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/Data/Set/Pointwise/SMul.lean
|
Set.preimage_smul_inv₀
|
[] |
[
1030,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1029,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieModuleEquiv.coe_to_linearEquiv
|
[] |
[
1023,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1022,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Complex.ofReal_cpow_of_nonpos
|
[
{
"state_after": "case inl\ny : ℂ\nhx : 0 ≤ 0\n⊢ ↑0 ^ y = (-↑0) ^ y * exp (↑π * I * y)\n\ncase inr\nx : ℝ\nhx : x ≤ 0\ny : ℂ\nhlt : x < 0\n⊢ ↑x ^ y = (-↑x) ^ y * exp (↑π * I * y)",
"state_before": "x : ℝ\nhx : x ≤ 0\ny : ℂ\n⊢ ↑x ^ y = (-↑x) ^ y * exp (↑π * I * y)",
"tactic": "rcases hx.eq_or_lt with (rfl | hlt)"
},
{
"state_after": "case inr\nx : ℝ\nhx : x ≤ 0\ny : ℂ\nhlt : x < 0\nhne : ↑x ≠ 0\n⊢ ↑x ^ y = (-↑x) ^ y * exp (↑π * I * y)",
"state_before": "case inr\nx : ℝ\nhx : x ≤ 0\ny : ℂ\nhlt : x < 0\n⊢ ↑x ^ y = (-↑x) ^ y * exp (↑π * I * y)",
"tactic": "have hne : (x : ℂ) ≠ 0 := ofReal_ne_zero.mpr hlt.ne"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nhx : x ≤ 0\ny : ℂ\nhlt : x < 0\nhne : ↑x ≠ 0\n⊢ ↑x ^ y = (-↑x) ^ y * exp (↑π * I * y)",
"tactic": "rw [cpow_def_of_ne_zero hne, cpow_def_of_ne_zero (neg_ne_zero.2 hne), ← exp_add, ← add_mul, log,\n log, abs.map_neg, arg_ofReal_of_neg hlt, ← ofReal_neg,\n arg_ofReal_of_nonneg (neg_nonneg.2 hx), ofReal_zero, MulZeroClass.zero_mul, add_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl\ny : ℂ\nhx : 0 ≤ 0\n⊢ ↑0 ^ y = (-↑0) ^ y * exp (↑π * I * y)",
"tactic": "rcases eq_or_ne y 0 with (rfl | hy) <;> simp [*]"
}
] |
[
258,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.replicate_succ
|
[] |
[
894,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
9
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.tendsto_div_nhds_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.598952\nE : Type ?u.598955\nF : Type ?u.598958\nG : Type ?u.598961\nE' : Type ?u.598964\nF' : Type ?u.598967\nG' : Type ?u.598970\nE'' : Type ?u.598973\nF'' : Type ?u.598976\nG'' : Type ?u.598979\nR : Type ?u.598982\nR' : Type ?u.598985\n𝕜 : Type u_2\n𝕜' : Type ?u.598991\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf g : α → 𝕜\nh : f =o[l] g\n⊢ (fun x => f x / g x) =o[l] fun _x => 1",
"tactic": "calc\n (fun x => f x / g x) =o[l] fun x => g x / g x := by\n simpa only [div_eq_mul_inv] using h.mul_isBigO (isBigO_refl _ _)\n _ =O[l] fun _x => (1 : 𝕜) := isBigO_of_le _ fun x => by simp [div_self_le_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.598952\nE : Type ?u.598955\nF : Type ?u.598958\nG : Type ?u.598961\nE' : Type ?u.598964\nF' : Type ?u.598967\nG' : Type ?u.598970\nE'' : Type ?u.598973\nF'' : Type ?u.598976\nG'' : Type ?u.598979\nR : Type ?u.598982\nR' : Type ?u.598985\n𝕜 : Type u_2\n𝕜' : Type ?u.598991\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf g : α → 𝕜\nh : f =o[l] g\n⊢ (fun x => f x / g x) =o[l] fun x => g x / g x",
"tactic": "simpa only [div_eq_mul_inv] using h.mul_isBigO (isBigO_refl _ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.598952\nE : Type ?u.598955\nF : Type ?u.598958\nG : Type ?u.598961\nE' : Type ?u.598964\nF' : Type ?u.598967\nG' : Type ?u.598970\nE'' : Type ?u.598973\nF'' : Type ?u.598976\nG'' : Type ?u.598979\nR : Type ?u.598982\nR' : Type ?u.598985\n𝕜 : Type u_2\n𝕜' : Type ?u.598991\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng✝ : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf g : α → 𝕜\nh : f =o[l] g\nx : α\n⊢ ‖g x / g x‖ ≤ ‖1‖",
"tactic": "simp [div_self_le_one]"
}
] |
[
1821,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1815,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae
|
[
{
"state_after": "ι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\n⊢ HasFDerivAtFilter (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca)\n (a, b) (𝓝[s] a ×ˢ 𝓝[t] b)",
"state_before": "ι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\n⊢ HasFDerivWithinAt (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca)\n (s ×ˢ t) (a, b)",
"tactic": "rw [HasFDerivWithinAt, nhdsWithin_prod_eq]"
},
{
"state_after": "ι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\n⊢ HasFDerivAtFilter (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca)\n (a, b) (𝓝[s] a ×ˢ 𝓝[t] b)",
"state_before": "ι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\n⊢ HasFDerivAtFilter (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca)\n (a, b) (𝓝[s] a ×ˢ 𝓝[t] b)",
"tactic": "have :=\n integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hf hmeas_a hmeas_b ha hb\n (tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[s] a)) tendsto_fst\n (tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[t] b)) tendsto_snd"
},
{
"state_after": "case refine'_1\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\n⊢ ∀ (x : ℝ × ℝ),\n ((∫ (x : ℝ) in x.fst..x.snd, f x) - ∫ (x : ℝ) in a..b, f x) - ((x.snd - b) • cb - (x.fst - a) • ca) =\n (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) x - (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (a, b) -\n ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x - (a, b))\n\ncase refine'_2\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\n⊢ (fun t => ‖t.fst - a‖ + ‖t.snd - b‖) =O[𝓝[s] a ×ˢ 𝓝[t] b] fun x' => x' - (a, b)",
"state_before": "ι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\n⊢ HasFDerivAtFilter (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca)\n (a, b) (𝓝[s] a ×ˢ 𝓝[t] b)",
"tactic": "refine' (this.congr_left _).trans_isBigO _"
},
{
"state_after": "case refine'_1\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\nx : ℝ × ℝ\n⊢ ((∫ (x : ℝ) in x.fst..x.snd, f x) - ∫ (x : ℝ) in a..b, f x) - ((x.snd - b) • cb - (x.fst - a) • ca) =\n (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) x - (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (a, b) -\n ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x - (a, b))",
"state_before": "case refine'_1\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\n⊢ ∀ (x : ℝ × ℝ),\n ((∫ (x : ℝ) in x.fst..x.snd, f x) - ∫ (x : ℝ) in a..b, f x) - ((x.snd - b) • cb - (x.fst - a) • ca) =\n (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) x - (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (a, b) -\n ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x - (a, b))",
"tactic": "intro x"
},
{
"state_after": "case refine'_1\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\nx : ℝ × ℝ\n⊢ x.snd • cb - b • cb - (x.fst • ca - a • ca) = x.snd • cb - x.fst • ca - (b • cb - a • ca)",
"state_before": "case refine'_1\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\nx : ℝ × ℝ\n⊢ ((∫ (x : ℝ) in x.fst..x.snd, f x) - ∫ (x : ℝ) in a..b, f x) - ((x.snd - b) • cb - (x.fst - a) • ca) =\n (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) x - (fun p => ∫ (x : ℝ) in p.fst..p.snd, f x) (a, b) -\n ↑(smulRight (snd ℝ ℝ ℝ) cb - smulRight (fst ℝ ℝ ℝ) ca) (x - (a, b))",
"tactic": "simp [sub_smul]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\nx : ℝ × ℝ\n⊢ x.snd • cb - b • cb - (x.fst • ca - a • ca) = x.snd • cb - x.fst • ca - (b • cb - a • ca)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type ?u.1718694\n𝕜 : Type ?u.1718697\nE : Type u_1\nF : Type ?u.1718703\nA : Type ?u.1718706\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\na b z : ℝ\nu v ua ub va vb : ι → ℝ\ninst✝³ : FTCFilter a la la'\ninst✝² : FTCFilter b lb lb'\nhf : IntervalIntegrable f volume a b\ns t : Set ℝ\ninst✝¹ : FTCFilter a (𝓝[s] a) la\ninst✝ : FTCFilter b (𝓝[t] b) lb\nhmeas_a : StronglyMeasurableAtFilter f la\nhmeas_b : StronglyMeasurableAtFilter f lb\nha : Tendsto f (la ⊓ Measure.ae volume) (𝓝 ca)\nhb : Tendsto f (lb ⊓ Measure.ae volume) (𝓝 cb)\nthis :\n (fun t =>\n ((∫ (x : ℝ) in t.fst..t.snd, f x) - ∫ (x : ℝ) in a..b, f x) -\n ((t.snd - b) • cb - (t.fst - a) • ca)) =o[𝓝[s] a ×ˢ 𝓝[t] b]\n fun t => ‖t.fst - a‖ + ‖t.snd - b‖\n⊢ (fun t => ‖t.fst - a‖ + ‖t.snd - b‖) =O[𝓝[s] a ×ˢ 𝓝[t] b] fun x' => x' - (a, b)",
"tactic": "exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left"
}
] |
[
842,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
829,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
closure_one_eq
|
[] |
[
1135,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1134,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.card_Ico_eq_card_Icc_sub_one
|
[] |
[
227,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.PosDef.add
|
[] |
[
954,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
953,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.characteristic_of_normal
|
[
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\n⊢ Characteristic ↑P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\n⊢ Characteristic ↑P",
"tactic": "haveI := Sylow.subsingleton_of_normal P h"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\n⊢ ∀ (ϕ : G ≃* G), Subgroup.map (MulEquiv.toMonoidHom ϕ) ↑P = ↑P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\n⊢ Characteristic ↑P",
"tactic": "rw [characteristic_iff_map_eq]"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\nΦ : G ≃* G\n⊢ Subgroup.map (MulEquiv.toMonoidHom Φ) ↑P = ↑P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\n⊢ ∀ (ϕ : G ≃* G), Subgroup.map (MulEquiv.toMonoidHom ϕ) ↑P = ↑P",
"tactic": "intro Φ"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\nΦ : G ≃* G\n⊢ ↑(Φ • P) = ↑P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\nΦ : G ≃* G\n⊢ Subgroup.map (MulEquiv.toMonoidHom Φ) ↑P = ↑P",
"tactic": "show (Φ • P).toSubgroup = P.toSubgroup"
},
{
"state_after": "case e_self\nG : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\nΦ : G ≃* G\n⊢ Φ • P = P",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\nΦ : G ≃* G\n⊢ ↑(Φ • P) = ↑P",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_self\nG : Type u\nα : Type v\nβ : Type w\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : Normal ↑P\nthis : Subsingleton (Sylow p G)\nΦ : G ≃* G\n⊢ Φ • P = P",
"tactic": "simp"
}
] |
[
720,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.add_le_add_iff_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.75944\nβ : Type ?u.75947\nγ : Type ?u.75950\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\n⊢ a + ↑0 ≤ b + ↑0 ↔ a ≤ b",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.75944\nβ : Type ?u.75947\nγ : Type ?u.75950\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nn : ℕ\n⊢ a + ↑(n + 1) ≤ b + ↑(n + 1) ↔ a ≤ b",
"tactic": "simp only [nat_cast_succ, add_succ, add_succ, succ_le_succ_iff, add_le_add_iff_right]"
}
] |
[
141,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.mem_pmem
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7041\nδ : Type ?u.7044\np : α → Prop\nf : (a : α) → p a → β\nx : Option α\na : α\nh : ∀ (a : α), a ∈ x → p a\nha✝ : a ∈ x\nha : x = some a\n⊢ pmap f x h = some (f a (_ : p a))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7041\nδ : Type ?u.7044\np : α → Prop\nf : (a : α) → p a → β\nx : Option α\na : α\nh : ∀ (a : α), a ∈ x → p a\nha : a ∈ x\n⊢ f a (_ : p a) ∈ pmap f x h",
"tactic": "rw [mem_def] at ha ⊢"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7041\nδ : Type ?u.7044\np : α → Prop\nf : (a : α) → p a → β\na : α\nh : ∀ (a_1 : α), a_1 ∈ some a → p a_1\nha : a ∈ some a\n⊢ pmap f (some a) h = some (f a (_ : p a))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7041\nδ : Type ?u.7044\np : α → Prop\nf : (a : α) → p a → β\nx : Option α\na : α\nh : ∀ (a : α), a ∈ x → p a\nha✝ : a ∈ x\nha : x = some a\n⊢ pmap f x h = some (f a (_ : p a))",
