt₁ t₂ : List α\np : t₁ ~ t₂\nh : a ∈ t₁\n⊢ List.bagInter (a :: l) t₁ = List.bagInter (a :: l) t₂",
"tactic": "simp [h, p.subset h, IH (p.erase _)]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁✝ t₂✝ : List α\np✝ : t₁✝ ~ t₂✝\na : α\nl : List α\nIH : ∀ {t₁ t₂ : List α}, t₁ ~ t₂ → List.bagInter l t₁ = List.bagInter l t₂\nt₁ t₂ : List α\np : t₁ ~ t₂\nh : ¬a ∈ t₁\n⊢ List.bagInter (a :: l) t₁ = List.bagInter (a :: l) t₂",
"tactic": "simp [h, mt p.mem_iff.2 h, IH p]"
}
]
| [
853,
37
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
848,
1
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|
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Theory.model_singleton_iff | [
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.522501\nP : Type ?u.522504\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nT : Theory L\nφ : Sentence L\n⊢ M ⊨ {φ} ↔ M ⊨ φ",
"tactic": "simp"
}
]
| [
858,
87
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
858,
1
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|
Mathlib/Topology/UniformSpace/UniformConvergence.lean | tendstoLocallyUniformlyOn_iff_forall_tendsto | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nx✝³ : α\nx✝² : x✝³ ∈ s\nx✝¹ : Set (β × β)\nx✝ : x✝¹ ∈ 𝓤 β\n⊢ (∃ t, t ∈ 𝓝[s] x✝³ ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ x✝¹) ↔\n ∃ t, t ∈ 𝓝[s] x✝³ ∧ ∀ᶠ (x : ι) in p, ∀ (y : α), y ∈ t → (x, y) ∈ (fun y => (f y.snd, F y.fst y.snd)) ⁻¹' x✝¹",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nx✝³ : α\nx✝² : x✝³ ∈ s\nx✝¹ : Set (β × β)\nx✝ : x✝¹ ∈ 𝓤 β\n⊢ (∃ t, t ∈ 𝓝[s] x✝³ ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ x✝¹) ↔\n x✝¹ ∈ map (fun y => (f y.snd, F y.fst y.snd)) (p ×ˢ 𝓝[s] x✝³)",
"tactic": "rw [mem_map, mem_prod_iff_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nx✝³ : α\nx✝² : x✝³ ∈ s\nx✝¹ : Set (β × β)\nx✝ : x✝¹ ∈ 𝓤 β\n⊢ (∃ t, t ∈ 𝓝[s] x✝³ ∧ ∀ᶠ (n : ι) in p, ∀ (y : α), y ∈ t → (f y, F n y) ∈ x✝¹) ↔\n ∃ t, t ∈ 𝓝[s] x✝³ ∧ ∀ᶠ (x : ι) in p, ∀ (y : α), y ∈ t → (x, y) ∈ (fun y => (f y.snd, F y.fst y.snd)) ⁻¹' x✝¹",
"tactic": "rfl"
}
]
| [
616,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
1
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|
Mathlib/Analysis/Calculus/Deriv/Add.lean | HasDerivAt.sub | []
| [
306,
12
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
8
]
|
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.erase_same | [
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝ : (i : ι) → Zero (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni✝ i : ι\nf : Π₀ (i : ι), β i\n⊢ ↑(erase i f) i = 0",
"tactic": "simp"
}
]
| [
753,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
753,
1
]
|
Mathlib/Data/Multiset/Nodup.lean | Multiset.nodup_iff_ne_cons_cons | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1277\nγ : Type ?u.1280\nr : α → α → Prop\ns✝ t✝ : Multiset α\na✝ : α\ns : Multiset α\nh : ∀ (a : α) (t : Multiset α), s ≠ a ::ₘ a ::ₘ t\na : α\nle : a ::ₘ a ::ₘ 0 ≤ s\nt : Multiset α\ns_eq : s = a ::ₘ a ::ₘ 0 + t\n⊢ s = a ::ₘ a ::ₘ t",
"tactic": "rwa [cons_add, cons_add, zero_add] at s_eq"
}
]
| [
82,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
]
|
Mathlib/RingTheory/DedekindDomain/Ideal.lean | FractionalIdeal.mul_left_le_iff | [
{
"state_after": "case h.e'_1.a\nR : Type ?u.702098\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ J * I ≤ J * I' ↔ I * J ≤ I' * J",
"state_before": "R : Type ?u.702098\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ J * I ≤ J * I' ↔ I ≤ I'",
"tactic": "convert mul_right_le_iff hJ using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1.a\nR : Type ?u.702098\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nJ : FractionalIdeal A⁰ K\nhJ : J ≠ 0\nI I' : FractionalIdeal A⁰ K\n⊢ J * I ≤ J * I' ↔ I * J ≤ I' * J",
"tactic": "simp only [mul_comm]"
}
]
| [
562,
92
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
561,
1
]
|
Mathlib/Order/SymmDiff.lean | bihimp_himp_left | []
| [
603,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
602,
1
]
|
Mathlib/SetTheory/ZFC/Basic.lean | PSet.nonempty_def | []
| [
360,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
