file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Tactic/CategoryTheory/Elementwise.lean
|
Tactic.Elementwise.forget_hom_Type
|
[] |
[
43,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Finset.exists_sum_fiber_lt_of_sum_fiber_nonneg_of_sum_lt_nsmul
|
[] |
[
157,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Units.conj_pow
|
[] |
[
1241,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1237,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.completeTheory.mem_or_not_mem
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nM : Type w\ninst✝ : Structure L M\nφ : Sentence L\n⊢ φ ∈ completeTheory L M ∨ Formula.not φ ∈ completeTheory L M",
"tactic": "simp_rw [completeTheory, Set.mem_setOf_eq, Sentence.Realize, Formula.realize_not, or_not]"
}
] |
[
505,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Algebra/Parity.lean
|
even_bit0
|
[] |
[
272,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
9
] |
Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean
|
not_intervalIntegrable_of_sub_inv_isBigO_punctured
|
[
{
"state_after": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ ¬IntervalIntegrable f volume a b",
"state_before": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\n⊢ ¬IntervalIntegrable f volume a b",
"tactic": "have A : ∀ᶠ x in 𝓝[≠] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x := by\n filter_upwards [self_mem_nhdsWithin] with x hx\n simpa using ((hasDerivAt_id x).sub_const c).log (sub_ne_zero.2 hx)"
},
{
"state_after": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\nB : Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[{c}ᶜ] c) atTop\n⊢ ¬IntervalIntegrable f volume a b",
"state_before": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ ¬IntervalIntegrable f volume a b",
"tactic": "have B : Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[≠] c) atTop := by\n refine' tendsto_abs_atBot_atTop.comp (Real.tendsto_log_nhdsWithin_zero.comp _)\n rw [← sub_self c]\n exact ((hasDerivAt_id c).sub_const c).tendsto_punctured_nhds one_ne_zero"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\nB : Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[{c}ᶜ] c) atTop\n⊢ ¬IntervalIntegrable f volume a b",
"tactic": "exact not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured\n (A.mono fun x hx => hx.differentiableAt) B\n (hf.congr' (A.mono fun x hx => hx.deriv.symm) EventuallyEq.rfl) hne hc"
},
{
"state_after": "case h\nE : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nx : ℝ\nhx : x ∈ {c}ᶜ\n⊢ HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x",
"state_before": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\n⊢ ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x",
"tactic": "filter_upwards [self_mem_nhdsWithin] with x hx"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nx : ℝ\nhx : x ∈ {c}ᶜ\n⊢ HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x",
"tactic": "simpa using ((hasDerivAt_id x).sub_const c).log (sub_ne_zero.2 hx)"
},
{
"state_after": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ Tendsto (fun x => x - c) (𝓝[{c}ᶜ] c) (𝓝[{0}ᶜ] 0)",
"state_before": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ Tendsto (fun x => ‖Real.log (x - c)‖) (𝓝[{c}ᶜ] c) atTop",
"tactic": "refine' tendsto_abs_atBot_atTop.comp (Real.tendsto_log_nhdsWithin_zero.comp _)"
},
{
"state_after": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ Tendsto (fun x => x - c) (𝓝[{c}ᶜ] c) (𝓝[{c - c}ᶜ] (c - c))",
"state_before": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ Tendsto (fun x => x - c) (𝓝[{c}ᶜ] c) (𝓝[{0}ᶜ] 0)",
"tactic": "rw [← sub_self c]"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.29629\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x => (x - c)⁻¹) =O[𝓝[{c}ᶜ] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[{c}ᶜ] c, HasDerivAt (fun x => Real.log (x - c)) (x - c)⁻¹ x\n⊢ Tendsto (fun x => x - c) (𝓝[{c}ᶜ] c) (𝓝[{c - c}ᶜ] (c - c))",
"tactic": "exact ((hasDerivAt_id c).sub_const c).tendsto_punctured_nhds one_ne_zero"
}
] |
[
155,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.sum.map_inv
|
[] |
[
970,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
969,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_eq_factors_multiset
|
[
{
"state_after": "case h\nn p : ℕ\n⊢ ↑(factorization n) p = ↑(↑Multiset.toFinsupp ↑(factors n)) p",
"state_before": "n : ℕ\n⊢ factorization n = ↑Multiset.toFinsupp ↑(factors n)",
"tactic": "ext p"
},
{
"state_after": "case h\nn p : ℕ\n⊢ ↑(factorization n) p = count p (factors n)",
"state_before": "case h\nn p : ℕ\n⊢ ↑(factorization n) p = ↑(↑Multiset.toFinsupp ↑(factors n)) p",
"tactic": "simp only [Multiset.toFinsupp_apply, Multiset.mem_coe, Multiset.coe_nodup, Multiset.coe_count]"
},
{
"state_after": "no goals",
"state_before": "case h\nn p : ℕ\n⊢ ↑(factorization n) p = count p (factors n)",
"tactic": "rw [@factors_count_eq n p]"
}
] |
[
94,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.map_comp
|
[] |
[
754,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/Data/PFunctor/Univariate/Basic.lean
|
PFunctor.liftp_iff
|
[
{
"state_after": "case mp\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\n⊢ Liftp p x → ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)\n\ncase mpr\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\n⊢ (∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)) → Liftp p x",
"state_before": "P : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\n⊢ Liftp p x ↔ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)",
"tactic": "constructor"
},
{
"state_after": "case mpr.intro.intro.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\na : P.A\nf : B P a → α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : B P a), p (f i)\n⊢ Liftp p x",
"state_before": "case mpr\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\n⊢ (∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)) → Liftp p x",
"tactic": "rintro ⟨a, f, xeq, pf⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\na : P.A\nf : B P a → α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : B P a), p (f i)\n⊢ Subtype.val <$> { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } } = x",
"state_before": "case mpr.intro.intro.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\na : P.A\nf : B P a → α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : B P a), p (f i)\n⊢ Liftp p x",
"tactic": "use ⟨a, fun i => ⟨f i, pf i⟩⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\na : P.A\nf : B P a → α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : B P a), p (f i)\n⊢ Subtype.val <$> { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } } = { fst := a, snd := f }",
"state_before": "case mpr.intro.intro.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\na : P.A\nf : B P a → α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : B P a), p (f i)\n⊢ Subtype.val <$> { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } } = x",
"tactic": "rw [xeq]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\na : P.A\nf : B P a → α\nxeq : x = { fst := a, snd := f }\npf : ∀ (i : B P a), p (f i)\n⊢ Subtype.val <$> { fst := a, snd := fun i => { val := f i, property := (_ : p (f i)) } } = { fst := a, snd := f }",
"tactic": "rfl"
},
{
"state_after": "case mp.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)",
"state_before": "case mp\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\n⊢ Liftp p x → ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)",
"tactic": "rintro ⟨y, hy⟩"
},
{
"state_after": "case mp.intro.mk\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\na : P.A\nf : B P a → Subtype p\nh : y = { fst := a, snd := f }\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)",
"state_before": "case mp.intro\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)",
"tactic": "cases' h : y with a f"
},
{
"state_after": "case mp.intro.mk\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\na : P.A\nf : B P a → Subtype p\nh : y = { fst := a, snd := f }\n⊢ x = { fst := a, snd := fun i => ↑(f i) }",
"state_before": "case mp.intro.mk\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\na : P.A\nf : B P a → Subtype p\nh : y = { fst := a, snd := f }\n⊢ ∃ a f, x = { fst := a, snd := f } ∧ ∀ (i : B P a), p (f i)",
"tactic": "refine' ⟨a, fun i => (f i).val, _, fun i => (f i).property⟩"
},
{
"state_after": "case mp.intro.mk\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\na : P.A\nf : B P a → Subtype p\nh : y = { fst := a, snd := f }\n⊢ { fst := a, snd := Subtype.val ∘ f } = { fst := a, snd := fun i => ↑(f i) }",
"state_before": "case mp.intro.mk\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\na : P.A\nf : B P a → Subtype p\nh : y = { fst := a, snd := f }\n⊢ x = { fst := a, snd := fun i => ↑(f i) }",
"tactic": "rw [← hy, h, PFunctor.map_eq]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.mk\nP : PFunctor\nα : Type u\np : α → Prop\nx : Obj P α\ny : Obj P (Subtype p)\nhy : Subtype.val <$> y = x\na : P.A\nf : B P a → Subtype p\nh : y = { fst := a, snd := f }\n⊢ { fst := a, snd := Subtype.val ∘ f } = { fst := a, snd := fun i => ↑(f i) }",
"tactic": "congr"
}
] |
[
197,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.proj_apply
|
[] |
[
1247,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1246,
1
] |
Mathlib/Analysis/MeanInequalities.lean
|
Real.young_inequality_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\na b p q : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhpq : IsConjugateExponent p q\n⊢ a * b ≤ a ^ p / p + b ^ q / q",
"tactic": "simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using\n geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg\n (rpow_nonneg_of_nonneg ha p) (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj"
}
] |
[
259,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/CategoryTheory/Functor/EpiMono.lean
|
CategoryTheory.Functor.preservesMonomorphisms_of_adjunction
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : D\nf : X ⟶ Y\nhf : Mono f\nZ : C\ng h : Z ⟶ G.obj X\nH : g ≫ G.map f = h ≫ G.map f\n⊢ g = h",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : D\nf : X ⟶ Y\nhf : Mono f\n⊢ ∀ {Z : C} (g h : Z ⟶ G.obj X), g ≫ G.map f = h ≫ G.map f → g = h",