"tactic": "subst ha"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.7041\nδ : Type ?u.7044\np : α → Prop\nf : (a : α) → p a → β\na : α\nh : ∀ (a_1 : α), a_1 ∈ some a → p a_1\nha : a ∈ some a\n⊢ pmap f (some a) h = some (f a (_ : p a))",
"tactic": "rfl"
}
] |
[
180,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.exists_eq_spanSingleton_mul
|
[
{
"state_after": "case intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\n⊢ ∃ a aI, a ≠ 0 ∧ I = spanSingleton R₁⁰ (↑(algebraMap R₁ K) a)⁻¹ * ↑aI",
"state_before": "R : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\n⊢ ∃ a aI, a ≠ 0 ∧ I = spanSingleton R₁⁰ (↑(algebraMap R₁ K) a)⁻¹ * ↑aI",
"tactic": "obtain ⟨a_inv, nonzero, ha⟩ := I.isFractional"
},
{
"state_after": "case intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\n⊢ ∃ a aI, a ≠ 0 ∧ I = spanSingleton R₁⁰ (↑(algebraMap R₁ K) a)⁻¹ * ↑aI",
"state_before": "case intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\n⊢ ∃ a aI, a ≠ 0 ∧ I = spanSingleton R₁⁰ (↑(algebraMap R₁ K) a)⁻¹ * ↑aI",
"tactic": "have nonzero := mem_nonZeroDivisors_iff_ne_zero.mp nonzero"
},
{
"state_after": "case intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\n⊢ ∃ a aI, a ≠ 0 ∧ I = spanSingleton R₁⁰ (↑(algebraMap R₁ K) a)⁻¹ * ↑aI",
"state_before": "case intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\n⊢ ∃ a aI, a ≠ 0 ∧ I = spanSingleton R₁⁰ (↑(algebraMap R₁ K) a)⁻¹ * ↑aI",
"tactic": "have map_a_nonzero : algebraMap R₁ K a_inv ≠ 0 :=\n mt IsFractionRing.to_map_eq_zero_iff.mp nonzero"
},
{
"state_after": "case intro.intro.refine'_1\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\n⊢ x ∈ I →\n ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'\n\ncase intro.intro.refine'_2\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\n⊢ (∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y') →\n x ∈ I",
"state_before": "case intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\n⊢ ∃ a aI, a ≠ 0 ∧ I = spanSingleton R₁⁰ (↑(algebraMap R₁ K) a)⁻¹ * ↑aI",
"tactic": "refine'\n ⟨a_inv,\n Submodule.comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (algebraMap R₁ K a_inv) * I),\n nonzero, ext fun x => Iff.trans ⟨_, _⟩ mem_singleton_mul.symm⟩"
},
{
"state_after": "case intro.intro.refine'_1\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\n⊢ ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'",
"state_before": "case intro.intro.refine'_1\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\n⊢ x ∈ I →\n ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'",
"tactic": "intro hx"
},
{
"state_after": "case intro.intro.refine'_1.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = a_inv • x\n⊢ ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'",
"state_before": "case intro.intro.refine'_1\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\n⊢ ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'",
"tactic": "obtain ⟨x', hx'⟩ := ha x hx"
},
{
"state_after": "case intro.intro.refine'_1.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * x\n⊢ ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'",
"state_before": "case intro.intro.refine'_1.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = a_inv • x\n⊢ ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'",
"tactic": "rw [Algebra.smul_def] at hx'"
},
{
"state_after": "case intro.intro.refine'_1.intro.refine'_1\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * x\n⊢ ∃ y', y' ∈ I ∧ ↑(Algebra.linearMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * y'\n\ncase intro.intro.refine'_1.intro.refine'_2\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * x\n⊢ x = (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x'",
"state_before": "case intro.intro.refine'_1.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * x\n⊢ ∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y'",
"tactic": "refine' ⟨algebraMap R₁ K x', (mem_coeIdeal _).mpr ⟨x', mem_singleton_mul.mpr _, rfl⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1.intro.refine'_1\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * x\n⊢ ∃ y', y' ∈ I ∧ ↑(Algebra.linearMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * y'",
"tactic": "exact ⟨x, hx, hx'⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1.intro.refine'_2\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\nhx : x ∈ I\nx' : R₁\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * x\n⊢ x = (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x'",
"tactic": "rw [hx', ← mul_assoc, inv_mul_cancel map_a_nonzero, one_mul]"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\ny : K\nhy : y ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * y ∈ I",
"state_before": "case intro.intro.refine'_2\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx : K\n⊢ (∃ y',\n y' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)) ∧\n x = (↑(algebraMap R₁ K) a_inv)⁻¹ * y') →\n x ∈ I",
"tactic": "rintro ⟨y, hy, rfl⟩"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx' : R₁\nhx' : x' ∈ comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)\nhy : ↑(algebraMap R₁ K) x' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x' ∈ I",
"state_before": "case intro.intro.refine'_2.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\ny : K\nhy : y ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * y ∈ I",
"tactic": "obtain ⟨x', hx', rfl⟩ := (mem_coeIdeal _).mp hy"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro.intro.intro.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx' : R₁\nhx'✝ : x' ∈ comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)\nhy : ↑(algebraMap R₁ K) x' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\ny' : K\nhy' : y' ∈ I\nhx' : ↑(Algebra.linearMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * y'\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x' ∈ I",
"state_before": "case intro.intro.refine'_2.intro.intro.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx' : R₁\nhx' : x' ∈ comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)\nhy : ↑(algebraMap R₁ K) x' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x' ∈ I",
"tactic": "obtain ⟨y', hy', hx'⟩ := mem_singleton_mul.mp hx'"
},
{
"state_after": "case intro.intro.refine'_2.intro.intro.intro.intro.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx' : R₁\nhx'✝ : x' ∈ comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)\nhy : ↑(algebraMap R₁ K) x' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\ny' : K\nhy' : y' ∈ I\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * y'\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x' ∈ I",
"state_before": "case intro.intro.refine'_2.intro.intro.intro.intro.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx' : R₁\nhx'✝ : x' ∈ comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)\nhy : ↑(algebraMap R₁ K) x' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\ny' : K\nhy' : y' ∈ I\nhx' : ↑(Algebra.linearMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * y'\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x' ∈ I",
"tactic": "rw [Algebra.linearMap_apply] at hx'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2.intro.intro.intro.intro.intro.intro\nR : Type ?u.1535657\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1535864\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\ninst✝¹ : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI : FractionalIdeal R₁⁰ K\na_inv : R₁\nnonzero✝ : a_inv ∈ R₁⁰\nha : ∀ (b : K), b ∈ ↑I → IsInteger R₁ (a_inv • b)\nnonzero : a_inv ≠ 0\nmap_a_nonzero : ↑(algebraMap R₁ K) a_inv ≠ 0\nx' : R₁\nhx'✝ : x' ∈ comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I)\nhy : ↑(algebraMap R₁ K) x' ∈ ↑(comap (Algebra.linearMap R₁ K) ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) a_inv) * I))\ny' : K\nhy' : y' ∈ I\nhx' : ↑(algebraMap R₁ K) x' = ↑(algebraMap R₁ K) a_inv * y'\n⊢ (↑(algebraMap R₁ K) a_inv)⁻¹ * ↑(algebraMap R₁ K) x' ∈ I",
"tactic": "rwa [hx', ← mul_assoc, inv_mul_cancel map_a_nonzero, one_mul]"
}
] |
[
1509,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1489,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.continuous_sin
|
[] |
[
93,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.coe_coe
|
[] |
[
136,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_iUnion_apply_ae
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.302552\nγ : Type ?u.302555\nδ : Type ?u.302558\nι : Type u_1\nR : Type ?u.302564\nR' : Type ?u.302567\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\nhm : ∀ (i : ι), NullMeasurableSet (s i)\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑μ (⋃ (i : ι), t ∩ s i) = ∑' (i : ι), ↑↑μ (t ∩ s i)",
"state_before": "α : Type u_2\nβ : Type ?u.302552\nγ : Type ?u.302555\nδ : Type ?u.302558\nι : Type u_1\nR : Type ?u.302564\nR' : Type ?u.302567\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\nhm : ∀ (i : ι), NullMeasurableSet (s i)\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(restrict μ (⋃ (i : ι), s i)) t = ∑' (i : ι), ↑↑(restrict μ (s i)) t",
"tactic": "simp only [restrict_apply, ht, inter_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.302552\nγ : Type ?u.302555\nδ : Type ?u.302558\nι : Type u_1\nR : Type ?u.302564\nR' : Type ?u.302567\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\nhm : ∀ (i : ι), NullMeasurableSet (s i)\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑μ (⋃ (i : ι), t ∩ s i) = ∑' (i : ι), ↑↑μ (t ∩ s i)",
"tactic": "exact\n measure_iUnion₀ (hd.mono fun i j h => h.mono (inter_subset_right _ _) (inter_subset_right _ _))\n fun i => ht.nullMeasurableSet.inter (hm i)"
}
] |
[
1761,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1755,
1
] |
Mathlib/Order/CompleteLattice.lean
|
toDual_sInf
|
[] |
[
430,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.JordanDecomposition.neg_posPart
|
[] |
[
114,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/LinearAlgebra/SModEq.lean
|
SModEq.refl
|
[] |
[
68,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
11
] |
Std/Data/List/Basic.lean
|
List.takeD_succ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nn : Nat\nl : List α\na : α\n⊢ takeD (n + 1) l a = Option.getD (head? l) a :: takeD n (tail l) a",
"tactic": "simp [takeD]"
}
] |
[
590,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
589,
9
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Ioo_ae_eq_Ico
|
[] |
[
3348,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3347,
1
] |
src/lean/Init/Data/Nat/SOM.lean
|
Nat.SOM.Expr.toPoly_denote
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\ne : Expr\n⊢ Poly.denote ctx (toPoly e) = denote ctx e",
"tactic": "induction e with\n| num k =>\n simp!; by_cases h : k == 0 <;> simp! [*]\n simp [eq_of_beq h]\n| var v => simp!\n| add a b => simp! [Poly.add_denote, *]\n| mul a b => simp! [Poly.mul_denote, *]"
},
{
"state_after": "case num\nctx : Context\nk : Nat\n⊢ Poly.denote ctx (bif k == 0 then [] else [(k, [])]) = k",
"state_before": "case num\nctx : Context\nk : Nat\n⊢ Poly.denote ctx (toPoly (num k)) = denote ctx (num k)",
"tactic": "simp!"
},
{
"state_after": "case num.inl\nctx : Context\nk : Nat\nh : (k == 0) = true\n⊢ 0 = k",
"state_before": "case num\nctx : Context\nk : Nat\n⊢ Poly.denote ctx (bif k == 0 then [] else [(k, [])]) = k",
"tactic": "by_cases h : k == 0 <;> simp! [*]"
},
{
"state_after": "no goals",
"state_before": "case num.inl\nctx : Context\nk : Nat\nh : (k == 0) = true\n⊢ 0 = k",
"tactic": "simp [eq_of_beq h]"
},
{
"state_after": "no goals",
"state_before": "case var\nctx : Context\nv : Var\n⊢ Poly.denote ctx (toPoly (var v)) = denote ctx (var v)",
"tactic": "simp!"