1
]
|
Mathlib/Algebra/Order/Module.lean | BddAbove.smul_of_nonpos | []
| [
218,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
217,
1
]
|
Mathlib/Data/W/Cardinal.lean | WType.cardinal_mk_le_of_le | [
{
"state_after": "case h\nα : Type u\nβ : α → Type u\nκ : Cardinal\nhκ✝ : (sum fun a => κ ^ (#β a)) ≤ κ\nγ : Type u\nhκ : (sum fun a => (#γ) ^ (#β a)) ≤ (#γ)\n⊢ (#WType β) ≤ (#γ)",
"state_before": "α : Type u\nβ : α → Type u\nκ : Cardinal\nhκ : (sum fun a => κ ^ (#β a)) ≤ κ\n⊢ (#WType β) ≤ κ",
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},
{
"state_after": "case h\nα : Type u\nβ : α → Type u\nκ : Cardinal\nhκ✝ : (sum fun a => κ ^ (#β a)) ≤ κ\nγ : Type u\nhκ : Nonempty ((i : α) × (β i → γ) ↪ γ)\n⊢ (#WType β) ≤ (#γ)",
"state_before": "case h\nα : Type u\nβ : α → Type u\nκ : Cardinal\nhκ✝ : (sum fun a => κ ^ (#β a)) ≤ κ\nγ : Type u\nhκ : (sum fun a => (#γ) ^ (#β a)) ≤ (#γ)\n⊢ (#WType β) ≤ (#γ)",
"tactic": "simp only [Cardinal.power_def, ← Cardinal.mk_sigma, Cardinal.le_def] at hκ"
},
{
"state_after": "case h.intro\nα : Type u\nβ : α → Type u\nκ : Cardinal\nhκ✝ : (sum fun a => κ ^ (#β a)) ≤ κ\nγ : Type u\nhκ : (i : α) × (β i → γ) ↪ γ\n⊢ (#WType β) ≤ (#γ)",
"state_before": "case h\nα : Type u\nβ : α → Type u\nκ : Cardinal\nhκ✝ : (sum fun a => κ ^ (#β a)) ≤ κ\nγ : Type u\nhκ : Nonempty ((i : α) × (β i → γ) ↪ γ)\n⊢ (#WType β) ≤ (#γ)",
"tactic": "cases' hκ with hκ"
},
{
"state_after": "no goals",
"state_before": "case h.intro\nα : Type u\nβ : α → Type u\nκ : Cardinal\nhκ✝ : (sum fun a => κ ^ (#β a)) ≤ κ\nγ : Type u\nhκ : (i : α) × (β i → γ) ↪ γ\n⊢ (#WType β) ≤ (#γ)",
"tactic": "exact Cardinal.mk_le_of_injective (elim_injective _ hκ.1 hκ.2)"
}
]
| [
55,
65
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
]
|
Mathlib/RingTheory/RingInvo.lean | RingInvo.coe_ringEquiv | []
| [
110,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
]
|
Mathlib/Topology/Semicontinuous.lean | ContinuousOn.upperSemicontinuousOn | []
| [
820,
82
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
819,
1
]
|
Mathlib/Tactic/Ring/Basic.lean | Mathlib.Tactic.Ring.add_congr | [
{
"state_after": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na' b' : R\n⊢ a' + b' = a' + b'",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na a' b b' c : R\nx✝² : a = a'\nx✝¹ : b = b'\nx✝ : a' + b' = c\n⊢ a + b = c",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na' b' : R\n⊢ a' + b' = a' + b'",
"tactic": "rfl"
}
]
| [
962,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
961,
1
]
|
Mathlib/CategoryTheory/Sites/CoverLifting.lean | CategoryTheory.RanIsSheafOfCoverLifting.gluedLimitCone_π_app | []
| [
214,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
]
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Complex.sin_pi_div_two | [
{
"state_after": "⊢ sin (↑π / 2) = sin ↑(π / 2)",
"state_before": "⊢ sin (↑π / 2) = ↑(Real.sin (π / 2))",
"tactic": "rw [ofReal_sin]"
},
{
"state_after": "no goals",
"state_before": "⊢ sin (↑π / 2) = sin ↑(π / 2)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "⊢ ↑(Real.sin (π / 2)) = 1",
"tactic": "simp"
}
]
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1118,
21
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1115,
1
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|
Mathlib/GroupTheory/Congruence.lean | Con.lift_apply_mk' | [
{
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"tactic": "rw [c.eq.2 h]"
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{
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"tactic": "ext x"
},
{
"state_after": "case h.mk\nM : Type u_1\nN : Type ?u.60876\nP : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ : M →* P\nf : Con.Quotient c →* P\nx : Con.Quotient c\na✝ : M\n⊢ ↑(lift c (MonoidHom.comp f (mk' c)) (_ : ∀ (x y : M), ↑c x y → ↑f ↑x = ↑f ↑y)) (Quot.mk r a✝) = ↑f (Quot.mk r a✝)",
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"tactic": "rcases x with ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nM : Type u_1\nN : Type ?u.60876\nP : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ : M →* P\nf : Con.Quotient c →* P\nx : Con.Quotient c\na✝ : M\n⊢ ↑(lift c (MonoidHom.comp f (mk' c)) (_ : ∀ (x y : M), ↑c x y → ↑f ↑x = ↑f ↑y)) (Quot.mk r a✝) = ↑f (Quot.mk r a✝)",
