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/GroupTheory/Submonoid/Operations.lean
|
MonoidHom.map_mclosure
|
[] |
[
1097,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1093,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.SignedMeasure.toMeasureOfLEZero_apply
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.712523\nm : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : restrict s i ≤ restrict 0 i\nhi₁ : MeasurableSet i\nhj₁ : MeasurableSet j\n⊢ ↑{ val := ↑(-s) (i ∩ j), property := (_ : 0 ≤ ↑(-s) (i ∩ j)) } =\n ↑{ val := -↑s (i ∩ j), property := (_ : 0 ≤ -↑s (i ∩ j)) }\n\ncase hj₁\nα : Type u_1\nβ : Type ?u.712523\nm : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : restrict s i ≤ restrict 0 i\nhi₁ : MeasurableSet i\nhj₁ : MeasurableSet j\n⊢ MeasurableSet j",
"state_before": "α : Type u_1\nβ : Type ?u.712523\nm : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : restrict s i ≤ restrict 0 i\nhi₁ : MeasurableSet i\nhj₁ : MeasurableSet j\n⊢ ↑↑(toMeasureOfLEZero s i hi₁ hi) j = ↑{ val := -↑s (i ∩ j), property := (_ : 0 ≤ -↑s (i ∩ j)) }",
"tactic": "erw [toMeasureOfZeroLE_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.712523\nm : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : restrict s i ≤ restrict 0 i\nhi₁ : MeasurableSet i\nhj₁ : MeasurableSet j\n⊢ ↑{ val := ↑(-s) (i ∩ j), property := (_ : 0 ≤ ↑(-s) (i ∩ j)) } =\n ↑{ val := -↑s (i ∩ j), property := (_ : 0 ≤ -↑s (i ∩ j)) }",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case hj₁\nα : Type u_1\nβ : Type ?u.712523\nm : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : restrict s i ≤ restrict 0 i\nhi₁ : MeasurableSet i\nhj₁ : MeasurableSet j\n⊢ MeasurableSet j",
"tactic": "assumption"
}
] |
[
1381,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1375,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_add_right'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.903143\nγ : Type ?u.903146\nδ : Type ?u.903149\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g\n⊢ (∫⁻ (a : α), f a + g a ∂μ) = (∫⁻ (a : α), f a ∂μ) + ∫⁻ (a : α), g a ∂μ",
"tactic": "simpa only [add_comm] using lintegral_add_left' hg f"
}
] |
[
590,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
588,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp'
|
[
{
"state_after": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\n⊢ Tendsto (fun b => dist (f b) f_lim) fi (𝓝 0) ↔ Tendsto (fun n => snorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\n⊢ Tendsto f fi (𝓝 f_lim) ↔ Tendsto (fun n => snorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)",
"tactic": "rw [tendsto_iff_dist_tendsto_zero]"
},
{
"state_after": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\n⊢ Tendsto (fun b => ENNReal.toReal (snorm (↑↑(f b) - ↑↑f_lim) p μ)) fi (𝓝 0) ↔\n Tendsto (fun n => snorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\n⊢ Tendsto (fun b => dist (f b) f_lim) fi (𝓝 0) ↔ Tendsto (fun n => snorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)",
"tactic": "simp_rw [dist_def]"
},
{
"state_after": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\nn : ι\n⊢ snorm (↑↑(f n) - ↑↑f_lim) p μ ≠ ⊤",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\n⊢ Tendsto (fun b => ENNReal.toReal (snorm (↑↑(f b) - ↑↑f_lim) p μ)) fi (𝓝 0) ↔\n Tendsto (fun n => snorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)",
"tactic": "rw [← ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff (fun n => ?_) ENNReal.zero_ne_top]"
},
{
"state_after": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\nn : ι\n⊢ snorm (↑↑(f n - f_lim)) p μ ≠ ⊤",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\nn : ι\n⊢ snorm (↑↑(f n) - ↑↑f_lim) p μ ≠ ⊤",
"tactic": "rw [snorm_congr_ae (Lp.coeFn_sub _ _).symm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8121841\nG : Type ?u.8121844\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : { x // x ∈ Lp E p }\nn : ι\n⊢ snorm (↑↑(f n - f_lim)) p μ ≠ ⊤",
"tactic": "exact Lp.snorm_ne_top _"
}
] |
[
1240,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1233,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
dist_self_mul_right
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.636975\n𝕜 : Type ?u.636978\nα : Type ?u.636981\nι : Type ?u.636984\nκ : Type ?u.636987\nE : Type u_1\nF : Type ?u.636993\nG : Type ?u.636996\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ dist a (a * b) = ‖b‖",
"tactic": "rw [← dist_one_left, ← dist_mul_left a 1 b, mul_one]"
}
] |
[
1403,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1402,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.setToFun_congr_measure
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1651275\nG : Type ?u.1651278\n𝕜 : Type ?u.1651281\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ⊤\nhc' : c' ≠ ⊤\nhμ_le : μ ≤ c • μ'\nhμ'_le : μ' ≤ c' • μ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ' T C'\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T hT f = setToFun μ' T hT' f\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1651275\nG : Type ?u.1651278\n𝕜 : Type ?u.1651281\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ⊤\nhc' : c' ≠ ⊤\nhμ_le : μ ≤ c • μ'\nhμ'_le : μ' ≤ c' • μ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ' T C'\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T hT f = setToFun μ' T hT' f",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1651275\nG : Type ?u.1651278\n𝕜 : Type ?u.1651281\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ⊤\nhc' : c' ≠ ⊤\nhμ_le : μ ≤ c • μ'\nhμ'_le : μ' ≤ c' • μ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ' T C'\nf : α → E\n⊢ setToFun μ T hT f = setToFun μ' T hT' f",
"tactic": "by_cases hf : Integrable f μ"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1651275\nG : Type ?u.1651278\n𝕜 : Type ?u.1651281\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ⊤\nhc' : c' ≠ ⊤\nhμ_le : μ ≤ c • μ'\nhμ'_le : μ' ≤ c' • μ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ' T C'\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T hT f = setToFun μ' T hT' f",