},
{
"state_after": "no goals",
"state_before": "case add\nctx : Context\na b : Expr\na_ih✝ : Poly.denote ctx (toPoly a) = denote ctx a\nb_ih✝ : Poly.denote ctx (toPoly b) = denote ctx b\n⊢ Poly.denote ctx (toPoly (add a b)) = denote ctx (add a b)",
"tactic": "simp! [Poly.add_denote, *]"
},
{
"state_after": "no goals",
"state_before": "case mul\nctx : Context\na b : Expr\na_ih✝ : Poly.denote ctx (toPoly a) = denote ctx a\nb_ih✝ : Poly.denote ctx (toPoly b) = denote ctx b\n⊢ Poly.denote ctx (toPoly (mul a b)) = denote ctx (mul a b)",
"tactic": "simp! [Poly.mul_denote, *]"
}
] |
[
178,
42
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
171,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.insert_subset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\n⊢ insert a s ⊆ t ↔ a ∈ t ∧ s ⊆ t",
"tactic": "simp only [subset_def, mem_insert_iff, or_imp, forall_and, forall_eq]"
}
] |
[
1156,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1155,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.map_mk'
|
[
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nL R : ListBlank Γ\n⊢ map f (mk' L R) = mk' (ListBlank.map f L) (ListBlank.map f R)",
"tactic": "simp only [Tape.mk', Tape.map, ListBlank.head_map, eq_self_iff_true, and_self_iff,\n ListBlank.tail_map]"
}
] |
[
724,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
721,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
TopHom.dual_id
|
[] |
[
727,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
726,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.Tendsto.const_mul_atBot
|
[] |
[
1208,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1206,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.not_dvd_of_degree_lt
|
[] |
[
208,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
LocalHomeomorph.contDiffAt_symm
|
[
{
"state_after": "case h0\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nhf : ContDiffAt 𝕜 0 (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ContDiffAt 𝕜 0 (↑(LocalHomeomorph.symm f)) a\n\ncase hsuc\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ContDiffAt 𝕜 (↑(Nat.succ n)) (↑(LocalHomeomorph.symm f)) a\n\ncase htop\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 ⊤ (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ContDiffAt 𝕜 ⊤ (↑(LocalHomeomorph.symm f)) a",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ContDiffAt 𝕜 n (↑(LocalHomeomorph.symm f)) a",
"tactic": "induction' n using ENat.nat_induction with n IH Itop"
},
{
"state_after": "case h0\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nhf : ContDiffAt 𝕜 0 (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ∃ u, u ∈ 𝓝 a ∧ ContinuousOn (↑(LocalHomeomorph.symm f)) u",
"state_before": "case h0\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nhf : ContDiffAt 𝕜 0 (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ContDiffAt 𝕜 0 (↑(LocalHomeomorph.symm f)) a",
"tactic": "rw [contDiffAt_zero]"
},
{
"state_after": "no goals",
"state_before": "case h0\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nhf : ContDiffAt 𝕜 0 (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ∃ u, u ∈ 𝓝 a ∧ ContinuousOn (↑(LocalHomeomorph.symm f)) u",
"tactic": "exact ⟨f.target, IsOpen.mem_nhds f.open_target ha, f.continuous_invFun⟩"
},
{
"state_after": "case hsuc.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\n⊢ ContDiffAt 𝕜 (↑(Nat.succ n)) (↑(LocalHomeomorph.symm f)) a",
"state_before": "case hsuc\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ContDiffAt 𝕜 (↑(Nat.succ n)) (↑(LocalHomeomorph.symm f)) a",
"tactic": "obtain ⟨f', ⟨u, hu, hff'⟩, hf'⟩ := contDiffAt_succ_iff_hasFDerivAt.mp hf"
},
{
"state_after": "case hsuc.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\n⊢ ∃ f', (∃ u, u ∈ 𝓝 a ∧ ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) (f' x) x) ∧ ContDiffAt 𝕜 (↑n) f' a",
"state_before": "case hsuc.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\n⊢ ContDiffAt 𝕜 (↑(Nat.succ n)) (↑(LocalHomeomorph.symm f)) a",
"tactic": "apply contDiffAt_succ_iff_hasFDerivAt.mpr"
},
{
"state_after": "case hsuc.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\n⊢ ∃ f', (∃ u, u ∈ 𝓝 a ∧ ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) (f' x) x) ∧ ContDiffAt 𝕜 (↑n) f' a",
"state_before": "case hsuc.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\n⊢ ∃ f', (∃ u, u ∈ 𝓝 a ∧ ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) (f' x) x) ∧ ContDiffAt 𝕜 (↑n) f' a",
"tactic": "have eq_f₀' : f' (f.symm a) = f₀' := (hff' (f.symm a) (mem_of_mem_nhds hu)).unique hf₀'"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\n⊢ ∃ u,\n u ∈ 𝓝 a ∧\n ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x\n\ncase hsuc.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\n⊢ ContDiffAt 𝕜 (↑n) (inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) a",
"state_before": "case hsuc.intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\n⊢ ∃ f', (∃ u, u ∈ 𝓝 a ∧ ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) (f' x) x) ∧ ContDiffAt 𝕜 (↑n) f' a",
"tactic": "refine' ⟨inverse ∘ f' ∘ f.symm, _, _⟩"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\n⊢ ∃ u,\n u ∈ 𝓝 a ∧\n ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\n⊢ ∃ u,\n u ∈ 𝓝 a ∧\n ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "have h_nhds : { y : E | ∃ e : E ≃L[𝕜] F, ↑e = f' y } ∈ 𝓝 (f.symm a) := by\n have hf₀' := f₀'.nhds\n rw [← eq_f₀'] at hf₀'\n exact hf'.continuousAt.preimage_mem_nhds hf₀'"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\n⊢ ∃ u,\n u ∈ 𝓝 a ∧\n ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\n⊢ ∃ u,\n u ∈ 𝓝 a ∧\n ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "obtain ⟨t, htu, ht, htf⟩ := mem_nhds_iff.mp (Filter.inter_mem hu h_nhds)"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\n⊢ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t ∈ 𝓝 a ∧\n ∀ (x : F),\n x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t →\n HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\n⊢ ∃ u,\n u ∈ 𝓝 a ∧\n ∀ (x : F), x ∈ u → HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "use f.target ∩ f.symm ⁻¹' t"
},
{
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"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\n⊢ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t ∈ 𝓝 a ∧\n ∀ (x : F),\n x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t →\n HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "refine' ⟨IsOpen.mem_nhds _ _, _⟩"
},
{
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"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_3\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\n⊢ ∀ (x : F),\n x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t →\n HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "intro x hx"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_3.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\n⊢ HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_3\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\n⊢ HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "obtain ⟨hxu, e, he⟩ := htu hx.2"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_3.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\nh_deriv : HasFDerivAt (↑f) (↑e) (↑(LocalHomeomorph.symm f) x)\n⊢ HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_3.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\n⊢ HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "have h_deriv : HasFDerivAt f (e : E →L[𝕜] F) (f.symm x) := by\n rw [he]\n exact hff' (f.symm x) hxu"
},
{
"state_after": "case h.e'_10\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\nh_deriv : HasFDerivAt (↑f) (↑e) (↑(LocalHomeomorph.symm f) x)\n⊢ (inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x = ↑(ContinuousLinearEquiv.symm e)",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_3.intro.intro\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\nh_deriv : HasFDerivAt (↑f) (↑e) (↑(LocalHomeomorph.symm f) x)\n⊢ HasFDerivAt (↑(LocalHomeomorph.symm f)) ((inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x) x",
"tactic": "convert f.hasFDerivAt_symm hx.1 h_deriv"
},
{
"state_after": "no goals",
"state_before": "case h.e'_10\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\nh_deriv : HasFDerivAt (↑f) (↑e) (↑(LocalHomeomorph.symm f) x)\n⊢ (inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) x = ↑(ContinuousLinearEquiv.symm e)",
"tactic": "simp [← he]"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀'✝ : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nhf₀' : range ContinuousLinearEquiv.toContinuousLinearMap ∈ 𝓝 ↑f₀'\n⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\n⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)",
"tactic": "have hf₀' := f₀'.nhds"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀'✝ : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nhf₀' : range ContinuousLinearEquiv.toContinuousLinearMap ∈ 𝓝 (f' (↑(LocalHomeomorph.symm f) a))\n⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀'✝ : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nhf₀' : range ContinuousLinearEquiv.toContinuousLinearMap ∈ 𝓝 ↑f₀'\n⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)",
"tactic": "rw [← eq_f₀'] at hf₀'"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀'✝ : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nhf₀' : range ContinuousLinearEquiv.toContinuousLinearMap ∈ 𝓝 (f' (↑(LocalHomeomorph.symm f) a))\n⊢ {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)",
"tactic": "exact hf'.continuousAt.preimage_mem_nhds hf₀'"
},
{
"state_after": "no goals",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_1\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\n⊢ IsOpen (f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t)",
"tactic": "exact f.preimage_open_of_open_symm ht"
},
{
"state_after": "no goals",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_1.intro.intro.intro.refine'_2\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\n⊢ a ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t",
"tactic": "exact mem_inter ha (mem_preimage.mpr htf)"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\n⊢ HasFDerivAt (↑f) (f' (↑(LocalHomeomorph.symm f) x)) (↑(LocalHomeomorph.symm f) x)",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\n⊢ HasFDerivAt (↑f) (↑e) (↑(LocalHomeomorph.symm f) x)",
"tactic": "rw [he]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t✝ u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_nhds : {y | ∃ e, ↑e = f' y} ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nt : Set E\nhtu : t ⊆ u ∩ {y | ∃ e, ↑e = f' y}\nht : IsOpen t\nhtf : ↑(LocalHomeomorph.symm f) a ∈ t\nx : F\nhx : x ∈ f.target ∩ ↑(LocalHomeomorph.symm f) ⁻¹' t\nhxu : ↑(LocalHomeomorph.symm f) x ∈ u\ne : E ≃L[𝕜] F\nhe : ↑e = f' (↑(LocalHomeomorph.symm f) x)\n⊢ HasFDerivAt (↑f) (f' (↑(LocalHomeomorph.symm f) x)) (↑(LocalHomeomorph.symm f) x)",
"tactic": "exact hff' (f.symm x) hxu"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_deriv₁ : ContDiffAt 𝕜 (↑n) inverse (f' (↑(LocalHomeomorph.symm f) a))\n⊢ ContDiffAt 𝕜 (↑n) (inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) a",
"state_before": "case hsuc.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\n⊢ ContDiffAt 𝕜 (↑n) (inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) a",
"tactic": "have h_deriv₁ : ContDiffAt 𝕜 n inverse (f' (f.symm a)) := by\n rw [eq_f₀']\n exact contDiffAt_map_inverse _"
},
{
"state_after": "case hsuc.intro.intro.intro.intro.refine'_2\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_deriv₁ : ContDiffAt 𝕜 (↑n) inverse (f' (↑(LocalHomeomorph.symm f) a))\nh_deriv₂ : ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\n⊢ ContDiffAt 𝕜 (↑n) (inverse ∘ f' ∘ ↑(LocalHomeomorph.symm f)) a",
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"tactic": "have h_deriv₂ : ContDiffAt 𝕜 n f.symm a := by\n refine' IH (hf.of_le _)\n norm_cast\n exact Nat.le_succ n"
},
{
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"tactic": "exact (h_deriv₁.comp _ hf').comp _ h_deriv₂"
},
{
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"tactic": "rw [eq_f₀']"
},
{
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"tactic": "exact contDiffAt_map_inverse _"
},
{
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"tactic": "refine' IH (hf.of_le _)"
},
{
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"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u✝ : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\nIH : ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(LocalHomeomorph.symm f) a)\nf' : E → E →L[𝕜] F\nhf' : ContDiffAt 𝕜 (↑n) f' (↑(LocalHomeomorph.symm f) a)\nu : Set E\nhu : u ∈ 𝓝 (↑(LocalHomeomorph.symm f) a)\nhff' : ∀ (x : E), x ∈ u → HasFDerivAt (↑f) (f' x) x\neq_f₀' : f' (↑(LocalHomeomorph.symm f) a) = ↑f₀'\nh_deriv₁ : ContDiffAt 𝕜 (↑n) inverse (f' (↑(LocalHomeomorph.symm f) a))\n⊢ n ≤ Nat.succ n",
"tactic": "exact Nat.le_succ n"
},
{
"state_after": "case htop\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 ⊤ (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a",
"state_before": "case htop\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 ⊤ (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ContDiffAt 𝕜 ⊤ (↑(LocalHomeomorph.symm f)) a",
"tactic": "refine' contDiffAt_top.mpr _"
},
{
"state_after": "case htop\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 ⊤ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\n⊢ ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a",