"tactic": "rfl"
}
]
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956,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
954,
1
]
|
Mathlib/Data/Set/Lattice.lean | Set.image2_iUnion₂_left | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort ?u.249089\nι₂ : Sort ?u.249092\nκ : ι → Sort u_5\nκ₁ : ι → Sort ?u.249102\nκ₂ : ι → Sort ?u.249107\nκ' : ι' → Sort ?u.249112\nf : α → β → γ\ns✝ : Set α\nt✝ : Set β\ns : (i : ι) → κ i → Set α\nt : Set β\n⊢ image2 f (⋃ (i : ι) (j : κ i), s i j) t = ⋃ (i : ι) (j : κ i), image2 f (s i j) t",
"tactic": "simp_rw [image2_iUnion_left]"
}
]
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1887,
101
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1886,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | CategoryTheory.Limits.ι_colimitConstInitial_hom | [
{
"state_after": "case w\nC✝ : Type u₁\ninst✝³ : Category C✝\nJ : Type u_1\ninst✝² : Category J\nC : Type u_3\ninst✝¹ : Category C\ninst✝ : HasInitial C\nj : J\n⊢ ∀ (j_1 : Discrete PEmpty),\n colimit.ι (Functor.empty C) j_1 ≫ colimit.ι ((Functor.const J).obj (⊥_ C)) j ≫ colimitConstInitial.hom =\n colimit.ι (Functor.empty C) j_1 ≫ initial.to (⊥_ C)",
"state_before": "C✝ : Type u₁\ninst✝³ : Category C✝\nJ : Type u_1\ninst✝² : Category J\nC : Type u_3\ninst✝¹ : Category C\ninst✝ : HasInitial C\nj : J\n⊢ colimit.ι ((Functor.const J).obj (⊥_ C)) j ≫ colimitConstInitial.hom = initial.to (⊥_ C)",
"tactic": "apply Limits.colimit.hom_ext"
},
{
"state_after": "no goals",
"state_before": "case w\nC✝ : Type u₁\ninst✝³ : Category C✝\nJ : Type u_1\ninst✝² : Category J\nC : Type u_3\ninst✝¹ : Category C\ninst✝ : HasInitial C\nj : J\n⊢ ∀ (j_1 : Discrete PEmpty),\n colimit.ι (Functor.empty C) j_1 ≫ colimit.ι ((Functor.const J).obj (⊥_ C)) j ≫ colimitConstInitial.hom =\n colimit.ι (Functor.empty C) j_1 ≫ initial.to (⊥_ C)",
"tactic": "aesop_cat"
}
]
| [
494,
65
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
491,
1
]
|
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.pow_fst | []
| [
323,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
]
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero | [
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x ≠ 0 ∨ y = 0\n⊢ ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y = 0\n⊢ ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖",
"tactic": "rw [← neg_eq_zero, ← inner_neg_right] at h"
},
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x ≠ 0 ∨ -y = 0\n⊢ ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x ≠ 0 ∨ y = 0\n⊢ ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖",
"tactic": "rw [← neg_eq_zero] at h0"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x ≠ 0 ∨ -y = 0\n⊢ ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖",
"tactic": "rw [sub_eq_add_neg, norm_div_cos_angle_add_of_inner_eq_zero h h0]"
}
]
| [
336,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
332,
1
]
|
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Ioc_eq_empty_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1802\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\n⊢ Ioc a b = ∅ ↔ ¬a < b",
"tactic": "rw [← coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff]"
}
]
| [
80,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | CategoryTheory.Limits.coequalizer.existsUnique | []
| [
1011,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1009,
1
]
|
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | HasFDerivAt.rpow_const | []
| [
459,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
457,
1
]
|
Mathlib/Data/Nat/Choose/Factorization.lean | Nat.factorization_choose_le_log | [
{
"state_after": "case pos\np n k : ℕ\nh : ↑(factorization (choose n k)) p = 0\n⊢ ↑(factorization (choose n k)) p ≤ log p n\n\ncase neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"state_before": "p n k : ℕ\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"tactic": "by_cases h : (choose n k).factorization p = 0"
},
{
