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Mathlib/Data/Finset/Basic.lean | Finset.inter_left_idem | [] | [
1732,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1731,
1
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Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | Set.Finite.lt_cInf_iff | [] | [
75,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
] |
Mathlib/Dynamics/PeriodicPts.lean | Function.minimalPeriod_eq_zero_iff_nmem_periodicPts | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.22504\nf fa : α → α\nfb : β → β\nx y : α\nm n : ℕ\n⊢ minimalPeriod f x = 0 ↔ ¬x ∈ periodicPts f",
"tactic": "rw [← minimalPeriod_pos_iff_mem_periodicPts, not_lt, nonpos_iff_eq_zero]"
}
] | [
320,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
319,
1
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Mathlib/Analysis/Convex/Exposed.lean | isExposed_empty | [
{
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"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderedRing 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA B C : Set E\nX : Finset E\nx : E\nx✝ : Set.Nonempty ∅\nw✝ : E\nhx : w✝ ∈ ∅\n⊢ ∃ l, ∅ = {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}",
"tactic": "exfalso"
},
{
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"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderedRing 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA B C : Set E\nX : Finset E\nx : E\nx✝ : Set.Nonempty ∅\nw✝ : E\nhx : w✝ ∈ ∅\n⊢ False",
"tactic": "exact hx"
}
] | [
85,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
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Mathlib/Topology/UniformSpace/Basic.lean | Uniform.continuousOn_iff'_right | [
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.199215\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nf : β → α\ns : Set β\n⊢ ContinuousOn f s ↔ ∀ (b : β), b ∈ s → Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α)",
"tactic": "simp [ContinuousOn, continuousWithinAt_iff'_right]"
}
] | [
1951,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1949,
1
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Mathlib/CategoryTheory/Limits/Opposites.lean | CategoryTheory.Limits.pullbackIsoUnopPushout_inv_snd | [
{
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"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u₂\ninst✝² : Category J\nX✝ : Type v₂\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasPullback f g\ninst✝ : HasPushout f.op g.op\n⊢ (PushoutCocone.unop (colimit.cocone (span f.op g.op))).π.app WalkingCospan.right = pushout.inr.unop",
"tactic": "simp"
}
] | [
644,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
641,
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Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.IsThreeCycle.orderOf | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ g : Perm α\nht : IsThreeCycle g\n⊢ _root_.orderOf g = 3",
"tactic": "rw [← lcm_cycleType, ht.cycleType, Multiset.lcm_singleton, normalize_eq]"
}
] | [
614,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
613,
1
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Mathlib/Topology/Sheaves/PUnit.lean | TopCat.Presheaf.isSheaf_on_pUnit_of_isTerminal | [] | [
57,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
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Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean | Antitone.ge_of_tendsto | [] | [
274,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
271,
1
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Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.count_injective_image | [
{
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"tactic": "by_cases hs : s.Finite"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.388219\nδ : Type ?u.388222\nι : Type ?u.388225\nR : Type ?u.388228\nR' : Type ?u.388231\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\ninst✝ : MeasurableSingletonClass β\nf : β → α\nhf : Injective f\ns : Set β\nhs : ¬Set.Finite s\n⊢ ↑↑count (f '' s) = ⊤",
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"tactic": "rw [count_apply_infinite hs]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.388219\nδ : Type ?u.388222\nι : Type ?u.388225\nR : Type ?u.388228\nR' : Type ?u.388231\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\ninst✝ : MeasurableSingletonClass β\nf : β → α\nhf : Injective f\ns : Set β\nhs : ¬Set.Finite (f '' s)\n⊢ ↑↑count (f '' s) = ⊤",
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"tactic": "rw [← finite_image_iff <| hf.injOn _] at hs"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.388219\nδ : Type ?u.388222\nι : Type ?u.388225\nR : Type ?u.388228\nR' : Type ?u.388231\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\ninst✝ : MeasurableSingletonClass β\nf : β → α\nhf : Injective f\ns : Set β\nhs : ¬Set.Finite (f '' s)\n⊢ ↑↑count (f '' s) = ⊤",
"tactic": "rw [count_apply_infinite hs]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.388219\nδ : Type ?u.388222\nι : Type ?u.388225\nR : Type ?u.388228\nR' : Type ?u.388231\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSingletonClass α\ninst✝ : MeasurableSingletonClass β\nf : β → α\nhf : Injective f\ns : Set β\nhs : Set.Finite s\n⊢ ↑↑count (f '' s) = ↑↑count s",
"tactic": "exact count_injective_image' hf hs.measurableSet (Finite.image f hs).measurableSet"
}
] | [
2339,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2333,
1
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Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.map_isPath_of_injective | [
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nhinj : Injective ↑f\nhp : IsPath p\n⊢ IsPath (Walk.map f p)",
"tactic": "induction p with\n| nil => simp\n| cons _ _ ih =>\n rw [Walk.cons_isPath_iff] at hp\n simp [ih hp.1]\n intro x hx hf\n cases hinj hf\n exact hp.2 hx"
},
{
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"tactic": "simp"
},
{
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"tactic": "rw [Walk.cons_isPath_iff] at hp"
},
{
"state_after": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nhinj : Injective ↑f\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : IsPath p✝ → IsPath (Walk.map f p✝)\nhp : IsPath p✝ ∧ ¬u✝ ∈ support p✝\n⊢ ∀ (x : V), x ∈ support p✝ → ¬↑f x = ↑f u✝",
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"tactic": "simp [ih hp.1]"
},
{
"state_after": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nhinj : Injective ↑f\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : IsPath p✝ → IsPath (Walk.map f p✝)\nhp : IsPath p✝ ∧ ¬u✝ ∈ support p✝\nx : V\nhx : x ∈ support p✝\nhf : ↑f x = ↑f u✝\n⊢ False",
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"tactic": "intro x hx hf"
},
{
"state_after": "case cons.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nhinj : Injective ↑f\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : IsPath p✝ → IsPath (Walk.map f p✝)\nhp : IsPath p✝ ∧ ¬u✝ ∈ support p✝\nhx : u✝ ∈ support p✝\nhf : ↑f u✝ = ↑f u✝\n⊢ False",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nhinj : Injective ↑f\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : IsPath p✝ → IsPath (Walk.map f p✝)\nhp : IsPath p✝ ∧ ¬u✝ ∈ support p✝\nx : V\nhx : x ∈ support p✝\nhf : ↑f x = ↑f u✝\n⊢ False",
"tactic": "cases hinj hf"
},
{
"state_after": "no goals",
"state_before": "case cons.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nhinj : Injective ↑f\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : IsPath p✝ → IsPath (Walk.map f p✝)\nhp : IsPath p✝ ∧ ¬u✝ ∈ support p✝\nhx : u✝ ∈ support p✝\nhf : ↑f u✝ = ↑f u✝\n⊢ False",
"tactic": "exact hp.2 hx"
}
] | [
1538,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1529,
1
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Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.sup_not_succ_of_ne_sup | [
{
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"state_before": "α : Type ?u.283171\nβ : Type ?u.283174\nγ : Type ?u.283177\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nhf : ∀ (i : ι), f i ≠ sup f\na : Ordinal\nhao : a < sup f\n⊢ succ a < sup f",
"tactic": "by_contra' hoa"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.283171\nβ : Type ?u.283174\nγ : Type ?u.283177\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u\nf : ι → Ordinal\nhf : ∀ (i : ι), f i ≠ sup f\na : Ordinal\nhao : a < sup f\nhoa : sup f ≤ succ a\n⊢ False",
"tactic": "exact\n hao.not_le (sup_le fun i => le_of_lt_succ <| (lt_of_le_of_ne (le_sup _ _) (hf i)).trans_le hoa)"
}
] | [
1289,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1285,
1
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Std/Data/String/Lemmas.lean | String.Pos.ext | [] | [
93,
56
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
92,
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src/lean/Init/Data/Nat/Basic.lean | Nat.lt_succ_of_le | [] | [
213,
15
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
212,
1
] |
Mathlib/Order/SuccPred/Basic.lean | Covby.succ_eq | [] | [
429,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
428,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean | PFunctor.M.ext | [
{
"state_after": "case H\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\n⊢ ∀ (i : ℕ), MIntl.approx x i = MIntl.approx y i",
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"tactic": "apply ext'"
},
{
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"tactic": "intro i"
},
{
"state_after": "case H.zero\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\n⊢ MIntl.approx x zero = MIntl.approx y zero\n\ncase H.succ\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ MIntl.approx x (succ i) = MIntl.approx y (succ i)",
"state_before": "case H\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\n⊢ MIntl.approx x i = MIntl.approx y i",
"tactic": "induction' i with i i_ih"
},
{