"tactic": "intro Z g h H"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : D\nf : X ⟶ Y\nhf : Mono f\nZ : C\ng h : Z ⟶ G.obj X\nH : ↑(Adjunction.homEquiv adj Z Y).symm (g ≫ G.map f) = ↑(Adjunction.homEquiv adj Z Y).symm (h ≫ G.map f)\n⊢ g = h",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : D\nf : X ⟶ Y\nhf : Mono f\nZ : C\ng h : Z ⟶ G.obj X\nH : g ≫ G.map f = h ≫ G.map f\n⊢ g = h",
"tactic": "replace H := congr_arg (adj.homEquiv Z Y).symm H"
},
{
"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 : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : D\nf : X ⟶ Y\nhf : Mono f\nZ : C\ng h : Z ⟶ G.obj X\nH : ↑(Adjunction.homEquiv adj Z Y).symm (g ≫ G.map f) = ↑(Adjunction.homEquiv adj Z Y).symm (h ≫ G.map f)\n⊢ g = h",
"tactic": "rwa [adj.homEquiv_naturality_right_symm, adj.homEquiv_naturality_right_symm, cancel_mono,\n Equiv.apply_eq_iff_eq] at H"
}
] |
[
200,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.compl_empty
|
[] |
[
183,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.sumComm_symm
|
[] |
[
344,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/CategoryTheory/Monoidal/Free/Basic.lean
|
CategoryTheory.FreeMonoidalCategory.mk_comp
|
[] |
[
190,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.normalizedFactors_pow
|
[
{
"state_after": "case zero\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\n⊢ normalizedFactors (x ^ Nat.zero) = Nat.zero • normalizedFactors x\n\ncase succ\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\nih : normalizedFactors (x ^ n) = n • normalizedFactors x\n⊢ normalizedFactors (x ^ Nat.succ n) = Nat.succ n • normalizedFactors x",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\n⊢ normalizedFactors (x ^ n) = n • normalizedFactors x",
"tactic": "induction' n with n ih"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\nih : normalizedFactors (x ^ n) = n • normalizedFactors x\nh0 : x = 0\n⊢ normalizedFactors (x ^ Nat.succ n) = Nat.succ n • normalizedFactors x\n\ncase neg\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\nih : normalizedFactors (x ^ n) = n • normalizedFactors x\nh0 : ¬x = 0\n⊢ normalizedFactors (x ^ Nat.succ n) = Nat.succ n • normalizedFactors x",
"state_before": "case succ\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\nih : normalizedFactors (x ^ n) = n • normalizedFactors x\n⊢ normalizedFactors (x ^ Nat.succ n) = Nat.succ n • normalizedFactors x",
"tactic": "by_cases h0 : x = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\nih : normalizedFactors (x ^ n) = n • normalizedFactors x\nh0 : ¬x = 0\n⊢ normalizedFactors (x ^ Nat.succ n) = Nat.succ n • normalizedFactors x",
"tactic": "rw [pow_succ, succ_nsmul, normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\n⊢ normalizedFactors (x ^ Nat.zero) = Nat.zero • normalizedFactors x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\nx : α\nn : ℕ\nih : normalizedFactors (x ^ n) = n • normalizedFactors x\nh0 : x = 0\n⊢ normalizedFactors (x ^ Nat.succ n) = Nat.succ n • normalizedFactors x",
"tactic": "simp [h0, zero_pow n.succ_pos, smul_zero]"
}
] |
[
697,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.induction_on'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nP : MvPolynomial σ R → Prop\np : MvPolynomial σ R\nh1 : ∀ (u : σ →₀ ℕ) (a : R), P (↑(monomial u) a)\nh2 : ∀ (p q : MvPolynomial σ R), P p → P q → P (p + q)\nthis : P (↑(monomial 0) 0)\n⊢ P 0",
"tactic": "rwa [monomial_zero] at this"
}
] |
[
420,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.card_replicate
|
[] |
[
914,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
913,
9
] |
Mathlib/Data/Set/Function.lean
|
Set.image_restrict
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1357\nι : Sort ?u.1360\nπ : α → Type ?u.1365\nf : α → β\ns t : Set α\n⊢ restrict s f '' (Subtype.val ⁻¹' t) = f '' (t ∩ s)",
"tactic": "rw [restrict_eq, image_comp, image_preimage_eq_inter_range, Subtype.range_coe]"
}
] |
[
79,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.liftNC_one
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type u_2\ninst✝³ : NonAssocSemiring R\ninst✝² : Semiring k\ninst✝¹ : One G\ng_hom : Type u_1\ninst✝ : OneHomClass g_hom G R\nf : k →+* R\ng : g_hom\n⊢ ↑(liftNC ↑f ↑g) 1 = 1",
"tactic": "simp [one_def]"
}
] |
[
247,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.coeff_eq_zero_of_lt_order
|
[
{
"state_after": "case inl\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\ni : Γ\nhi : i < order 0\n⊢ coeff 0 i = 0\n\ncase inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhi : i < order x\nhx : x ≠ 0\n⊢ coeff x i = 0",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhi : i < order x\n⊢ coeff x i = 0",
"tactic": "rcases eq_or_ne x 0 with (rfl | hx)"
},
{
"state_after": "case inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhx : x ≠ 0\nhi : coeff x i ≠ 0\n⊢ ¬i < order x",
"state_before": "case inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhi : i < order x\nhx : x ≠ 0\n⊢ coeff x i = 0",
"tactic": "contrapose! hi"
},
{
"state_after": "case inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhx : x ≠ 0\nhi : i ∈ support x\n⊢ ¬i < order x",
"state_before": "case inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhx : x ≠ 0\nhi : coeff x i ≠ 0\n⊢ ¬i < order x",
"tactic": "rw [← mem_support] at hi"
},
{
"state_after": "case inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhx : x ≠ 0\nhi : i ∈ support x\n⊢ ¬i < Set.IsWf.min (_ : Set.IsWf (support x)) (_ : Set.Nonempty (support x))",
"state_before": "case inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhx : x ≠ 0\nhi : i ∈ support x\n⊢ ¬i < order x",
"tactic": "rw [order_of_ne hx]"
},
{
"state_after": "no goals",
"state_before": "case inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\nx : HahnSeries Γ R\ni : Γ\nhx : x ≠ 0\nhi : i ∈ support x\n⊢ ¬i < Set.IsWf.min (_ : Set.IsWf (support x)) (_ : Set.Nonempty (support x))",
"tactic": "exact Set.IsWf.not_lt_min _ _ hi"
},
{
"state_after": "no goals",
"state_before": "case inl\nΓ : Type u_1\nR : Type u_2\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\na b : Γ\nr : R\ninst✝ : Zero Γ\ni : Γ\nhi : i < order 0\n⊢ coeff 0 i = 0",
"tactic": "simp"
}
] |
[
259,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Topology/Algebra/Order/ProjIcc.lean
|
continuous_IccExtend_iff
|
[] |
[
53,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Analysis/NormedSpace/Extend.lean
|
LinearMap.extendTo𝕜'_apply_re
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : Module ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →ₗ[ℝ] ℝ\nx : F\n⊢ ↑re (↑(extendTo𝕜' fr) x) = ↑fr x",
"tactic": "simp only [extendTo𝕜'_apply, map_sub, MulZeroClass.zero_mul, MulZeroClass.mul_zero, sub_zero,\n isROrC_simps]"
}
] |
[
93,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.inter_subset_left
|
[] |
[
1586,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1586,
1
] |
Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
|
Multiset.Nat.antidiagonalTuple_two
|
[] |
[
221,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.darts_nodup_of_support_nodup
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nh : List.Nodup (support p)\n⊢ List.Nodup (darts p)",
"tactic": "induction p with\n| nil => simp\n| cons _ p' ih =>\n simp only [darts_cons, support_cons, List.nodup_cons] at h ⊢\n exact ⟨fun h' => h.1 (dart_fst_mem_support_of_mem_darts p' h'), ih h.2⟩"
},
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ : V\nh : List.Nodup (support nil)\n⊢ List.Nodup (darts nil)",
"tactic": "simp"
},
{
"state_after": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np' : Walk G v✝ w✝\nih : List.Nodup (support p') → List.Nodup (darts p')\nh : ¬u✝ ∈ support p' ∧ List.Nodup (support p')\n⊢ ¬{ toProd := (u✝, v✝), is_adj := h✝ } ∈ darts p' ∧ List.Nodup (darts p')",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np' : Walk G v✝ w✝\nih : List.Nodup (support p') → List.Nodup (darts p')\nh : List.Nodup (support (cons h✝ p'))\n⊢ List.Nodup (darts (cons h✝ p'))",
"tactic": "simp only [darts_cons, support_cons, List.nodup_cons] at h ⊢"
},
{
"state_after": "no goals",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np' : Walk G v✝ w✝\nih : List.Nodup (support p') → List.Nodup (darts p')\nh : ¬u✝ ∈ support p' ∧ List.Nodup (support p')\n⊢ ¬{ toProd := (u✝, v✝), is_adj := h✝ } ∈ darts p' ∧ List.Nodup (darts p')",
"tactic": "exact ⟨fun h' => h.1 (dart_fst_mem_support_of_mem_darts p' h'), ih h.2⟩"
}
] |
[
819,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_mul_eq_iff_eq_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.54779\nβ : Type ?u.54782\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : a⁻¹ * b = c\n⊢ b = a * c",
"tactic": "rw [← h, mul_inv_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.54779\nβ : Type ?u.54782\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : b = a * c\n⊢ a⁻¹ * b = c",
"tactic": "rw [h, inv_mul_cancel_left]"
}
] |
[
705,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
704,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.isLUB_l_image
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.9528\na✝ a₁ a₂ : α\nb✝ b₁ b₂ : β\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\ns : Set α\na : α\nh : IsLUB s a\nb : β\nhb : b ∈ upperBounds (l '' s)\n⊢ u b ∈ upperBounds s",
"tactic": "rwa [gc.upperBounds_l_image] at hb"
}
] |
[
135,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.mem_iUnion₂
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.42510\nβ : Type ?u.42513\nγ : Type u_1\nι : Sort u_2\nι' : Sort ?u.42522\nι₂ : Sort ?u.42525\nκ : ι → Sort u_3\nκ₁ : ι → Sort ?u.42535\nκ₂ : ι → Sort ?u.42540\nκ' : ι' → Sort ?u.42545\nx : γ\ns : (i : ι) → κ i → Set γ\n⊢ (x ∈ ⋃ (i : ι) (j : κ i), s i j) ↔ ∃ i j, x ∈ s i j",
"tactic": "simp_rw [mem_iUnion]"
}
] |
[
151,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.polar_smul_right_of_tower
|
[
{
"state_after": "no goals",
"state_before": "S : Type u_3\nR : Type u_1\nR₁ : Type ?u.194332\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : CommRing R₁\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nQ : QuadraticForm R M\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\na : S\nx y : M\n⊢ polar (↑Q) x (a • y) = a • polar (↑Q) x y",