"tactic": "exact setToFun_congr_measure_of_integrable c' hc' hμ'_le hT hT' f hf"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1651275\nG : Type ?u.1651278\n𝕜 : Type ?u.1651281\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ⊤\nhc' : c' ≠ ⊤\nhμ_le : μ ≤ c • μ'\nhμ'_le : μ' ≤ c' • μ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ' T C'\nf : α → E\nhf : ¬Integrable f\nh_int : ∀ (g : α → E), ¬Integrable g → ¬Integrable g\n⊢ setToFun μ T hT f = setToFun μ' T hT' f",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1651275\nG : Type ?u.1651278\n𝕜 : Type ?u.1651281\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ⊤\nhc' : c' ≠ ⊤\nhμ_le : μ ≤ c • μ'\nhμ'_le : μ' ≤ c' • μ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ' T C'\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T hT f = setToFun μ' T hT' f",
"tactic": "have h_int : ∀ g : α → E, ¬Integrable g μ → ¬Integrable g μ' := fun g =>\n mt fun h => h.of_measure_le_smul _ hc hμ_le"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1651275\nG : Type ?u.1651278\n𝕜 : Type ?u.1651281\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ⊤\nhc' : c' ≠ ⊤\nhμ_le : μ ≤ c • μ'\nhμ'_le : μ' ≤ c' • μ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ' T C'\nf : α → E\nhf : ¬Integrable f\nh_int : ∀ (g : α → E), ¬Integrable g → ¬Integrable g\n⊢ setToFun μ T hT f = setToFun μ' T hT' f",
"tactic": "simp_rw [setToFun_undef _ hf, setToFun_undef _ (h_int f hf)]"
}
] |
[
1649,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1640,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.memℒp_const_iff
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1273342\nG : Type ?u.1273345\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\n⊢ Memℒp (fun x => c) p ↔ snorm (fun x => c) p μ < ⊤",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1273342\nG : Type ?u.1273345\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\n⊢ Memℒp (fun x => c) p ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "rw [← snorm_const_lt_top_iff hp_ne_zero hp_ne_top]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1273342\nG : Type ?u.1273345\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\n⊢ Memℒp (fun x => c) p ↔ snorm (fun x => c) p μ < ⊤",
"tactic": "exact ⟨fun h => h.2, fun h => ⟨aestronglyMeasurable_const, h⟩⟩"
}
] |
[
345,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
HasStrictFDerivAt.localInverse_unique
|
[] |
[
659,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/Algebra/Star/Pointwise.lean
|
Set.star_subset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns✝ t✝ : Set α\na : α\ninst✝ : InvolutiveStar α\ns t : Set α\n⊢ s⋆ ⊆ t ↔ s ⊆ t⋆",
"tactic": "rw [← star_subset_star, star_star]"
}
] |
[
111,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
Set.pairwise_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10052\nγ : Type ?u.10055\nι : Type ?u.10058\nι' : Type ?u.10061\nr p q : α → α → Prop\nf g : ι → α\ns t u : Set α\na b : α\n⊢ Set.Pairwise (insert a s) r ↔ Set.Pairwise s r ∧ ∀ (b : α), b ∈ s → a ≠ b → r a b ∧ r b a",
"tactic": "simp only [insert_eq, pairwise_union, pairwise_singleton, true_and_iff, mem_singleton_iff,\n forall_eq]"
}
] |
[
157,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Data/List/Intervals.lean
|
List.Ico.filter_lt_of_succ_bot
|
[
{
"state_after": "n m : ℕ\nhnm : n < m\nr : min m (n + 1) = n + 1\n⊢ filter (fun x => decide (x < n + 1)) (Ico n m) = [n]",
"state_before": "n m : ℕ\nhnm : n < m\n⊢ filter (fun x => decide (x < n + 1)) (Ico n m) = [n]",
"tactic": "have r : min m (n + 1) = n + 1 := (@inf_eq_right _ _ m (n + 1)).mpr hnm"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\nhnm : n < m\nr : min m (n + 1) = n + 1\n⊢ filter (fun x => decide (x < n + 1)) (Ico n m) = [n]",
"tactic": "simp [filter_lt n m (n + 1), r]"
}
] |
[
218,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Data/FunLike/Equiv.lean
|
EquivLike.injective_comp
|
[] |
[
183,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Order/Synonym.lean
|
OrderDual.lt_toDual
|
[] |
[
117,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_ge_iff
|
[] |
[
69,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseRange.equalizer
|
[] |
[
364,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.mem_nhds_iff
|
[] |
[
108,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.sub_nonpos_of_le
|
[
{
"state_after": "a b : Int\nh✝ : a ≤ b\nh : a + -b ≤ b + -b\n⊢ a - b ≤ 0",
"state_before": "a b : Int\nh : a ≤ b\n⊢ a - b ≤ 0",
"tactic": "have h := Int.add_le_add_right h (-b)"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh✝ : a ≤ b\nh : a + -b ≤ b + -b\n⊢ a - b ≤ 0",
"tactic": "rwa [Int.add_right_neg] at h"
}
] |
[
928,
31
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
926,
11
] |
Mathlib/GroupTheory/Nilpotent.lean
|
upperCentralSeries_nilpotencyClass
|
[] |
[
372,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_comp_mul_deriv
|
[] |
[
1502,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1499,
1
] |
Mathlib/Algebra/Group/Semiconj.lean
|
SemiconjBy.pow_right
|
[
{
"state_after": "case zero\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\n⊢ SemiconjBy a (x ^ Nat.zero) (y ^ Nat.zero)\n\ncase succ\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : ℕ\nih : SemiconjBy a (x ^ n) (y ^ n)\n⊢ SemiconjBy a (x ^ Nat.succ n) (y ^ Nat.succ n)",
"state_before": "M : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : ℕ\n⊢ SemiconjBy a (x ^ n) (y ^ n)",
"tactic": "induction' n with n ih"
},
{
"state_after": "case zero\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\n⊢ SemiconjBy a 1 1",
"state_before": "case zero\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\n⊢ SemiconjBy a (x ^ Nat.zero) (y ^ Nat.zero)",
"tactic": "rw [pow_zero, pow_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\n⊢ SemiconjBy a 1 1",
"tactic": "exact SemiconjBy.one_right _"
},
{
"state_after": "case succ\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : ℕ\nih : SemiconjBy a (x ^ n) (y ^ n)\n⊢ SemiconjBy a (x * x ^ n) (y * y ^ n)",
"state_before": "case succ\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : ℕ\nih : SemiconjBy a (x ^ n) (y ^ n)\n⊢ SemiconjBy a (x ^ Nat.succ n) (y ^ Nat.succ n)",
"tactic": "rw [pow_succ, pow_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nM : Type u_1\ninst✝ : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : ℕ\nih : SemiconjBy a (x ^ n) (y ^ n)\n⊢ SemiconjBy a (x * x ^ n) (y * y ^ n)",
"tactic": "exact h.mul_right ih"
}
] |
[
175,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Data/List/Basic.lean
|