"state_after": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"state_before": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"tactic": "have hp : p.Prime := Not.imp_symm (choose n k).factorization_eq_zero_of_non_prime h"
},
{
"state_after": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhkn : k ≤ n\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"state_before": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"tactic": "have hkn : k ≤ n := by\n refine' le_of_not_lt fun hnk => h _\n simp [choose_eq_zero_of_lt hnk]"
},
{
"state_after": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhkn : k ≤ n\n⊢ Part.get (multiplicity p (choose n k)) (_ : multiplicity.Finite p (choose n k)) ≤ log p n",
"state_before": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhkn : k ≤ n\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"tactic": "rw [factorization_def _ hp, @padicValNat_def _ ⟨hp⟩ _ (choose_pos hkn)]"
},
{
"state_after": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhkn : k ≤ n\n⊢ Finset.card (Finset.filter (fun i => p ^ i ≤ k % p ^ i + (n - k) % p ^ i) (Finset.Ico 1 (log p n + 1))) ≤ log p n",
"state_before": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhkn : k ≤ n\n⊢ Part.get (multiplicity p (choose n k)) (_ : multiplicity.Finite p (choose n k)) ≤ log p n",
"tactic": "simp only [hp.multiplicity_choose hkn (lt_add_one _), PartENat.get_natCast]"
},
{
"state_after": "no goals",
"state_before": "case neg\np n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhkn : k ≤ n\n⊢ Finset.card (Finset.filter (fun i => p ^ i ≤ k % p ^ i + (n - k) % p ^ i) (Finset.Ico 1 (log p n + 1))) ≤ log p n",
"tactic": "refine (Finset.card_filter_le _ _).trans (le_of_eq (Nat.card_Ico _ _))"
},
{
"state_after": "no goals",
"state_before": "case pos\np n k : ℕ\nh : ↑(factorization (choose n k)) p = 0\n⊢ ↑(factorization (choose n k)) p ≤ log p n",
"tactic": "simp [h]"
},
{
"state_after": "p n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhnk : n < k\n⊢ ↑(factorization (choose n k)) p = 0",
"state_before": "p n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\n⊢ k ≤ n",
"tactic": "refine' le_of_not_lt fun hnk => h _"
},
{
"state_after": "no goals",
"state_before": "p n k : ℕ\nh : ¬↑(factorization (choose n k)) p = 0\nhp : Prime p\nhnk : n < k\n⊢ ↑(factorization (choose n k)) p = 0",
"tactic": "simp [choose_eq_zero_of_lt hnk]"
}
]
| [
50,
73
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
]
|
Mathlib/Data/Stream/Init.lean | Stream'.tail_inits | [
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (initsCore [head s] (tail s)) = initsCore [head s, head (tail s)] (tail (tail s))",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (inits s) = initsCore [head s, head (tail s)] (tail (tail s))",
"tactic": "unfold inits"
},
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail ([head s] :: initsCore ([head s] ++ [head (tail s)]) (tail (tail s))) =\n initsCore [head s, head (tail s)] (tail (tail s))",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (initsCore [head s] (tail s)) = initsCore [head s, head (tail s)] (tail (tail s))",
"tactic": "rw [inits_core_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail ([head s] :: initsCore ([head s] ++ [head (tail s)]) (tail (tail s))) =\n initsCore [head s, head (tail s)] (tail (tail s))",
"tactic": "rfl"
}
]
| [
694,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
691,
1
]
|
Mathlib/Data/Nat/Cast/Basic.lean | Nat.cast_tsub | [
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nm n : ℕ\nh : m ≤ n\n⊢ ↑(m - n) = ↑m - ↑n\n\ncase inr\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nm n : ℕ\nh : n ≤ m\n⊢ ↑(m - n) = ↑m - ↑n",
"state_before": "α : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nm n : ℕ\n⊢ ↑(m - n) = ↑m - ↑n",
"tactic": "cases' le_total m n with h h"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nm n : ℕ\nh : m ≤ n\n⊢ ↑m ≤ ↑n",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nm n : ℕ\nh : m ≤ n\n⊢ ↑(m - n) = ↑m - ↑n",
"tactic": "rw [tsub_eq_zero_of_le h, cast_zero, tsub_eq_zero_of_le]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nm n : ℕ\nh : m ≤ n\n⊢ ↑m ≤ ↑n",