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"state_before": "case H.zero\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\n⊢ MIntl.approx x zero = MIntl.approx y zero",
"tactic": "cases x.approx 0"
},
{
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"tactic": "cases y.approx 0"
},
{
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"tactic": "constructor"
},
{
"state_after": "case H.succ.hx\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree' i x x\n\ncase H.succ.hy\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree' i x y\n\ncase H.succ.hrec\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ ∀ (ps : Path F), i = length ps → iselect ps x = iselect ps y",
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"tactic": "apply ext_aux x y x"
},
{
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"tactic": "introv H'"
},
{
"state_after": "case H.succ.hrec\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), head (isubtree ps x) = head (isubtree ps y)\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\nps : Path F\nH' : i = length ps\n⊢ iselect ps x = iselect ps y",
"state_before": "case H.succ.hrec\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\nps : Path F\nH' : i = length ps\n⊢ iselect ps x = iselect ps y",
"tactic": "dsimp only [iselect] at H"
},
{
"state_after": "case H.succ.hrec.refl\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), head (isubtree ps x) = head (isubtree ps y)\nps : Path F\ni_ih : MIntl.approx x (length ps) = MIntl.approx y (length ps)\n⊢ iselect ps x = iselect ps y",
"state_before": "case H.succ.hrec\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), head (isubtree ps x) = head (isubtree ps y)\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\nps : Path F\nH' : i = length ps\n⊢ iselect ps x = iselect ps y",
"tactic": "cases H'"
},
{
"state_after": "no goals",
"state_before": "case H.succ.hrec.refl\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), head (isubtree ps x) = head (isubtree ps y)\nps : Path F\ni_ih : MIntl.approx x (length ps) = MIntl.approx y (length ps)\n⊢ iselect ps x = iselect ps y",
"tactic": "apply H ps"
},
{
"state_after": "case H.succ.hx\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree (MIntl.approx x i) (MIntl.approx x (i + 1))",
"state_before": "case H.succ.hx\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree' i x x",
"tactic": "rw [← agree_iff_agree']"
},
{
"state_after": "no goals",
"state_before": "case H.succ.hx\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree (MIntl.approx x i) (MIntl.approx x (i + 1))",
"tactic": "apply x.consistent"
},
{
"state_after": "case H.succ.hy\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree (MIntl.approx y i) (MIntl.approx y (i + 1))",
"state_before": "case H.succ.hy\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree' i x y",
"tactic": "rw [← agree_iff_agree', i_ih]"
},
{
"state_after": "no goals",
"state_before": "case H.succ.hy\nF : PFunctor\nX : Type ?u.33413\nf : X → Obj F X\ninst✝ : Inhabited (M F)\nx y : M F\nH : ∀ (ps : Path F), iselect ps x = iselect ps y\ni : ℕ\ni_ih : MIntl.approx x i = MIntl.approx y i\n⊢ Agree (MIntl.approx y i) (MIntl.approx y (i + 1))",
"tactic": "apply y.consistent"
}
] | [
646,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
631,
1
] |
Mathlib/Algebra/Order/Kleene.lean | kstar_mul_kstar | [] | [
267,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
] |
Mathlib/Data/Complex/Basic.lean | Complex.mul_conj | [
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ (z * ↑(starRingEnd ℂ) z).re = (↑(↑normSq z)).re ∧ (z * ↑(starRingEnd ℂ) z).im = (↑(↑normSq z)).im",
"tactic": "simp [normSq, mul_comm, sub_eq_neg_add, add_comm, ofReal']"
}
] | [
672,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
671,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean | AlgEquiv.coe_fun_injective | [] | [
161,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/Data/Int/GCD.lean | Int.pow_dvd_pow_iff | [
{
"state_after": "m n : ℤ\nk : ℕ\nk0 : 0 < k\nh : m ^ k ∣ n ^ k\n⊢ m ∣ n",
"state_before": "m n : ℤ\nk : ℕ\nk0 : 0 < k\n⊢ m ^ k ∣ n ^ k ↔ m ∣ n",
"tactic": "refine' ⟨fun h => _, fun h => pow_dvd_pow_of_dvd h _⟩"
},
{
"state_after": "no goals",
"state_before": "m n : ℤ\nk : ℕ\nk0 : 0 < k\nh : m ^ k ∣ n ^ k\n⊢ m ∣ n",
"tactic": "rwa [← natAbs_dvd_natAbs, ← Nat.pow_dvd_pow_iff k0, ← Int.natAbs_pow, ← Int.natAbs_pow,\n natAbs_dvd_natAbs]"
}
] | [
388,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
385,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean | PadicSeq.norm_zero_iff | [
{
"state_after": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\n⊢ norm f = 0 → f ≈ 0\n\ncase mpr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\n⊢ f ≈ 0 → norm f = 0",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\n⊢ norm f = 0 ↔ f ≈ 0",
"tactic": "constructor"
},
{
"state_after": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : norm f = 0\n⊢ f ≈ 0",
"state_before": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\n⊢ norm f = 0 → f ≈ 0",
"tactic": "intro h"
},
{
"state_after": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : norm f = 0\nhf : ¬f ≈ 0\n⊢ False",
"state_before": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : norm f = 0\n⊢ f ≈ 0",
"tactic": "by_contra hf"
},
{
"state_after": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) = 0\nhf : ¬f ≈ 0\n⊢ False",
"state_before": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : norm f = 0\nhf : ¬f ≈ 0\n⊢ False",
"tactic": "unfold norm at h"
},
{
"state_after": "case mp.inl\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nh✝ : f ≈ 0\nh : 0 = 0\n⊢ False\n\ncase mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\n⊢ False",
"state_before": "case mp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) = 0\nhf : ¬f ≈ 0\n⊢ False",
"tactic": "split_ifs at h"
},
{
"state_after": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\n⊢ False",
"state_before": "case mp.inl\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : ¬f ≈ 0\nh✝ : f ≈ 0\nh : 0 = 0\n⊢ False\n\ncase mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\n⊢ False",
"tactic": "contradiction"
},
{
"state_after": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\n⊢ f ≈ 0",
"state_before": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\n⊢ False",
"tactic": "apply hf"
},
{
"state_after": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → padicNorm p (↑(f - 0) j) < ε",
"state_before": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\n⊢ f ≈ 0",
"tactic": "intro ε hε"
},
{
"state_after": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\n⊢ ∀ (j : ℕ), j ≥ stationaryPoint hf → padicNorm p (↑(f - 0) j) < ε",
"state_before": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → padicNorm p (↑(f - 0) j) < ε",
"tactic": "exists stationaryPoint hf"
},
{
"state_after": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\nj : ℕ\nhj : j ≥ stationaryPoint hf\n⊢ padicNorm p (↑(f - 0) j) < ε",
"state_before": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\n⊢ ∀ (j : ℕ), j ≥ stationaryPoint hf → padicNorm p (↑(f - 0) j) < ε",
"tactic": "intro j hj"
},
{
"state_after": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\nj : ℕ\nhj : j ≥ stationaryPoint hf\nheq : padicNorm p (↑f j) = padicNorm p (↑f (stationaryPoint hf))\n⊢ padicNorm p (↑(f - 0) j) < ε",
"state_before": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\nj : ℕ\nhj : j ≥ stationaryPoint hf\n⊢ padicNorm p (↑(f - 0) j) < ε",
"tactic": "have heq := stationaryPoint_spec hf le_rfl hj"
},
{
"state_after": "no goals",
"state_before": "case mp.inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h✝ : ¬f ≈ 0\nh : padicNorm p (↑f (stationaryPoint h✝)) = 0\nε : ℚ\nhε : ε > 0\nj : ℕ\nhj : j ≥ stationaryPoint hf\nheq : padicNorm p (↑f j) = padicNorm p (↑f (stationaryPoint hf))\n⊢ padicNorm p (↑(f - 0) j) < ε",
"tactic": "simpa [h, heq]"
},
{
"state_after": "case mpr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f ≈ 0\n⊢ norm f = 0",
"state_before": "case mpr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\n⊢ f ≈ 0 → norm f = 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f ≈ 0\n⊢ norm f = 0",
"tactic": "simp [norm, h]"
}
] | [
137,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/Data/ZMod/Basic.lean | ZMod.valMinAbs_def_pos | [
{
"state_after": "case zero\ninst✝ : NeZero Nat.zero\nx : ZMod Nat.zero\n⊢ valMinAbs x = if val x ≤ Nat.zero / 2 then ↑(val x) else ↑(val x) - ↑Nat.zero\n\ncase succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\nx : ZMod (Nat.succ n✝)\n⊢ valMinAbs x = if val x ≤ Nat.succ n✝ / 2 then ↑(val x) else ↑(val x) - ↑(Nat.succ n✝)",
"state_before": "n : ℕ\ninst✝ : NeZero n\nx : ZMod n\n⊢ valMinAbs x = if val x ≤ n / 2 then ↑(val x) else ↑(val x) - ↑n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\ninst✝ : NeZero Nat.zero\nx : ZMod Nat.zero\n⊢ valMinAbs x = if val x ≤ Nat.zero / 2 then ↑(val x) else ↑(val x) - ↑Nat.zero",
"tactic": "cases NeZero.ne 0 rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nn✝ : ℕ\ninst✝ : NeZero (Nat.succ n✝)\nx : ZMod (Nat.succ n✝)\n⊢ valMinAbs x = if val x ≤ Nat.succ n✝ / 2 then ↑(val x) else ↑(val x) - ↑(Nat.succ n✝)",
"tactic": "rfl"
}
] | [
914,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
910,
1
] |
Mathlib/CategoryTheory/Functor/Category.lean | CategoryTheory.NatTrans.exchange | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G H I✝ : C ⥤ D\nI J K : D ⥤ E\nα : F ⟶ G\nβ : G ⟶ H\nγ : I ⟶ J\nδ : J ⟶ K\n⊢ (α ≫ β) ◫ (γ ≫ δ) = (α ◫ γ) ≫ β ◫ δ",
"tactic": "aesop_cat"
}
] | [
133,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.mul_injective_of_disjoint | [
{
"state_after": "G : Type u_1\nG' : Type ?u.729648\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.729657\ninst✝¹ : AddGroup A\nN : Type ?u.729663\ninst✝ : Group N\nH₁ H₂ : Subgroup G\nh : Disjoint H₁ H₂\nx y : { x // x ∈ H₁ } × { x // x ∈ H₂ }\nhxy : (fun g => ↑g.fst * ↑g.snd) x = (fun g => ↑g.fst * ↑g.snd) y\n⊢ x = y",