List.insertNth_removeNth_of_le
|
[] |
[
1638,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1632,
1
] |
Mathlib/NumberTheory/VonMangoldt.lean
|
Nat.ArithmeticFunction.vonMangoldt_nonneg
|
[
{
"state_after": "n : ℕ\n⊢ 0 ≤ if IsPrimePow n then Real.log ↑(minFac n) else 0",
"state_before": "n : ℕ\n⊢ 0 ≤ ↑Λ n",
"tactic": "rw [vonMangoldt_apply]"
},
{
"state_after": "case inl\nn : ℕ\nh✝ : IsPrimePow n\n⊢ 0 ≤ Real.log ↑(minFac n)\n\ncase inr\nn : ℕ\nh✝ : ¬IsPrimePow n\n⊢ 0 ≤ 0",
"state_before": "n : ℕ\n⊢ 0 ≤ if IsPrimePow n then Real.log ↑(minFac n) else 0",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case inr\nn : ℕ\nh✝ : ¬IsPrimePow n\n⊢ 0 ≤ 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inl\nn : ℕ\nh✝ : IsPrimePow n\n⊢ 0 ≤ Real.log ↑(minFac n)",
"tactic": "exact Real.log_nonneg (one_le_cast.2 (Nat.minFac_pos n))"
}
] |
[
86,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Computability/Language.lean
|
Language.iSup_mul
|
[] |
[
219,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Order/Basic.lean
|
LT.lt.false
|
[] |
[
308,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
11
] |
Mathlib/Algebra/Star/Free.lean
|
FreeMonoid.star_one
|
[] |
[
42,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.IsQuadratic.inv
|
[
{
"state_after": "case h\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\n⊢ ↑χ⁻¹ ↑x = ↑χ ↑x",
"state_before": "R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\n⊢ χ⁻¹ = χ",
"tactic": "ext x"
},
{
"state_after": "case h\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"state_before": "case h\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\n⊢ ↑χ⁻¹ ↑x = ↑χ ↑x",
"tactic": "rw [inv_apply_eq_inv]"
},
{
"state_after": "case h.inl\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₀ : ↑χ ↑x = 0\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x\n\ncase h.inr.inl\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₁ : ↑χ ↑x = 1\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x\n\ncase h.inr.inr\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₂ : ↑χ ↑x = -1\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"state_before": "case h\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"tactic": "rcases hχ x with (h₀ | h₁ | h₂)"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₀ : ↑χ ↑x = 0\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"tactic": "rw [h₀, Ring.inverse_zero]"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inl\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₁ : ↑χ ↑x = 1\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"tactic": "rw [h₁, Ring.inverse_one]"
},
{
"state_after": "case h.inr.inr\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₂ : ↑χ ↑x = -1\nthis : -1 = ↑(-1)\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"state_before": "case h.inr.inr\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₂ : ↑χ ↑x = -1\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"tactic": "have : (-1 : R') = (-1 : R'ˣ) := by rw [Units.val_neg, Units.val_one]"
},
{
"state_after": "case h.inr.inr\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₂ : ↑χ ↑x = -1\nthis : -1 = ↑(-1)\n⊢ ↑(-1)⁻¹ = ↑(-1)",
"state_before": "case h.inr.inr\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₂ : ↑χ ↑x = -1\nthis : -1 = ↑(-1)\n⊢ Ring.inverse (↑χ ↑x) = ↑χ ↑x",
"tactic": "rw [h₂, this, Ring.inverse_unit (-1 : R'ˣ)]"
},
{
"state_after": "no goals",
"state_before": "case h.inr.inr\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₂ : ↑χ ↑x = -1\nthis : -1 = ↑(-1)\n⊢ ↑(-1)⁻¹ = ↑(-1)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nx : Rˣ\nh₂ : ↑χ ↑x = -1\n⊢ -1 = ↑(-1)",
"tactic": "rw [Units.val_neg, Units.val_one]"
}
] |
[
500,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.is_glb_mem
|
[
{
"state_after": "F : Type ?u.408857\nα : Type u_1\nβ : Type ?u.408863\nγ : Type ?u.408866\nι : Type ?u.408869\nκ : Type ?u.408872\ninst✝ : LinearOrder α\ni : α\ns : Finset α\nhis : IsGLB (↑s) i\nhs : Finset.Nonempty s\n⊢ i ∈ ↑s",
"state_before": "F : Type ?u.408857\nα : Type u_1\nβ : Type ?u.408863\nγ : Type ?u.408866\nι : Type ?u.408869\nκ : Type ?u.408872\ninst✝ : LinearOrder α\ni : α\ns : Finset α\nhis : IsGLB (↑s) i\nhs : Finset.Nonempty s\n⊢ i ∈ s",
"tactic": "rw [← mem_coe]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.408857\nα : Type u_1\nβ : Type ?u.408863\nγ : Type ?u.408866\nι : Type ?u.408869\nκ : Type ?u.408872\ninst✝ : LinearOrder α\ni : α\ns : Finset α\nhis : IsGLB (↑s) i\nhs : Finset.Nonempty s\n⊢ i ∈ ↑s",
"tactic": "exact ((is_glb_iff_is_least i s hs).mp his).1"
}
] |
[
1728,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1725,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.Compacts.coe_prod
|
[] |
[
200,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean
|
Euclidean.mem_ball_self
|
[] |
[
82,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasurableEmbedding.restrict_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1222190\nδ : Type ?u.1222193\nι : Type ?u.1222196\nR : Type ?u.1222199\nR' : Type ?u.1222202\nm0 : MeasurableSpace α\nm1 : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nμ : MeasureTheory.Measure α\ns t : Set β\nht : MeasurableSet t\n⊢ ↑↑(Measure.restrict (Measure.map f μ) s) t = ↑↑(Measure.map f (Measure.restrict μ (f ⁻¹' s))) t",
"tactic": "simp [hf.map_apply, ht, hf.measurable ht]"
}
] |
[
4215,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4213,
1
] |
Mathlib/Algebra/Free.lean
|
FreeSemigroup.lift_of_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semigroup β\nf : α → β\nx : α\ny : FreeSemigroup α\n⊢ ↑(↑lift f) (of x * y) = f x * ↑(↑lift f) y",
"tactic": "rw [map_mul, lift_of]"
}
] |
[
554,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/Logic/Basic.lean
|
congr_fun₂
|
[] |
[
593,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Std/Data/List/Lemmas.lean
|
List.map_filterMap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → Option β\ng : β → γ\nl : List α\n⊢ map g (filterMap f l) = filterMap (fun x => Option.map g (f x)) l",
"tactic": "simp only [← filterMap_eq_map, filterMap_filterMap, Option.map_eq_bind]"
}
] |
[
1184,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1182,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsGreatest.insert