"state_before": "case htop\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n (↑f) (↑(LocalHomeomorph.symm f) a)\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 ⊤ (↑f) (↑(LocalHomeomorph.symm f) a)\n⊢ ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a",
"tactic": "intro n"
},
{
"state_after": "no goals",
"state_before": "case htop\n𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.2744530\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf✝ f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : CompleteSpace E\nf : LocalHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.target\nhf₀' : HasFDerivAt (↑f) (↑f₀') (↑(LocalHomeomorph.symm f) a)\nhf✝ : ContDiffAt 𝕜 n✝ (↑f) (↑(LocalHomeomorph.symm f) a)\nItop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑f) (↑(LocalHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a\nhf : ContDiffAt 𝕜 ⊤ (↑f) (↑(LocalHomeomorph.symm f) a)\nn : ℕ\n⊢ ContDiffAt 𝕜 (↑n) (↑(LocalHomeomorph.symm f)) a",
"tactic": "exact Itop n (contDiffAt_top.mp hf n)"
}
] |
[
1827,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1783,
1
] |
Mathlib/Topology/Maps.lean
|
Inducing.isClosed_preimage
|
[] |
[
171,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
isConnected_Ico
|
[] |
[
494,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.map_op_mul
|
[
{
"state_after": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ map (↑(opLinearEquiv R)) (M * N) ≤ map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M\n\ncase a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M ≤ map (↑(opLinearEquiv R)) (M * N)",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ map (↑(opLinearEquiv R)) (M * N) = map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ M * N ≤ comap (↑(opLinearEquiv R)) (map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M)",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ map (↑(opLinearEquiv R)) (M * N) ≤ map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M",
"tactic": "simp_rw [map_le_iff_le_comap]"
},
{
"state_after": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ m * n ∈ comap (↑(opLinearEquiv R)) (map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M)",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ M * N ≤ comap (↑(opLinearEquiv R)) (map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M)",
"tactic": "refine' mul_le.2 fun m hm n hn => _"
},
{
"state_after": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ ↑↑(opLinearEquiv R) (m * n) ∈\n comap (↑(LinearEquiv.symm (opLinearEquiv R))) N * comap (↑(LinearEquiv.symm (opLinearEquiv R))) M",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ m * n ∈ comap (↑(opLinearEquiv R)) (map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M)",
"tactic": "rw [mem_comap, map_equiv_eq_comap_symm, map_equiv_eq_comap_symm]"
},
{
"state_after": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ op n * op m ∈ comap (↑(LinearEquiv.symm (opLinearEquiv R))) N * comap (↑(LinearEquiv.symm (opLinearEquiv R))) M",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ ↑↑(opLinearEquiv R) (m * n) ∈\n comap (↑(LinearEquiv.symm (opLinearEquiv R))) N * comap (↑(LinearEquiv.symm (opLinearEquiv R))) M",
"tactic": "show op n * op m ∈ _"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : m ∈ M\nn : A\nhn : n ∈ N\n⊢ op n * op m ∈ comap (↑(LinearEquiv.symm (opLinearEquiv R))) N * comap (↑(LinearEquiv.symm (opLinearEquiv R))) M",
"tactic": "exact mul_mem_mul hn hm"
},
{
"state_after": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : op m ∈ map (↑(opLinearEquiv R)) N\nn : A\nhn : op n ∈ map (↑(opLinearEquiv R)) M\n⊢ op m * op n ∈ map (↑(opLinearEquiv R)) (M * N)",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ map (↑(opLinearEquiv R)) N * map (↑(opLinearEquiv R)) M ≤ map (↑(opLinearEquiv R)) (M * N)",
"tactic": "refine' mul_le.2 (MulOpposite.rec' fun m hm => MulOpposite.rec' fun n hn => _)"
},
{
"state_after": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : ↑(LinearEquiv.symm (opLinearEquiv R)) (op m) ∈ N\nn : A\nhn : ↑(LinearEquiv.symm (opLinearEquiv R)) (op n) ∈ M\n⊢ ↑(LinearEquiv.symm (opLinearEquiv R)) (op m * op n) ∈ M * N",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : op m ∈ map (↑(opLinearEquiv R)) N\nn : A\nhn : op n ∈ map (↑(opLinearEquiv R)) M\n⊢ op m * op n ∈ map (↑(opLinearEquiv R)) (M * N)",
"tactic": "rw [Submodule.mem_map_equiv] at hm hn⊢"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm✝ n✝ m : A\nhm : ↑(LinearEquiv.symm (opLinearEquiv R)) (op m) ∈ N\nn : A\nhn : ↑(LinearEquiv.symm (opLinearEquiv R)) (op n) ∈ M\n⊢ ↑(LinearEquiv.symm (opLinearEquiv R)) (op m * op n) ∈ M * N",
"tactic": "exact mul_mem_mul hn hm"
}
] |
[
293,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
DifferentiableWithinAt.prod
|
[] |
[
100,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : cos x = 1\nn : ℤ\nhn : ↑n * π = x\nhn0 : n % 2 = 0\n⊢ ↑(n / 2) * (2 * π) = x",
"tactic": "rwa [← mul_assoc, ← @Int.cast_two ℝ, ← Int.cast_mul,\n Int.ediv_mul_cancel ((Int.dvd_iff_emod_eq_zero _ _).2 hn0)]"
},
{
"state_after": "x : ℝ\nh : cos x = 1\nn : ℤ\nhn : ↑(n / 2) * (2 * π) + π = x\nhn1 : n % 2 = 1\n⊢ ↑(n / 2) * (2 * π) = x",
"state_before": "x : ℝ\nh : cos x = 1\nn : ℤ\nhn : ↑n * π = x\nhn1 : n % 2 = 1\n⊢ ↑(n / 2) * (2 * π) = x",
"tactic": "rw [← Int.emod_add_ediv n 2, hn1, Int.cast_add, Int.cast_one, add_mul, one_mul, add_comm,\n mul_comm (2 : ℤ), Int.cast_mul, mul_assoc, Int.cast_two] at hn"
},
{
"state_after": "x : ℝ\nn : ℤ\nh : -1 = 1\nhn : ↑(n / 2) * (2 * π) + π = x\nhn1 : n % 2 = 1\n⊢ ↑(n / 2) * (2 * π) = x",
"state_before": "x : ℝ\nh : cos x = 1\nn : ℤ\nhn : ↑(n / 2) * (2 * π) + π = x\nhn1 : n % 2 = 1\n⊢ ↑(n / 2) * (2 * π) = x",
"tactic": "rw [← hn, cos_int_mul_two_pi_add_pi] at h"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nn : ℤ\nh : -1 = 1\nhn : ↑(n / 2) * (2 * π) + π = x\nhn1 : n % 2 = 1\n⊢ ↑(n / 2) * (2 * π) = x",
"tactic": "exact absurd h (by norm_num)"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nn : ℤ\nh : -1 = 1\nhn : ↑(n / 2) * (2 * π) + π = x\nhn1 : n % 2 = 1\n⊢ ¬-1 = 1",
"tactic": "norm_num"
}
] |
[
542,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.min_eq_right
|
[
{
"state_after": "a b : Int\nh : b ≤ a\n⊢ min b a = b",
"state_before": "a b : Int\nh : b ≤ a\n⊢ min a b = b",
"tactic": "rw [Int.min_comm a b]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh : b ≤ a\n⊢ min b a = b",
"tactic": "exact Int.min_eq_left h"
}
] |
[
706,
49
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
705,
11
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.coe_toHomeomorph_symm
|
[] |
[
518,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/MeasureTheory/Function/Jacobian.lean
|
MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul
|
[
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\n⊢ (∫ (x : E) in f.target, g x ∂μ) = ∫ (x : E) in f.source, abs (ContinuousLinearMap.det (f' x)) • g (↑f x) ∂μ",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\n⊢ (∫ (x : E) in f.target, g x ∂μ) = ∫ (x : E) in f.source, abs (ContinuousLinearMap.det (f' x)) • g (↑f x) ∂μ",
"tactic": "have : f '' f.source = f.target := LocalEquiv.image_source_eq_target f.toLocalEquiv"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\n⊢ (∫ (x : E) in ↑f '' f.source, g x ∂μ) = ∫ (x : E) in f.source, abs (ContinuousLinearMap.det (f' x)) • g (↑f x) ∂μ",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\n⊢ (∫ (x : E) in f.target, g x ∂μ) = ∫ (x : E) in f.source, abs (ContinuousLinearMap.det (f' x)) • g (↑f x) ∂μ",
"tactic": "rw [← this]"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\n⊢ ∀ (x : E), x ∈ f.source → HasFDerivWithinAt (↑f) (f' x) f.source x",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\n⊢ (∫ (x : E) in ↑f '' f.source, g x ∂μ) = ∫ (x : E) in f.source, abs (ContinuousLinearMap.det (f' x)) • g (↑f x) ∂μ",
"tactic": "apply integral_image_eq_integral_abs_det_fderiv_smul μ f.open_source.measurableSet _ f.injOn"
},
{
"state_after": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\nx : E\nhx : x ∈ f.source\n⊢ HasFDerivWithinAt (↑f) (f' x) f.source x",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\n⊢ ∀ (x : E), x ∈ f.source → HasFDerivWithinAt (↑f) (f' x) f.source x",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\nF : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ns : Set E\nf✝ : E → E\nf' : E → E →L[ℝ] E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : IsAddHaarMeasure μ\ninst✝ : CompleteSpace F\nf : LocalHomeomorph E E\nhf' : ∀ (x : E), x ∈ f.source → HasFDerivAt (↑f) (f' x) x\ng : E → F\nthis : ↑f '' f.source = f.target\nx : E\nhx : x ∈ f.source\n⊢ HasFDerivWithinAt (↑f) (f' x) f.source x",
"tactic": "exact (hf' x hx).hasFDerivWithinAt"
}
] |
[
1266,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1259,
1
] |
Mathlib/CategoryTheory/Abelian/RightDerived.lean
|
CategoryTheory.Functor.rightDerived_map_eq
|
[
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (injectiveResolutions C ⋙\n mapHomotopyCategory F (ComplexShape.up ℕ) ⋙ HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n f =\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).mapIso\n (HomotopyCategory.isoOfHomotopyEquiv\n (mapHomotopyEquiv F\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q))) ≪≫\n (HomotopyCategory.homologyFactors D (ComplexShape.up ℕ) n).app\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex)).hom ≫\n (homologyFunctor D (ComplexShape.up ℕ) n).map ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) ≫\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).mapIso\n (HomotopyCategory.isoOfHomotopyEquiv\n (mapHomotopyEquiv F\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P))) ≪≫\n (HomotopyCategory.homologyFactors D (ComplexShape.up ℕ) n).app\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex)).inv",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (rightDerived F n).map f =\n (rightDerivedObjIso F n Q).hom ≫\n (homologyFunctor D (ComplexShape.up ℕ) n).map ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) ≫\n (rightDerivedObjIso F n P).inv",
"tactic": "dsimp only [Functor.rightDerived, Functor.rightDerivedObjIso]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom)) ≫\n 𝟙\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).obj\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).obj\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex)))) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n =\n 0)\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (_ :\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right =\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right) ≫\n 𝟙 (HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (injectiveResolutions C ⋙\n mapHomotopyCategory F (ComplexShape.up ℕ) ⋙ HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n f =\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).mapIso\n (HomotopyCategory.isoOfHomotopyEquiv\n (mapHomotopyEquiv F\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q))) ≪≫\n (HomotopyCategory.homologyFactors D (ComplexShape.up ℕ) n).app\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex)).hom ≫\n (homologyFunctor D (ComplexShape.up ℕ) n).map ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) ≫\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).mapIso\n (HomotopyCategory.isoOfHomotopyEquiv\n (mapHomotopyEquiv F\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P))) ≪≫\n (HomotopyCategory.homologyFactors D (ComplexShape.up ℕ) n).app\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex)).inv",
"tactic": "dsimp"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom)) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n =\n 0)\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (_ :\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right =\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom)) ≫\n 𝟙\n ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).obj\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).obj\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex)))) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n =\n 0)\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (_ :\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right =\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right) ≫\n 𝟙 (HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"tactic": "simp only [Category.comp_id, Category.id_comp]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom)) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g)) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom)) ≫\n homology.map\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj Q.cocomplex) n =\n 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).obj P.cocomplex) n =\n 0)\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n)\n (_ :\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right =\n (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g) n).right) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"tactic": "rw [← homologyFunctor_map, HomotopyCategory.homologyFunctor_map_factors]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv)))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom)) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map g)) ≫\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"tactic": "simp only [← Functor.map_comp]"
},
{
"state_after": "case e_a\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f) =\n (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f)) =\n (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map\n ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv)))",