"tactic": "rw [← IsScalarTower.algebraMap_smul R a y, polar_smul_right, Algebra.smul_def]"
}
] |
[
336,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.union_diff_cancel_right
|
[] |
[
1838,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1837,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
Commute.orderOf_mul_eq_mul_orderOf_of_coprime
|
[
{
"state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nh : _root_.Commute x y\nhco : coprime (orderOf x) (orderOf y)\n⊢ minimalPeriod ((fun x_1 => x * x_1) ∘ fun x => y * x) 1 = orderOf x * orderOf y",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nh : _root_.Commute x y\nhco : coprime (orderOf x) (orderOf y)\n⊢ orderOf (x * y) = orderOf x * orderOf y",
"tactic": "rw [orderOf, ← comp_mul_left]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nh : _root_.Commute x y\nhco : coprime (orderOf x) (orderOf y)\n⊢ minimalPeriod ((fun x_1 => x * x_1) ∘ fun x => y * x) 1 = orderOf x * orderOf y",
"tactic": "exact h.function_commute_mul_left.minimalPeriod_of_comp_eq_mul_of_coprime hco"
}
] |
[
412,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
|
eq_div_iff
|
[] |
[
96,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.coe_monoidHom_trans
|
[] |
[
534,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
Matrix.blockDiagonal_tsum
|
[
{
"state_after": "case pos\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : Summable f\n⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)\n\ncase neg\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : ¬Summable f\n⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)",
"state_before": "X : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\n⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)",
"tactic": "by_cases hf : Summable f"
},
{
"state_after": "no goals",
"state_before": "case pos\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : Summable f\n⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)",
"tactic": "exact hf.hasSum.matrix_blockDiagonal.tsum_eq.symm"
},
{
"state_after": "case neg\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal (f x)\n⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)",
"state_before": "case neg\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : ¬Summable f\n⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)",
"tactic": "have hft := summable_matrix_blockDiagonal.not.mpr hf"
},
{
"state_after": "case neg\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal (f x)\n⊢ blockDiagonal 0 = 0",
"state_before": "case neg\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal (f x)\n⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)",
"tactic": "rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft]"
},
{
"state_after": "no goals",
"state_before": "case neg\nX : Type u_5\nα : Type ?u.94238\nl : Type ?u.94241\nm : Type u_3\nn : Type u_4\np : Type u_1\nS : Type ?u.94253\nR : Type u_2\nm' : l → Type ?u.94261\nn' : l → Type ?u.94266\ninst✝⁵ : Semiring α\ninst✝⁴ : AddCommMonoid R\ninst✝³ : TopologicalSpace R\ninst✝² : Module α R\ninst✝¹ : DecidableEq p\ninst✝ : T2Space R\nf : X → p → Matrix m n R\nhf : ¬Summable f\nhft : ¬Summable fun x => blockDiagonal (f x)\n⊢ blockDiagonal 0 = 0",
"tactic": "exact blockDiagonal_zero"
}
] |
[
405,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.sign_permCongr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\ne : α ≃ β\np : Perm α\n⊢ ∀ (x : β), ↑e.symm (↑(↑(permCongr e) p) x) = ↑p (↑e.symm x)",
"tactic": "simp"
}
] |
[
767,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.zneg_toZNumNeg
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.446609\nn : Num\n⊢ -toZNumNeg n = toZNum n",
"tactic": "cases n <;> rfl"
}
] |
[
778,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
778,
1
] |
Mathlib/Data/List/Basic.lean
|
List.get_reverse
|
[] |
[
1458,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1456,
9
] |
Mathlib/Algebra/Invertible.lean
|
mul_invOf_self_assoc'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na b : α\nx✝ : Invertible a\n⊢ a * (⅟a * b) = b",
"tactic": "rw [← mul_assoc, mul_invOf_self, one_mul]"
}
] |
[
128,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Order/Basic.lean
|
le_of_forall_le
|
[] |
[
529,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
528,
1
] |
Mathlib/Algebra/Opposites.lean
|
AddOpposite.unop_inv
|
[] |
[
417,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.fract_eq_zero_or_add_one_sub_ceil
|
[
{
"state_after": "case inl\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a = 0\n⊢ fract a = 0 ∨ fract a = a + 1 - ↑⌈a⌉\n\ncase inr\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ fract a = 0 ∨ fract a = a + 1 - ↑⌈a⌉",
"state_before": "F : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\n⊢ fract a = 0 ∨ fract a = a + 1 - ↑⌈a⌉",
"tactic": "cases' eq_or_ne (fract a) 0 with ha ha"
},
{
"state_after": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ fract a = a + 1 - ↑⌈a⌉",
"state_before": "case inr\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ fract a = 0 ∨ fract a = a + 1 - ↑⌈a⌉",
"tactic": "right"
},
{
"state_after": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ↑⌈a⌉ = ↑⌊a⌋ + 1",
"state_before": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ fract a = a + 1 - ↑⌈a⌉",
"tactic": "suffices (⌈a⌉ : α) = ⌊a⌋ + 1 by\n rw [this, ← self_sub_fract]\n abel"
},
{
"state_after": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ⌈a⌉ = ⌊a⌋ + 1",
"state_before": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ↑⌈a⌉ = ↑⌊a⌋ + 1",
"tactic": "norm_cast"
},
{
"state_after": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ↑(⌊a⌋ + 1) - 1 < a ∧ a ≤ ↑(⌊a⌋ + 1)",
"state_before": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ⌈a⌉ = ⌊a⌋ + 1",
"tactic": "rw [ceil_eq_iff]"
},
{
"state_after": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ↑(⌊a⌋ + 1) - 1 < a",
"state_before": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ↑(⌊a⌋ + 1) - 1 < a ∧ a ≤ ↑(⌊a⌋ + 1)",
"tactic": "refine' ⟨_, _root_.le_of_lt <| by simp⟩"
},
{
"state_after": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ 0 < fract a",
"state_before": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ ↑(⌊a⌋ + 1) - 1 < a",
"tactic": "rw [cast_add, cast_one, add_tsub_cancel_right, ← self_sub_fract a, sub_lt_self_iff]"
},
{
"state_after": "no goals",
"state_before": "case inr.h\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ 0 < fract a",
"tactic": "exact ha.symm.lt_of_le (fract_nonneg a)"
},
{
"state_after": "no goals",
"state_before": "case inl\nF : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a = 0\n⊢ fract a = 0 ∨ fract a = a + 1 - ↑⌈a⌉",
"tactic": "exact Or.inl ha"
},
{
"state_after": "F : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\nthis : ↑⌈a⌉ = ↑⌊a⌋ + 1\n⊢ fract a = a + 1 - (a - fract a + 1)",
"state_before": "F : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\nthis : ↑⌈a⌉ = ↑⌊a⌋ + 1\n⊢ fract a = a + 1 - ↑⌈a⌉",
"tactic": "rw [this, ← self_sub_fract]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\nthis : ↑⌈a⌉ = ↑⌊a⌋ + 1\n⊢ fract a = a + 1 - (a - fract a + 1)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.229164\nα : Type u_1\nβ : Type ?u.229170\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a : α\nha : fract a ≠ 0\n⊢ a < ↑(⌊a⌋ + 1)",
"tactic": "simp"
}
] |
[
1248,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1237,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
SubmonoidClass.finsupp_prod_mem
|
[] |
[
205,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
ContDiffOn.congr_mono
|
[] |
[
717,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/RingTheory/ChainOfDivisors.lean
|
DivisorChain.eq_pow_second_of_chain_of_has_chain
|
[
{
"state_after": "case intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ q = c 1 ^ n",
"state_before": "M : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\n⊢ q = c 1 ^ n",
"tactic": "obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq"
},
{
"state_after": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ n = ↑i",
"state_before": "case intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ q = c 1 ^ n",
"tactic": "convert hi'"
},
{
"state_after": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ Nat.succ n ≤ Nat.succ ↑i",
"state_before": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ n = ↑i",
"tactic": "refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)"
},
{
"state_after": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ Finset.image c Finset.univ ⊆ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"state_before": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ Nat.succ n ≤ Nat.succ ↑i",
"tactic": "calc\n n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm\n _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr (h₁.injective.injOn _)).symm\n _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=\n (Finset.card_le_of_subset ?_)\n _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le\n _ = i + 1 := Finset.card_fin _"
},
{
"state_after": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nr : Associates M\nhr : r ∈ Finset.image c Finset.univ\n⊢ r ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"state_before": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\n⊢ Finset.image c Finset.univ ⊆ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"tactic": "intro r hr"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"state_before": "case h.e'_3.h.e'_6\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nr : Associates M\nhr : r ∈ Finset.image c Finset.univ\n⊢ r ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"tactic": "obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ q\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"state_before": "case h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"tactic": "have := h₂.2 ⟨j, rfl⟩"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"state_before": "case h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ q\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"tactic": "rw [hi'] at this"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro.refine_2\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nh : ∃ i_1, i_1 ≤ ↑i ∧ Associated (c j) (c 1 ^ i_1)\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ\n\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Prime (c 1)",