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t : Set α\na✝ b✝ : α\ninst✝ : LinearOrder γ\na b : γ\ns : Set γ\nhs : IsGreatest s b\n⊢ IsGreatest ({a} ∪ s) (max a b)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t : Set α\na✝ b✝ : α\ninst✝ : LinearOrder γ\na b : γ\ns : Set γ\nhs : IsGreatest s b\n⊢ IsGreatest (Insert.insert a s) (max a b)",
"tactic": "rw [insert_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns✝ t : Set α\na✝ b✝ : α\ninst✝ : LinearOrder γ\na b : γ\ns : Set γ\nhs : IsGreatest s b\n⊢ IsGreatest ({a} ∪ s) (max a b)",
"tactic": "exact isGreatest_singleton.union hs"
}
] |
[
961,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
958,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.max_le_of_forall_le
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.111970\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nl✝ l : List α\na : α\nh✝ : ∀ (x : α), x ∈ l → x ≤ a\nh : ∀ (x : α), x ∈ [] → x ≤ a\n⊢ foldr max ⊥ [] ≤ a\n\ncase cons\nα : Type u_1\nβ : Type ?u.111970\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nl✝¹ l✝ : List α\na : α\nh✝ : ∀ (x : α), x ∈ l✝ → x ≤ a\ny : α\nl : List α\nIH : (∀ (x : α), x ∈ l → x ≤ a) → foldr max ⊥ l ≤ a\nh : ∀ (x : α), x ∈ y :: l → x ≤ a\n⊢ foldr max ⊥ (y :: l) ≤ a",
"state_before": "α : Type u_1\nβ : Type ?u.111970\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nl✝ l : List α\na : α\nh : ∀ (x : α), x ∈ l → x ≤ a\n⊢ foldr max ⊥ l ≤ a",
"tactic": "induction' l with y l IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.111970\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nl✝ l : List α\na : α\nh✝ : ∀ (x : α), x ∈ l → x ≤ a\nh : ∀ (x : α), x ∈ [] → x ≤ a\n⊢ foldr max ⊥ [] ≤ a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.111970\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nl✝¹ l✝ : List α\na : α\nh✝ : ∀ (x : α), x ∈ l✝ → x ≤ a\ny : α\nl : List α\nIH : (∀ (x : α), x ∈ l → x ≤ a) → foldr max ⊥ l ≤ a\nh : ∀ (x : α), x ∈ y :: l → x ≤ a\n⊢ foldr max ⊥ (y :: l) ≤ a",
"tactic": "simpa [h y (mem_cons_self _ _)] using IH fun x hx => h x <| mem_cons_of_mem _ hx"
}
] |
[
419,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_comap_of_surjective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.258330\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : α → β\nhf : Surjective f\nl : Filter β\n⊢ range f ∈ l",
"tactic": "simp only [hf.range_eq, univ_mem]"
}
] |
[
2275,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2273,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.IsProper.top_not_mem
|
[] |
[
274,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.map_add
|
[] |
[
209,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
11
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Iic_diff_Iic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.62385\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\n⊢ Iic b \\ Iic a = Ioc a b",
"tactic": "rw [diff_eq, compl_Iic, inter_comm, Ioi_inter_Iic]"
}
] |
[
1097,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1096,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContinuousLinearMap.contDiff
|
[] |
[
156,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.vsub_eq_empty
|
[] |
[
628,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.mul_inv
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.417122\ninst✝ : CommMonoid P\nf : M →* N\nh : ∀ (y : { x // x ∈ S }), IsUnit (↑f ↑y)\nx₁ x₂ : M\ny₁ y₂ : { x // x ∈ S }\n⊢ ↑f x₁ * ↑(↑(IsUnit.liftRight (MonoidHom.restrict f S) h) y₁)⁻¹ =\n ↑f x₂ * ↑(↑(IsUnit.liftRight (MonoidHom.restrict f S) h) y₂)⁻¹ ↔\n ↑f (x₁ * ↑y₂) = ↑f (x₂ * ↑y₁)",
"tactic": "rw [mul_inv_right h, mul_assoc, mul_comm _ (f y₂), ← mul_assoc, mul_inv_left h, mul_comm x₂,\n f.map_mul, f.map_mul]"
}
] |
[
649,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.Pairwise.sublists'
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\n⊢ Pairwise (Lex (swap R)) (List.sublists' l) ∧\n Pairwise (fun a_1 b => Lex (swap R) (a :: a_1) (a :: b)) (List.sublists' l) ∧\n ∀ (a_1 : List α), a_1 <+ l → ∀ (b x : List α), x <+ l → a :: x = b → Lex (swap R) a_1 b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\n⊢ Pairwise (Lex (swap R)) (List.sublists' (a :: l))",
"tactic": "simp only [sublists'_cons, pairwise_append, pairwise_map, mem_sublists', mem_map, exists_imp,\n and_imp]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\n⊢ ∀ (a_1 : List α), a_1 <+ l → ∀ (b x : List α), x <+ l → a :: x = b → Lex (swap R) a_1 b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\n⊢ Pairwise (Lex (swap R)) (List.sublists' l) ∧\n Pairwise (fun a_1 b => Lex (swap R) (a :: a_1) (a :: b)) (List.sublists' l) ∧\n ∀ (a_1 : List α), a_1 <+ l → ∀ (b x : List α), x <+ l → a :: x = b → Lex (swap R) a_1 b",
"tactic": "refine' ⟨H₂.sublists', H₂.sublists'.imp fun l₁ => Lex.cons l₁, _⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\nl₁ : List α\nsl₁ : l₁ <+ l\nl₂ : List α\na✝ : l₂ <+ l\n⊢ Lex (swap R) l₁ (a :: l₂)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\n⊢ ∀ (a_1 : List α), a_1 <+ l → ∀ (b x : List α), x <+ l → a :: x = b → Lex (swap R) a_1 b",
"tactic": "rintro l₁ sl₁ x l₂ _ rfl"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\nl₂ : List α\na✝ : l₂ <+ l\nsl₁ : [] <+ l\n⊢ Lex (swap R) [] (a :: l₂)\n\ncase cons\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\nl₂ : List α\na✝ : l₂ <+ l\nb : α\nl₁ : List α\nsl₁ : b :: l₁ <+ l\n⊢ Lex (swap R) (b :: l₁) (a :: l₂)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\nl₁ : List α\nsl₁ : l₁ <+ l\nl₂ : List α\na✝ : l₂ <+ l\n⊢ Lex (swap R) l₁ (a :: l₂)",
"tactic": "cases' l₁ with b l₁"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\nl₂ : List α\na✝ : l₂ <+ l\nb : α\nl₁ : List α\nsl₁ : b :: l₁ <+ l\n⊢ Lex (swap R) (b :: l₁) (a :: l₂)",
"tactic": "exact Lex.rel (H₁ _ <| sl₁.subset <| mem_cons_self _ _)"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\na : α\nl : List α\nH₁ : ∀ (a' : α), a' ∈ l → R a a'\nH₂ : Pairwise R l\nl₂ : List α\na✝ : l₂ <+ l\nsl₁ : [] <+ l\n⊢ Lex (swap R) [] (a :: l₂)",
"tactic": "constructor"
}
] |
[
363,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_const'
|
[
{
"state_after": "case inl\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂0, f x = f y\n⊢ AEMeasurable f\n\ncase inr\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nμ ν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\n⊢ AEMeasurable f",