"tactic": "exact mono_cast h"
},
{
"state_after": "case inr.intro\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn m : ℕ\nh : n ≤ m + n\n⊢ ↑(m + n - n) = ↑(m + n) - ↑n",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nm n : ℕ\nh : n ≤ m\n⊢ ↑(m - n) = ↑m - ↑n",
"tactic": "rcases le_iff_exists_add'.mp h with ⟨m, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.21410\ninst✝³ : CanonicallyOrderedCommSemiring α\ninst✝² : Sub α\ninst✝¹ : OrderedSub α\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nn m : ℕ\nh : n ≤ m + n\n⊢ ↑(m + n - n) = ↑(m + n) - ↑n",
"tactic": "rw [add_tsub_cancel_right, cast_add, add_tsub_cancel_right]"
}
]
| [
169,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
]
|
Mathlib/Order/BooleanAlgebra.lean | inf_sdiff_distrib_left | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.39786\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\na b c : α\n⊢ a ⊓ b \\ c = (a ⊓ b) \\ (a ⊓ c)",
"tactic": "rw [sdiff_inf, sdiff_eq_bot_iff.2 inf_le_left, bot_sup_eq, inf_sdiff_assoc]"
}
]
| [
454,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
453,
1
]
|
Mathlib/Data/Multiset/Basic.lean | Multiset.filter_union | []
| [
2059,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2058,
1
]
|
Mathlib/LinearAlgebra/Matrix/ToLin.lean | Matrix.mulVecLin_submatrix | []
| [
234,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
]
|
Mathlib/RingTheory/WittVector/Basic.lean | WittVector.ghostFun_int_cast | [
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nS : Type ?u.755378\nT : Type ?u.755381\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.755396\nβ : Type ?u.755399\nx y : 𝕎 R\ni : ℤ\n⊢ WittVector.ghostFun (Int.castDef i) = ↑i",
"tactic": "cases i <;> simp [*, Int.castDef, ghostFun_nat_cast, ghostFun_neg, -Pi.coe_nat, -Pi.coe_int]"
}
]
| [
213,
97
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
9
]
|
Mathlib/GroupTheory/GroupAction/SubMulAction.lean | SubMulAction.neg_mem | [
{
"state_after": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nhx : x ∈ p\n⊢ -1 • x ∈ p",
"state_before": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nhx : x ∈ p\n⊢ -x ∈ p",
"tactic": "rw [← neg_one_smul R]"
},
{
"state_after": "no goals",
"state_before": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nhx : x ∈ p\n⊢ -1 • x ∈ p",
"tactic": "exact p.smul_mem _ hx"
}
]
| [
360,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
358,
1
]
|
Mathlib/RingTheory/Ideal/Operations.lean | RingHom.ker_ne_top | []
| [
2023,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2022,
1
]
|
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.spanSingleton_eq_spanSingleton | [
{
"state_after": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1308502\ninst✝⁴ : CommRing R₁\nK : Type ?u.1308508\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : NoZeroSMulDivisors R P\nx y : P\n⊢ { val := span R {x}, property := (_ : IsFractional S (span R {x})) } =\n { val := span R {y}, property := (_ : IsFractional S (span R {y})) } ↔\n span R {x} = span R {y}",
"state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1308502\ninst✝⁴ : CommRing R₁\nK : Type ?u.1308508\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : NoZeroSMulDivisors R P\nx y : P\n⊢ spanSingleton S x = spanSingleton S y ↔ ∃ z, z • x = y",
"tactic": "rw [← Submodule.span_singleton_eq_span_singleton, spanSingleton, spanSingleton]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1308502\ninst✝⁴ : CommRing R₁\nK : Type ?u.1308508\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : NoZeroSMulDivisors R P\nx y : P\n⊢ { val := span R {x}, property := (_ : IsFractional S (span R {x})) } =\n { val := span R {y}, property := (_ : IsFractional S (span R {y})) } ↔\n span R {x} = span R {y}",
"tactic": "exact Subtype.mk_eq_mk"
}
]
| [
1326,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1323,
1
]
|
Mathlib/Data/Polynomial/Degree/Lemmas.lean | Polynomial.natDegree_iterate_comp | [
{