"state_before": "case h.e'_3.h.e'_6.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ",
"tactic": "have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nu : ℕ\nhu : u ≤ ↑i\nhu' : Associated (c j) (c 1 ^ u)\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ\n\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Prime (c 1)",
"state_before": "case h.e'_3.h.e'_6.intro.intro.refine_2\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nh : ∃ i_1, i_1 ≤ ↑i ∧ Associated (c j) (c 1 ^ i_1)\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ\n\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Prime (c 1)",
"tactic": "rcases h with ⟨u, hu, hu'⟩"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nu : ℕ\nhu : u ≤ ↑i\nhu' : Associated (c j) (c 1 ^ u)\n⊢ c 1 ^ ↑↑u = c j\n\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Prime (c 1)",
"state_before": "case h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nu : ℕ\nhu : u ≤ ↑i\nhu' : Associated (c j) (c 1 ^ u)\n⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ\n\ncase h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Prime (c 1)",
"tactic": "refine' Finset.mem_image.mpr ⟨u, Finset.mem_univ _, _⟩"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nu : ℕ\nhu : u ≤ ↑i\nhu' : c j = c 1 ^ u\n⊢ c 1 ^ ↑↑u = c j",
"state_before": "case h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nu : ℕ\nhu : u ≤ ↑i\nhu' : Associated (c j) (c 1 ^ u)\n⊢ c 1 ^ ↑↑u = c j",
"tactic": "rw [associated_iff_eq] at hu'"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\nu : ℕ\nhu : u ≤ ↑i\nhu' : c j = c 1 ^ u\n⊢ c 1 ^ ↑↑u = c j",
"tactic": "rw [Fin.val_cast_of_lt (Nat.lt_succ_of_le hu), hu']"
},
{
"state_after": "case h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Irreducible (c 1)",
"state_before": "case h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Prime (c 1)",
"tactic": "rw [← irreducible_iff_prime]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6.intro.intro.refine_1\nM : Type u_1\ninst✝¹ : CancelCommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhq : q ≠ 0\ni : Fin (n + 1)\nhi' : q = c 1 ^ ↑i\nj : Fin (n + 1)\nhr : c j ∈ Finset.image c Finset.univ\nthis : c j ≤ c 1 ^ ↑i\n⊢ Irreducible (c 1)",
"tactic": "exact second_of_chain_is_irreducible hn h₁ (@h₂) hq"
}
] |
[
214,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.superpolynomialDecay_iff_abs_isBoundedUnder
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"state_before": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\n⊢ SuperpolynomialDecay l k f ↔ ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)",
"tactic": "refine'\n ⟨fun h z => Tendsto.isBoundedUnder_le (Tendsto.abs (h z)), fun h =>\n (superpolynomialDecay_iff_abs_tendsto_zero l k f).2 fun z => _⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"state_before": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"tactic": "obtain ⟨m, hm⟩ := h (z + 1)"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"tactic": "have h1 : Tendsto (fun _ : α => (0 : β)) l (𝓝 0) := tendsto_const_nhds"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"tactic": "have h2 : Tendsto (fun a : α => |(k a)⁻¹| * m) l (𝓝 0) :=\n MulZeroClass.zero_mul m ▸\n Tendsto.mul_const m ((tendsto_zero_iff_abs_tendsto_zero _).1 hk.inv_tendsto_atTop)"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\n⊢ ∀ᶠ (x : α) in l, (fun x x_1 => x ≤ x_1) (abs (k x ^ (z + 1) * f x)) m → abs (k x ^ z * f x) ≤ abs (k x)⁻¹ * m",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\n⊢ Tendsto (fun a => abs (k a ^ z * f a)) l (𝓝 0)",
"tactic": "refine'\n tendsto_of_tendsto_of_tendsto_of_le_of_le' h1 h2 (eventually_of_forall fun x => abs_nonneg _)\n ((eventually_map.1 hm).mp _)"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\nx : α\nhk0 : k x ≠ 0\nhx : (fun x x_1 => x ≤ x_1) (abs (k x ^ (z + 1) * f x)) m\n⊢ abs (k x ^ z * f x) ≤ abs (k x)⁻¹ * m",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\n⊢ ∀ᶠ (x : α) in l, (fun x x_1 => x ≤ x_1) (abs (k x ^ (z + 1) * f x)) m → abs (k x ^ z * f x) ≤ abs (k x)⁻¹ * m",
"tactic": "refine' (hk.eventually_ne_atTop 0).mono fun x hk0 hx => _"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\nx : α\nhk0 : k x ≠ 0\nhx : (fun x x_1 => x ≤ x_1) (abs (k x ^ (z + 1) * f x)) m\n⊢ abs (k x ^ z * f x) = abs (k x)⁻¹ * abs (k x ^ (z + 1) * f x)",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\nx : α\nhk0 : k x ≠ 0\nhx : (fun x x_1 => x ≤ x_1) (abs (k x ^ (z + 1) * f x)) m\n⊢ abs (k x ^ z * f x) ≤ abs (k x)⁻¹ * m",
"tactic": "refine' Eq.trans_le _ (mul_le_mul_of_nonneg_left hx <| abs_nonneg (k x)⁻¹)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => abs (k a ^ z * f a)\nz : ℕ\nm : β\nhm : ∀ᶠ (x : β) in Filter.map (fun a => abs (k a ^ (z + 1) * f a)) l, (fun x x_1 => x ≤ x_1) x m\nh1 : Tendsto (fun x => 0) l (𝓝 0)\nh2 : Tendsto (fun a => abs (k a)⁻¹ * m) l (𝓝 0)\nx : α\nhk0 : k x ≠ 0\nhx : (fun x x_1 => x ≤ x_1) (abs (k x ^ (z + 1) * f x)) m\n⊢ abs (k x ^ z * f x) = abs (k x)⁻¹ * abs (k x ^ (z + 1) * f x)",
"tactic": "rw [← abs_mul, ← mul_assoc, pow_succ, ← mul_assoc, inv_mul_cancel hk0, one_mul]"
}
] |
[
241,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.Colorable.chromaticNumber_le_of_forall_imp
|
[
{
"state_after": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nV' : Type u_1\nG' : SimpleGraph V'\nm : ℕ\nhc : Colorable G' m\nh : ∀ (n : ℕ), Colorable G' n → Colorable G n\n⊢ chromaticNumber G' ∈ {n | Colorable G n}",
"state_before": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nV' : Type u_1\nG' : SimpleGraph V'\nm : ℕ\nhc : Colorable G' m\nh : ∀ (n : ℕ), Colorable G' n → Colorable G n\n⊢ chromaticNumber G ≤ chromaticNumber G'",
"tactic": "apply csInf_le chromaticNumber_bddBelow"
},
{
"state_after": "case a\nV : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nV' : Type u_1\nG' : SimpleGraph V'\nm : ℕ\nhc : Colorable G' m\nh : ∀ (n : ℕ), Colorable G' n → Colorable G n\n⊢ Colorable G' (chromaticNumber G')",
"state_before": "V : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nV' : Type u_1\nG' : SimpleGraph V'\nm : ℕ\nhc : Colorable G' m\nh : ∀ (n : ℕ), Colorable G' n → Colorable G n\n⊢ chromaticNumber G' ∈ {n | Colorable G n}",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case a\nV : Type u\nG : SimpleGraph V\nα : Type v\nC : Coloring G α\nV' : Type u_1\nG' : SimpleGraph V'\nm : ℕ\nhc : Colorable G' m\nh : ∀ (n : ℕ), Colorable G' n → Colorable G n\n⊢ Colorable G' (chromaticNumber G')",
"tactic": "apply colorable_chromaticNumber hc"
}
] |
[
330,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
IsIntegralClosure.algebraMap_mk'
|
[] |
[
868,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
867,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_seq_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.289749\nι : Sort x\nf : Filter (α → β)\ng : Filter α\ns : Set β\n⊢ s ∈ seq f g ↔ ∃ u, u ∈ f ∧ ∃ t, t ∈ g ∧ Set.seq u t ⊆ s",
"tactic": "simp only [mem_seq_def, seq_subset, exists_prop, iff_self_iff]"
}
] |
[
2618,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2616,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.isFractional_span_iff
|
[
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s\n⊢ IsInteger R 0",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s\n⊢ IsInteger R (a • 0)",
"tactic": "rw [smul_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s\n⊢ IsInteger R 0",
"tactic": "exact isInteger_zero"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s\nx y : P\nhx : IsInteger R (a • x)\nhy : IsInteger R (a • y)\n⊢ IsInteger R (a • x + a • y)",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s\nx y : P\nhx : IsInteger R (a • x)\nhy : IsInteger R (a • y)\n⊢ IsInteger R (a • (x + y))",
"tactic": "rw [smul_add]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s\nx y : P\nhx : IsInteger R (a • x)\nhy : IsInteger R (a • y)\n⊢ IsInteger R (a • x + a • y)",
"tactic": "exact isInteger_add hx hy"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns✝ : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s✝ → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s✝ → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s✝\ns : R\nx : P\nhx : IsInteger R (a • x)\n⊢ IsInteger R (s • a • x)",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns✝ : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s✝ → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s✝ → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s✝\ns : R\nx : P\nhx : IsInteger R (a • x)\n⊢ IsInteger R (a • s • x)",
"tactic": "rw [smul_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.659673\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.660126\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\ns✝ : Set P\nx✝ : ∃ a, a ∈ S ∧ ∀ (b : P), b ∈ s✝ → IsInteger R (a • b)\na : R\na_mem : a ∈ S\nh : ∀ (b : P), b ∈ s✝ → IsInteger R (a • b)\nb : P\nhb : b ∈ span R s✝\ns : R\nx : P\nhx : IsInteger R (a • x)\n⊢ IsInteger R (s • a • x)",
"tactic": "exact isInteger_smul hx"
}
] |
[
855,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
842,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.powerset_aux'_perm
|
[
{
"state_after": "case nil\nα : Type u_1\nl₁ l₂ : List α\n⊢ powersetAux' [] ~ powersetAux' []\n\ncase cons\nα : Type u_1\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\np : l₁ ~ l₂\nIH : powersetAux' l₁ ~ powersetAux' l₂\n⊢ powersetAux' (a :: l₁) ~ powersetAux' (a :: l₂)\n\ncase swap\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ powersetAux' (b :: a :: l) ~ powersetAux' (a :: b :: l)\n\ncase trans\nα : Type u_1\nl₁✝ l₂✝ l₁ l₂ l₃ : List α\na✝¹ : l₁ ~ l₂\na✝ : l₂ ~ l₃\nIH₁ : powersetAux' l₁ ~ powersetAux' l₂\nIH₂ : powersetAux' l₂ ~ powersetAux' l₃\n⊢ powersetAux' l₁ ~ powersetAux' l₃",
"state_before": "α : Type u_1\nl₁ l₂ : List α\np : l₁ ~ l₂\n⊢ powersetAux' l₁ ~ powersetAux' l₂",