"state_before": "G : Type u_1\nG' : Type ?u.729648\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.729657\ninst✝¹ : AddGroup A\nN : Type ?u.729663\ninst✝ : Group N\nH₁ H₂ : Subgroup G\nh : Disjoint H₁ H₂\n⊢ Injective fun g => ↑g.fst * ↑g.snd",
"tactic": "intro x y hxy"
},
{
"state_after": "G : Type u_1\nG' : Type ?u.729648\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.729657\ninst✝¹ : AddGroup A\nN : Type ?u.729663\ninst✝ : Group N\nH₁ H₂ : Subgroup G\nh : Disjoint H₁ H₂\nx y : { x // x ∈ H₁ } × { x // x ∈ H₂ }\nhxy✝ : (↑y.fst)⁻¹ * ↑x.fst * ↑x.snd = ↑y.snd\nhxy : (↑y.fst)⁻¹ * ↑x.fst * (↑x.snd * (↑y.snd)⁻¹) = 1\n⊢ x = y",
"state_before": "G : Type u_1\nG' : Type ?u.729648\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.729657\ninst✝¹ : AddGroup A\nN : Type ?u.729663\ninst✝ : Group N\nH₁ H₂ : Subgroup G\nh : Disjoint H₁ H₂\nx y : { x // x ∈ H₁ } × { x // x ∈ H₂ }\nhxy : (fun g => ↑g.fst * ↑g.snd) x = (fun g => ↑g.fst * ↑g.snd) y\n⊢ x = y",
"tactic": "rw [← inv_mul_eq_iff_eq_mul, ← mul_assoc, ← mul_inv_eq_one, mul_assoc] at hxy"
},
{
"state_after": "G : Type u_1\nG' : Type ?u.729648\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.729657\ninst✝¹ : AddGroup A\nN : Type ?u.729663\ninst✝ : Group N\nH₁ H₂ : Subgroup G\nh : Disjoint H₁ H₂\nx y : { x // x ∈ H₁ } × { x // x ∈ H₂ }\nhxy✝ : (↑y.fst)⁻¹ * ↑x.fst * ↑x.snd = ↑y.snd\nhxy : ↑(y.fst⁻¹ * x.fst) = 1 ∧ ↑(x.snd * y.snd⁻¹) = 1\n⊢ x = y",
"state_before": "G : Type u_1\nG' : Type ?u.729648\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.729657\ninst✝¹ : AddGroup A\nN : Type ?u.729663\ninst✝ : Group N\nH₁ H₂ : Subgroup G\nh : Disjoint H₁ H₂\nx y : { x // x ∈ H₁ } × { x // x ∈ H₂ }\nhxy✝ : (↑y.fst)⁻¹ * ↑x.fst * ↑x.snd = ↑y.snd\nhxy : (↑y.fst)⁻¹ * ↑x.fst * (↑x.snd * (↑y.snd)⁻¹) = 1\n⊢ x = y",
"tactic": "replace hxy := disjoint_iff_mul_eq_one.mp h (y.1⁻¹ * x.1).prop (x.2 * y.2⁻¹).prop hxy"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.729648\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.729657\ninst✝¹ : AddGroup A\nN : Type ?u.729663\ninst✝ : Group N\nH₁ H₂ : Subgroup G\nh : Disjoint H₁ H₂\nx y : { x // x ∈ H₁ } × { x // x ∈ H₂ }\nhxy✝ : (↑y.fst)⁻¹ * ↑x.fst * ↑x.snd = ↑y.snd\nhxy : ↑(y.fst⁻¹ * x.fst) = 1 ∧ ↑(x.snd * y.snd⁻¹) = 1\n⊢ x = y",
"tactic": "rwa [coe_mul, coe_mul, coe_inv, coe_inv, inv_mul_eq_one, mul_inv_eq_one, ← Subtype.ext_iff, ←\n Subtype.ext_iff, eq_comm, ← Prod.ext_iff] at hxy"
}
] | [
3667,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3661,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean | deriv_clm_apply | [] | [
413,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
411,
1
] |
Mathlib/Data/Set/Basic.lean | Set.Nontrivial.nonempty | [] | [
2524,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2522,
11
] |
Mathlib/Order/MinMax.lean | min_max_distrib_right | [] | [
128,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean | HomologicalComplex.Hom.prev_eq | [
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC C₁ C₂ C₃ : HomologicalComplex V c\nf : Hom C₁ C₂\nj : ι\nw : ComplexShape.Rel c (ComplexShape.prev c j) j\n⊢ prev f j = (xPrevIso C₁ w).hom ≫ HomologicalComplex.Hom.f f (ComplexShape.prev c j) ≫ (xPrevIso C₂ w).inv",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC C₁ C₂ C₃ : HomologicalComplex V c\nf : Hom C₁ C₂\ni j : ι\nw : ComplexShape.Rel c i j\n⊢ prev f j = (xPrevIso C₁ w).hom ≫ HomologicalComplex.Hom.f f i ≫ (xPrevIso C₂ w).inv",
"tactic": "obtain rfl := c.prev_eq' w"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC C₁ C₂ C₃ : HomologicalComplex V c\nf : Hom C₁ C₂\nj : ι\nw : ComplexShape.Rel c (ComplexShape.prev c j) j\n⊢ prev f j = (xPrevIso C₁ w).hom ≫ HomologicalComplex.Hom.f f (ComplexShape.prev c j) ≫ (xPrevIso C₂ w).inv",
"tactic": "simp only [xPrevIso, eqToIso_refl, Iso.refl_hom, Iso.refl_inv, comp_id, id_comp]"
}
] | [
530,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
527,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean | Unitization.fst_add | [] | [
224,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
223,
1
] |
Mathlib/Algebra/Lie/Submodule.lean | LieHom.range_coeSubmodule | [] | [
986,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
985,
1
] |
Mathlib/GroupTheory/SchurZassenhaus.lean | Subgroup.SchurZassenhausInduction.step0 | [
{
"state_after": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\ninst✝ : Normal ⊥\nh1 : Nat.coprime (Fintype.card { x // x ∈ ⊥ }) (index ⊥)\nh3 : ∀ (H : Subgroup G), ¬IsComplement' ⊥ H\n⊢ False",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nN : Subgroup G\ninst✝ : Normal N\nh1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\nh3 : ∀ (H : Subgroup G), ¬IsComplement' N H\n⊢ N ≠ ⊥",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝² : Group G\ninst✝¹ : Fintype G\nh2 :\n ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n Fintype.card G' < Fintype.card G →\n ∀ {N' : Subgroup G'} [inst_2 : Normal N'],\n Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'\ninst✝ : Normal ⊥\nh1 : Nat.coprime (Fintype.card { x // x ∈ ⊥ }) (index ⊥)\nh3 : ∀ (H : Subgroup G), ¬IsComplement' ⊥ H\n⊢ False",
"tactic": "exact h3 ⊤ isComplement'_bot_top"
}
] | [
173,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
9
] |
Mathlib/Data/Finset/Lattice.lean | Finset.min'_lt_of_mem_erase_min' | [] | [
1494,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1492,
1
] |
Mathlib/LinearAlgebra/Matrix/Orthogonal.lean | Matrix.HasOrthogonalRows.hasOrthogonalCols | [] | [
71,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.coe_toReal_le | [
{
"state_after": "case pos\nx : EReal\nh : x ≠ ⊥\nh' : x = ⊤\n⊢ ↑(toReal x) ≤ x\n\ncase neg\nx : EReal\nh : x ≠ ⊥\nh' : ¬x = ⊤\n⊢ ↑(toReal x) ≤ x",
"state_before": "x : EReal\nh : x ≠ ⊥\n⊢ ↑(toReal x) ≤ x",
"tactic": "by_cases h' : x = ⊤"
},
{
"state_after": "no goals",
"state_before": "case pos\nx : EReal\nh : x ≠ ⊥\nh' : x = ⊤\n⊢ ↑(toReal x) ≤ x",
"tactic": "simp only [h', le_top]"
},
{
"state_after": "no goals",
"state_before": "case neg\nx : EReal\nh : x ≠ ⊥\nh' : ¬x = ⊤\n⊢ ↑(toReal x) ≤ x",
"tactic": "simp only [le_refl, coe_toReal h' h]"
}
] | [
427,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
424,
1
] |
Mathlib/Algebra/Algebra/Hom.lean | AlgHom.map_prod | [] | [
468,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
466,
11
] |
Mathlib/RingTheory/Ideal/Basic.lean | Ideal.add_mem_iff_left | [] | [
660,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
659,
11
] |
Mathlib/Order/BooleanAlgebra.lean | IsCompl.compl_eq_iff | [] | [
667,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
666,
1
] |
Mathlib/Data/Bool/Basic.lean | Bool.not_eq_not | [
{
"state_after": "no goals",
"state_before": "⊢ ∀ {a b : Bool}, ¬a = !b ↔ a = b",
"tactic": "decide"
}
] | [
211,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
] |
Mathlib/Topology/Separation.lean | Inducing.embedding | [] | [
204,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
11
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean | Complex.UnitDisc.coe_ne_one | [] | [
62,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
] |
Mathlib/Topology/Order/Basic.lean | isOpen_gt' | [] | [
908,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
907,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_groupoid | [] | [
539,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
537,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.limsup_le_of_le | [] | [
443,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
440,
1
] |
Mathlib/Data/Quot.lean | Trunc.induction_on | [] | [
500,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
499,
11
] |
Mathlib/Data/List/Intervals.lean | List.Ico.filter_le_of_le | [
{
"state_after": "case inl\nn m l : ℕ\nhnl : n ≤ l\nhlm : l ≤ m\n⊢ filter (fun x => decide (l ≤ x)) (Ico n m) = Ico l m\n\ncase inr\nn m l : ℕ\nhnl : n ≤ l\nhml : m ≤ l\n⊢ filter (fun x => decide (l ≤ x)) (Ico n m) = Ico l m",
"state_before": "n m l : ℕ\nhnl : n ≤ l\n⊢ filter (fun x => decide (l ≤ x)) (Ico n m) = Ico l m",
"tactic": "cases' le_total l m with hlm hml"
},
{
"state_after": "no goals",
"state_before": "case inl\nn m l : ℕ\nhnl : n ≤ l\nhlm : l ≤ m\n⊢ filter (fun x => decide (l ≤ x)) (Ico n m) = Ico l m",
"tactic": "rw [← append_consecutive hnl hlm, filter_append, filter_le_of_top_le (le_refl l),\n filter_le_of_le_bot (le_refl l), nil_append]"
},
{
"state_after": "no goals",
"state_before": "case inr\nn m l : ℕ\nhnl : n ≤ l\nhml : m ≤ l\n⊢ filter (fun x => decide (l ≤ x)) (Ico n m) = Ico l m",
"tactic": "rw [eq_nil_of_le hml, filter_le_of_top_le hml]"
}
] | [
205,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
200,
1
] |
Mathlib/Analysis/Calculus/Dslope.lean | continuousWithinAt_dslope_of_ne | [
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : b ≠ a\nhc : ContinuousWithinAt f s b\n⊢ ContinuousWithinAt (dslope f a) s b",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : b ≠ a\n⊢ ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b",
"tactic": "refine' ⟨ContinuousWithinAt.of_dslope, fun hc => _⟩"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : b ≠ a\nhc : ContinuousWithinAt f s b\n⊢ ContinuousWithinAt (slope f a) s b",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : b ≠ a\nhc : ContinuousWithinAt f s b\n⊢ ContinuousWithinAt (dslope f a) s b",
"tactic": "simp only [dslope, continuousWithinAt_update_of_ne h]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nh : b ≠ a\nhc : ContinuousWithinAt f s b\n⊢ ContinuousWithinAt (slope f a) s b",
"tactic": "exact ((continuousWithinAt_id.sub continuousWithinAt_const).inv₀ (sub_ne_zero.2 h)).smul\n (hc.sub continuousWithinAt_const)"
}
] | [
114,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Order/Height.lean | Set.exists_chain_of_le_chainHeight | [