"state_before": "ι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nμ ν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\n⊢ AEMeasurable f",
"tactic": "rcases eq_or_ne μ 0 with (rfl | hμ)"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂0, f x = f y\n⊢ AEMeasurable f",
"tactic": "exact aemeasurable_zero_measure"
},
{
"state_after": "case inr\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nμ ν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\nthis : NeBot (ae μ)\n⊢ AEMeasurable f",
"state_before": "case inr\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nμ ν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\n⊢ AEMeasurable f",
"tactic": "haveI := ae_neBot.2 hμ"
},
{
"state_after": "case inr.intro\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nμ ν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\nthis : NeBot (ae μ)\nx : α\nhx : ∀ᵐ (y : α) ∂μ, f x = f y\n⊢ AEMeasurable f",
"state_before": "case inr\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nμ ν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\nthis : NeBot (ae μ)\n⊢ AEMeasurable f",
"tactic": "rcases h.exists with ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nι : Type ?u.3045534\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3045543\nδ : Type ?u.3045546\nR : Type ?u.3045549\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nf g : α → β\nμ ν : MeasureTheory.Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\nthis : NeBot (ae μ)\nx : α\nhx : ∀ᵐ (y : α) ∂μ, f x = f y\n⊢ AEMeasurable f",
"tactic": "exact ⟨const α (f x), measurable_const, EventuallyEq.symm hx⟩"
}
] |
[
249,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.cast_addNat
|
[] |
[
1464,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1463,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_union_distrib
|
[] |
[
531,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
uniformEmbedding_subtypeEmb
|
[] |
[
266,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Order/Lattice.lean
|
le_of_inf_le_sup_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DistribLattice α\nx y z : α\nh₁ : x ⊓ z ≤ y ⊓ z\nh₂ : x ⊔ z ≤ y ⊔ z\n⊢ y ⊓ z ⊔ x = (y ⊔ x) ⊓ (x ⊔ z)",
"tactic": "rw [sup_inf_right, @sup_comm _ _ x]"
}
] |
[
792,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
785,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
smul_one_smul
|
[
{
"state_after": "no goals",
"state_before": "M✝ : Type ?u.26534\nN✝ : Type ?u.26537\nG : Type ?u.26540\nA : Type ?u.26543\nB : Type ?u.26546\nα : Type u_3\nβ : Type ?u.26552\nγ : Type ?u.26555\nδ : Type ?u.26558\nM : Type u_1\nN : Type u_2\ninst✝⁴ : Monoid N\ninst✝³ : SMul M N\ninst✝² : MulAction N α\ninst✝¹ : SMul M α\ninst✝ : IsScalarTower M N α\nx : M\ny : α\n⊢ (x • 1) • y = x • y",
"tactic": "rw [smul_assoc, one_smul]"
}
] |
[
643,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_lt
|
[] |
[
328,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.erase_ne
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.164124\nγ : Type ?u.164127\nι : Type ?u.164130\nM : Type u_2\nM' : Type ?u.164136\nN : Type ?u.164139\nP : Type ?u.164142\nG : Type ?u.164145\nH : Type ?u.164148\nR : Type ?u.164151\nS : Type ?u.164154\ninst✝ : Zero M\na a' : α\nf : α →₀ M\nh : a' ≠ a\n⊢ ↑(erase a f) a' = ↑f a'",
"tactic": "classical simp only [erase, coe_mk, h, ite_false]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.164124\nγ : Type ?u.164127\nι : Type ?u.164130\nM : Type u_2\nM' : Type ?u.164136\nN : Type ?u.164139\nP : Type ?u.164142\nG : Type ?u.164145\nH : Type ?u.164148\nR : Type ?u.164151\nS : Type ?u.164154\ninst✝ : Zero M\na a' : α\nf : α →₀ M\nh : a' ≠ a\n⊢ ↑(erase a f) a' = ↑f a'",
"tactic": "simp only [erase, coe_mk, h, ite_false]"
}
] |
[
649,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
648,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderEmbedding.monotone
|
[] |
[
666,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean
|
Polynomial.Chebyshev.complex_ofReal_eval_T
|
[] |
[
56,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
CFilter.ofEquiv_val
|
[
{
"state_after": "case mk\nα : Type u_3\nβ : Type ?u.5122\nσ : Type u_1\nτ : Type u_2\ninst✝ : PartialOrder α\nF : CFilter α σ\nE : σ ≃ τ\na : τ\nf✝ : σ → α\npt✝ : σ\ninf✝ : σ → σ → σ\ninf_le_left✝ : ∀ (a b : σ), f✝ (inf✝ a b) ≤ f✝ a\ninf_le_right✝ : ∀ (a b : σ), f✝ (inf✝ a b) ≤ f✝ b\n⊢ f (ofEquiv E { f := f✝, pt := pt✝, inf := inf✝, inf_le_left := inf_le_left✝, inf_le_right := inf_le_right✝ }) a =\n f { f := f✝, pt := pt✝, inf := inf✝, inf_le_left := inf_le_left✝, inf_le_right := inf_le_right✝ } (↑E.symm a)",
"state_before": "α : Type u_3\nβ : Type ?u.5122\nσ : Type u_1\nτ : Type u_2\ninst✝ : PartialOrder α\nF✝ : CFilter α σ\nE : σ ≃ τ\nF : CFilter α σ\na : τ\n⊢ f (ofEquiv E F) a = f F (↑E.symm a)",
"tactic": "cases F"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_3\nβ : Type ?u.5122\nσ : Type u_1\nτ : Type u_2\ninst✝ : PartialOrder α\nF : CFilter α σ\nE : σ ≃ τ\na : τ\nf✝ : σ → α\npt✝ : σ\ninf✝ : σ → σ → σ\ninf_le_left✝ : ∀ (a b : σ), f✝ (inf✝ a b) ≤ f✝ a\ninf_le_right✝ : ∀ (a b : σ), f✝ (inf✝ a b) ≤ f✝ b\n⊢ f (ofEquiv E { f := f✝, pt := pt✝, inf := inf✝, inf_le_left := inf_le_left✝, inf_le_right := inf_le_right✝ }) a =\n f { f := f✝, pt := pt✝, inf := inf✝, inf_le_left := inf_le_left✝, inf_le_right := inf_le_right✝ } (↑E.symm a)",
"tactic": "rfl"
}
] |
[
78,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.inr_isoPushout_hom
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nh : IsPushout f g inl inr\ninst✝ : HasPushout f g\n⊢ inr ≫ (isoPushout h).hom = pushout.inr",
"tactic": "simp [← Iso.eq_comp_inv]"
}
] |
[
440,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
439,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
nhds_translation_mul_inv
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns : Set α\nx✝ : α\nx : G\n⊢ 𝓝 (1 * x⁻¹⁻¹) = 𝓝 x",
"tactic": "simp"
}
] |
[
815,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
814,
1
] |
Mathlib/Data/List/Basic.lean
|
List.foldr_eta
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.232631\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\n⊢ ∀ (l : List α), foldr cons [] l = l",
"tactic": "simp only [foldr_self_append, append_nil, forall_const]"
}
] |
[