"tactic": "congr 1"
},
{
"state_after": "case e_a.h\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ Homotopy ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (injectiveResolution.desc f))\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"state_before": "case e_a\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (mapHomotopyCategory F (ComplexShape.up ℕ)).map ((injectiveResolutions C).map f) =\n (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"tactic": "apply HomotopyCategory.eq_of_homotopy"
},
{
"state_after": "case e_a.h.h\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ Homotopy (injectiveResolution.desc f)\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv)",
"state_before": "case e_a.h\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ Homotopy ((mapHomologicalComplex F (ComplexShape.up ℕ)).map (injectiveResolution.desc f))\n ((mapHomologicalComplex F (ComplexShape.up ℕ)).map\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv))",
"tactic": "apply Functor.mapHomotopy"
},
{
"state_after": "case e_a.h.h.g_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (Nonempty.some (_ : Nonempty (InjectiveResolution Y))).ι ≫ injectiveResolution.desc f =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι\n\ncase e_a.h.h.h_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (Nonempty.some (_ : Nonempty (InjectiveResolution Y))).ι ≫\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι",
"state_before": "case e_a.h.h\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ Homotopy (injectiveResolution.desc f)\n ((InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv)",
"tactic": "apply InjectiveResolution.descHomotopy f"
},
{
"state_after": "no goals",
"state_before": "case e_a.h.h.g_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (Nonempty.some (_ : Nonempty (InjectiveResolution Y))).ι ≫ injectiveResolution.desc f =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι",
"tactic": "simp"
},
{
"state_after": "case e_a.h.h.h_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ Q.ι ≫ g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι",
"state_before": "case e_a.h.h.h_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (Nonempty.some (_ : Nonempty (InjectiveResolution Y))).ι ≫\n (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution Y))) Q).hom ≫\n g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι",
"tactic": "simp only [InjectiveResolution.homotopyEquiv_hom_ι_assoc]"
},
{
"state_after": "case e_a.h.h.h_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (CochainComplex.single₀ C).map f ≫\n P.ι ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι",
"state_before": "case e_a.h.h.h_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ Q.ι ≫ g ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι",
"tactic": "rw [← Category.assoc, w, Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "case e_a.h.h.h_comm\nC : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Additive F\nn : ℕ\nX Y : C\nf : Y ⟶ X\nP : InjectiveResolution X\nQ : InjectiveResolution Y\ng : Q.cocomplex ⟶ P.cocomplex\nw : Q.ι ≫ g = (CochainComplex.single₀ C).map f ≫ P.ι\n⊢ (CochainComplex.single₀ C).map f ≫\n P.ι ≫ (InjectiveResolution.homotopyEquiv (Nonempty.some (_ : Nonempty (InjectiveResolution X))) P).inv =\n (CochainComplex.single₀ C).map f ≫ (Nonempty.some (_ : Nonempty (InjectiveResolution X))).ι",
"tactic": "simp only [InjectiveResolution.homotopyEquiv_inv_ι]"
}
] |
[
127,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.bit_lt_bit
|
[] |
[
716,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/Algebra/Group/Semiconj.lean
|
SemiconjBy.one_right
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝ : MulOneClass M\na : M\n⊢ SemiconjBy a 1 1",
"tactic": "rw [SemiconjBy, mul_one, one_mul]"
}
] |
[
93,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.center_toSubring
|
[] |
[
1360,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1358,
1
] |
Mathlib/Init/Logic.lean
|
ExistsUnique.exists
|
[] |
[
245,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.symm_trans_apply
|
[] |
[
924,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
MonoidHom.mem_mker
|
[] |
[
1166,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1165,
1
] |
Mathlib/CategoryTheory/GradedObject.lean
|
CategoryTheory.GradedObject.comapEq_symm
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nβ γ : Type w\nf g : β → γ\nh : f = g\n⊢ comapEq C (_ : g = f) = (comapEq C h).symm",
"tactic": "aesop_cat"
}
] |
[
108,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
norm_algebraMap_nNReal
|
[] |
[
526,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.coe_mk
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Fintype/Powerset.lean
|
Fintype.card_finset_len
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Fintype α\nk : ℕ\n⊢ card { s // Finset.card s = k } = Nat.choose (card α) k",
"tactic": "simp [Fintype.subtype_card, Finset.card_univ]"
}
] |
[
57,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.exists_pow_neg_lt_rat
|
[
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑p ^ (-↑k) < ↑ε\n⊢ ∃ k, ↑p ^ (-↑k) < ε",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\n⊢ ∃ k, ↑p ^ (-↑k) < ε",
"tactic": "obtain ⟨k, hk⟩ := @exists_pow_neg_lt p _ ε (by exact_mod_cast hε)"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑p ^ (-↑k) < ↑ε\n⊢ ↑p ^ (-↑k) < ε",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑p ^ (-↑k) < ↑ε\n⊢ ∃ k, ↑p ^ (-↑k) < ε",
"tactic": "use k"
},
{
"state_after": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑↑p ^ (-↑k) < ↑ε\n⊢ ↑p ^ (-↑k) < ε",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑p ^ (-↑k) < ↑ε\n⊢ ↑p ^ (-↑k) < ε",
"tactic": "rw [show (p : ℝ) = (p : ℚ) by simp] at hk"
},
{
"state_after": "no goals",
"state_before": "case intro\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑↑p ^ (-↑k) < ↑ε\n⊢ ↑p ^ (-↑k) < ε",
"tactic": "exact_mod_cast hk"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\n⊢ 0 < ↑ε",
"tactic": "exact_mod_cast hε"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑p ^ (-↑k) < ↑ε\n⊢ ↑p = ↑↑p",
"tactic": "simp"
}
] |
[
355,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
sq_eq_sq_iff_eq_or_eq_neg
|
[] |
[
297,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
IndexedPartition.eq
|
[] |
[
387,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.EventuallyLE.le_sup_of_le_right
|
[] |
[
1792,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1790,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.liftOfRightInverse_comp_apply
|
[] |
[
3359,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3356,
1
] |
Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean
|
Polynomial.annIdealGenerator_aeval_eq_zero
|
[] |
[
140,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.dist_extend_extend
|
[
{
"state_after": "case refine'_1\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : δ\n⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤\n max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))\n\ncase refine'_2\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\n⊢ dist g₁ g₂ ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)\n\ncase refine'_3\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\n⊢ dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"state_before": "F : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\n⊢ dist (extend f g₁ h₁) (extend f g₂ h₂) = max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"tactic": "refine' le_antisymm ((dist_le <| le_max_iff.2 <| Or.inl dist_nonneg).2 fun x => _) (max_le _ _)"
},
{
"state_after": "case refine'_1.inl.intro\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤\n max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))\n\ncase refine'_1.inr\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : δ\nhx : ¬∃ y, ↑f y = x\n⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤\n max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"state_before": "case refine'_1\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : δ\n⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤\n max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"tactic": "rcases _root_.em (∃ y, f y = x) with (⟨x, rfl⟩ | hx)"
},
{
"state_after": "case refine'_1.inl.intro\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑g₁ x) (↑g₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"state_before": "case refine'_1.inl.intro\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤\n max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"tactic": "simp only [extend_apply]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inl.intro\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑g₁ x) (↑g₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"tactic": "exact (dist_coe_le_dist x).trans (le_max_left _ _)"
},
{
"state_after": "case refine'_1.inr\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : δ\nhx : ¬∃ y, ↑f y = x\n⊢ dist (↑h₁ x) (↑h₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"state_before": "case refine'_1.inr\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : δ\nhx : ¬∃ y, ↑f y = x\n⊢ dist (↑(extend f g₁ h₁) x) (↑(extend f g₂ h₂) x) ≤\n max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"tactic": "simp only [extend_apply' hx]"
},
{
"state_after": "case refine'_1.inr.intro\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : { x // x ∈ range ↑fᶜ }\n⊢ dist (↑h₁ ↑x) (↑h₂ ↑x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"state_before": "case refine'_1.inr\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : δ\nhx : ¬∃ y, ↑f y = x\n⊢ dist (↑h₁ x) (↑h₂ x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"tactic": "lift x to (range fᶜ : Set δ) using hx"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr.intro\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : { x // x ∈ range ↑fᶜ }\n⊢ dist (↑h₁ ↑x) (↑h₂ ↑x) ≤ max (dist g₁ g₂) (dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)))",
"tactic": "calc\n dist (h₁ x) (h₂ x) = dist (h₁.restrict (range fᶜ) x) (h₂.restrict (range fᶜ) x) := rfl\n _ ≤ dist (h₁.restrict (range fᶜ)) (h₂.restrict (range fᶜ)) := (dist_coe_le_dist x)\n _ ≤ _ := le_max_right _ _"
},
{
"state_after": "case refine'_2\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑g₁ x) (↑g₂ x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"state_before": "case refine'_2\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\n⊢ dist g₁ g₂ ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"tactic": "refine' (dist_le dist_nonneg).2 fun x => _"
},
{
"state_after": "case refine'_2\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"state_before": "case refine'_2\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑g₁ x) (↑g₂ x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"tactic": "rw [← extend_apply f g₁ h₁, ← extend_apply f g₂ h₂]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist (↑(extend f g₁ h₁) (↑f x)) (↑(extend f g₂ h₂) (↑f x)) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"tactic": "exact dist_coe_le_dist _"
},
{
"state_after": "case refine'_3\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : ↑(range ↑fᶜ)\n⊢ dist (↑(restrict h₁ (range ↑fᶜ)) x) (↑(restrict h₂ (range ↑fᶜ)) x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"state_before": "case refine'_3\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\n⊢ dist (restrict h₁ (range ↑fᶜ)) (restrict h₂ (range ↑fᶜ)) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"tactic": "refine' (dist_le dist_nonneg).2 fun x => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nF : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : ↑(range ↑fᶜ)\n⊢ dist (↑(restrict h₁ (range ↑fᶜ)) x) (↑(restrict h₂ (range ↑fᶜ)) x) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)",
"tactic": "calc\n dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by\n rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]\n _ ≤ _ := dist_coe_le_dist _"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.548018\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g : α →ᵇ β\nx✝ : α\nC : ℝ\nδ : Type u_1\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : ↑(range ↑fᶜ)\n⊢ dist (↑h₁ ↑x) (↑h₂ ↑x) = dist (↑(extend f g₁ h₁) ↑x) (↑(extend f g₂ h₂) ↑x)",
"tactic": "rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]"
}
] |
[
504,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean
|
rieszContentAux_mono
|
[] |
[
68,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.bind_congr
|
[] |
[
1035,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1028,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.ext
|
[] |
[
126,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.zero_comp
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q r : R[X]\n⊢ comp 0 p = 0",
"tactic": "rw [← C_0, C_comp]"
}
] |
[
568,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Analysis/LocallyConvex/Polar.lean
|
LinearMap.polar_gc
|
[] |
[
87,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.imClm_coe
|
[] |
[
922,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
921,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\n⊢ ∃ fs, (∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))) ∧ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(fs n) x = 0",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\n⊢ ∃ fs, (∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))) ∧ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(fs n) x = 0",