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : (⨆ (l : ↑(subchain s)), ↑(length ↑l)) = ⊤\n⊢ ∃ l, l ∈ subchain s ∧ length l = n\n\ncase inr\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : (⨆ (l : ↑(subchain s)), ↑(length ↑l)) < ⊤\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"state_before": "α : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "cases' (le_top : s.chainHeight ≤ ⊤).eq_or_lt with ha ha <;>\n rw [chainHeight_eq_iSup_subtype] at ha"
},
{
"state_after": "case inl.intro.intro.intro.mk.intro\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : (⨆ (l : ↑(subchain s)), ↑(length ↑l)) = ⊤\nl : List α\nh₁ : Chain' (fun x x_1 => x < x_1) l\nh₂ : ∀ (i : α), i ∈ l → i ∈ s\nh₃ :\n ¬(fun x => ↑(length ↑x)) { val := l, property := (_ : Chain' (fun x x_1 => x < x_1) l ∧ ∀ (i : α), i ∈ l → i ∈ s) } ≤\n n\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : (⨆ (l : ↑(subchain s)), ↑(length ↑l)) = ⊤\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "obtain ⟨_, ⟨⟨l, h₁, h₂⟩, rfl⟩, h₃⟩ :=\n not_bddAbove_iff'.mp ((WithTop.iSup_coe_eq_top _).mp ha) n"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro.intro.mk.intro\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : (⨆ (l : ↑(subchain s)), ↑(length ↑l)) = ⊤\nl : List α\nh₁ : Chain' (fun x x_1 => x < x_1) l\nh₂ : ∀ (i : α), i ∈ l → i ∈ s\nh₃ :\n ¬(fun x => ↑(length ↑x)) { val := l, property := (_ : Chain' (fun x x_1 => x < x_1) l ∧ ∀ (i : α), i ∈ l → i ∈ s) } ≤\n n\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "exact ⟨l.take n, ⟨h₁.take _, fun x h ↦ h₂ _ <| take_subset _ _ h⟩,\n (l.length_take n).trans <| min_eq_left <| le_of_not_ge h₃⟩"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : BddAbove (range fun l => length ↑l)\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : (⨆ (l : ↑(subchain s)), ↑(length ↑l)) < ⊤\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "rw [ENat.iSup_coe_lt_top] at ha"
},
{
"state_after": "case inr.intro.mk.intro\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : BddAbove (range fun l => length ↑l)\nl : List α\nh₁ : Chain' (fun x x_1 => x < x_1) l\nh₂ : ∀ (i : α), i ∈ l → i ∈ s\ne : length l = sSup (range fun l => length ↑l)\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : BddAbove (range fun l => length ↑l)\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "obtain ⟨⟨l, h₁, h₂⟩, e : l.length = _⟩ := Nat.sSup_mem (Set.range_nonempty _) ha"
},
{
"state_after": "case inr.intro.mk.intro\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : BddAbove (range fun l => length ↑l)\nl : List α\nh₁ : Chain' (fun x x_1 => x < x_1) l\nh₂ : ∀ (i : α), i ∈ l → i ∈ s\ne : length l = sSup (range fun l => length ↑l)\n⊢ n ≤ length l",
"state_before": "case inr.intro.mk.intro\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : BddAbove (range fun l => length ↑l)\nl : List α\nh₁ : Chain' (fun x x_1 => x < x_1) l\nh₂ : ∀ (i : α), i ∈ l → i ∈ s\ne : length l = sSup (range fun l => length ↑l)\n⊢ ∃ l, l ∈ subchain s ∧ length l = n",
"tactic": "refine'\n ⟨l.take n, ⟨h₁.take _, fun x h ↦ h₂ _ <| take_subset _ _ h⟩,\n (l.length_take n).trans <| min_eq_left <| _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.mk.intro\nα : Type u_1\nβ : Type ?u.3758\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl✝ : List α\na : α\nn : ℕ\nhn : ↑n ≤ chainHeight s\nha : BddAbove (range fun l => length ↑l)\nl : List α\nh₁ : Chain' (fun x x_1 => x < x_1) l\nh₂ : ∀ (i : α), i ∈ l → i ∈ s\ne : length l = sSup (range fun l => length ↑l)\n⊢ n ≤ length l",
"tactic": "rwa [e, ← Nat.cast_le (α := ℕ∞), sSup_range, ENat.coe_iSup ha, ← chainHeight_eq_iSup_subtype]"
}
] | [
109,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | CategoryTheory.Groupoid.Free.lift_unique | [
{
"state_after": "case hΦ\nV : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Quotient.functor redStep ⋙ Φ = Paths.lift (Quiver.Symmetrify.lift φ)",
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"tactic": "apply Quotient.lift_unique"
},
{
"state_after": "case hΦ.hΦ\nV : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor = Quiver.Symmetrify.lift φ",
"state_before": "case hΦ\nV : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Quotient.functor redStep ⋙ Φ = Paths.lift (Quiver.Symmetrify.lift φ)",
"tactic": "apply Paths.lift_unique"
},
{
"state_after": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Quiver.Symmetrify.of ⋙q (Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor) = φ\n\nV : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ ∀ {X Y : Quiver.Symmetrify V} (f : X ⟶ Y),\n (Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map (Quiver.reverse f) =\n Quiver.reverse ((Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map f)",
"state_before": "case hΦ.hΦ\nV : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor = Quiver.Symmetrify.lift φ",
"tactic": "fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _"
},
{
"state_after": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Quiver.Symmetrify.of ⋙q (Paths.of ⋙q ((Quotient.functor redStep).toPrefunctor ⋙q Φ.toPrefunctor)) = φ",
"state_before": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Quiver.Symmetrify.of ⋙q (Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor) = φ",
"tactic": "rw [← Functor.toPrefunctor_comp]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ Quiver.Symmetrify.of ⋙q (Paths.of ⋙q ((Quotient.functor redStep).toPrefunctor ⋙q Φ.toPrefunctor)) = φ",
"tactic": "exact hΦ"
},
{
"state_after": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\n⊢ (Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map (Quiver.reverse f) =\n Quiver.reverse ((Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map f)",
"state_before": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\n⊢ ∀ {X Y : Quiver.Symmetrify V} (f : X ⟶ Y),\n (Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map (Quiver.reverse f) =\n Quiver.reverse ((Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map f)",
"tactic": "rintro X Y f"
},
{
"state_after": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\n⊢ Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath (Quiver.reverse f))) =\n Quiver.reverse (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))",
"state_before": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\n⊢ (Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map (Quiver.reverse f) =\n Quiver.reverse ((Paths.of ⋙q (Quotient.functor redStep ⋙ Φ).toPrefunctor).map f)",
"tactic": "simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map, inv_eq_inv]"
},
{
"state_after": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\n⊢ Φ.map (inv ((Quotient.functor redStep).map (Quiver.Hom.toPath f))) =\n inv (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))",
"state_before": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\n⊢ Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath (Quiver.reverse f))) =\n Quiver.reverse (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))",
"tactic": "change Φ.map (inv ((Quotient.functor redStep).toPrefunctor.map f.toPath)) =\n inv (Φ.map ((Quotient.functor redStep).toPrefunctor.map f.toPath))"
},
{
"state_after": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\nthis :\n Φ.map (CategoryTheory.inv ((Quotient.functor redStep).map (Quiver.Hom.toPath f))) =\n CategoryTheory.inv (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))\n⊢ Φ.map (inv ((Quotient.functor redStep).map (Quiver.Hom.toPath f))) =\n inv (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))",
"state_before": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\n⊢ Φ.map (inv ((Quotient.functor redStep).map (Quiver.Hom.toPath f))) =\n inv (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))",
"tactic": "have := Functor.map_inv Φ ((Quotient.functor redStep).toPrefunctor.map f.toPath)"
},
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ✝ φ : V ⥤q V'\nΦ : FreeGroupoid V ⥤ V'\nhΦ : of V ⋙q Φ.toPrefunctor = φ\nX Y : Quiver.Symmetrify V\nf : X ⟶ Y\nthis :\n Φ.map (CategoryTheory.inv ((Quotient.functor redStep).map (Quiver.Hom.toPath f))) =\n CategoryTheory.inv (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))\n⊢ Φ.map (inv ((Quotient.functor redStep).map (Quiver.Hom.toPath f))) =\n inv (Φ.map ((Quotient.functor redStep).map (Quiver.Hom.toPath f)))",
"tactic": "convert this <;> simp only [inv_eq_inv]"
}
] | [
188,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.tendsto_atTop_add_right_of_le' | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.153094\nι' : Type ?u.153097\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.153106\ninst✝ : OrderedAddCommGroup β\nl : Filter α\nf g : α → β\nC : β\nhf : Tendsto f l atTop\nhg : ∀ᶠ (x : α) in l, C ≤ g x\n⊢ ∀ᶠ (x : α) in l, (fun x => -g x) x ≤ -C",
"tactic": "simp [hg]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.153094\nι' : Type ?u.153097\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.153106\ninst✝ : OrderedAddCommGroup β\nl : Filter α\nf g : α → β\nC : β\nhf : Tendsto f l atTop\nhg : ∀ᶠ (x : α) in l, C ≤ g x\n⊢ Tendsto (fun x => (fun x => f x + g x) x + (fun x => -g x) x) l atTop",
"tactic": "simp [hf]"
}
] | [
791,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
788,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean | MvPolynomial.degreeOf_lt_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.284872\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nn : σ\nf : MvPolynomial σ R\nd : ℕ\nh : 0 < d\n⊢ degreeOf n f < d ↔ ∀ (m : σ →₀ ℕ), m ∈ support f → ↑m n < d",
"tactic": "rwa [degreeOf_eq_sup n f, Finset.sup_lt_iff]"
}
] | [
507,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
505,
1
] |
Mathlib/Data/Seq/Seq.lean | Stream'.Seq.enum_nil | [] | [
624,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
623,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.tsum_eq_iSup_nat' | [] | [
843,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
838,
11
] |
Mathlib/Algebra/Hom/Centroid.lean | CentroidHom.coe_one | [] | [
299,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
298,
1
] |
Mathlib/Order/Monotone/Basic.lean | MonotoneOn.dual | [] | [
230,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