"tactic": "induction' p with a l₁ l₂ p IH a b l l₁ l₂ l₃ _ _ IH₁ IH₂"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nl₁ l₂ : List α\n⊢ powersetAux' [] ~ powersetAux' []",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type u_1\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\np : l₁ ~ l₂\nIH : powersetAux' l₁ ~ powersetAux' l₂\n⊢ powersetAux' l₁ ++ List.map (cons a) (powersetAux' l₁) ~ powersetAux' l₂ ++ List.map (cons a) (powersetAux' l₂)",
"state_before": "case cons\nα : Type u_1\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\np : l₁ ~ l₂\nIH : powersetAux' l₁ ~ powersetAux' l₂\n⊢ powersetAux' (a :: l₁) ~ powersetAux' (a :: l₂)",
"tactic": "simp only [powersetAux'_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nl₁✝ l₂✝ : List α\na : α\nl₁ l₂ : List α\np : l₁ ~ l₂\nIH : powersetAux' l₁ ~ powersetAux' l₂\n⊢ powersetAux' l₁ ++ List.map (cons a) (powersetAux' l₁) ~ powersetAux' l₂ ++ List.map (cons a) (powersetAux' l₂)",
"tactic": "exact IH.append (IH.map _)"
},
{
"state_after": "case swap\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ powersetAux' l ++\n (List.map (cons a) (powersetAux' l) ++\n (List.map (cons b) (powersetAux' l) ++ List.map (cons b ∘ cons a) (powersetAux' l))) ~\n powersetAux' l ++\n (List.map (cons b) (powersetAux' l) ++\n (List.map (cons a) (powersetAux' l) ++ List.map (cons a ∘ cons b) (powersetAux' l)))",
"state_before": "case swap\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ powersetAux' (b :: a :: l) ~ powersetAux' (a :: b :: l)",
"tactic": "simp only [powersetAux'_cons, map_append, List.map_map, append_assoc]"
},
{
"state_after": "case swap.a\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ List.map (cons a) (powersetAux' l) ++\n (List.map (cons b) (powersetAux' l) ++ List.map (cons b ∘ cons a) (powersetAux' l)) ~\n List.map (cons b) (powersetAux' l) ++\n (List.map (cons a) (powersetAux' l) ++ List.map (cons a ∘ cons b) (powersetAux' l))",
"state_before": "case swap\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ powersetAux' l ++\n (List.map (cons a) (powersetAux' l) ++\n (List.map (cons b) (powersetAux' l) ++ List.map (cons b ∘ cons a) (powersetAux' l))) ~\n powersetAux' l ++\n (List.map (cons b) (powersetAux' l) ++\n (List.map (cons a) (powersetAux' l) ++ List.map (cons a ∘ cons b) (powersetAux' l)))",
"tactic": "apply Perm.append_left"
},
{
"state_after": "case swap.a\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ List.map (cons a) (powersetAux' l) ++ List.map (cons b) (powersetAux' l) ++\n List.map (cons a ∘ cons b) (powersetAux' l) ~\n List.map (cons b) (powersetAux' l) ++ List.map (cons a) (powersetAux' l) ++\n List.map (cons a ∘ cons b) (powersetAux' l)",
"state_before": "case swap.a\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ List.map (cons a) (powersetAux' l) ++\n (List.map (cons b) (powersetAux' l) ++ List.map (cons b ∘ cons a) (powersetAux' l)) ~\n List.map (cons b) (powersetAux' l) ++\n (List.map (cons a) (powersetAux' l) ++ List.map (cons a ∘ cons b) (powersetAux' l))",
"tactic": "rw [← append_assoc, ← append_assoc,\n (by funext s; simp [cons_swap] : cons b ∘ cons a = cons a ∘ cons b)]"
},
{
"state_after": "no goals",
"state_before": "case swap.a\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ List.map (cons a) (powersetAux' l) ++ List.map (cons b) (powersetAux' l) ++\n List.map (cons a ∘ cons b) (powersetAux' l) ~\n List.map (cons b) (powersetAux' l) ++ List.map (cons a) (powersetAux' l) ++\n List.map (cons a ∘ cons b) (powersetAux' l)",
"tactic": "exact perm_append_comm.append_right _"
},
{
"state_after": "case h\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\ns : Multiset α\n⊢ (cons b ∘ cons a) s = (cons a ∘ cons b) s",
"state_before": "α : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ cons b ∘ cons a = cons a ∘ cons b",
"tactic": "funext s"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nl₁ l₂ : List α\na b : α\nl : List α\ns : Multiset α\n⊢ (cons b ∘ cons a) s = (cons a ∘ cons b) s",
"tactic": "simp [cons_swap]"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type u_1\nl₁✝ l₂✝ l₁ l₂ l₃ : List α\na✝¹ : l₁ ~ l₂\na✝ : l₂ ~ l₃\nIH₁ : powersetAux' l₁ ~ powersetAux' l₂\nIH₂ : powersetAux' l₂ ~ powersetAux' l₃\n⊢ powersetAux' l₁ ~ powersetAux' l₃",
"tactic": "exact IH₁.trans IH₂"
}
] |
[
74,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Algebra/Ring/Defs.lean
|
mul_two
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonAssocSemiring α\nn : α\n⊢ n * 1 + n * 1 = n + n",
"tactic": "rw [mul_one]"
}
] |
[
192,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.sSup_adj
|
[] |
[
390,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Topology/Constructions.lean
|
continuousAt_apply
|
[] |
[
1209,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1208,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
strictConvexOn_exp
|
[
{
"state_after": "⊢ ∀ {x y z : ℝ}, x ∈ univ → z ∈ univ → x < y → y < z → (exp y - exp x) / (y - x) < (exp z - exp y) / (z - y)",
"state_before": "⊢ StrictConvexOn ℝ univ exp",
"tactic": "apply strictConvexOn_of_slope_strict_mono_adjacent convex_univ"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ (exp y - exp x) / (y - x) < (exp z - exp y) / (z - y)",
"state_before": "⊢ ∀ {x y z : ℝ}, x ∈ univ → z ∈ univ → x < y → y < z → (exp y - exp x) / (y - x) < (exp z - exp y) / (z - y)",
"tactic": "rintro x y z - - hxy hyz"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ (exp y - exp x) / (y - x) < exp y\n\nx y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ exp y < (exp z - exp y) / (z - y)",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ (exp y - exp x) / (y - x) < (exp z - exp y) / (z - y)",
"tactic": "trans exp y"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\n⊢ (exp y - exp x) / (y - x) < exp y",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ (exp y - exp x) / (y - x) < exp y",
"tactic": "have h1 : 0 < y - x := by linarith"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ (exp y - exp x) / (y - x) < exp y",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\n⊢ (exp y - exp x) / (y - x) < exp y",
"tactic": "have h2 : x - y < 0 := by linarith"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y - exp x < exp y * (y - x)",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ (exp y - exp x) / (y - x) < exp y",
"tactic": "rw [div_lt_iff h1]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y - exp x < exp y * (y - x)",
"tactic": "calc\n exp y - exp x = exp y - exp y * exp (x - y) := by rw [← exp_add]; ring_nf\n _ = exp y * (1 - exp (x - y)) := by ring\n _ < exp y * -(x - y) := by gcongr; linarith [add_one_lt_exp_of_nonzero h2.ne]\n _ = exp y * (y - x) := by ring"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ 0 < y - x",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\n⊢ x - y < 0",
"tactic": "linarith"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y - exp x = exp y - exp (y + (x - y))",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y - exp x = exp y - exp y * exp (x - y)",
"tactic": "rw [← exp_add]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y - exp x = exp y - exp (y + (x - y))",
"tactic": "ring_nf"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y - exp y * exp (x - y) = exp y * (1 - exp (x - y))",
"tactic": "ring"
},
{
"state_after": "case bc\nx y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ 1 - exp (x - y) < -(x - y)",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y * (1 - exp (x - y)) < exp y * -(x - y)",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case bc\nx y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ 1 - exp (x - y) < -(x - y)",
"tactic": "linarith [add_one_lt_exp_of_nonzero h2.ne]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ exp y * -(x - y) = exp y * (y - x)",
"tactic": "ring"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp y < (exp z - exp y) / (z - y)",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ exp y < (exp z - exp y) / (z - y)",
"tactic": "have h1 : 0 < z - y := by linarith"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp y * (z - y) < exp z - exp y",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp y < (exp z - exp y) / (z - y)",
"tactic": "rw [lt_div_iff h1]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp y * (z - y) < exp z - exp y",
"tactic": "calc\n exp y * (z - y) < exp y * (exp (z - y) - 1) := by\n gcongr _ * ?_\n linarith [add_one_lt_exp_of_nonzero h1.ne']\n _ = exp (z - y) * exp y - exp y := by ring\n _ ≤ exp z - exp y := by rw [← exp_add]; ring_nf; rfl"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\n⊢ 0 < z - y",
"tactic": "linarith"
},
{
"state_after": "case bc\nx y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ z - y < exp (z - y) - 1",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp y * (z - y) < exp y * (exp (z - y) - 1)",
"tactic": "gcongr _ * ?_"
},
{
"state_after": "no goals",
"state_before": "case bc\nx y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ z - y < exp (z - y) - 1",
"tactic": "linarith [add_one_lt_exp_of_nonzero h1.ne']"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp y * (exp (z - y) - 1) = exp (z - y) * exp y - exp y",
"tactic": "ring"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp (z - y + y) - exp y ≤ exp z - exp y",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp (z - y) * exp y - exp y ≤ exp z - exp y",
"tactic": "rw [← exp_add]"
},
{
"state_after": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp z - exp y ≤ exp z - exp y",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp (z - y + y) - exp y ≤ exp z - exp y",
"tactic": "ring_nf"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n⊢ exp z - exp y ≤ exp z - exp y",
"tactic": "rfl"
}
] |
[
68,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.nilpotencyLength_eq_zero_iff
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ nilpotencyLength R L M = 0 ↔ Subsingleton M",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\n⊢ nilpotencyLength R L M = 0 ↔ Subsingleton M",
"tactic": "let s := {k | lowerCentralSeries R L M k = ⊥}"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\n⊢ nilpotencyLength R L M = 0 ↔ Subsingleton M",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ nilpotencyLength R L M = 0 ↔ Subsingleton M",