"state_after": "case zero\nR : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\n⊢ natDegree ((comp p^[Nat.zero]) q) = natDegree p ^ Nat.zero * natDegree q\n\ncase succ\nR : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nk : ℕ\nIH : natDegree ((comp p^[k]) q) = natDegree p ^ k * natDegree q\n⊢ natDegree ((comp p^[Nat.succ k]) q) = natDegree p ^ Nat.succ k * natDegree q",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nk : ℕ\n⊢ natDegree ((comp p^[k]) q) = natDegree p ^ k * natDegree q",
"tactic": "induction' k with k IH"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\n⊢ natDegree ((comp p^[Nat.zero]) q) = natDegree p ^ Nat.zero * natDegree q",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np q : R[X]\nk : ℕ\nIH : natDegree ((comp p^[k]) q) = natDegree p ^ k * natDegree q\n⊢ natDegree ((comp p^[Nat.succ k]) q) = natDegree p ^ Nat.succ k * natDegree q",
"tactic": "rw [Function.iterate_succ_apply', natDegree_comp, IH, pow_succ, mul_assoc]"
}
]
| [
361,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
1
]
|
Mathlib/Data/Set/Semiring.lean | SetSemiring.up_lt_up | []
| [
81,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
1
]
|
Mathlib/CategoryTheory/Sites/Whiskering.lean | CategoryTheory.GrothendieckTopology.Cover.multicospanComp_app_left | []
| [
68,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
]
|
Mathlib/RingTheory/IntegralClosure.lean | RingHom.isIntegralElem_of_isIntegralElem_comp | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type ?u.1572223\nB : Type ?u.1572226\nS : Type u_3\nT : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra A B\ninst✝² : Algebra R B\nf : R →+* S\ng : S →+* T\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : T\nh : IsIntegralElem (comp g f) x\np : R[X]\nhp : Monic p\nhp' : eval₂ (comp g f) x p = 0\n⊢ eval₂ g x (Polynomial.map f p) = 0",
"tactic": "rwa [← eval₂_map] at hp'"
}
]
| [
1065,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1062,
1
]
|
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.setToFun_finset_sum | [
{
"state_after": "case h.e'_2.h.e'_14.h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1436330\nG : Type ?u.1436333\n𝕜 : Type ?u.1436336\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns : Finset ι\nf : ι → α → E\nhf : ∀ (i : ι), i ∈ s → Integrable (f i)\na : α\n⊢ ∑ i in s, f i a = Finset.sum s (fun i => f i) a",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1436330\nG : Type ?u.1436333\n𝕜 : Type ?u.1436336\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns : Finset ι\nf : ι → α → E\nhf : ∀ (i : ι), i ∈ s → Integrable (f i)\n⊢ (setToFun μ T hT fun a => ∑ i in s, f i a) = ∑ i in s, setToFun μ T hT (f i)",
"tactic": "convert setToFun_finset_sum' hT s hf with a"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_14.h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1436330\nG : Type ?u.1436333\n𝕜 : Type ?u.1436336\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_4\ns : Finset ι\nf : ι → α → E\nhf : ∀ (i : ι), i ∈ s → Integrable (f i)\na : α\n⊢ ∑ i in s, f i a = Finset.sum s (fun i => f i) a",
"tactic": "simp"
}
]
| [
1395,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1392,
1
]
|
Mathlib/Data/Nat/PartENat.lean | PartENat.eq_zero_iff | []
| [
336,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
335,
8
]
|
Mathlib/Algebra/Hom/Freiman.lean | FreimanHom.mul_comp | []
| [
351,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
349,
1
]
|
Mathlib/Topology/LocallyFinite.lean | LocallyFinite.preimage_continuous | []
| [
199,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
196,
1
]
|
Mathlib/Control/Functor.lean | Functor.Comp.run_seq | []
| [
270,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
11
]
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | Real.hasSum_cos | [
{
"state_after": "no goals",
"state_before": "r : ℝ\n⊢ HasSum (fun n => (-1) ^ n * r ^ (2 * n) / ↑(2 * n)!) (cos r)",
"tactic": "exact_mod_cast Complex.hasSum_cos r"
}
]
| [
107,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
]
|
Mathlib/Data/Finsupp/Basic.lean | Finsupp.sum_option_index_smul | []
| [
862,
76
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
859,
1
]
|
Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.bounded_image | []