228,
11
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.cancel_coe | [] | [
1080,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1079,
1
] |
Mathlib/GroupTheory/Perm/Fin.lean | Equiv.Perm.decomposeFin_symm_apply_zero | [
{
"state_after": "no goals",
"state_before": "n : ℕ\np : Fin (n + 1)\ne : Perm (Fin n)\n⊢ ↑(↑decomposeFin.symm (p, e)) 0 = p",
"tactic": "simp [Equiv.Perm.decomposeFin]"
}
] | [
45,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
44,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | Real.volume_pi_Ico | [] | [
250,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean | Multiset.prod_dvd_prod_of_dvd | [
{
"state_after": "case h₁\nι : Type ?u.85910\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85919\ninst✝ : CommMonoid β\nS : Multiset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\n⊢ prod (map g1 0) ∣ prod (map g2 0)\n\ncase h₂\nι : Type ?u.85910\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85919\ninst✝ : CommMonoid β\nS : Multiset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\n⊢ ∀ {a : α} {s : Multiset α},\n a ∈ S → s ⊆ S → prod (map g1 s) ∣ prod (map g2 s) → prod (map g1 (insert a s)) ∣ prod (map g2 (insert a s))",
"state_before": "ι : Type ?u.85910\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85919\ninst✝ : CommMonoid β\nS : Multiset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\n⊢ prod (map g1 S) ∣ prod (map g2 S)",
"tactic": "apply Multiset.induction_on' S"
},
{
"state_after": "case h₂\nι : Type ?u.85910\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85919\ninst✝ : CommMonoid β\nS : Multiset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\na : α\nT : Multiset α\nhaS : a ∈ S\na✝ : T ⊆ S\nIH : prod (map g1 T) ∣ prod (map g2 T)\n⊢ prod (map g1 (insert a T)) ∣ prod (map g2 (insert a T))",
"state_before": "case h₂\nι : Type ?u.85910\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85919\ninst✝ : CommMonoid β\nS : Multiset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\n⊢ ∀ {a : α} {s : Multiset α},\n a ∈ S → s ⊆ S → prod (map g1 s) ∣ prod (map g2 s) → prod (map g1 (insert a s)) ∣ prod (map g2 (insert a s))",
"tactic": "intro a T haS _ IH"
},
{
"state_after": "no goals",
"state_before": "case h₂\nι : Type ?u.85910\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85919\ninst✝ : CommMonoid β\nS : Multiset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\na : α\nT : Multiset α\nhaS : a ∈ S\na✝ : T ⊆ S\nIH : prod (map g1 T) ∣ prod (map g2 T)\n⊢ prod (map g1 (insert a T)) ∣ prod (map g2 (insert a T))",
"tactic": "simp [mul_dvd_mul (h a haS) IH]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nι : Type ?u.85910\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.85919\ninst✝ : CommMonoid β\nS : Multiset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\n⊢ prod (map g1 0) ∣ prod (map g2 0)",
"tactic": "simp"
}
] | [
257,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
252,
1
] |
Mathlib/Logic/Denumerable.lean | Nat.Subtype.succ_le_of_lt | [
{
"state_after": "α : Type ?u.43600\nβ : Type ?u.43603\ns : Set ℕ\ninst✝¹ : Infinite ↑s\ninst✝ : DecidablePred fun x => x ∈ s\nx y : ↑s\nh : y < x\nhx : ∃ m, ↑y + m + 1 ∈ s\nk : ℕ\nhk : ↑x = ↑y + k + 1\nthis : Nat.find hx ≤ k\n⊢ ↑y + Nat.find hx + 1 ≤ ↑y + k + 1",
"state_before": "α : Type ?u.43600\nβ : Type ?u.43603\ns : Set ℕ\ninst✝¹ : Infinite ↑s\ninst✝ : DecidablePred fun x => x ∈ s\nx y : ↑s\nh : y < x\nhx : ∃ m, ↑y + m + 1 ∈ s\nk : ℕ\nhk : ↑x = ↑y + k + 1\nthis : Nat.find hx ≤ k\n⊢ ↑y + Nat.find hx + 1 ≤ ↑x",
"tactic": "rw [hk]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.43600\nβ : Type ?u.43603\ns : Set ℕ\ninst✝¹ : Infinite ↑s\ninst✝ : DecidablePred fun x => x ∈ s\nx y : ↑s\nh : y < x\nhx : ∃ m, ↑y + m + 1 ∈ s\nk : ℕ\nhk : ↑x = ↑y + k + 1\nthis : Nat.find hx ≤ k\n⊢ ↑y + Nat.find hx + 1 ≤ ↑y + k + 1",
"tactic": "exact add_le_add_right (add_le_add_left this _) _"
}
] | [
251,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
245,
1
] |
Std/Data/Int/DivMod.lean | Int.mul_fmod_right | [
{
"state_after": "no goals",
"state_before": "a b : Int\n⊢ fmod (a * b) a = 0",
"tactic": "rw [Int.mul_comm, mul_fmod_left]"
}
] | [
468,
35
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
467,
9
] |
Mathlib/Data/Nat/Sqrt.lean | Nat.sqrt_le_add | [
{
"state_after": "n : ℕ\n⊢ n ≤ succ (sqrt n) * sqrt n + sqrt n",
"state_before": "n : ℕ\n⊢ n ≤ sqrt n * sqrt n + sqrt n + sqrt n",
"tactic": "rw [← succ_mul]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ n ≤ succ (sqrt n) * sqrt n + sqrt n",
"tactic": "exact le_of_lt_succ (lt_succ_sqrt n)"
}
] | [
84,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/Data/Vector/Basic.lean | Vector.prod_set' | [
{
"state_after": "n : ℕ\nα : Type u_1\nβ : Type ?u.63254\nγ : Type ?u.63257\ninst✝ : CommGroup α\nv : Vector α n\ni : Fin n\na : α\n⊢ (List.prod (toList v) * if hn : ↑i < List.length (toList v) then (List.nthLe (toList v) (↑i) hn)⁻¹ * a else 1) =\n List.prod (toList v) * (get v i)⁻¹ * a",
"state_before": "n : ℕ\nα : Type u_1\nβ : Type ?u.63254\nγ : Type ?u.63257\ninst✝ : CommGroup α\nv : Vector α n\ni : Fin n\na : α\n⊢ List.prod (toList (set v i a)) = List.prod (toList v) * (get v i)⁻¹ * a",
"tactic": "refine' (List.prod_set' v.toList i a).trans _"
},
{
"state_after": "n : ℕ\nα : Type u_1\nβ : Type ?u.63254\nγ : Type ?u.63257\ninst✝ : CommGroup α\nv : Vector α n\ni : Fin n\na : α\n⊢ List.nthLe (toList v) ↑i (_ : ↑i < List.length (toList v)) =\n List.get (toList v) (↑(Fin.cast (_ : n = List.length (toList v))) i)",
"state_before": "n : ℕ\nα : Type u_1\nβ : Type ?u.63254\nγ : Type ?u.63257\ninst✝ : CommGroup α\nv : Vector α n\ni : Fin n\na : α\n⊢ (List.prod (toList v) * if hn : ↑i < List.length (toList v) then (List.nthLe (toList v) (↑i) hn)⁻¹ * a else 1) =\n List.prod (toList v) * (get v i)⁻¹ * a",
"tactic": "simp [get_eq_get, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\nβ : Type ?u.63254\nγ : Type ?u.63257\ninst✝ : CommGroup α\nv : Vector α n\ni : Fin n\na : α\n⊢ List.nthLe (toList v) ↑i (_ : ↑i < List.length (toList v)) =\n List.get (toList v) (↑(Fin.cast (_ : n = List.length (toList v))) i)",
"tactic": "rfl"
}
] | [
623,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
620,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean | one_lt_div' | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : α\n⊢ 1 < a / b ↔ b < a",
"tactic": "rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]"
}
] | [
891,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
890,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean | NormedAddGroupHom.norm_comp_le_of_le' | [
{
"state_after": "V : Type ?u.470292\nV₁ : Type u_3\nV₂ : Type u_1\nV₃ : Type u_2\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g✝ : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nC₁ C₂ C₃ : ℝ\nh : C₃ = C₂ * C₁\nhg : ‖g‖ ≤ C₂\nhf : ‖f‖ ≤ C₁\n⊢ ‖NormedAddGroupHom.comp g f‖ ≤ C₂ * C₁",
"state_before": "V : Type ?u.470292\nV₁ : Type u_3\nV₂ : Type u_1\nV₃ : Type u_2\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g✝ : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nC₁ C₂ C₃ : ℝ\nh : C₃ = C₂ * C₁\nhg : ‖g‖ ≤ C₂\nhf : ‖f‖ ≤ C₁\n⊢ ‖NormedAddGroupHom.comp g f‖ ≤ C₃",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "V : Type ?u.470292\nV₁ : Type u_3\nV₂ : Type u_1\nV₃ : Type u_2\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf g✝ : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nC₁ C₂ C₃ : ℝ\nh : C₃ = C₂ * C₁\nhg : ‖g‖ ≤ C₂\nhf : ‖f‖ ≤ C₁\n⊢ ‖NormedAddGroupHom.comp g f‖ ≤ C₂ * C₁",
"tactic": "exact norm_comp_le_of_le hg hf"
}
] | [
670,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
667,
1
] |
Mathlib/Algebra/Lie/SkewAdjoint.lean | mem_skewAdjointMatricesLieSubalgebra | [] | [
128,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/RingTheory/PowerBasis.lean | PowerBasis.dim_le_natDegree_of_root | [
{
"state_after": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval pb.gen) p = 0\nhlt : natDegree p < pb.dim\n⊢ p = 0",
"state_before": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval pb.gen) p = 0\n⊢ pb.dim ≤ natDegree p",
"tactic": "refine' le_of_not_lt fun hlt => ne_zero _"
},
{
"state_after": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval pb.gen) p = 0\nhlt : natDegree p < pb.dim\n⊢ ∑ i : Fin pb.dim, ↑(monomial ↑i) (coeff p ↑i) = 0",
"state_before": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval pb.gen) p = 0\nhlt : natDegree p < pb.dim\n⊢ p = 0",
"tactic": "rw [p.as_sum_range' _ hlt, Finset.sum_range]"
},
{
"state_after": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval pb.gen) p = 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
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"tactic": "refine' Fintype.sum_eq_zero _ fun i => _"
},
{
"state_after": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\nroot : ∑ x : Fin pb.dim, coeff p ↑x • ↑pb.basis x = 0\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
"state_before": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nroot : ↑(aeval pb.gen) p = 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
"tactic": "simp_rw [aeval_eq_sum_range' hlt, Finset.sum_range, ← pb.basis_eq_pow] at root"
},
{
"state_after": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\nroot : ∑ x : Fin pb.dim, coeff p ↑x • ↑pb.basis x = 0\nthis : ∀ (i : Fin pb.dim), coeff p ↑i = 0\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
"state_before": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\nroot : ∑ x : Fin pb.dim, coeff p ↑x • ↑pb.basis x = 0\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
"tactic": "have := Fintype.linearIndependent_iff.1 pb.basis.linearIndependent _ root"
},
{
"state_after": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\nroot : ∑ x : Fin pb.dim, coeff p ↑x • ↑pb.basis x = 0\nthis : ∀ (i : Fin pb.dim), coeff p ↑i = 0\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