"tactic": "let g_seq_s : ℕ → @SimpleFunc α m β := fun n => (hf.approx n).restrict s"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\n⊢ ∃ fs, (∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))) ∧ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(fs n) x = 0",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\n⊢ ∃ fs, (∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))) ∧ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(fs n) x = 0",
"tactic": "have hg_eq : ∀ x ∈ s, ∀ n, g_seq_s n x = hf.approx n x := by\n intro x hx n\n rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_mem hx]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\n⊢ ∃ fs, (∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))) ∧ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(fs n) x = 0",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\n⊢ ∃ fs, (∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))) ∧ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(fs n) x = 0",
"tactic": "have hg_zero : ∀ (x) (_ : x ∉ s), ∀ n, g_seq_s n x = 0 := by\n intro x hx n\n rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_not_mem hx]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\n⊢ Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\n⊢ ∃ fs, (∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))) ∧ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(fs n) x = 0",
"tactic": "refine' ⟨g_seq_s, fun x => _, hg_zero⟩"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : ¬x ∈ s\n⊢ Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\n⊢ Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))",
"tactic": "by_cases hx : x ∈ s"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nx : α\nhx : x ∈ s\nn : ℕ\n⊢ ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\n⊢ ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x",
"tactic": "intro x hx n"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nx : α\nhx : x ∈ s\nn : ℕ\n⊢ ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x",
"tactic": "rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_mem hx]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nx : α\nhx : ¬x ∈ s\nn : ℕ\n⊢ ↑(g_seq_s n) x = 0",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\n⊢ ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0",
"tactic": "intro x hx n"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nx : α\nhx : ¬x ∈ s\nn : ℕ\n⊢ ↑(g_seq_s n) x = 0",
"tactic": "rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_not_mem hx]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n => ↑(StronglyMeasurable.approx hf n) x) atTop (𝓝 (f x))",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))",
"tactic": "simp_rw [hg_eq x hx]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : x ∈ s\n⊢ Tendsto (fun n => ↑(StronglyMeasurable.approx hf n) x) atTop (𝓝 (f x))",
"tactic": "exact hf.tendsto_approx x"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : ¬x ∈ s\n⊢ Tendsto (fun n => 0) atTop (𝓝 0)",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : ¬x ∈ s\n⊢ Tendsto (fun n => ↑(g_seq_s n) x) atTop (𝓝 (f x))",
"tactic": "simp_rw [hg_zero x hx, hf_zero x hx]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.228351\nι : Type ?u.228354\ninst✝² : Countable ι\nf✝ g : α → β\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\ns : Set α\nf : α → β\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhf_zero : ∀ (x : α), ¬x ∈ s → f x = 0\ng_seq_s : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (StronglyMeasurable.approx hf n) s\nhg_eq : ∀ (x : α), x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = ↑(StronglyMeasurable.approx hf n) x\nhg_zero : ∀ (x : α), ¬x ∈ s → ∀ (n : ℕ), ↑(g_seq_s n) x = 0\nx : α\nhx : ¬x ∈ s\n⊢ Tendsto (fun n => 0) atTop (𝓝 0)",
"tactic": "exact tendsto_const_nhds"
}
] |
[
904,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
887,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
ContinuousMap.coe_toLp
|
[] |
[
1738,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1736,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.mul_apply
|
[] |
[
676,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
675,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
expNegInvGlue.zero_of_nonpos
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ≤ 0\n⊢ expNegInvGlue x = 0",
"tactic": "simp [expNegInvGlue, hx]"
}
] |
[
67,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
Isometry.uniformEmbedding
|
[] |
[
203,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
11
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_comm
|
[] |
[
1148,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1147,
1
] |
Mathlib/Algebra/ModEq.lean
|
AddCommGroup.nsmul_modEq_nsmul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ n • b - n • a = m • n • p ↔ b - a = m • p",
"tactic": "rw [← smul_sub, smul_comm, smul_right_inj hn]"
}
] |
[
178,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_sdiff_add_card_eq_card
|
[] |
[
437,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/Data/Finset/Powerset.lean
|
Finset.powerset_nonempty
|
[] |
[
61,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/LinearAlgebra/Matrix/Diagonal.lean
|
Matrix.diagonal_comp_stdBasis
|
[] |
[
45,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.comp_assoc
|
[] |
[
463,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.inl_eq_prod
|
[] |
[
199,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Algebra/Group/UniqueProds.lean
|
UniqueMul.iff_existsUnique
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.2277\ninst✝¹ : Mul G\ninst✝ : Mul H\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nx✝ : UniqueMul A B a0 b0\n⊢ ∀ (y : (a0, b0) ∈ A ×ˢ B), (fun x => (a0, b0).fst * (a0, b0).snd = a0 * b0) y → y = (_ : (a0, b0) ∈ A ×ˢ B)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.2277\ninst✝¹ : Mul G\ninst✝ : Mul H\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nx✝ : UniqueMul A B a0 b0\n⊢ ∀ (y : G × G), (fun ab => ∃! x, ab.fst * ab.snd = a0 * b0) y → y = (a0, b0)",
"tactic": "simpa"
},
{
"state_after": "case mk\nG : Type u_1\nH : Type ?u.2277\ninst✝¹ : Mul G\ninst✝ : Mul H\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nh : ∃! ab x, ab.fst * ab.snd = a0 * b0\nx1 x2 : G\nh✝ : (x1, x2) ∈ A ×ˢ B\na✝ : (x1, x2).fst * (x1, x2).snd = a0 * b0\nJ : ∀ (y : G × G), y ∈ A ×ˢ B → y.fst * y.snd = a0 * b0 → y = (x1, x2)\nx y : G\nhx : x ∈ A\nhy : y ∈ B\nl : x * y = a0 * b0\n⊢ x = a0 ∧ y = b0",
"state_before": "G : Type u_1\nH : Type ?u.2277\ninst✝¹ : Mul G\ninst✝ : Mul H\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nh : ∃! ab x, ab.fst * ab.snd = a0 * b0\n⊢ ∀ (x : G × G),\n x ∈ A ×ˢ B →\n x.fst * x.snd = a0 * b0 → (∀ (y : G × G), y ∈ A ×ˢ B → y.fst * y.snd = a0 * b0 → y = x) → UniqueMul A B a0 b0",
"tactic": "rintro ⟨x1, x2⟩ _ _ J x y hx hy l"
},
{
"state_after": "case mk.intro\nG : Type u_1\nH : Type ?u.2277\ninst✝¹ : Mul G\ninst✝ : Mul H\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nh : ∃! ab x, ab.fst * ab.snd = a0 * b0\nx y : G\nhx : x ∈ A\nhy : y ∈ B\nl : x * y = a0 * b0\nh✝ : (a0, b0) ∈ A ×ˢ B\na✝ : (a0, b0).fst * (a0, b0).snd = a0 * b0\nJ : ∀ (y : G × G), y ∈ A ×ˢ B → y.fst * y.snd = a0 * b0 → y = (a0, b0)\n⊢ x = a0 ∧ y = b0",
"state_before": "case mk\nG : Type u_1\nH : Type ?u.2277\ninst✝¹ : Mul G\ninst✝ : Mul H\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nh : ∃! ab x, ab.fst * ab.snd = a0 * b0\nx1 x2 : G\nh✝ : (x1, x2) ∈ A ×ˢ B\na✝ : (x1, x2).fst * (x1, x2).snd = a0 * b0\nJ : ∀ (y : G × G), y ∈ A ×ˢ B → y.fst * y.snd = a0 * b0 → y = (x1, x2)\nx y : G\nhx : x ∈ A\nhy : y ∈ B\nl : x * y = a0 * b0\n⊢ x = a0 ∧ y = b0",
"tactic": "rcases Prod.mk.inj_iff.mp (J (a0, b0) (Finset.mk_mem_product aA bB) rfl) with ⟨rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nG : Type u_1\nH : Type ?u.2277\ninst✝¹ : Mul G\ninst✝ : Mul H\nA B : Finset G\na0 b0 : G\naA : a0 ∈ A\nbB : b0 ∈ B\nh : ∃! ab x, ab.fst * ab.snd = a0 * b0\nx y : G\nhx : x ∈ A\nhy : y ∈ B\nl : x * y = a0 * b0\nh✝ : (a0, b0) ∈ A ×ˢ B\na✝ : (a0, b0).fst * (a0, b0).snd = a0 * b0\nJ : ∀ (y : G × G), y ∈ A ×ˢ B → y.fst * y.snd = a0 * b0 → y = (a0, b0)\n⊢ x = a0 ∧ y = b0",
"tactic": "exact Prod.mk.inj_iff.mp (J (x, y) (Finset.mk_mem_product hx hy) l)"
}
] |
[
93,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/MeasureTheory/Measure/OpenPos.lean
|
MeasureTheory.Measure.interior_eq_empty_of_null
|
[] |
[
88,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.Balanced.del
|
[
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nc : RBColor\nn : Nat\ncut : α → Ordering\nt : RBNode α\n⊢ DelProp (isBlack nil) (del cut nil) 0",
"tactic": "exact ⟨_, .nil⟩"
},
{
"state_after": "case black\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ RedRed True (del cut (node black a v✝ b)) n",
"state_before": "case black\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ DelProp (isBlack (node black a v✝ b)) (del cut (node black a v✝ b)) (n + 1)",
"tactic": "refine ⟨_, rfl, ?_⟩"
},
{
"state_after": "case black\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ RedRed True\n (match cut v✝ with\n | Ordering.lt =>\n match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b\n | Ordering.gt =>\n match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b)\n | Ordering.eq => append a b)\n n",
"state_before": "case black\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ RedRed True (del cut (node black a v✝ b)) n",
"tactic": "unfold del"
},
{
"state_after": "case black.h_1\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ RedRed True\n (match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b)\n n\n\ncase black.h_2\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ RedRed True\n (match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b))\n n\n\ncase black.h_3\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ RedRed True (append a b) n",
"state_before": "case black\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ RedRed True\n (match cut v✝ with\n | Ordering.lt =>\n match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b\n | Ordering.gt =>\n match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b)\n | Ordering.eq => append a b)\n n",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case black.h_1\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ RedRed True\n (match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b)\n n",
"tactic": "exact match a, n, iha with\n| .nil, _, ⟨c, ha⟩ | .node red .., _, ⟨c, ha⟩ => .redred ⟨⟩ ha hb\n| .node black .., _, ⟨n, rfl, ha⟩ => (hb.balLeft ha).imp fun _ => ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case black.h_2\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ RedRed True\n (match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b))\n n",
"tactic": "exact match b, n, ihb with\n| .nil, _, ⟨c, hb⟩ | .node .red .., _, ⟨c, hb⟩ => .redred ⟨⟩ ha hb\n| .node black .., _, ⟨n, rfl, hb⟩ => (ha.balRight hb).imp fun _ => ⟨⟩"
},
{
"state_after": "no goals",
"state_before": "case black.h_3\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nc₁✝ : RBColor\nn : Nat\nb : RBNode α\nc₂✝ : RBColor\nv✝ : α\nha : Balanced a c₁✝ n\nhb : Balanced b c₂✝ n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ RedRed True (append a b) n",
"tactic": "exact (ha.append hb).imp fun _ => ⟨⟩"
},
{
"state_after": "case red\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ DelProp (isBlack (node red a v✝ b))\n (match cut v✝ with\n | Ordering.lt =>\n match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b\n | Ordering.gt =>\n match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b)\n | Ordering.eq => append a b)\n n",
"state_before": "case red\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ DelProp (isBlack (node red a v✝ b)) (del cut (node red a v✝ b)) n",
"tactic": "unfold del"
},
{
"state_after": "case red.h_1\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ DelProp (isBlack (node red a v✝ b))\n (match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b)\n n\n\ncase red.h_2\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ DelProp (isBlack (node red a v✝ b))\n (match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b))\n n\n\ncase red.h_3\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ DelProp (isBlack (node red a v✝ b)) (append a b) n",
"state_before": "case red\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\n⊢ DelProp (isBlack (node red a v✝ b))\n (match cut v✝ with\n | Ordering.lt =>\n match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b\n | Ordering.gt =>\n match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b)\n | Ordering.eq => append a b)\n n",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case red.h_1\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ DelProp (isBlack (node red a v✝ b))\n (match isBlack a with\n | black => balLeft (del cut a) v✝ b\n | red => node red (del cut a) v✝ b)\n n",
"tactic": "exact match a, n, iha with\n| .nil, _, _ => ⟨_, .red ha hb⟩\n| .node black .., _, ⟨n, rfl, ha⟩ => (hb.balLeft ha).of_false (fun.)"
},
{
"state_after": "no goals",
"state_before": "case red.h_2\nα : Type u_1\nc : RBColor\nn✝ : Nat\ncut : α → Ordering\nt a : RBNode α\nn : Nat\nb : RBNode α\nv✝ : α\nha : Balanced a black n\nhb : Balanced b black n\niha : DelProp (isBlack a) (del cut a) n\nihb : DelProp (isBlack b) (del cut b) n\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ DelProp (isBlack (node red a v✝ b))\n (match isBlack b with\n | black => balRight a v✝ (del cut b)\n | red => node red a v✝ (del cut b))\n n",
"tactic": "exact match b, n, ihb with\n| .nil, _, _ => ⟨_, .red ha hb⟩\n| .node black .., _, ⟨n, rfl, hb⟩ => (ha.balRight hb).of_false (fun.)"