| [
124,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
]
|
Mathlib/Data/Nat/Factors.lean | Nat.prod_factors | [
{
"state_after": "no goals",
"state_before": "⊢ 0 ≠ 0 → List.prod (factors 0) = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "⊢ 1 ≠ 0 → List.prod (factors 1) = 1",
"tactic": "simp"
},
{
"state_after": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\nh₁ : (k + 2) / m ≠ 0\n⊢ List.prod (factors (k + 2)) = k + 2",
"state_before": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\n⊢ List.prod (factors (k + 2)) = k + 2",
"tactic": "have h₁ : (k + 2) / m ≠ 0 := fun h => by\n have : (k + 2) = 0 * m := (Nat.div_eq_iff_eq_mul_left (minFac_pos _) (minFac_dvd _)).1 h\n rw [zero_mul] at this; exact (show k + 2 ≠ 0 by simp) this"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\nh₁ : (k + 2) / m ≠ 0\n⊢ List.prod (factors (k + 2)) = k + 2",
"tactic": "rw [factors, List.prod_cons, prod_factors h₁, Nat.mul_div_cancel' (minFac_dvd _)]"
},
{
"state_after": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis✝ : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0 * m\n⊢ False",
"state_before": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\n⊢ False",
"tactic": "have : (k + 2) = 0 * m := (Nat.div_eq_iff_eq_mul_left (minFac_pos _) (minFac_dvd _)).1 h"
},
{
"state_after": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis✝ : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0\n⊢ False",
"state_before": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis✝ : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0 * m\n⊢ False",
"tactic": "rw [zero_mul] at this"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis✝ : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0\n⊢ False",
"tactic": "exact (show k + 2 ≠ 0 by simp) this"
},
{
"state_after": "no goals",
"state_before": "k : ℕ\nx✝ : k + 2 ≠ 0\nm : ℕ := minFac (k + 2)\nthis✝ : (k + 2) / m < k + 2\nh : (k + 2) / m = 0\nthis : k + 2 = 0\n⊢ k + 2 ≠ 0",
"tactic": "simp"
}
]
| [
82,
88
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
]
|
Mathlib/Data/Rel.lean | Rel.mem_preimage | []
| [
180,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
]
|
Mathlib/Data/List/Rdrop.lean | List.rtakeWhile_suffix | [
{
"state_after": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ takeWhile p (reverse l) <+: reverse l",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ rtakeWhile p l <:+ l",
"tactic": "rw [← reverse_prefix, rtakeWhile, reverse_reverse]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ takeWhile p (reverse l) <+: reverse l",
"tactic": "exact takeWhile_prefix _"
}
]
| [
218,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
1
]
|
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.nfpBFamily_le_fp | []
| [
319,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
317,
1
]
|
Mathlib/Topology/FiberBundle/Trivialization.lean | Trivialization.coordChange_apply_snd | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.54393\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.54404\nZ : Type u_3\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\nproj : Z → B\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace (TotalSpace E)\ne : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace E\nb : B\ny : E b\ne₁ e₂ : Trivialization F proj\np : Z\nh : proj p ∈ e₁.baseSet\n⊢ coordChange e₁ e₂ (proj p) (↑e₁ p).snd = (↑e₂ p).snd",
"tactic": "rw [coordChange, e₁.symm_apply_mk_proj (e₁.mem_source.2 h)]"
}
]
| [
679,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
677,
1
]
|
Mathlib/Analysis/Convex/Basic.lean | Monotone.convex_ge | []
| [
389,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
388,
1
]
|
Mathlib/RingTheory/PrincipalIdealDomain.lean | Submodule.IsPrincipal.prime_generator_of_isPrime | [
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : CommRing R\ninst✝¹ : Module R M\nS : Ideal R\ninst✝ : IsPrincipal S\nis_prime : Ideal.IsPrime S\nne_bot : S ≠ ⊥\nx✝¹ x✝ : R\n⊢ generator S ∣ x✝¹ * x✝ → generator S ∣ x✝¹ ∨ generator S ∣ x✝",
"tactic": "simpa only [← mem_iff_generator_dvd S] using is_prime.2"
}
]
| [