"state_before": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\nroot : ∑ x : Fin pb.dim, coeff p ↑x • ↑pb.basis x = 0\nthis : ∀ (i : Fin pb.dim), coeff p ↑i = 0\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
"tactic": "dsimp only at this"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.115843\nS : Type u_2\nT : Type ?u.115849\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring S\ninst✝⁶ : Algebra R S\nA : Type u_1\nB : Type ?u.116155\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : Algebra A B\nK : Type ?u.116577\ninst✝¹ : Field K\ninst✝ : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p ≠ 0\nhlt : natDegree p < pb.dim\ni : Fin pb.dim\nroot : ∑ x : Fin pb.dim, coeff p ↑x • ↑pb.basis x = 0\nthis : ∀ (i : Fin pb.dim), coeff p ↑i = 0\n⊢ ↑(monomial ↑i) (coeff p ↑i) = 0",
"tactic": "rw [this, monomial_zero_right]"
}
] | [
182,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.Raised.add_left | [
{
"state_after": "case inl\nα : Type ?u.244533\nk m : ℕ\n⊢ Raised (k + m) (k + m)\n\ncase inr\nα : Type ?u.244533\nk n : ℕ\n⊢ Raised (k + n) (k + (n + 1))",
"state_before": "α : Type ?u.244533\nk n m : ℕ\nH : Raised n m\n⊢ Raised (k + n) (k + m)",
"tactic": "rcases H with (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.244533\nk m : ℕ\n⊢ Raised (k + m) (k + m)",
"tactic": "exact Or.inl rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type ?u.244533\nk n : ℕ\n⊢ Raised (k + n) (k + (n + 1))",
"tactic": "exact Or.inr rfl"
}
] | [
803,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
800,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean | AlgEquiv.coe_ringEquiv' | [] | [
200,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean | MulHom.srange_top_iff_surjective | [
{
"state_after": "no goals",
"state_before": "M : Type u_2\nN✝ : Type ?u.61232\nP : Type ?u.61235\nσ : Type ?u.61238\ninst✝³ : Mul M\ninst✝² : Mul N✝\ninst✝¹ : Mul P\nS : Subsemigroup M\nN : Type u_1\ninst✝ : Mul N\nf : M →ₙ* N\n⊢ ↑(srange f) = ↑⊤ ↔ Set.range ↑f = Set.univ",
"tactic": "rw [coe_srange, coe_top]"
}
] | [
785,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
783,
1
] |
Mathlib/Analysis/Complex/Schwarz.lean | Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div' | [] | [
140,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.eventually_atBot_curry | [] | [
1496,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1494,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.toReal_injective | [
{
"state_after": "θ ψ : Angle\nh : toReal θ = toReal ψ\n⊢ θ = ψ",
"state_before": "⊢ Function.Injective toReal",
"tactic": "intro θ ψ h"
},
{
"state_after": "case h\nψ : Angle\nx✝ : ℝ\nh : toReal ↑x✝ = toReal ψ\n⊢ ↑x✝ = ψ",
"state_before": "θ ψ : Angle\nh : toReal θ = toReal ψ\n⊢ θ = ψ",
"tactic": "induction θ using Real.Angle.induction_on"
},
{
"state_after": "case h.h\nx✝¹ x✝ : ℝ\nh : toReal ↑x✝¹ = toReal ↑x✝\n⊢ ↑x✝¹ = ↑x✝",
"state_before": "case h\nψ : Angle\nx✝ : ℝ\nh : toReal ↑x✝ = toReal ψ\n⊢ ↑x✝ = ψ",
"tactic": "induction ψ using Real.Angle.induction_on"
},
{
"state_after": "no goals",
"state_before": "case h.h\nx✝¹ x✝ : ℝ\nh : toReal ↑x✝¹ = toReal ↑x✝\n⊢ ↑x✝¹ = ↑x✝",
"tactic": "simpa [toReal_coe, toIocMod_eq_toIocMod, zsmul_eq_mul, mul_comm _ (2 * π), ←\n angle_eq_iff_two_pi_dvd_sub, eq_comm] using h"
}
] | [
545,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
540,
1
] |
Mathlib/Topology/StoneCech.lean | denseRange_pure | [] | [
131,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Std/Data/AssocList.lean | Std.AssocList.toList_toAssocList | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nl : AssocList α β\n⊢ List.toAssocList (toList l) = l",
"tactic": "induction l <;> simp [*]"
}
] | [
238,
27
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
237,
9
] |
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | Asymptotics.IsEquivalent.mul | [] | [
287,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
286,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasStrictDerivAt.hasDerivAt | [] | [
304,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | Polynomial.cyclotomic'_zero | [
{
"state_after": "no goals",
"state_before": "R✝ : Type ?u.32552\ninst✝³ : CommRing R✝\ninst✝² : IsDomain R✝\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ cyclotomic' 0 R = 1",
"tactic": "simp only [cyclotomic', Finset.prod_empty, primitiveRoots_zero]"
}
] | [
78,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Std/Data/Int/Lemmas.lean | Int.subNatNat_self | [
{
"state_after": "no goals",
"state_before": "m : Nat\n⊢ subNatNat (succ m) (succ m) = 0",
"tactic": "rw [subNatNat_of_sub_eq_zero (Nat.sub_self ..), Nat.sub_self, ofNat_zero]"
}
] | [
293,
91
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
291,
1
] |
Mathlib/Data/Nat/Digits.lean | Nat.ofDigits_digits_append_digits | [
{
"state_after": "no goals",
"state_before": "n✝ b m n : ℕ\n⊢ ofDigits b (digits b n ++ digits b m) = n + b ^ List.length (digits b n) * m",
"tactic": "rw [ofDigits_append, ofDigits_digits, ofDigits_digits]"
}
] | [
429,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
427,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean | Matrix.mul_transvection_apply_same | [
{
"state_after": "no goals",
"state_before": "n : Type u_1\np : Type ?u.14412\nR : Type u₂\n𝕜 : Type ?u.14417\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\na : n\nc : R\nM : Matrix n n R\n⊢ (M ⬝ transvection i j c) a j = M a j + c * M a i",
"tactic": "simp [transvection, Matrix.mul_add, mul_comm]"
}
] | [
131,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Data/Set/Intervals/Group.lean | Set.add_mem_Icc_iff_right | [] | [
85,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Std/Data/Int/Lemmas.lean | Int.zero_lt_one | [] | [
654,
53
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
654,
11
] |
Mathlib/Data/Fintype/Basic.lean | Finset.compl_eq_univ_iff | [] | [
198,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean | lift.SubfieldWithHom.maximalSubfieldWithHom_is_maximal | [] | [
290,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.equivFunOnFintype_single | [
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ns : Finset ι\nx✝ : (i : ↑↑s) → β ↑i\ni✝ : ι\ninst✝ : Fintype ι\ni : ι\nm : β i\nx : ι\n⊢ ↑equivFunOnFintype (single i m) x = Pi.single i m x",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ns : Finset ι\nx : (i : ↑↑s) → β ↑i\ni✝ : ι\ninst✝ : Fintype ι\ni : ι\nm : β i\n⊢ ↑equivFunOnFintype (single i m) = Pi.single i m",
"tactic": "ext x"
},
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ns : Finset ι\nx✝ : (i : ↑↑s) → β ↑i\ni✝ : ι\ninst✝ : Fintype ι\ni : ι\nm : β i\nx : ι\n⊢ ↑(single i m) x = if h : x = i then (_ : i = x) ▸ m else 0",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ns : Finset ι\nx✝ : (i : ↑↑s) → β ↑i\ni✝ : ι\ninst✝ : Fintype ι\ni : ι\nm : β i\nx : ι\n⊢ ↑equivFunOnFintype (single i m) x = Pi.single i m x",
"tactic": "dsimp [Pi.single, Function.update]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ns : Finset ι\nx✝ : (i : ↑↑s) → β ↑i\ni✝ : ι\ninst✝ : Fintype ι\ni : ι\nm : β i\nx : ι\n⊢ ↑(single i m) x = if h : x = i then (_ : i = x) ▸ m else 0",
"tactic": "simp [Dfinsupp.single_eq_pi_single, @eq_comm _ i]"
}
] | [
731,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
727,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | SymmetricRel.inter | [
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.17571\nU V : Set (α × α)\nhU : SymmetricRel U\nhV : SymmetricRel V\n⊢ SymmetricRel (U ∩ V)",
"tactic": "rw [SymmetricRel, preimage_inter, hU.eq, hV.eq]"
}
] | [
240,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
239,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean | MeasureTheory.Lp.ext | [
{
"state_after": "case mk\nα : Type u_1\nE : Type u_2\nF : Type ?u.358556\nG : Type ?u.358559\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : { x // x ∈ Lp E p }\nval✝ : α →ₘ[μ] E\nproperty✝ : val✝ ∈ Lp E p\nh : ↑↑{ val := val✝, property := property✝ } =ᵐ[μ] ↑↑g\n⊢ { val := val✝, property := property✝ } = g",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.358556\nG : Type ?u.358559\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : { x // x ∈ Lp E p }\nh : ↑↑f =ᵐ[μ] ↑↑g\n⊢ f = g",
"tactic": "cases f"
},
{
"state_after": "case mk.mk\nα : Type u_1\nE : Type u_2\nF : Type ?u.358556\nG : Type ?u.358559\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nval✝¹ : α →ₘ[μ] E\nproperty✝¹ : val✝¹ ∈ Lp E p\nval✝ : α →ₘ[μ] E\nproperty✝ : val✝ ∈ Lp E p\nh : ↑↑{ val := val✝¹, property := property✝¹ } =ᵐ[μ] ↑↑{ val := val✝, property := property✝ }\n⊢ { val := val✝¹, property := property✝¹ } = { val := val✝, property := property✝ }",
"state_before": "case mk\nα : Type u_1\nE : Type u_2\nF : Type ?u.358556\nG : Type ?u.358559\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : { x // x ∈ Lp E p }\nval✝ : α →ₘ[μ] E\nproperty✝ : val✝ ∈ Lp E p\nh : ↑↑{ val := val✝, property := property✝ } =ᵐ[μ] ↑↑g\n⊢ { val := val✝, property := property✝ } = g",
"tactic": "cases g"
},
{
"state_after": "case mk.mk\nα : Type u_1\nE : Type u_2\nF : Type ?u.358556\nG : Type ?u.358559\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nval✝¹ : α →ₘ[μ] E\nproperty✝¹ : val✝¹ ∈ Lp E p\nval✝ : α →ₘ[μ] E\nproperty✝ : val✝ ∈ Lp E p\nh : ↑↑{ val := val✝¹, property := property✝¹ } =ᵐ[μ] ↑↑{ val := val✝, property := property✝ }\n⊢ val✝¹ = val✝",
"state_before": "case mk.mk\nα : Type u_1\nE : Type u_2\nF : Type ?u.358556\nG : Type ?u.358559\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nval✝¹ : α →ₘ[μ] E\nproperty✝¹ : val✝¹ ∈ Lp E p\nval✝ : α →ₘ[μ] E\nproperty✝ : val✝ ∈ Lp E p\nh : ↑↑{ val := val✝¹, property := property✝¹ } =ᵐ[μ] ↑↑{ val := val✝, property := property✝ }\n⊢ { val := val✝¹, property := property✝¹ } = { val := val✝, property := property✝ }",