2460,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2459,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.map_add
|
[] |
[
194,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/GroupTheory/GroupAction/Prod.lean
|
Prod.pow_mk
|
[] |
[
107,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
sub_sq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.128156\nM : Type ?u.128159\ninst✝ : CommRing R\na b : R\n⊢ (a - b) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2",
"tactic": "rw [sub_eq_add_neg, add_sq, neg_sq, mul_neg, ← sub_eq_add_neg]"
}
] |
[
284,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
Module.End.map_generalizedEigenrange_le
|
[
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nμ : K\nn : ℕ\n⊢ Submodule.map f (LinearMap.range ((f - ↑(algebraMap K (End K V)) μ) ^ n)) =\n LinearMap.range (f * (f - ↑(algebraMap K (End K V)) μ) ^ n)",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nμ : K\nn : ℕ\n⊢ Submodule.map f (generalizedEigenrange f μ n) = LinearMap.range (f * (f - ↑(algebraMap K (End K V)) μ) ^ n)",
"tactic": "rw [generalizedEigenrange]"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nμ : K\nn : ℕ\n⊢ Submodule.map f (LinearMap.range ((f - ↑(algebraMap K (End K V)) μ) ^ n)) =\n LinearMap.range (f * (f - ↑(algebraMap K (End K V)) μ) ^ n)",
"tactic": "exact (LinearMap.range_comp _ _).symm"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nμ : K\nn : ℕ\n⊢ LinearMap.range (f * (f - ↑(algebraMap K (End K V)) μ) ^ n) =\n LinearMap.range ((f - ↑(algebraMap K (End K V)) μ) ^ n * f)",
"tactic": "rw [Algebra.mul_sub_algebraMap_pow_commutes]"
}
] |
[
458,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.mapAlgEquiv_trans
|
[
{
"state_after": "case h.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_4\na a' a₁ a₂ : R\ne✝ : ℕ\ns : σ →₀ ℕ\ninst✝⁶ : CommSemiring R\nA₁ : Type u_1\nA₂ : Type u_2\nA₃ : Type u_3\ninst✝⁵ : CommSemiring A₁\ninst✝⁴ : CommSemiring A₂\ninst✝³ : CommSemiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nf : A₂ ≃ₐ[R] A₃\na✝ : MvPolynomial σ A₁\nm✝ : σ →₀ ℕ\n⊢ coeff m✝ (↑(AlgEquiv.trans (mapAlgEquiv σ e) (mapAlgEquiv σ f)) a✝) =\n coeff m✝ (↑(mapAlgEquiv σ (AlgEquiv.trans e f)) a✝)",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_4\na a' a₁ a₂ : R\ne✝ : ℕ\ns : σ →₀ ℕ\ninst✝⁶ : CommSemiring R\nA₁ : Type u_1\nA₂ : Type u_2\nA₃ : Type u_3\ninst✝⁵ : CommSemiring A₁\ninst✝⁴ : CommSemiring A₂\ninst✝³ : CommSemiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nf : A₂ ≃ₐ[R] A₃\n⊢ AlgEquiv.trans (mapAlgEquiv σ e) (mapAlgEquiv σ f) = mapAlgEquiv σ (AlgEquiv.trans e f)",
"tactic": "ext"
},
{
"state_after": "case h.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_4\na a' a₁ a₂ : R\ne✝ : ℕ\ns : σ →₀ ℕ\ninst✝⁶ : CommSemiring R\nA₁ : Type u_1\nA₂ : Type u_2\nA₃ : Type u_3\ninst✝⁵ : CommSemiring A₁\ninst✝⁴ : CommSemiring A₂\ninst✝³ : CommSemiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nf : A₂ ≃ₐ[R] A₃\na✝ : MvPolynomial σ A₁\nm✝ : σ →₀ ℕ\n⊢ coeff m✝ (↑(map (RingHom.comp ↑f ↑e)) a✝) = coeff m✝ (↑(map ↑(AlgEquiv.trans e f)) a✝)",
"state_before": "case h.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_4\na a' a₁ a₂ : R\ne✝ : ℕ\ns : σ →₀ ℕ\ninst✝⁶ : CommSemiring R\nA₁ : Type u_1\nA₂ : Type u_2\nA₃ : Type u_3\ninst✝⁵ : CommSemiring A₁\ninst✝⁴ : CommSemiring A₂\ninst✝³ : CommSemiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nf : A₂ ≃ₐ[R] A₃\na✝ : MvPolynomial σ A₁\nm✝ : σ →₀ ℕ\n⊢ coeff m✝ (↑(AlgEquiv.trans (mapAlgEquiv σ e) (mapAlgEquiv σ f)) a✝) =\n coeff m✝ (↑(mapAlgEquiv σ (AlgEquiv.trans e f)) a✝)",
"tactic": "simp only [AlgEquiv.trans_apply, mapAlgEquiv_apply, map_map]"
},
{
"state_after": "no goals",
"state_before": "case h.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_4\na a' a₁ a₂ : R\ne✝ : ℕ\ns : σ →₀ ℕ\ninst✝⁶ : CommSemiring R\nA₁ : Type u_1\nA₂ : Type u_2\nA₃ : Type u_3\ninst✝⁵ : CommSemiring A₁\ninst✝⁴ : CommSemiring A₂\ninst✝³ : CommSemiring A₃\ninst✝² : Algebra R A₁\ninst✝¹ : Algebra R A₂\ninst✝ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nf : A₂ ≃ₐ[R] A₃\na✝ : MvPolynomial σ A₁\nm✝ : σ →₀ ℕ\n⊢ coeff m✝ (↑(map (RingHom.comp ↑f ↑e)) a✝) = coeff m✝ (↑(map ↑(AlgEquiv.trans e f)) a✝)",
"tactic": "rfl"
}
] |
[
151,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Data/QPF/Multivariate/Basic.lean
|
MvQPF.liftpPreservation_iff_uniform
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\n⊢ LiftPPreservation ↔ IsUniform",
"tactic": "rw [← suppPreservation_iff_liftpPreservation, suppPreservation_iff_isUniform]"
}
] |
[
286,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.memℒp_def
|
[] |
[
119,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Computability/DFA.lean
|
DFA.mem_accepts
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nσ : Type v\nM : DFA α σ\nx : List α\n⊢ x ∈ accepts M ↔ evalFrom M M.start x ∈ M.accept",
"tactic": "rfl"
}
] |
[
101,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sSup_Prop_eq
|
[] |
[
1722,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1721,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.C_eq_algebraMap
|
[] |
[
1086,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1085,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.le_iff_cmp
|
[
{
"state_after": "α : Type ?u.484320\nm n : Num\n⊢ Ordering.swap (cmp m n) = Ordering.lt ↔ cmp m n = Ordering.gt",
"state_before": "α : Type ?u.484320\nm n : Num\n⊢ cmp n m = Ordering.lt ↔ cmp m n = Ordering.gt",
"tactic": "rw [← cmp_swap]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.484320\nm n : Num\n⊢ Ordering.swap (cmp m n) = Ordering.lt ↔ cmp m n = Ordering.gt",
"tactic": "cases cmp m n <;> exact by decide"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.484320\nm n : Num\n⊢ Ordering.swap Ordering.gt = Ordering.lt ↔ Ordering.gt = Ordering.gt",
"tactic": "decide"
}
] |
[
885,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
884,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.fderiv_integral
|
[] |
[
745,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
differentiableWithinAt_fst
|
[] |
[
197,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.bliminf_eq
|
[] |
[
394,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