143,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
]
|
Mathlib/Algebra/Quaternion.lean | Cardinal.mk_quaternion | []
| [
1458,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1457,
1
]
|
Mathlib/Data/Finsupp/Defs.lean | Finsupp.coeFn_injective | []
| [
153,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigOWith.triangle | []
| [
1162,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1159,
1
]
|
Mathlib/Topology/MetricSpace/Basic.lean | Metric.Bounded.subset_ball_lt | [
{
"state_after": "case intro\nα : Type u\nβ : Type v\nX : Type ?u.489346\nι : Type ?u.489349\ninst✝ : PseudoMetricSpace α\nx : α\ns t : Set α\nr✝ : ℝ\nh : Bounded s\na : ℝ\nc : α\nr : ℝ\nhr : s ⊆ closedBall c r\n⊢ ∃ r, a < r ∧ s ⊆ closedBall c r",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.489346\nι : Type ?u.489349\ninst✝ : PseudoMetricSpace α\nx : α\ns t : Set α\nr : ℝ\nh : Bounded s\na : ℝ\nc : α\n⊢ ∃ r, a < r ∧ s ⊆ closedBall c r",
"tactic": "rcases h.subset_ball c with ⟨r, hr⟩"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nX : Type ?u.489346\nι : Type ?u.489349\ninst✝ : PseudoMetricSpace α\nx : α\ns t : Set α\nr✝ : ℝ\nh : Bounded s\na : ℝ\nc : α\nr : ℝ\nhr : s ⊆ closedBall c r\n⊢ s ⊆ closedBall c (max r (a + 1))",
"state_before": "case intro\nα : Type u\nβ : Type v\nX : Type ?u.489346\nι : Type ?u.489349\ninst✝ : PseudoMetricSpace α\nx : α\ns t : Set α\nr✝ : ℝ\nh : Bounded s\na : ℝ\nc : α\nr : ℝ\nhr : s ⊆ closedBall c r\n⊢ ∃ r, a < r ∧ s ⊆ closedBall c r",
"tactic": "refine' ⟨max r (a + 1), lt_of_lt_of_le (by linarith) (le_max_right _ _), _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nX : Type ?u.489346\nι : Type ?u.489349\ninst✝ : PseudoMetricSpace α\nx : α\ns t : Set α\nr✝ : ℝ\nh : Bounded s\na : ℝ\nc : α\nr : ℝ\nhr : s ⊆ closedBall c r\n⊢ s ⊆ closedBall c (max r (a + 1))",
"tactic": "exact hr.trans (closedBall_subset_closedBall (le_max_left _ _))"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.489346\nι : Type ?u.489349\ninst✝ : PseudoMetricSpace α\nx : α\ns t : Set α\nr✝ : ℝ\nh : Bounded s\na : ℝ\nc : α\nr : ℝ\nhr : s ⊆ closedBall c r\n⊢ a < a + 1",
"tactic": "linarith"
}
]
| [
2352,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2348,
1
]
|
Mathlib/Data/QPF/Univariate/Basic.lean | Qpf.Fix.rec_unique | [
{
"state_after": "case h\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nh : Fix F → α\nhyp : ∀ (x : F (Fix F)), h (mk x) = g (h <$> x)\nx : Fix F\n⊢ rec g x = h x",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nh : Fix F → α\nhyp : ∀ (x : F (Fix F)), h (mk x) = g (h <$> x)\n⊢ rec g = h",
"tactic": "ext x"
},
{
"state_after": "case h.h\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nh : Fix F → α\nhyp : ∀ (x : F (Fix F)), h (mk x) = g (h <$> x)\nx : Fix F\n⊢ ∀ (x : F (Fix F)), rec g <$> x = (fun x => h x) <$> x → rec g (mk x) = h (mk x)",
"state_before": "case h\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nh : Fix F → α\nhyp : ∀ (x : F (Fix F)), h (mk x) = g (h <$> x)\nx : Fix F\n⊢ rec g x = h x",
"tactic": "apply Fix.ind_rec"
},
{
"state_after": "case h.h\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nh : Fix F → α\nhyp : ∀ (x : F (Fix F)), h (mk x) = g (h <$> x)\nx✝ : Fix F\nx : F (Fix F)\nhyp' : rec g <$> x = (fun x => h x) <$> x\n⊢ rec g (mk x) = h (mk x)",
"state_before": "case h.h\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nh : Fix F → α\nhyp : ∀ (x : F (Fix F)), h (mk x) = g (h <$> x)\nx : Fix F\n⊢ ∀ (x : F (Fix F)), rec g <$> x = (fun x => h x) <$> x → rec g (mk x) = h (mk x)",
"tactic": "intro x hyp'"
},
{
"state_after": "no goals",
"state_before": "case h.h\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nh : Fix F → α\nhyp : ∀ (x : F (Fix F)), h (mk x) = g (h <$> x)\nx✝ : Fix F\nx : F (Fix F)\nhyp' : rec g <$> x = (fun x => h x) <$> x\n⊢ rec g (mk x) = h (mk x)",
"tactic": "rw [hyp, ← hyp', Fix.rec_eq]"
}
]
| [
330,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
325,
1
]
|
Mathlib/FieldTheory/Normal.lean | Normal.splits | []
| [
57,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
1
]
|
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