"tactic": "simp only [Subtype.mk_eq_mk]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u_1\nE : Type u_2\nF : Type ?u.358556\nG : Type ?u.358559\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nval✝¹ : α →ₘ[μ] E\nproperty✝¹ : val✝¹ ∈ Lp E p\nval✝ : α →ₘ[μ] E\nproperty✝ : val✝ ∈ Lp E p\nh : ↑↑{ val := val✝¹, property := property✝¹ } =ᵐ[μ] ↑↑{ val := val✝, property := property✝ }\n⊢ val✝¹ = val✝",
"tactic": "exact AEEqFun.ext h"
}
] | [
166,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean | expSeries_eq_expSeries | [
{
"state_after": "no goals",
"state_before": "𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nn : ℕ\nx : 𝔸\n⊢ (↑(expSeries 𝕂 𝔸 n) fun x_1 => x) = ↑(expSeries 𝕂' 𝔸 n) fun x_1 => x",
"tactic": "rw [expSeries_apply_eq, expSeries_apply_eq, inv_nat_cast_smul_eq 𝕂 𝕂']"
}
] | [
655,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
653,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean | ContinuousMap.addEquivBoundedOfCompact_apply | [] | [
97,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/RingTheory/MvPolynomial/Basic.lean | MvPolynomial.mem_restrictTotalDegree | [
{
"state_after": "σ : Type u\nR : Type v\ninst✝ : CommRing R\np✝ m : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ ∀ (b : σ →₀ ℕ), b ∈ support p → (Finsupp.sum b fun x e => e) ≤ m",
"state_before": "σ : Type u\nR : Type v\ninst✝ : CommRing R\np✝ m : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ totalDegree p ≤ m",
"tactic": "rw [totalDegree, Finset.sup_le_iff]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u\nR : Type v\ninst✝ : CommRing R\np✝ m : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ ∀ (b : σ →₀ ℕ), b ∈ support p → (Finsupp.sum b fun x e => e) ≤ m",
"tactic": "rfl"
}
] | [
93,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
90,
1
] |
Mathlib/Data/PNat/Basic.lean | PNat.mod_add_div' | [
{
"state_after": "m k : ℕ+\n⊢ ↑(mod m k) + ↑k * div m k = ↑m",
"state_before": "m k : ℕ+\n⊢ ↑(mod m k) + div m k * ↑k = ↑m",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "m k : ℕ+\n⊢ ↑(mod m k) + ↑k * div m k = ↑m",
"tactic": "exact mod_add_div _ _"
}
] | [
371,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Complex.abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le | [
{
"state_after": "a b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\n⊢ a * ((Real.exp z.re + Real.exp (-z.re)) * Real.cos (Abs.abs z.im)) ≤ a * (Real.exp (Abs.abs z.re) * Real.cos b)",
"state_before": "a b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\n⊢ ↑abs (exp (↑a * (exp z + exp (-z)))) ≤ Real.exp (a * Real.cos b * Real.exp (Abs.abs z.re))",
"tactic": "simp only [abs_exp, Real.exp_le_exp, ofReal_mul_re, add_re, exp_re, neg_im, Real.cos_neg, ←\n add_mul, mul_assoc, mul_comm (Real.cos b), neg_re, ← Real.cos_abs z.im]"
},
{
"state_after": "a b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ a * ((Real.exp z.re + Real.exp (-z.re)) * Real.cos (Abs.abs z.im)) ≤ a * (Real.exp (Abs.abs z.re) * Real.cos b)",
"state_before": "a b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\n⊢ a * ((Real.exp z.re + Real.exp (-z.re)) * Real.cos (Abs.abs z.im)) ≤ a * (Real.exp (Abs.abs z.re) * Real.cos b)",
"tactic": "have : Real.exp (|z.re|) ≤ Real.exp z.re + Real.exp (-z.re) :=\n apply_abs_le_add_of_nonneg (fun x => (Real.exp_pos x).le) z.re"
},
{
"state_after": "case refine'_1\na b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ Real.cos b ≤ Real.cos (Abs.abs z.im)\n\ncase refine'_2\na b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ 0 ≤ Real.cos b",
"state_before": "a b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ a * ((Real.exp z.re + Real.exp (-z.re)) * Real.cos (Abs.abs z.im)) ≤ a * (Real.exp (Abs.abs z.re) * Real.cos b)",
"tactic": "refine' mul_le_mul_of_nonpos_left (mul_le_mul this _ _ ((Real.exp_pos _).le.trans this)) ha"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\na b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ Real.cos b ≤ Real.cos (Abs.abs z.im)",
"tactic": "exact\n Real.cos_le_cos_of_nonneg_of_le_pi (_root_.abs_nonneg _)\n (hb.trans <| half_le_self <| Real.pi_pos.le) hz"
},
{
"state_after": "case refine'_2\na b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ -(π / 2) ≤ b",
"state_before": "case refine'_2\na b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ 0 ≤ Real.cos b",
"tactic": "refine' Real.cos_nonneg_of_mem_Icc ⟨_, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\na b : ℝ\nha : a ≤ 0\nz : ℂ\nhz : Abs.abs z.im ≤ b\nhb : b ≤ π / 2\nthis : Real.exp (Abs.abs z.re) ≤ Real.exp z.re + Real.exp (-z.re)\n⊢ -(π / 2) ≤ b",
"tactic": "exact (neg_nonpos.2 <| Real.pi_div_two_pos.le).trans ((_root_.abs_nonneg _).trans hz)"
}
] | [
1419,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1407,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.closure_cycle_adjacent_swap | [
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"tactic": "let H := closure ({σ, swap x (σ x)} : Set (Perm α))"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"tactic": "have h3 : σ ∈ H := subset_closure (Set.mem_insert σ _)"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"tactic": "have h4 : swap x (σ x) ∈ H := subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"tactic": "have step3 : ∀ y : α, swap x y ∈ H := by\n intro y\n have hx : x ∈ (⊤ : Finset α) := Finset.mem_univ x\n rw [← h2, mem_support] at hx\n have hy : y ∈ (⊤ : Finset α) := Finset.mem_univ y\n rw [← h2, mem_support] at hy\n cases' IsCycle.exists_pow_eq h1 hx hy with n hn\n rw [← hn]\n exact step2 n"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\nstep4 : ∀ (y z : α), swap y z ∈ H\n⊢ {σ | IsSwap σ} ⊆ ↑(closure {σ, swap x (↑σ x)})",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\nstep4 : ∀ (y z : α), swap y z ∈ H\n⊢ closure {σ, swap x (↑σ x)} = ⊤",
"tactic": "rw [eq_top_iff, ← closure_isSwap, closure_le]"
},
{
"state_after": "case intro.intro.intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\nstep4 : ∀ (y z : α), swap y z ∈ H\nτ : Perm α\ny z : α\nleft✝ : y ≠ z\nh6 : τ = swap y z\n⊢ τ ∈ ↑(closure {σ, swap x (↑σ x)})",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\nstep4 : ∀ (y z : α), swap y z ∈ H\n⊢ {σ | IsSwap σ} ⊆ ↑(closure {σ, swap x (↑σ x)})",
"tactic": "rintro τ ⟨y, z, _, h6⟩"
},
{
"state_after": "case intro.intro.intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\nstep4 : ∀ (y z : α), swap y z ∈ H\nτ : Perm α\ny z : α\nleft✝ : y ≠ z\nh6 : τ = swap y z\n⊢ swap y z ∈ ↑(closure {σ, swap x (↑σ x)})",
"state_before": "case intro.intro.intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\nstep4 : ∀ (y z : α), swap y z ∈ H\nτ : Perm α\ny z : α\nleft✝ : y ≠ z\nh6 : τ = swap y z\n⊢ τ ∈ ↑(closure {σ, swap x (↑σ x)})",
"tactic": "rw [h6]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\nstep4 : ∀ (y z : α), swap y z ∈ H\nτ : Perm α\ny z : α\nleft✝ : y ≠ z\nh6 : τ = swap y z\n⊢ swap y z ∈ ↑(closure {σ, swap x (↑σ x)})",
"tactic": "exact step4 y z"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\n⊢ swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\n⊢ ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H",
"tactic": "intro n"
},
{
"state_after": "case zero\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\n⊢ swap (↑(σ ^ Nat.zero) x) (↑(σ ^ (Nat.zero + 1)) x) ∈ H\n\ncase succ\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\nih : swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap (↑(σ ^ Nat.succ n) x) (↑(σ ^ (Nat.succ n + 1)) x) ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\n⊢ swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\n⊢ swap (↑(σ ^ Nat.zero) x) (↑(σ ^ (Nat.zero + 1)) x) ∈ H",
"tactic": "exact subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))"
},
{
"state_after": "case h.e'_4\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\nih : swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap (↑(σ ^ Nat.succ n) x) (↑(σ ^ (Nat.succ n + 1)) x) = σ * swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) * σ⁻¹",
"state_before": "case succ\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\nih : swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap (↑(σ ^ Nat.succ n) x) (↑(σ ^ (Nat.succ n + 1)) x) ∈ H",
"tactic": "convert H.mul_mem (H.mul_mem h3 ih) (H.inv_mem h3)"
},
{
"state_after": "case h.e'_4\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\nih : swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap (↑(σ * σ ^ n) x) (↑(σ * (σ * σ ^ n)) x) = swap (↑σ (↑(σ ^ n) x)) (↑σ (↑(σ * σ ^ n) x))",
"state_before": "case h.e'_4\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\nih : swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap (↑(σ ^ Nat.succ n) x) (↑(σ ^ (Nat.succ n + 1)) x) = σ * swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) * σ⁻¹",
"tactic": "simp_rw [mul_swap_eq_swap_mul, mul_inv_cancel_right, pow_succ]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nn : ℕ\nih : swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap (↑(σ * σ ^ n) x) (↑(σ * (σ * σ ^ n)) x) = swap (↑σ (↑(σ ^ n) x)) (↑σ (↑(σ * σ ^ n) x))",
"tactic": "rfl"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\n⊢ swap x (↑(σ ^ n) x) ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H",
"tactic": "intro n"
},
{
"state_after": "case zero\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap x (↑(σ ^ Nat.zero) x) ∈ H\n\ncase succ\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\n⊢ swap x (↑(σ ^ n) x) ∈ H",
"tactic": "induction' n with n ih"
},
{
"state_after": "case zero\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ Equiv.refl α ∈ closure {σ, swap x (↑σ x)}",
"state_before": "case zero\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ swap x (↑(σ ^ Nat.zero) x) ∈ H",
"tactic": "simp only [Nat.zero_eq, pow_zero, coe_one, id_eq, swap_self, Set.mem_singleton_iff]"
},
{
"state_after": "no goals",
"state_before": "case zero\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\n⊢ Equiv.refl α ∈ closure {σ, swap x (↑σ x)}",