MeasurableSpace.measurableSet_injective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11608\nγ : Type ?u.11611\nδ : Type ?u.11614\nδ' : Type ?u.11617\nι : Sort ?u.11620\ns t u : Set α\nMeasurableSet'✝¹ : Set α → Prop\nmeasurableSet_empty✝¹ : MeasurableSet'✝¹ ∅\nmeasurableSet_compl✝¹ : ∀ (s : Set α), MeasurableSet'✝¹ s → MeasurableSet'✝¹ (sᶜ)\nmeasurableSet_iUnion✝¹ : ∀ (f : ℕ → Set α), (∀ (i : ℕ), MeasurableSet'✝¹ (f i)) → MeasurableSet'✝¹ (⋃ (i : ℕ), f i)\nMeasurableSet'✝ : Set α → Prop\nmeasurableSet_empty✝ : MeasurableSet'✝ ∅\nmeasurableSet_compl✝ : ∀ (s : Set α), MeasurableSet'✝ s → MeasurableSet'✝ (sᶜ)\nmeasurableSet_iUnion✝ : ∀ (f : ℕ → Set α), (∀ (i : ℕ), MeasurableSet'✝ (f i)) → MeasurableSet'✝ (⋃ (i : ℕ), f i)\nx✝ : MeasurableSet = MeasurableSet\n⊢ { MeasurableSet' := MeasurableSet'✝¹, measurableSet_empty := measurableSet_empty✝¹,\n measurableSet_compl := measurableSet_compl✝¹, measurableSet_iUnion := measurableSet_iUnion✝¹ } =\n { MeasurableSet' := MeasurableSet'✝, measurableSet_empty := measurableSet_empty✝,\n measurableSet_compl := measurableSet_compl✝, measurableSet_iUnion := measurableSet_iUnion✝ }",
"tactic": "congr"
}
] |
[
257,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/LinearAlgebra/AdicCompletion.lean
|
IsPrecomplete.prec
|
[] |
[
72,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisCoinsertion.l_injective
|
[] |
[
785,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
self_sub_toIcoDiv_zsmul
|
[] |
[
116,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Std/Data/List/Lemmas.lean
|
List.length_erase_of_mem
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\n⊢ length (eraseP (fun b => decide (a = b)) l) = pred (length l)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\n⊢ length (List.erase l a) = pred (length l)",
"tactic": "rw [erase_eq_eraseP]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\n⊢ length (eraseP (fun b => decide (a = b)) l) = pred (length l)",
"tactic": "exact length_eraseP_of_mem h (decide_eq_true rfl)"
}
] |
[
1063,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1061,
9
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_eq_nhds
|
[] |
[
221,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
9
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.lt_sup_iff
|
[
{
"state_after": "case mp\nF : Type ?u.212488\nα : Type u_1\nβ : Type ?u.212494\nγ : Type ?u.212497\nι : Type u_2\nκ : Type ?u.212503\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\n⊢ a < sup s f → ∃ b, b ∈ s ∧ a < f b\n\ncase mpr\nF : Type ?u.212488\nα : Type u_1\nβ : Type ?u.212494\nγ : Type ?u.212497\nι : Type u_2\nκ : Type ?u.212503\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\n⊢ (∃ b, b ∈ s ∧ a < f b) → a < sup s f",
"state_before": "F : Type ?u.212488\nα : Type u_1\nβ : Type ?u.212494\nγ : Type ?u.212497\nι : Type u_2\nκ : Type ?u.212503\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\n⊢ a < sup s f ↔ ∃ b, b ∈ s ∧ a < f b",
"tactic": "apply Iff.intro"
},
{
"state_after": "case mp.cons\nF : Type ?u.212488\nα : Type u_1\nβ : Type ?u.212494\nγ : Type ?u.212497\nι : Type u_2\nκ : Type ?u.212503\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nc : ι\nt : Finset ι\nhc : ¬c ∈ t\nih : a < sup t f → ∃ b, b ∈ t ∧ a < f b\n⊢ a < f c ∨ a < sup t f → ∃ b, b ∈ cons c t hc ∧ a < f b",
"state_before": "case mp.cons\nF : Type ?u.212488\nα : Type u_1\nβ : Type ?u.212494\nγ : Type ?u.212497\nι : Type u_2\nκ : Type ?u.212503\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nc : ι\nt : Finset ι\nhc : ¬c ∈ t\nih : a < sup t f → ∃ b, b ∈ t ∧ a < f b\n⊢ a < sup (cons c t hc) f → ∃ b, b ∈ cons c t hc ∧ a < f b",
"tactic": "rw [sup_cons, lt_sup_iff]"
},
{
"state_after": "no goals",
"state_before": "case mp.cons\nF : Type ?u.212488\nα : Type u_1\nβ : Type ?u.212494\nγ : Type ?u.212497\nι : Type u_2\nκ : Type ?u.212503\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\nc : ι\nt : Finset ι\nhc : ¬c ∈ t\nih : a < sup t f → ∃ b, b ∈ t ∧ a < f b\n⊢ a < f c ∨ a < sup t f → ∃ b, b ∈ cons c t hc ∧ a < f b",
"tactic": "exact fun\n| Or.inl h => ⟨c, mem_cons.2 (Or.inl rfl), h⟩\n| Or.inr h => let ⟨b, hb, hlt⟩ := ih h; ⟨b, mem_cons.2 (Or.inr hb), hlt⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr\nF : Type ?u.212488\nα : Type u_1\nβ : Type ?u.212494\nγ : Type ?u.212497\nι : Type u_2\nκ : Type ?u.212503\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\n⊢ (∃ b, b ∈ s ∧ a < f b) → a < sup s f",
"tactic": "exact fun ⟨b, hb, hlt⟩ => lt_of_lt_of_le hlt (le_sup hb)"
}
] |
[
701,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
11
] |
Mathlib/Topology/OmegaCompletePartialOrder.lean
|
Scott.isωSup_iff_isLUB
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Preorder α\nc : Chain α\nx : α\n⊢ IsωSup c x ↔ IsLUB (range ↑c) x",
"tactic": "simp [IsωSup, IsLUB, IsLeast, upperBounds, lowerBounds]"
}
] |
[
45,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
LinearIsometryEquiv.toAffineIsometryEquiv_toAffineIsometry
|
[] |
[
431,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
contDiffOn_zero
|
[
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\n⊢ ContDiffOn 𝕜 0 f s",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\n⊢ ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s",
"tactic": "refine' ⟨fun H => H.continuousOn, fun H => _⟩"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\n⊢ ContDiffOn 𝕜 0 f s",
"tactic": "intro x hx m hm"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u",
"tactic": "have : (m : ℕ∞) = 0 := le_antisymm hm bot_le"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn 0 f p u",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u",
"tactic": "rw [this]"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ HasFTaylorSeriesUpToOn 0 f (ftaylorSeriesWithin 𝕜 f s) (insert x s)",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ ∃ u, u ∈ 𝓝[insert x s] x ∧ ∃ p, HasFTaylorSeriesUpToOn 0 f p u",
"tactic": "refine' ⟨insert x s, self_mem_nhdsWithin, ftaylorSeriesWithin 𝕜 f s, _⟩"
},
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ ContinuousOn f (insert x s) ∧\n ∀ (x_1 : E), x_1 ∈ insert x s → ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin 𝕜 f s x_1 0) = f x_1",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ HasFTaylorSeriesUpToOn 0 f (ftaylorSeriesWithin 𝕜 f s) (insert x s)",
"tactic": "rw [hasFTaylorSeriesUpToOn_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ ContinuousOn f (insert x s) ∧\n ∀ (x_1 : E), x_1 ∈ insert x s → ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin 𝕜 f s x_1 0) = f x_1",