}
] |
[
454,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
432,
11
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift'_lift_assoc
|
[] |
[
358,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.zigzag_obj_of_zigzag
|
[] |
[
290,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Analysis/Calculus/Dslope.lean
|
dslope_sub_smul_of_ne
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf✝ : 𝕜 → E\na b : 𝕜\ns : Set 𝕜\nf : 𝕜 → E\nh : b ≠ a\n⊢ dslope (fun x => (x - a) • f x) a b = f b",
"tactic": "rw [dslope_of_ne _ h, slope_sub_smul _ h.symm]"
}
] |
[
77,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.smul_mem_iff
|
[] |
[
391,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.mem_nthRootsFinset
|
[] |
[
640,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
8
] |
Mathlib/Data/Multiset/Sum.lean
|
Multiset.Nodup.disjSum
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx✝ : α ⊕ β\nhs✝ : Nodup s\nht✝ : Nodup t\nx : α ⊕ β\nhs : x ∈ Multiset.map inl s\nht : x ∈ Multiset.map inr t\n⊢ False",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx : α ⊕ β\nhs : Nodup s\nht : Nodup t\n⊢ Nodup (disjSum s t)",
"tactic": "refine' ((hs.map inl_injective).add_iff <| ht.map inr_injective).2 fun x hs ht => _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx✝ : α ⊕ β\nhs✝ : Nodup s\nht✝ : Nodup t\nx : α ⊕ β\nhs : ∃ a, a ∈ s ∧ inl a = x\nht : ∃ a, a ∈ t ∧ inr a = x\n⊢ False",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx✝ : α ⊕ β\nhs✝ : Nodup s\nht✝ : Nodup t\nx : α ⊕ β\nhs : x ∈ Multiset.map inl s\nht : x ∈ Multiset.map inr t\n⊢ False",
"tactic": "rw [Multiset.mem_map] at hs ht"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na✝ : α\nb : β\nx : α ⊕ β\nhs : Nodup s\nht✝ : Nodup t\na : α\nleft✝ : a ∈ s\nht : ∃ a_1, a_1 ∈ t ∧ inr a_1 = inl a\n⊢ False",
"state_before": "α : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na : α\nb : β\nx✝ : α ⊕ β\nhs✝ : Nodup s\nht✝ : Nodup t\nx : α ⊕ β\nhs : ∃ a, a ∈ s ∧ inl a = x\nht : ∃ a, a ∈ t ∧ inr a = x\n⊢ False",
"tactic": "obtain ⟨a, _, rfl⟩ := hs"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na✝ : α\nb✝ : β\nx : α ⊕ β\nhs : Nodup s\nht : Nodup t\na : α\nleft✝¹ : a ∈ s\nb : β\nleft✝ : b ∈ t\nh : inr b = inl a\n⊢ False",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na✝ : α\nb : β\nx : α ⊕ β\nhs : Nodup s\nht✝ : Nodup t\na : α\nleft✝ : a ∈ s\nht : ∃ a_1, a_1 ∈ t ∧ inr a_1 = inl a\n⊢ False",
"tactic": "obtain ⟨b, _, h⟩ := ht"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ns : Multiset α\nt : Multiset β\ns₁ s₂ : Multiset α\nt₁ t₂ : Multiset β\na✝ : α\nb✝ : β\nx : α ⊕ β\nhs : Nodup s\nht : Nodup t\na : α\nleft✝¹ : a ∈ s\nb : β\nleft✝ : b ∈ t\nh : inr b = inl a\n⊢ False",
"tactic": "exact inr_ne_inl h"
}
] |
[
112,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
11
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
ApproximatesLinearOn.image_mem_nhds
|
[
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.309590\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.309693\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nx : E\nhs : s ∈ 𝓝 x\nhc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹\nt : Set E\nhts : t ⊆ s\nht : IsOpen t\nxt : x ∈ t\n⊢ f '' s ∈ 𝓝 (f x)",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.309590\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.309693\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nx : E\nhs : s ∈ 𝓝 x\nhc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹\n⊢ f '' s ∈ 𝓝 (f x)",
"tactic": "obtain ⟨t, hts, ht, xt⟩ : ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := _root_.mem_nhds_iff.1 hs"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.309590\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.309693\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nx : E\nhs : s ∈ 𝓝 x\nhc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹\nt : Set E\nhts : t ⊆ s\nht : IsOpen t\nxt : x ∈ t\nthis : f '' t ∈ 𝓝 (f x)\n⊢ f '' s ∈ 𝓝 (f x)",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.309590\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.309693\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nx : E\nhs : s ∈ 𝓝 x\nhc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹\nt : Set E\nhts : t ⊆ s\nht : IsOpen t\nxt : x ∈ t\n⊢ f '' s ∈ 𝓝 (f x)",
"tactic": "have := IsOpen.mem_nhds ((hf.mono_set hts).open_image f'symm ht hc) (mem_image_of_mem _ xt)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.309590\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.309693\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\nf'symm : ContinuousLinearMap.NonlinearRightInverse f'\nx : E\nhs : s ∈ 𝓝 x\nhc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹\nt : Set E\nhts : t ⊆ s\nht : IsOpen t\nxt : x ∈ t\nthis : f '' t ∈ 𝓝 (f x)\n⊢ f '' s ∈ 𝓝 (f x)",
"tactic": "exact mem_of_superset this (image_subset _ hts)"
}
] |
[
342,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.lim_im
|
[
{
"state_after": "f : CauSeq ℂ ↑abs\n⊢ CauSeq.lim (cauSeqIm f) = (↑(CauSeq.lim (cauSeqRe f)) + ↑(CauSeq.lim (cauSeqIm f)) * I).im",
"state_before": "f : CauSeq ℂ ↑abs\n⊢ CauSeq.lim (cauSeqIm f) = (CauSeq.lim f).im",
"tactic": "rw [lim_eq_lim_im_add_lim_re]"
},
{
"state_after": "no goals",
"state_before": "f : CauSeq ℂ ↑abs\n⊢ CauSeq.lim (cauSeqIm f) = (↑(CauSeq.lim (cauSeqRe f)) + ↑(CauSeq.lim (cauSeqIm f)) * I).im",
"tactic": "simp [ofReal']"
}
] |
[
1303,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1302,
1
] |
Mathlib/CategoryTheory/Quotient.lean
|
CategoryTheory.Quotient.functor_map_eq_iff
|
[
{
"state_after": "case mp\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ (functor r).map f = (functor r).map f' → r f f'\n\ncase mpr\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ r f f' → (functor r).map f = (functor r).map f'",
"state_before": "C : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ (functor r).map f = (functor r).map f' ↔ r f f'",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ EqvGen (CompClosure r) f f' → r f f'",
"state_before": "case mp\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ (functor r).map f = (functor r).map f' → r f f'",
"tactic": "erw [Quot.eq]"
},
{
"state_after": "case mp\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh✝ : Congruence r\nX Y : C\nf f' : X ⟶ Y\nh : EqvGen (CompClosure r) f f'\n⊢ r f f'",
"state_before": "case mp\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ EqvGen (CompClosure r) f f' → r f f'",
"tactic": "intro h"
},
{
"state_after": "case mp.rel\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' m m' : X ⟶ Y\nhm : CompClosure r m m'\n⊢ r m m'\n\ncase mp.refl\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ : X ⟶ Y\n⊢ r x✝ x✝\n\ncase mp.symm\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ y✝ : X ⟶ Y\na✝ : EqvGen (CompClosure r) x✝ y✝\na_ih✝ : r x✝ y✝\n⊢ r y✝ x✝\n\ncase mp.trans\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ y✝ z✝ : X ⟶ Y\na✝¹ : EqvGen (CompClosure r) x✝ y✝\na✝ : EqvGen (CompClosure r) y✝ z✝\na_ih✝¹ : r x✝ y✝\na_ih✝ : r y✝ z✝\n⊢ r x✝ z✝",
"state_before": "case mp\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh✝ : Congruence r\nX Y : C\nf f' : X ⟶ Y\nh : EqvGen (CompClosure r) f f'\n⊢ r f f'",
"tactic": "induction' h with m m' hm"
},
{
"state_after": "case mp.rel.intro\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\na✝ b✝ : C\nf✝ : X ⟶ a✝\nm₁✝ m₂✝ : a✝ ⟶ b✝\ng✝ : b✝ ⟶ Y\nh✝ : r m₁✝ m₂✝\n⊢ r (f✝ ≫ m₁✝ ≫ g✝) (f✝ ≫ m₂✝ ≫ g✝)",
"state_before": "case mp.rel\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' m m' : X ⟶ Y\nhm : CompClosure r m m'\n⊢ r m m'",
"tactic": "cases hm"
},
{
"state_after": "case mp.rel.intro.a\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\na✝ b✝ : C\nf✝ : X ⟶ a✝\nm₁✝ m₂✝ : a✝ ⟶ b✝\ng✝ : b✝ ⟶ Y\nh✝ : r m₁✝ m₂✝\n⊢ r (m₁✝ ≫ g✝) (m₂✝ ≫ g✝)",
"state_before": "case mp.rel.intro\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\na✝ b✝ : C\nf✝ : X ⟶ a✝\nm₁✝ m₂✝ : a✝ ⟶ b✝\ng✝ : b✝ ⟶ Y\nh✝ : r m₁✝ m₂✝\n⊢ r (f✝ ≫ m₁✝ ≫ g✝) (f✝ ≫ m₂✝ ≫ g✝)",
"tactic": "apply Congruence.compLeft"
},
{
"state_after": "case mp.rel.intro.a.a\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\na✝ b✝ : C\nf✝ : X ⟶ a✝\nm₁✝ m₂✝ : a✝ ⟶ b✝\ng✝ : b✝ ⟶ Y\nh✝ : r m₁✝ m₂✝\n⊢ r m₁✝ m₂✝",
"state_before": "case mp.rel.intro.a\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\na✝ b✝ : C\nf✝ : X ⟶ a✝\nm₁✝ m₂✝ : a✝ ⟶ b✝\ng✝ : b✝ ⟶ Y\nh✝ : r m₁✝ m₂✝\n⊢ r (m₁✝ ≫ g✝) (m₂✝ ≫ g✝)",
"tactic": "apply Congruence.compRight"
},
{
"state_after": "no goals",
"state_before": "case mp.rel.intro.a.a\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\na✝ b✝ : C\nf✝ : X ⟶ a✝\nm₁✝ m₂✝ : a✝ ⟶ b✝\ng✝ : b✝ ⟶ Y\nh✝ : r m₁✝ m₂✝\n⊢ r m₁✝ m₂✝",
"tactic": "assumption"
},
{
"state_after": "case mp.refl\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ : X ⟶ Y\nthis : IsEquiv (X ⟶ Y) r\n⊢ r x✝ x✝",
"state_before": "case mp.refl\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ : X ⟶ Y\n⊢ r x✝ x✝",
"tactic": "haveI := (h.isEquiv : IsEquiv _ (@r X Y))"
},
{
"state_after": "no goals",
"state_before": "case mp.refl\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ : X ⟶ Y\nthis : IsEquiv (X ⟶ Y) r\n⊢ r x✝ x✝",
"tactic": "apply refl"
},
{
"state_after": "case mp.symm.a\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ y✝ : X ⟶ Y\na✝ : EqvGen (CompClosure r) x✝ y✝\na_ih✝ : r x✝ y✝\n⊢ r x✝ y✝",
"state_before": "case mp.symm\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ y✝ : X ⟶ Y\na✝ : EqvGen (CompClosure r) x✝ y✝\na_ih✝ : r x✝ y✝\n⊢ r y✝ x✝",
"tactic": "apply symm"
},
{
"state_after": "no goals",
"state_before": "case mp.symm.a\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ y✝ : X ⟶ Y\na✝ : EqvGen (CompClosure r) x✝ y✝\na_ih✝ : r x✝ y✝\n⊢ r x✝ y✝",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "case mp.trans\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' x✝ y✝ z✝ : X ⟶ Y\na✝¹ : EqvGen (CompClosure r) x✝ y✝\na✝ : EqvGen (CompClosure r) y✝ z✝\na_ih✝¹ : r x✝ y✝\na_ih✝ : r y✝ z✝\n⊢ r x✝ z✝",
"tactic": "apply _root_.trans <;> assumption"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type u_1\ninst✝ : Category C\nr : HomRel C\nh : Congruence r\nX Y : C\nf f' : X ⟶ Y\n⊢ r f f' → (functor r).map f = (functor r).map f'",
"tactic": "apply Quotient.sound"
}
] |
[
160,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.isUnit_nonsing_inv_det_iff
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.252005\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\n⊢ IsUnit (det A⁻¹) ↔ IsUnit (det A)",
"tactic": "rw [Matrix.det_nonsing_inv, isUnit_ring_inverse]"
}
] |
[
422,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Involutive.rightInverse
|
[] |
[
893,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
893,
11
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
hasSum_zero_iff_of_nonneg
|
[
{
"state_after": "case refine'_1\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ f i\nhf' : HasSum f 0\n⊢ f = 0\n\ncase refine'_2\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ f i\n⊢ f = 0 → HasSum f 0",
"state_before": "ι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ f i\n⊢ HasSum f 0 ↔ f = 0",
"tactic": "refine' ⟨fun hf' => _, _⟩"
},
{
"state_after": "case refine'_1.h\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ f i\nhf' : HasSum f 0\ni : ι\n⊢ f i = OfNat.ofNat 0 i",
"state_before": "case refine'_1\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ f i\nhf' : HasSum f 0\n⊢ f = 0",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.h\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ f i\nhf' : HasSum f 0\ni : ι\n⊢ f i = OfNat.ofNat 0 i",
"tactic": "exact (hf i).antisymm' (le_hasSum hf' _ fun j _ => hf j)"
},
{
"state_after": "case refine'_2\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\ng : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ OfNat.ofNat 0 i\n⊢ HasSum 0 0",
"state_before": "case refine'_2\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ f i\n⊢ f = 0 → HasSum f 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type u_2\nκ : Type ?u.35269\nα : Type u_1\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\ng : ι → α\na a₁ a₂ : α\nhf : ∀ (i : ι), 0 ≤ OfNat.ofNat 0 i\n⊢ HasSum 0 0",
"tactic": "exact hasSum_zero"
}
] |
[
154,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.ModEq.trans
|