"tactic": "convert H.one_mem"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : x = ↑(σ ^ n) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H\n\ncase neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"state_before": "case succ\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"tactic": "by_cases h5 : x = (σ ^ n) x"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : x = ↑(σ ^ (n + 1)) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H\n\ncase neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : ¬x = ↑(σ ^ (n + 1)) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"state_before": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"tactic": "by_cases h6 : x = (σ ^ (n + 1) : Perm α) x"
},
{
"state_after": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : ¬x = ↑(σ ^ (n + 1)) x\n⊢ swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) * swap x (↑(σ ^ n) x) * swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H",
"state_before": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : ¬x = ↑(σ ^ (n + 1)) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"tactic": "rw [swap_comm, ← swap_mul_swap_mul_swap h5 h6]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : ¬x = ↑(σ ^ (n + 1)) x\n⊢ swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) * swap x (↑(σ ^ n) x) * swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H",
"tactic": "exact H.mul_mem (H.mul_mem (step1 n) ih) (step1 n)"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : x = ↑(σ ^ n) x\n⊢ swap x (↑σ x) ∈ H",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : x = ↑(σ ^ n) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"tactic": "rw [pow_succ, mul_apply, ← h5]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : x = ↑(σ ^ n) x\n⊢ swap x (↑σ x) ∈ H",
"tactic": "exact h4"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : x = ↑(σ ^ (n + 1)) x\n⊢ Equiv.refl α ∈ H",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : x = ↑(σ ^ (n + 1)) x\n⊢ swap x (↑(σ ^ Nat.succ n) x) ∈ H",
"tactic": "rw [← h6, swap_self]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nn : ℕ\nih : swap x (↑(σ ^ n) x) ∈ H\nh5 : ¬x = ↑(σ ^ n) x\nh6 : x = ↑(σ ^ (n + 1)) x\n⊢ Equiv.refl α ∈ H",
"tactic": "exact H.one_mem"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\n⊢ swap x y ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\n⊢ ∀ (y : α), swap x y ∈ H",
"tactic": "intro y"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : x ∈ ⊤\n⊢ swap x y ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\n⊢ swap x y ∈ H",
"tactic": "have hx : x ∈ (⊤ : Finset α) := Finset.mem_univ x"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\n⊢ swap x y ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : x ∈ ⊤\n⊢ swap x y ∈ H",
"tactic": "rw [← h2, mem_support] at hx"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : y ∈ ⊤\n⊢ swap x y ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\n⊢ swap x y ∈ H",
"tactic": "have hy : y ∈ (⊤ : Finset α) := Finset.mem_univ y"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : ↑σ y ≠ y\n⊢ swap x y ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : y ∈ ⊤\n⊢ swap x y ∈ H",
"tactic": "rw [← h2, mem_support] at hy"
},
{
"state_after": "case intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : ↑σ y ≠ y\nn : ℕ\nhn : ↑(σ ^ n) x = y\n⊢ swap x y ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : ↑σ y ≠ y\n⊢ swap x y ∈ H",
"tactic": "cases' IsCycle.exists_pow_eq h1 hx hy with n hn"
},
{
"state_after": "case intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : ↑σ y ≠ y\nn : ℕ\nhn : ↑(σ ^ n) x = y\n⊢ swap x (↑(σ ^ n) x) ∈ H",
"state_before": "case intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : ↑σ y ≠ y\nn : ℕ\nhn : ↑(σ ^ n) x = y\n⊢ swap x y ∈ H",
"tactic": "rw [← hn]"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\ny : α\nhx : ↑σ x ≠ x\nhy : ↑σ y ≠ y\nn : ℕ\nhn : ↑(σ ^ n) x = y\n⊢ swap x (↑(σ ^ n) x) ∈ H",
"tactic": "exact step2 n"
},
{
"state_after": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\n⊢ swap y z ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\n⊢ ∀ (y z : α), swap y z ∈ H",
"tactic": "intro y z"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : z = x\n⊢ swap y z ∈ H\n\ncase neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\n⊢ swap y z ∈ H",
"state_before": "ι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\n⊢ swap y z ∈ H",
"tactic": "by_cases h5 : z = x"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : z = y\n⊢ swap y z ∈ H\n\ncase neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : ¬z = y\n⊢ swap y z ∈ H",
"state_before": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\n⊢ swap y z ∈ H",
"tactic": "by_cases h6 : z = y"
},
{
"state_after": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : ¬z = y\n⊢ swap x y * swap x z * swap x y ∈ H",
"state_before": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : ¬z = y\n⊢ swap y z ∈ H",
"tactic": "rw [← swap_mul_swap_mul_swap h5 h6, swap_comm z x]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : ¬z = y\n⊢ swap x y * swap x z * swap x y ∈ H",
"tactic": "exact H.mul_mem (H.mul_mem (step3 y) (step3 z)) (step3 y)"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : z = x\n⊢ swap x y ∈ H",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : z = x\n⊢ swap y z ∈ H",
"tactic": "rw [h5, swap_comm]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : z = x\n⊢ swap x y ∈ H",
"tactic": "exact step3 y"
},
{
"state_after": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : z = y\n⊢ Equiv.refl α ∈ H",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : z = y\n⊢ swap y z ∈ H",
"tactic": "rw [h6, swap_self]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.2796652\nα : Type u_1\nβ : Type ?u.2796658\ninst✝² : DecidableEq α\ninst✝¹ : Finite β\ninst✝ : Fintype α\nσ : Perm α\nh1 : IsCycle σ\nh2 : support σ = ⊤\nx : α\nH : Subgroup (Perm α) := closure {σ, swap x (↑σ x)}\nh3 : σ ∈ H\nh4 : swap x (↑σ x) ∈ H\nstep1 : ∀ (n : ℕ), swap (↑(σ ^ n) x) (↑(σ ^ (n + 1)) x) ∈ H\nstep2 : ∀ (n : ℕ), swap x (↑(σ ^ n) x) ∈ H\nstep3 : ∀ (y : α), swap x y ∈ H\ny z : α\nh5 : ¬z = x\nh6 : z = y\n⊢ Equiv.refl α ∈ H",
"tactic": "exact H.one_mem"
}
] | [
1665,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1618,
1
] |
Mathlib/RingTheory/Adjoin/FG.lean | Subalgebra.fg_of_submodule_fg | [
{
"state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nh : Submodule.FG ⊤\ns : Finset A\nhs : span R ↑s = ⊤\n⊢ ↑s ⊆ ↑(↑toSubmodule (Algebra.adjoin R ↑s))",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nh : Submodule.FG ⊤\ns : Finset A\nhs : span R ↑s = ⊤\n⊢ ↑toSubmodule (Algebra.adjoin R ↑s) = ↑toSubmodule ⊤",
"tactic": "rw [Algebra.top_toSubmodule, eq_top_iff, ← hs, span_le]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nh : Submodule.FG ⊤\ns : Finset A\nhs : span R ↑s = ⊤\n⊢ ↑s ⊆ ↑(↑toSubmodule (Algebra.adjoin R ↑s))",
"tactic": "exact Algebra.subset_adjoin"
}
] | [
130,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
] |
Mathlib/FieldTheory/Subfield.lean | SubfieldClass.coe_rat_mem | [
{
"state_after": "no goals",
"state_before": "K : Type u\nL : Type v\nM : Type w\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Field M\nS : Type u_1\ninst✝ : SetLike S K\nh : SubfieldClass S K\ns : S\nx : ℚ\n⊢ ↑x ∈ s",
"tactic": "simpa only [Rat.cast_def] using div_mem (coe_int_mem s x.num) (coe_nat_mem s x.den)"
}
] | [
95,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Computability/Halting.lean | Nat.Partrec'.part_iff | [] | [
416,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
415,
1
] |
Mathlib/RingTheory/LaurentSeries.lean | PowerSeries.coe_zero | [] | [
189,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | Real.tan_arcsin | [
{
"state_after": "x : ℝ\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"state_before": "x : ℝ\n⊢ tan (arcsin x) = x / sqrt (1 - x ^ 2)",
"tactic": "rw [tan_eq_sin_div_cos, cos_arcsin]"
},
{
"state_after": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"state_before": "x : ℝ\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "by_cases hx₁ : -1 ≤ x"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)\n\ncase pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "swap"
},
{
"state_after": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "by_cases hx₂ : x ≤ 1"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)\n\ncase pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)\n\ncase neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "swap"
},
{
"state_after": "no goals",
"state_before": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "rw [sin_arcsin hx₁ hx₂]"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"state_before": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "have h : sqrt (1 - x ^ 2) = 0 := sqrt_eq_zero_of_nonpos (by nlinarith)"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / 0 = x / 0",
"state_before": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nx : ℝ\nhx₁ : ¬-1 ≤ x\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / 0 = x / 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx₁ : ¬-1 ≤ x\n⊢ 1 - x ^ 2 ≤ 0",
"tactic": "nlinarith"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"state_before": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "have h : sqrt (1 - x ^ 2) = 0 := sqrt_eq_zero_of_nonpos (by nlinarith)"
},
{
"state_after": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / 0 = x / 0",
"state_before": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / sqrt (1 - x ^ 2) = x / sqrt (1 - x ^ 2)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\nh : sqrt (1 - x ^ 2) = 0\n⊢ sin (arcsin x) / 0 = x / 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx₁ : -1 ≤ x\nhx₂ : ¬x ≤ 1\n⊢ 1 - x ^ 2 ≤ 0",
"tactic": "nlinarith"
}
] | [
332,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | mul_le_mul_left | [] | [
213,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
212,
1
] |
Mathlib/Data/Pi/Algebra.lean | Pi.inv_apply | [] | [
175,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
] |
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