"tactic": "have hs : s.Nonempty := by\n obtain ⟨k, hk⟩ := (by infer_instance : IsNilpotent R L M)\n exact ⟨k, hk⟩"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\n⊢ sInf s = 0 ↔ Subsingleton M",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\n⊢ nilpotencyLength R L M = 0 ↔ Subsingleton M",
"tactic": "change sInf s = 0 ↔ _"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\n⊢ sInf s = 0 ↔ lowerCentralSeries R L M 0 = ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\n⊢ sInf s = 0 ↔ Subsingleton M",
"tactic": "rw [← LieSubmodule.subsingleton_iff R L M, ← subsingleton_iff_bot_eq_top, ←\n lowerCentralSeries_zero, @eq_comm (LieSubmodule R L M)]"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\nh : lowerCentralSeries R L M 0 = ⊥\n⊢ sInf s = 0",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\n⊢ sInf s = 0 ↔ lowerCentralSeries R L M 0 = ⊥",
"tactic": "refine' ⟨fun h => h ▸ Nat.sInf_mem hs, fun h => _⟩"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\nh : lowerCentralSeries R L M 0 = ⊥\n⊢ 0 ∈ s ∨ s = ∅",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\nh : lowerCentralSeries R L M 0 = ⊥\n⊢ sInf s = 0",
"tactic": "rw [Nat.sInf_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nhs : Set.Nonempty s\nh : lowerCentralSeries R L M 0 = ⊥\n⊢ 0 ∈ s ∨ s = ∅",
"tactic": "exact Or.inl h"
},
{
"state_after": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ Set.Nonempty s",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ Set.Nonempty s",
"tactic": "obtain ⟨k, hk⟩ := (by infer_instance : IsNilpotent R L M)"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nR : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk✝ : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ Set.Nonempty s",
"tactic": "exact ⟨k, hk⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : IsNilpotent R L M\ns : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}\n⊢ IsNilpotent R L M",
"tactic": "infer_instance"
}
] |
[
272,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisCoinsertion.u_sSup_l_image
|
[] |
[
820,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_injective_left
|
[] |
[
229,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.eq_singleton_iff_nonempty_unique_mem
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\n⊢ s = {a} → Finset.Nonempty s ∧ ∀ (x : α), x ∈ s → x = a\n\ncase mpr\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\n⊢ (Finset.Nonempty s ∧ ∀ (x : α), x ∈ s → x = a) → s = {a}",
"state_before": "α : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\n⊢ s = {a} ↔ Finset.Nonempty s ∧ ∀ (x : α), x ∈ s → x = a",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns : Finset α\na✝ b a : α\n⊢ Finset.Nonempty {a} ∧ ∀ (x : α), x ∈ {a} → x = a",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\n⊢ s = {a} → Finset.Nonempty s ∧ ∀ (x : α), x ∈ s → x = a",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns : Finset α\na✝ b a : α\n⊢ Finset.Nonempty {a} ∧ ∀ (x : α), x ∈ {a} → x = a",
"tactic": "simp"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ s = {a}",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\n⊢ (Finset.Nonempty s ∧ ∀ (x : α), x ∈ s → x = a) → s = {a}",
"tactic": "rintro ⟨hne, h_uniq⟩"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ a ∈ s ∧ ∀ (x : α), x ∈ s → x = a",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ s = {a}",
"tactic": "rw [eq_singleton_iff_unique_mem]"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ a ∈ s",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ a ∈ s ∧ ∀ (x : α), x ∈ s → x = a",
"tactic": "refine' ⟨_, h_uniq⟩"
},
{
"state_after": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ Exists.choose hne ∈ s",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ a ∈ s",
"tactic": "rw [← h_uniq hne.choose hne.choose_spec]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nα : Type u_1\nβ : Type ?u.30920\nγ : Type ?u.30923\ns✝ : Finset α\na✝ b : α\ns : Finset α\na : α\nhne : Finset.Nonempty s\nh_uniq : ∀ (x : α), x ∈ s → x = a\n⊢ Exists.choose hne ∈ s",
"tactic": "exact hne.choose_spec"
}
] |
[
756,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
le_iff_forall_one_lt_lt_mul
|
[] |
[
1053,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1052,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.Adj.symm
|
[] |
[
115,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
11
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.Theory.isSatisfiable_empty
|
[] |
[
88,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.recBotCoe_bot
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
Submodule.coe_inner
|
[] |
[
1955,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1954,
1
] |
Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean
|
IsConformalMap.ne_zero
|
[
{
"state_after": "R : Type u_2\nM : Type ?u.245254\nN : Type u_3\nG : Type ?u.245260\nM' : Type u_1\ninst✝⁹ : NormedField R\ninst✝⁸ : SeminormedAddCommGroup M\ninst✝⁷ : SeminormedAddCommGroup N\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : NormedSpace R M\ninst✝⁴ : NormedSpace R N\ninst✝³ : NormedSpace R G\ninst✝² : NormedAddCommGroup M'\ninst✝¹ : NormedSpace R M'\nf : M →L[R] N\ng : N →L[R] G\nc : R\ninst✝ : Nontrivial M'\nhf' : IsConformalMap 0\n⊢ False",
"state_before": "R : Type u_2\nM : Type ?u.245254\nN : Type u_3\nG : Type ?u.245260\nM' : Type u_1\ninst✝⁹ : NormedField R\ninst✝⁸ : SeminormedAddCommGroup M\ninst✝⁷ : SeminormedAddCommGroup N\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : NormedSpace R M\ninst✝⁴ : NormedSpace R N\ninst✝³ : NormedSpace R G\ninst✝² : NormedAddCommGroup M'\ninst✝¹ : NormedSpace R M'\nf : M →L[R] N\ng : N →L[R] G\nc : R\ninst✝ : Nontrivial M'\nf' : M' →L[R] N\nhf' : IsConformalMap f'\n⊢ f' ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "case intro\nR : Type u_2\nM : Type ?u.245254\nN : Type u_3\nG : Type ?u.245260\nM' : Type u_1\ninst✝⁹ : NormedField R\ninst✝⁸ : SeminormedAddCommGroup M\ninst✝⁷ : SeminormedAddCommGroup N\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : NormedSpace R M\ninst✝⁴ : NormedSpace R N\ninst✝³ : NormedSpace R G\ninst✝² : NormedAddCommGroup M'\ninst✝¹ : NormedSpace R M'\nf : M →L[R] N\ng : N →L[R] G\nc : R\ninst✝ : Nontrivial M'\nhf' : IsConformalMap 0\na : M'\nha : a ≠ 0\n⊢ False",
"state_before": "R : Type u_2\nM : Type ?u.245254\nN : Type u_3\nG : Type ?u.245260\nM' : Type u_1\ninst✝⁹ : NormedField R\ninst✝⁸ : SeminormedAddCommGroup M\ninst✝⁷ : SeminormedAddCommGroup N\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : NormedSpace R M\ninst✝⁴ : NormedSpace R N\ninst✝³ : NormedSpace R G\ninst✝² : NormedAddCommGroup M'\ninst✝¹ : NormedSpace R M'\nf : M →L[R] N\ng : N →L[R] G\nc : R\ninst✝ : Nontrivial M'\nhf' : IsConformalMap 0\n⊢ False",
"tactic": "rcases exists_ne (0 : M') with ⟨a, ha⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_2\nM : Type ?u.245254\nN : Type u_3\nG : Type ?u.245260\nM' : Type u_1\ninst✝⁹ : NormedField R\ninst✝⁸ : SeminormedAddCommGroup M\ninst✝⁷ : SeminormedAddCommGroup N\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : NormedSpace R M\ninst✝⁴ : NormedSpace R N\ninst✝³ : NormedSpace R G\ninst✝² : NormedAddCommGroup M'\ninst✝¹ : NormedSpace R M'\nf : M →L[R] N\ng : N →L[R] G\nc : R\ninst✝ : Nontrivial M'\nhf' : IsConformalMap 0\na : M'\nha : a ≠ 0\n⊢ False",
"tactic": "exact ha (hf'.injective rfl)"
}
] |
[
103,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
|
MvQPF.wrepr_equiv
|
[
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α),\n (∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)) →\n WEquiv (wrepr (MvPFunctor.wMk (P F) a f' f)) (MvPFunctor.wMk (P F) a f' f)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\n⊢ WEquiv (wrepr x) x",
"tactic": "apply q.P.w_ind _ x"
},
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (wrepr (MvPFunctor.wMk (P F) a f' f)) (MvPFunctor.wMk (P F) a f' f)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α),\n (∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)) →\n WEquiv (wrepr (MvPFunctor.wMk (P F) a f' f)) (MvPFunctor.wMk (P F) a f' f)",
"tactic": "intro a f' f ih"
},
{
"state_after": "case a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (wrepr (MvPFunctor.wMk (P F) a f' f))\n (MvPFunctor.wMk' (P F) ((TypeVec.id ::: wrepr) <$$> { fst := a, snd := MvPFunctor.appendContents (P F) f' f }))\n\ncase a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (MvPFunctor.wMk' (P F) ((TypeVec.id ::: wrepr) <$$> { fst := a, snd := MvPFunctor.appendContents (P F) f' f }))\n (MvPFunctor.wMk (P F) a f' f)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (wrepr (MvPFunctor.wMk (P F) a f' f)) (MvPFunctor.wMk (P F) a f' f)",
"tactic": "apply WEquiv.trans _ (q.P.wMk' (appendFun id wrepr <$$> ⟨a, q.P.appendContents f' f⟩))"
},
{
"state_after": "case a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv\n (match { fst := a, snd := splitFun f' (wrepr ∘ f) } with\n | { fst := a, snd := f } => MvPFunctor.wMk (P F) a (dropFun f) (lastFun f))\n (MvPFunctor.wMk (P F) a f' f)",
"state_before": "case a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (MvPFunctor.wMk' (P F) ((TypeVec.id ::: wrepr) <$$> { fst := a, snd := MvPFunctor.appendContents (P F) f' f }))\n (MvPFunctor.wMk (P F) a f' f)",
"tactic": "rw [q.P.map_eq, MvPFunctor.wMk', appendFun_comp_splitFun, id_comp]"
},
{
"state_after": "case a.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (lastFun (splitFun f' (wrepr ∘ f)) x) (f x)",
"state_before": "case a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv\n (match { fst := a, snd := splitFun f' (wrepr ∘ f) } with\n | { fst := a, snd := f } => MvPFunctor.wMk (P F) a (dropFun f) (lastFun f))\n (MvPFunctor.wMk (P F) a f' f)",
"tactic": "apply WEquiv.ind"
},
{
"state_after": "no goals",
"state_before": "case a.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (lastFun (splitFun f' (wrepr ∘ f)) x) (f x)",
"tactic": "exact ih"
},
{
"state_after": "case a.h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ abs (MvPFunctor.wDest' (P F) (wrepr (MvPFunctor.wMk (P F) a f' f))) =\n abs\n (MvPFunctor.wDest' (P F)\n (MvPFunctor.wMk' (P F) ((TypeVec.id ::: wrepr) <$$> { fst := a, snd := MvPFunctor.appendContents (P F) f' f })))",
"state_before": "case a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ WEquiv (wrepr (MvPFunctor.wMk (P F) a f' f))\n (MvPFunctor.wMk' (P F) ((TypeVec.id ::: wrepr) <$$> { fst := a, snd := MvPFunctor.appendContents (P F) f' f }))",