[] |
[
67,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
11
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.isPell_star
|
[
{
"state_after": "no goals",
"state_before": "d x y : ℤ\n⊢ IsPell { re := x, im := y } ↔ IsPell (star { re := x, im := y })",
"tactic": "simp [IsPell, Zsqrtd.star_mk]"
}
] |
[
86,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.ker_isPGroup_of_injective
|
[] |
[
302,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Order/Basic.lean
|
lt_update_self_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type u_2\nr : α → α → Prop\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → Preorder (π i)\nx y : (i : ι) → π i\ni : ι\na b : π i\n⊢ x < update x i a ↔ x i < a",
"tactic": "simp [lt_iff_le_not_le]"
}
] |
[
894,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean
|
spectrum.nonempty_of_isAlgClosed_of_finiteDimensional
|
[
{
"state_after": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\n⊢ Set.Nonempty (σ a)",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\n⊢ Set.Nonempty (σ a)",
"tactic": "obtain ⟨p, ⟨h_mon, h_eval_p⟩⟩ := isIntegral_of_noetherian (IsNoetherian.iff_fg.2 I) a"
},
{
"state_after": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\nnu : ¬IsUnit (↑(aeval a) p)\n⊢ Set.Nonempty (σ a)",
"state_before": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\n⊢ Set.Nonempty (σ a)",
"tactic": "have nu : ¬IsUnit (aeval a p) := by rw [← aeval_def] at h_eval_p ; rw [h_eval_p]; simp"
},
{
"state_after": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\nnu : ¬IsUnit (↑(aeval a) (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots p))))\n⊢ Set.Nonempty (σ a)",
"state_before": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\nnu : ¬IsUnit (↑(aeval a) p)\n⊢ Set.Nonempty (σ a)",
"tactic": "rw [eq_prod_roots_of_monic_of_splits_id h_mon (IsAlgClosed.splits p)] at nu"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\nnu : ¬IsUnit (↑(aeval a) (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots p))))\nk : 𝕜\nhk : k ∈ σ a\nright✝ : eval k p = 0\n⊢ Set.Nonempty (σ a)",
"state_before": "case intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\nnu : ¬IsUnit (↑(aeval a) (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots p))))\n⊢ Set.Nonempty (σ a)",
"tactic": "obtain ⟨k, hk, _⟩ := exists_mem_of_not_isUnit_aeval_prod nu"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\nnu : ¬IsUnit (↑(aeval a) (Multiset.prod (Multiset.map (fun a => X - ↑C a) (roots p))))\nk : 𝕜\nhk : k ∈ σ a\nright✝ : eval k p = 0\n⊢ Set.Nonempty (σ a)",
"tactic": "exact ⟨k, hk⟩"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : ↑(aeval a) p = 0\n⊢ ¬IsUnit (↑(aeval a) p)",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : eval₂ ↑ₐ a p = 0\n⊢ ¬IsUnit (↑(aeval a) p)",
"tactic": "rw [← aeval_def] at h_eval_p"
},
{
"state_after": "𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : ↑(aeval a) p = 0\n⊢ ¬IsUnit 0",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : ↑(aeval a) p = 0\n⊢ ¬IsUnit (↑(aeval a) p)",
"tactic": "rw [h_eval_p]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nA : Type v\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring A\ninst✝² : Algebra 𝕜 A\ninst✝¹ : IsAlgClosed 𝕜\ninst✝ : Nontrivial A\nI : FiniteDimensional 𝕜 A\na : A\np : 𝕜[X]\nh_mon : Monic p\nh_eval_p : ↑(aeval a) p = 0\n⊢ ¬IsUnit 0",
"tactic": "simp"
}
] |
[
166,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/Regular/Basic.lean
|
isRightRegular_of_rightCancelSemigroup
|
[] |
[
349,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.isClosed_infty
|
[
{
"state_after": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ IsOpen (range some)",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ IsClosed {∞}",
"tactic": "rw [← compl_range_coe, isClosed_compl_iff]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ IsOpen (range some)",
"tactic": "exact isOpen_range_coe"
}
] |
[
284,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.le_antisymm
|
[
{
"state_after": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\n⊢ a = b",
"state_before": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\n⊢ a = b",
"tactic": "let ⟨n, hn⟩ := le.dest h₁"
},
{
"state_after": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\n⊢ a = b",
"state_before": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\n⊢ a = b",
"tactic": "let ⟨m, hm⟩ := le.dest h₂"
},
{
"state_after": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\nthis : a + ↑n = b\n⊢ a = b",
"state_before": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\n⊢ a = b",
"tactic": "have := hn"
},
{
"state_after": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\nthis : b + ↑(m + n) = b\n⊢ a = b",
"state_before": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\nthis : a + ↑n = b\n⊢ a = b",
"tactic": "rw [← hm, Int.add_assoc, ← ofNat_add] at this"
},
{
"state_after": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\nthis✝ : b + ↑(m + n) = b\nthis : m + n = 0\n⊢ a = b",
"state_before": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\nthis : b + ↑(m + n) = b\n⊢ a = b",
"tactic": "have := Int.ofNat.inj <| Int.add_left_cancel <| this.trans (Int.add_zero _).symm"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh₁ : a ≤ b\nh₂ : b ≤ a\nn : Nat\nhn : a + ↑n = b\nm : Nat\nhm : b + ↑m = a\nthis✝ : b + ↑(m + n) = b\nthis : m + n = 0\n⊢ a = b",
"tactic": "rw [← hn, Nat.eq_zero_of_add_eq_zero_left this, ofNat_zero, Int.add_zero a]"
}
] |
[
619,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
615,
11
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.extend_top
|
[] |
[
1353,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1352,
1
] |
Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
|
Geometry.SimplicialComplex.not_facet_iff_subface
|
[
{
"state_after": "case refine'_1\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nhs : s ∈ K.faces\nhs' : ¬(s ∈ K.faces ∧ ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t)\n⊢ ∃ t, t ∈ K.faces ∧ s ⊂ t\n\ncase refine'_2\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nhs : s ∈ K.faces\n⊢ (∃ t, t ∈ K.faces ∧ s ⊂ t) → ¬s ∈ facets K",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nhs : s ∈ K.faces\n⊢ ¬s ∈ facets K ↔ ∃ t, t ∈ K.faces ∧ s ⊂ t",
"tactic": "refine' ⟨fun hs' : ¬(_ ∧ _) => _, _⟩"
},
{
"state_after": "case refine'_1\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nhs : s ∈ K.faces\nhs' : s ∈ K.faces → Exists fun ⦃t⦄ => t ∈ K.faces ∧ s ⊆ t ∧ s ≠ t\n⊢ ∃ t, t ∈ K.faces ∧ s ⊂ t",
"state_before": "case refine'_1\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nhs : s ∈ K.faces\nhs' : ¬(s ∈ K.faces ∧ ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t)\n⊢ ∃ t, t ∈ K.faces ∧ s ⊂ t",
"tactic": "push_neg at hs'"
},
{
"state_after": "case refine'_1.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs : s ∈ K.faces\nhs' : s ∈ K.faces → Exists fun ⦃t⦄ => t ∈ K.faces ∧ s ⊆ t ∧ s ≠ t\nt : Finset E\nht : t ∈ K.faces ∧ s ⊆ t ∧ s ≠ t\n⊢ ∃ t, t ∈ K.faces ∧ s ⊂ t",
"state_before": "case refine'_1\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nhs : s ∈ K.faces\nhs' : s ∈ K.faces → Exists fun ⦃t⦄ => t ∈ K.faces ∧ s ⊆ t ∧ s ≠ t\n⊢ ∃ t, t ∈ K.faces ∧ s ⊂ t",
"tactic": "obtain ⟨t, ht⟩ := hs' hs"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs : s ∈ K.faces\nhs' : s ∈ K.faces → Exists fun ⦃t⦄ => t ∈ K.faces ∧ s ⊆ t ∧ s ≠ t\nt : Finset E\nht : t ∈ K.faces ∧ s ⊆ t ∧ s ≠ t\n⊢ ∃ t, t ∈ K.faces ∧ s ⊂ t",
"tactic": "exact ⟨t, ht.1, ⟨ht.2.1, fun hts => ht.2.2 (Subset.antisymm ht.2.1 hts)⟩⟩"
},
{
"state_after": "case refine'_2.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs✝ : s ∈ K.faces\nt : Finset E\nht : t ∈ K.faces ∧ s ⊂ t\nhs : s ∈ K.faces\nhs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t\n⊢ False",
"state_before": "case refine'_2\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nx : E\nhs : s ∈ K.faces\n⊢ (∃ t, t ∈ K.faces ∧ s ⊂ t) → ¬s ∈ facets K",
"tactic": "rintro ⟨t, ht⟩ ⟨hs, hs'⟩"
},
{
"state_after": "case refine'_2.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs✝ : s ∈ K.faces\nt : Finset E\nht : t ∈ K.faces ∧ s ⊂ t\nhs : s ∈ K.faces\nhs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t\nthis : s = t\n⊢ False",
"state_before": "case refine'_2.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs✝ : s ∈ K.faces\nt : Finset E\nht : t ∈ K.faces ∧ s ⊂ t\nhs : s ∈ K.faces\nhs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t\n⊢ False",
"tactic": "have := hs' ht.1 ht.2.1"
},
{
"state_after": "case refine'_2.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs✝ : s ∈ K.faces\nt : Finset E\nht : t ∈ K.faces ∧ t ⊂ t\nhs : s ∈ K.faces\nhs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t\nthis : s = t\n⊢ False",
"state_before": "case refine'_2.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs✝ : s ∈ K.faces\nt : Finset E\nht : t ∈ K.faces ∧ s ⊂ t\nhs : s ∈ K.faces\nhs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t\nthis : s = t\n⊢ False",
"tactic": "rw [this] at ht"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.69803\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t✝ : Finset E\nx : E\nhs✝ : s ∈ K.faces\nt : Finset E\nht : t ∈ K.faces ∧ t ⊂ t\nhs : s ∈ K.faces\nhs' : ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t\nthis : s = t\n⊢ False",
"tactic": "exact ht.2.2 (Subset.refl t)"
}
] |
[
211,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.toLinearIsometry_inj
|
[] |
[
619,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
list_prod_mem
|
[
{
"state_after": "case intro\nM : Type u_1\nA : Type ?u.11708\nB : Type u_2\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS : B\nl : List { x // x ∈ S }\n⊢ List.prod (List.map Subtype.val l) ∈ S",
"state_before": "M : Type u_1\nA : Type ?u.11708\nB : Type u_2\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS : B\nl : List M\nhl : ∀ (x : M), x ∈ l → x ∈ S\n⊢ List.prod l ∈ S",
"tactic": "lift l to List S using hl"
},
{
"state_after": "case intro\nM : Type u_1\nA : Type ?u.11708\nB : Type u_2\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS : B\nl : List { x // x ∈ S }\n⊢ ↑(List.prod l) ∈ S",
"state_before": "case intro\nM : Type u_1\nA : Type ?u.11708\nB : Type u_2\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS : B\nl : List { x // x ∈ S }\n⊢ List.prod (List.map Subtype.val l) ∈ S",
"tactic": "rw [← coe_list_prod]"
},
{
"state_after": "no goals",
"state_before": "case intro\nM : Type u_1\nA : Type ?u.11708\nB : Type u_2\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS : B\nl : List { x // x ∈ S }\n⊢ ↑(List.prod l) ∈ S",
"tactic": "exact l.prod.coe_prop"
}
] |
[
79,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1256704\nG : Type ?u.1256707\n𝕜 : Type ?u.1256710\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nx : { x // x ∈ simpleFunc E 1 μ }\n⊢ ‖x‖ = ↑1 * ‖↑(coeToLp α E ℝ) x‖",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1256704\nG : Type ?u.1256707\n𝕜 : Type ?u.1256710\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\n⊢ ‖setToL1 hT‖ ≤ ↑1 * ‖setToL1SCLM α E μ hT‖",
"tactic": "refine'\n ContinuousLinearMap.op_norm_extend_le (setToL1SCLM α E μ hT) (coeToLp α E ℝ)\n (simpleFunc.denseRange one_ne_top) fun x => le_of_eq _"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1256704\nG : Type ?u.1256707\n𝕜 : Type ?u.1256710\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nx : { x // x ∈ simpleFunc E 1 μ }\n⊢ ‖x‖ = ‖↑(coeToLp α E ℝ) x‖",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1256704\nG : Type ?u.1256707\n𝕜 : Type ?u.1256710\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nx : { x // x ∈ simpleFunc E 1 μ }\n⊢ ‖x‖ = ↑1 * ‖↑(coeToLp α E ℝ) x‖",
"tactic": "rw [NNReal.coe_one, one_mul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1256704\nG : Type ?u.1256707\n𝕜 : Type ?u.1256710\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nx : { x // x ∈ simpleFunc E 1 μ }\n⊢ ‖x‖ = ‖↑(coeToLp α E ℝ) x‖",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1256704\nG : Type ?u.1256707\n𝕜 : Type ?u.1256710\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\n⊢ ↑1 * ‖setToL1SCLM α E μ hT‖ = ‖setToL1SCLM α E μ hT‖",
"tactic": "rw [NNReal.coe_one, one_mul]"
}
] |
[
1222,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1213,
1
] |
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