"tactic": "apply wEquiv.abs'"
},
{
"state_after": "no goals",
"state_before": "case a.h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : MvPFunctor.W (P F) α\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\nih : ∀ (i : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (wrepr (f i)) (f i)\n⊢ abs (MvPFunctor.wDest' (P F) (wrepr (MvPFunctor.wMk (P F) a f' f))) =\n abs\n (MvPFunctor.wDest' (P F)\n (MvPFunctor.wMk' (P F) ((TypeVec.id ::: wrepr) <$$> { fst := a, snd := MvPFunctor.appendContents (P F) f' f })))",
"tactic": "rw [wrepr_wMk, q.P.wDest'_wMk', q.P.wDest'_wMk', abs_repr]"
}
] |
[
156,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/List/Rdrop.lean
|
List.rdropWhile_eq_nil_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nn : ℕ\n⊢ rdropWhile p l = [] ↔ ∀ (x : α), x ∈ l → p x = true",
"tactic": "simp [rdropWhile]"
}
] |
[
141,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
Subalgebra.bot_eq_top_iff_finrank_eq_one
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\nF : Type u_2\nE : Type u_1\ninst✝³ : Field F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\ninst✝ : Nontrivial E\n⊢ ⊥ = ⊤ ↔ finrank F E = 1",
"tactic": "rw [← finrank_top, ← subalgebra_top_finrank_eq_submodule_top_finrank,\n Subalgebra.finrank_eq_one_iff, eq_comm]"
}
] |
[
1428,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1425,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_coe_sign
|
[
{
"state_after": "no goals",
"state_before": "x : SignType\n⊢ ↑↑x = ↑x",
"tactic": "cases x <;> rfl"
}
] |
[
1122,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1122,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
lt_of_tsub_lt_tsub_right
|
[] |
[
415,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.compl_eq_univ_diff
|
[] |
[
1893,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1892,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.map_univ_equiv
|
[] |
[
270,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
IsSelfAdjoint.adjoint_eq
|
[] |
[
283,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.map_equiv_eq_comap_symm
|
[] |
[
1473,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1471,
1
] |
Mathlib/RingTheory/Artinian.lean
|
IsArtinian.bijective_of_injective_endomorphism
|
[] |
[
252,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean
|
CategoryTheory.InjectiveResolution.exact_ofCocomplex
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn : ℕ\n⊢ Exact\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n 0).d₀\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (0 + 1)).d₀",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn : ℕ\n⊢ Exact (HomologicalComplex.d (ofCocomplex Z) 0 (0 + 1)) (HomologicalComplex.d (ofCocomplex Z) (0 + 1) (0 + 2))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn : ℕ\n⊢ Exact\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n 0).d₀\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (0 + 1)).d₀",
"tactic": "apply exact_f_d"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn m : ℕ\n⊢ Exact\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1)).d₀\n (if m + 1 + 1 + 1 = m + 1 + 2 then\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1 + 1)).d₀\n else 0)",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn m : ℕ\n⊢ Exact (HomologicalComplex.d (ofCocomplex Z) (m + 1) (m + 1 + 1))\n (HomologicalComplex.d (ofCocomplex Z) (m + 1 + 1) (m + 1 + 2))",
"tactic": "simp only [ofCocomplex_X, ComplexShape.up_Rel, not_true, ofCocomplex_d,\n eqToHom_refl, Category.comp_id, dite_eq_ite, ite_true]"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn m : ℕ\n⊢ Exact\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1)).d₀\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1 + 1)).d₀",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn m : ℕ\n⊢ Exact\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1)).d₀\n (if m + 1 + 1 + 1 = m + 1 + 2 then\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1 + 1)).d₀\n else 0)",
"tactic": "erw [if_pos (c := m + 1 + 1 + 1 = m + 2 + 1) rfl]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nZ : C\nn m : ℕ\n⊢ Exact\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1)).d₀\n (CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))\n (d (d (Injective.ι Z)))\n (_ :\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.snd.snd ≫\n {\n fst :=\n syzygies\n { fst := under Z, snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n {\n fst :=\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd,\n snd :=\n (_ :\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd ≫\n d\n { fst := under Z,\n snd := { fst := syzygies (Injective.ι Z), snd := d (Injective.ι Z) } }.2.snd =\n 0) } }.snd.fst =\n 0)\n (fun t =>\n { fst := syzygies t.snd.snd.snd.snd.fst,\n snd :=\n { fst := d t.snd.snd.snd.snd.fst, snd := (_ : t.snd.snd.snd.snd.fst ≫ d t.snd.snd.snd.snd.fst = 0) } })\n (m + 1 + 1)).d₀",
"tactic": "apply exact_f_d"
}
] |
[
296,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Surjective.of_comp_iff
|
[] |
[
176,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Order/CompleteLattice.lean
|
unary_relation_sSup_iff
|
[
{
"state_after": "α✝ : Type ?u.195778\nβ : Type ?u.195781\nβ₂ : Type ?u.195784\nγ : Type ?u.195787\nι : Sort ?u.195790\nι' : Sort ?u.195793\nκ : ι → Sort ?u.195798\nκ' : ι' → Sort ?u.195803\nα : Type u_1\ns : Set (α → Prop)\na : α\n⊢ (⨆ (f : ↑s), ↑f a) ↔ ∃ r, r ∈ s ∧ r a",
"state_before": "α✝ : Type ?u.195778\nβ : Type ?u.195781\nβ₂ : Type ?u.195784\nγ : Type ?u.195787\nι : Sort ?u.195790\nι' : Sort ?u.195793\nκ : ι → Sort ?u.195798\nκ' : ι' → Sort ?u.195803\nα : Type u_1\ns : Set (α → Prop)\na : α\n⊢ sSup s a ↔ ∃ r, r ∈ s ∧ r a",
"tactic": "rw [sSup_apply]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.195778\nβ : Type ?u.195781\nβ₂ : Type ?u.195784\nγ : Type ?u.195787\nι : Sort ?u.195790\nι' : Sort ?u.195793\nκ : ι → Sort ?u.195798\nκ' : ι' → Sort ?u.195803\nα : Type u_1\ns : Set (α → Prop)\na : α\n⊢ (⨆ (f : ↑s), ↑f a) ↔ ∃ r, r ∈ s ∧ r a",
"tactic": "simp [← eq_iff_iff]"
}
] |
[
1784,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1781,
1
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.bot_trans
|
[
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : β ≃. γ\nx✝ : α\na✝ : γ\n⊢ a✝ ∈ ↑(PEquiv.trans ⊥ f) x✝ ↔ a✝ ∈ ↑⊥ x✝",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : β ≃. γ\n⊢ PEquiv.trans ⊥ f = ⊥",
"tactic": "ext"
},
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : β ≃. γ\nx✝ : α\na✝ : γ\n⊢ a✝ ∈ none ↔ a✝ ∈ none",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : β ≃. γ\nx✝ : α\na✝ : γ\n⊢ a✝ ∈ ↑(PEquiv.trans ⊥ f) x✝ ↔ a✝ ∈ ↑⊥ x✝",
"tactic": "dsimp [PEquiv.trans]"
},
{
"state_after": "no goals",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : β ≃. γ\nx✝ : α\na✝ : γ\n⊢ a✝ ∈ none ↔ a✝ ∈ none",
"tactic": "simp"
}
] |
[
318,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Algebra/Lie/Abelian.lean
|
LieAlgebra.abelian_of_le_center
|
[] |
[
278,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
AffineBasis.exists_affine_subbasis
|
[
{
"state_after": "case intro.intro.intro\nι : Type ?u.183706\nι' : Type ?u.183709\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : DivisionRing k\ninst✝ : Module k V\nt : Set P\nht : affineSpan k t = ⊤\ns : Set P\nhst : s ⊆ t\nh_tot : affineSpan k s = affineSpan k t\nh_ind : AffineIndependent k Subtype.val\n⊢ ∃ s x b, ↑b = Subtype.val",
"state_before": "ι : Type ?u.183706\nι' : Type ?u.183709\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : DivisionRing k\ninst✝ : Module k V\nt : Set P\nht : affineSpan k t = ⊤\n⊢ ∃ s x b, ↑b = Subtype.val",
"tactic": "obtain ⟨s, hst, h_tot, h_ind⟩ := exists_affineIndependent k V t"
},
{
"state_after": "case intro.intro.intro\nι : Type ?u.183706\nι' : Type ?u.183709\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : DivisionRing k\ninst✝ : Module k V\nt : Set P\nht : affineSpan k t = ⊤\ns : Set P\nhst : s ⊆ t\nh_tot : affineSpan k s = affineSpan k t\nh_ind : AffineIndependent k Subtype.val\n⊢ affineSpan k (range Subtype.val) = ⊤",
"state_before": "case intro.intro.intro\nι : Type ?u.183706\nι' : Type ?u.183709\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : DivisionRing k\ninst✝ : Module k V\nt : Set P\nht : affineSpan k t = ⊤\ns : Set P\nhst : s ⊆ t\nh_tot : affineSpan k s = affineSpan k t\nh_ind : AffineIndependent k Subtype.val\n⊢ ∃ s x b, ↑b = Subtype.val",
"tactic": "refine' ⟨s, hst, ⟨(↑), h_ind, _⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nι : Type ?u.183706\nι' : Type ?u.183709\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : DivisionRing k\ninst✝ : Module k V\nt : Set P\nht : affineSpan k t = ⊤\ns : Set P\nhst : s ⊆ t\nh_tot : affineSpan k s = affineSpan k t\nh_ind : AffineIndependent k Subtype.val\n⊢ affineSpan k (range Subtype.val) = ⊤",
"tactic": "rw [Subtype.range_coe, h_tot, ht]"
}
] |
[
324,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Topology/ShrinkingLemma.lean
|
exists_subset_iUnion_closure_subset
|
[
{
"state_after": "ι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis : Nonempty (PartialRefinement u s)\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i",
"state_before": "ι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i",
"tactic": "haveI : Nonempty (PartialRefinement u s) := ⟨⟨u, ∅, uo, us, False.elim, fun _ => rfl⟩⟩"
},
{
"state_after": "case intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i",
"state_before": "ι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i",
"tactic": "rcases zorn_nonempty_partialOrder this with ⟨v, hv⟩"
},
{
"state_after": "case intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝¹ : Nonempty (PartialRefinement u s)\nthis✝ :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\nthis : ∀ (i : ι), i ∈ v.carrier\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i\n\ncase this\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∀ (i : ι), i ∈ v.carrier",
"state_before": "case intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i",
"tactic": "suffices : ∀ i, i ∈ v.carrier"
},
{
"state_after": "case this\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∀ (i : ι), i ∈ v.carrier",
"state_before": "case intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝¹ : Nonempty (PartialRefinement u s)\nthis✝ :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\nthis : ∀ (i : ι), i ∈ v.carrier\n⊢ ∃ v, s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i\n\ncase this\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∀ (i : ι), i ∈ v.carrier",
"tactic": "exact ⟨v, v.subset_iUnion, fun i => v.isOpen _, fun i => v.closure_subset (this i)⟩"
},
{
"state_after": "case this\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∃ i, ¬i ∈ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v",
"state_before": "case this\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∀ (i : ι), i ∈ v.carrier",
"tactic": "contrapose! hv"
},
{
"state_after": "case this.intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : ¬i ∈ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v",
"state_before": "case this\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\nhv : ∃ i, ¬i ∈ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v",
"tactic": "rcases hv with ⟨i, hi⟩"
},
{
"state_after": "case this.intro.intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : ¬i ∈ v.carrier\nv' : PartialRefinement u s\nhlt : v < v'\n⊢ ∃ a, v ≤ a ∧ a ≠ v",
"state_before": "case this.intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : ¬i ∈ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v",
"tactic": "rcases v.exists_gt hs i hi with ⟨v', hlt⟩"
},
{
"state_after": "no goals",
"state_before": "case this.intro.intro\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ (x : X), x ∈ s → Set.Finite {i | x ∈ u i}\nus : s ⊆ ⋃ (i : ι), u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis :\n ∀ (c : Set (PartialRefinement u s)),\n IsChain (fun x x_1 => x ≤ x_1) c → Set.Nonempty c → ∃ ub, ∀ (v : PartialRefinement u s), v ∈ c → v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : ¬i ∈ v.carrier\nv' : PartialRefinement u s\nhlt : v < v'\n⊢ ∃ a, v ≤ a ∧ a ≠ v",
"tactic": "exact ⟨v', hlt.le, hlt.ne'⟩"
}
] |
[
234,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Data/Bool/Count.lean
|
List.Chain'.count_false_eq_count_true
|
[] |
[
79,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.mem_mk
|
[] |
[
188,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.lift_surjective_of_surjective
|
[] |
[
998,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
996,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.image_mem_nhds
|
[] |
[
376,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Algebra/MonoidAlgebra/Grading.lean
|
AddMonoidAlgebra.grade_eq_lsingle_range
|
[] |
[
85,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.toFinsupp_C
|
[] |
[
496,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.contDiffWithinAt_extend_coord_change
|
[
{
"state_after": "𝕜 : Type u_3\nE : Type u_4\nM : Type u_2\nH : Type u_1\nE' : Type ?u.176540\nM' : Type ?u.176543\nH' : Type ?u.176546\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : E\nhx : x ∈ (LocalEquiv.symm (extend f' I) ≫ extend f I).source\n⊢ (LocalEquiv.symm (extend f' I) ≫ extend f I).source ∈ 𝓝[range ↑I] x",
"state_before": "𝕜 : Type u_3\nE : Type u_4\nM : Type u_2\nH : Type u_1\nE' : Type ?u.176540\nM' : Type ?u.176543\nH' : Type ?u.176546\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : E\nhx : x ∈ (LocalEquiv.symm (extend f' I) ≫ extend f I).source\n⊢ ContDiffWithinAt 𝕜 ⊤ (↑(extend f I) ∘ ↑(LocalEquiv.symm (extend f' I))) (range ↑I) x",
"tactic": "apply (contDiffOn_extend_coord_change I hf hf' x hx).mono_of_mem"
},
{
"state_after": "𝕜 : Type u_3\nE : Type u_4\nM : Type u_2\nH : Type u_1\nE' : Type ?u.176540\nM' : Type ?u.176543\nH' : Type ?u.176546\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : E\nhx : x ∈ ↑I '' (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source\n⊢ ↑I '' (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source ∈ 𝓝[range ↑I] x",
"state_before": "𝕜 : Type u_3\nE : Type u_4\nM : Type u_2\nH : Type u_1\nE' : Type ?u.176540\nM' : Type ?u.176543\nH' : Type ?u.176546\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : E\nhx : x ∈ (LocalEquiv.symm (extend f' I) ≫ extend f I).source\n⊢ (LocalEquiv.symm (extend f' I) ≫ extend f I).source ∈ 𝓝[range ↑I] x",
"tactic": "rw [extend_coord_change_source] at hx ⊢"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\nE : Type u_4\nM : Type u_2\nH : Type u_1\nE' : Type ?u.176540\nM' : Type ?u.176543\nH' : Type ?u.176546\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nz : H\nhz : z ∈ (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source\n⊢ ↑I '' (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source ∈ 𝓝[range ↑I] ↑I z",
"state_before": "𝕜 : Type u_3\nE : Type u_4\nM : Type u_2\nH : Type u_1\nE' : Type ?u.176540\nM' : Type ?u.176543\nH' : Type ?u.176546\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nx : E\nhx : x ∈ ↑I '' (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source\n⊢ ↑I '' (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source ∈ 𝓝[range ↑I] x",
"tactic": "obtain ⟨z, hz, rfl⟩ := hx"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type u_3\nE : Type u_4\nM : Type u_2\nH : Type u_1\nE' : Type ?u.176540\nM' : Type ?u.176543\nH' : Type ?u.176546\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ninst✝ : ChartedSpace H M\nhf : f ∈ maximalAtlas I M\nhf' : f' ∈ maximalAtlas I M\nz : H\nhz : z ∈ (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source\n⊢ ↑I '' (LocalHomeomorph.symm f' ≫ₕ f).toLocalEquiv.source ∈ 𝓝[range ↑I] ↑I z",
"tactic": "exact I.image_mem_nhdsWithin ((LocalHomeomorph.open_source _).mem_nhds hz)"
}
] |
[
1009,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1003,
1
] |
Mathlib/Algebra/GeomSum.lean
|
Nat.pred_mul_geom_sum_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.218631\na b n : ℕ\n⊢ (b - 1) * ∑ i in range (succ n), a / b ^ i =\n ∑ i in range n, a / b ^ (i + 1) * b + a * b - (∑ i in range n, a / b ^ i + a / b ^ n)",
"tactic": "rw [tsub_mul, mul_comm, sum_mul, one_mul, sum_range_succ', sum_range_succ, pow_zero,\nNat.div_one]"
},
{
"state_after": "α : Type u\nβ : Type ?u.218631\na b n i : ℕ\nx✝ : i ∈ range n\n⊢ a / b ^ (i + 1) * b ≤ a / b ^ i",
"state_before": "α : Type u\nβ : Type ?u.218631\na b n : ℕ\n⊢ ∑ i in range n, a / b ^ (i + 1) * b + a * b - (∑ i in range n, a / b ^ i + a / b ^ n) ≤\n ∑ i in range n, a / b ^ i + a * b - (∑ i in range n, a / b ^ i + a / b ^ n)",
"tactic": "refine' tsub_le_tsub_right (add_le_add_right (sum_le_sum fun i _ => _) _) _"
},
{
"state_after": "α : Type u\nβ : Type ?u.218631\na b n i : ℕ\nx✝ : i ∈ range n\n⊢ a / (b ^ i * b) * b ≤ a / b ^ i",
"state_before": "α : Type u\nβ : Type ?u.218631\na b n i : ℕ\nx✝ : i ∈ range n\n⊢ a / b ^ (i + 1) * b ≤ a / b ^ i",
"tactic": "rw [pow_succ', mul_comm b]"
},
{
"state_after": "α : Type u\nβ : Type ?u.218631\na b n i : ℕ\nx✝ : i ∈ range n\n⊢ a / b ^ i / b * b ≤ a / b ^ i",
"state_before": "α : Type u\nβ : Type ?u.218631\na b n i : ℕ\nx✝ : i ∈ range n\n⊢ a / (b ^ i * b) * b ≤ a / b ^ i",
"tactic": "rw [← Nat.div_div_eq_div_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.218631\na b n i : ℕ\nx✝ : i ∈ range n\n⊢ a / b ^ i / b * b ≤ a / b ^ i",
"tactic": "exact Nat.div_mul_le_self _ _"
}
] |
[
420,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/Algebra/Order/Nonneg/Field.lean
|
Nonneg.coe_inv
|
[] |
[
46,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
11
] |
Mathlib/Order/Monotone/Basic.lean
|
monotoneOn_const
|
[] |
[
529,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
toLex_natCast
|
[] |
[
376,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mem_support_finset_sum
|
[] |
[
1197,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1193,
1
] |
Std/Data/RBMap/Alter.lean
|
Std.RBSet.ModifyWF.of_eq
|
[
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nf : α → α\nt : RBSet α cmp\nH : ∀ {x : α}, RBNode.find? cut t.val = some x → cmpEq cmp (f x) x\n⊢ OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nf : α → α\nt : RBSet α cmp\nH : ∀ {x : α}, RBNode.find? cut t.val = some x → cmpEq cmp (f x) x\n⊢ ModifyWF t cut f",
"tactic": "refine ⟨.modify ?_ t.2⟩"
},
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nf : α → α\nt : RBSet α cmp\n⊢ (∀ {x : α}, RBNode.find? cut t.val = some x → cmpEq cmp (f x) x) →\n OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nf : α → α\nt : RBSet α cmp\nH : ∀ {x : α}, RBNode.find? cut t.val = some x → cmpEq cmp (f x) x\n⊢ OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst",
"tactic": "revert H"
},
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nf : α → α\nt : RBSet α cmp\n⊢ (∀ {x : α}, root? (zoom cut t.val Path.root).fst = some x → cmpEq cmp (f x) x) →\n OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nf : α → α\nt : RBSet α cmp\n⊢ (∀ {x : α}, RBNode.find? cut t.val = some x → cmpEq cmp (f x) x) →\n OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst",
"tactic": "rw [find?_eq_zoom]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nf : α → α\nt : RBSet α cmp\n⊢ (∀ {x : α}, root? (zoom cut t.val Path.root).fst = some x → cmpEq cmp (f x) x) →\n OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val Path.root).fst",
"tactic": "cases (t.1.zoom cut).1 <;> intro H <;> [trivial; exact H rfl]"
}
] |
[
421,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
417,
1
] |
Mathlib/MeasureTheory/Covering/VitaliFamily.lean
|
VitaliFamily.FineSubfamilyOn.covering_disjoint_subtype
|
[] |
[
143,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.filter_smul
|
[] |
[
1575,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1573,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.hausdorffEdist_closure₂
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.53920\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t u : Set α\nΦ : α → β\n⊢ hausdorffEdist s (closure t) = hausdorffEdist s t",
"tactic": "simp [@hausdorffEdist_comm _ _ s _]"
}
] |
[
407,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/GroupTheory/GroupAction/Units.lean
|
Units.smul_isUnit
|
[] |
[
46,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
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