"tactic": "exact ⟨by rwa [insert_eq_of_mem hx], fun x _ => by simp [ftaylorSeriesWithin]⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\n⊢ ContinuousOn f (insert x s)",
"tactic": "rwa [insert_eq_of_mem hx]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝² x₀ : E\nc : F\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nH : ContinuousOn f s\nx✝¹ : E\nhx : x✝¹ ∈ s\nm : ℕ\nhm : ↑m ≤ 0\nthis : ↑m = 0\nx : E\nx✝ : x ∈ insert x✝¹ s\n⊢ ContinuousMultilinearMap.uncurry0 (ftaylorSeriesWithin 𝕜 f s x 0) = f x",
"tactic": "simp [ftaylorSeriesWithin]"
}
] |
[
966,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
959,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
exists_maximal_orthonormal
|
[
{
"state_after": "case refine_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nthis :\n ∃ m, m ∈ {b | Orthonormal 𝕜 Subtype.val} ∧ s ⊆ m ∧ ∀ (a : Set E), a ∈ {b | Orthonormal 𝕜 Subtype.val} → m ⊆ a → a = m\n⊢ ∃ w _hw, Orthonormal 𝕜 Subtype.val ∧ ∀ (u : Set E), u ⊇ w → Orthonormal 𝕜 Subtype.val → u = w\n\ncase refine_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\n⊢ ∀ (c : Set (Set E)),\n c ⊆ {b | Orthonormal 𝕜 Subtype.val} →\n IsChain (fun x x_1 => x ⊆ x_1) c →\n Set.Nonempty c → ∃ ub, ub ∈ {b | Orthonormal 𝕜 Subtype.val} ∧ ∀ (s : Set E), s ∈ c → s ⊆ ub",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\n⊢ ∃ w _hw, Orthonormal 𝕜 Subtype.val ∧ ∀ (u : Set E), u ⊇ w → Orthonormal 𝕜 Subtype.val → u = w",
"tactic": "have := zorn_subset_nonempty { b | Orthonormal 𝕜 (Subtype.val : b → E) } ?_ _ hs"
},
{
"state_after": "case refine_2.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nb : Set E\nbi : b ∈ {b | Orthonormal 𝕜 Subtype.val}\nsb : s ⊆ b\nh : ∀ (a : Set E), a ∈ {b | Orthonormal 𝕜 Subtype.val} → b ⊆ a → a = b\n⊢ ∃ w _hw, Orthonormal 𝕜 Subtype.val ∧ ∀ (u : Set E), u ⊇ w → Orthonormal 𝕜 Subtype.val → u = w\n\ncase refine_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\n⊢ ∀ (c : Set (Set E)),\n c ⊆ {b | Orthonormal 𝕜 Subtype.val} →\n IsChain (fun x x_1 => x ⊆ x_1) c →\n Set.Nonempty c → ∃ ub, ub ∈ {b | Orthonormal 𝕜 Subtype.val} ∧ ∀ (s : Set E), s ∈ c → s ⊆ ub",
"state_before": "case refine_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nthis :\n ∃ m, m ∈ {b | Orthonormal 𝕜 Subtype.val} ∧ s ⊆ m ∧ ∀ (a : Set E), a ∈ {b | Orthonormal 𝕜 Subtype.val} → m ⊆ a → a = m\n⊢ ∃ w _hw, Orthonormal 𝕜 Subtype.val ∧ ∀ (u : Set E), u ⊇ w → Orthonormal 𝕜 Subtype.val → u = w\n\ncase refine_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\n⊢ ∀ (c : Set (Set E)),\n c ⊆ {b | Orthonormal 𝕜 Subtype.val} →\n IsChain (fun x x_1 => x ⊆ x_1) c →\n Set.Nonempty c → ∃ ub, ub ∈ {b | Orthonormal 𝕜 Subtype.val} ∧ ∀ (s : Set E), s ∈ c → s ⊆ ub",
"tactic": "obtain ⟨b, bi, sb, h⟩ := this"
},
{
"state_after": "case refine_2.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nb : Set E\nbi : b ∈ {b | Orthonormal 𝕜 Subtype.val}\nsb : s ⊆ b\nh : ∀ (a : Set E), a ∈ {b | Orthonormal 𝕜 Subtype.val} → b ⊆ a → a = b\n⊢ ∀ (u : Set E), u ⊇ b → Orthonormal 𝕜 Subtype.val → u = b",
"state_before": "case refine_2.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nb : Set E\nbi : b ∈ {b | Orthonormal 𝕜 Subtype.val}\nsb : s ⊆ b\nh : ∀ (a : Set E), a ∈ {b | Orthonormal 𝕜 Subtype.val} → b ⊆ a → a = b\n⊢ ∃ w _hw, Orthonormal 𝕜 Subtype.val ∧ ∀ (u : Set E), u ⊇ w → Orthonormal 𝕜 Subtype.val → u = w",
"tactic": "refine' ⟨b, sb, bi, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_2.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nb : Set E\nbi : b ∈ {b | Orthonormal 𝕜 Subtype.val}\nsb : s ⊆ b\nh : ∀ (a : Set E), a ∈ {b | Orthonormal 𝕜 Subtype.val} → b ⊆ a → a = b\n⊢ ∀ (u : Set E), u ⊇ b → Orthonormal 𝕜 Subtype.val → u = b",
"tactic": "exact fun u hus hu => h u hu hus"
},
{
"state_after": "case refine_1.refine'_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nc : Set (Set E)\nhc : c ⊆ {b | Orthonormal 𝕜 Subtype.val}\ncc : IsChain (fun x x_1 => x ⊆ x_1) c\n_c0 : Set.Nonempty c\n⊢ ⋃₀ c ∈ {b | Orthonormal 𝕜 Subtype.val}\n\ncase refine_1.refine'_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nc : Set (Set E)\nhc : c ⊆ {b | Orthonormal 𝕜 Subtype.val}\ncc : IsChain (fun x x_1 => x ⊆ x_1) c\n_c0 : Set.Nonempty c\n⊢ ∀ (s : Set E), s ∈ c → s ⊆ ⋃₀ c",
"state_before": "case refine_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\n⊢ ∀ (c : Set (Set E)),\n c ⊆ {b | Orthonormal 𝕜 Subtype.val} →\n IsChain (fun x x_1 => x ⊆ x_1) c →\n Set.Nonempty c → ∃ ub, ub ∈ {b | Orthonormal 𝕜 Subtype.val} ∧ ∀ (s : Set E), s ∈ c → s ⊆ ub",
"tactic": "refine' fun c hc cc _c0 => ⟨⋃₀ c, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_1.refine'_1\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nc : Set (Set E)\nhc : c ⊆ {b | Orthonormal 𝕜 Subtype.val}\ncc : IsChain (fun x x_1 => x ⊆ x_1) c\n_c0 : Set.Nonempty c\n⊢ ⋃₀ c ∈ {b | Orthonormal 𝕜 Subtype.val}",
"tactic": "exact orthonormal_sUnion_of_directed cc.directedOn fun x xc => hc xc"
},
{
"state_after": "no goals",
"state_before": "case refine_1.refine'_2\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.2163317\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type ?u.2163366\ndec_ι : DecidableEq ι\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nc : Set (Set E)\nhc : c ⊆ {b | Orthonormal 𝕜 Subtype.val}\ncc : IsChain (fun x x_1 => x ⊆ x_1) c\n_c0 : Set.Nonempty c\n⊢ ∀ (s : Set E), s ∈ c → s ⊆ ⋃₀ c",
"tactic": "exact fun _ => Set.subset_sUnion_of_mem"
}
] |
[
968,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
959,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
InfHom.ext
|
[] |
[
543,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Std/Data/List/Lemmas.lean
|
List.infix_refl
|
[] |
[
1582,
70
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1582,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_equivMapOfInjective_apply
|
[] |
[
635,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
ApproximatesLinearOn.closedBall_subset_target
|
[] |
[
522,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.adj_comm
|
[] |
[
106,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.take_concat_get
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\ni : Nat\nh : i < length l\n⊢ l = concat (take i l) l[i] ++ drop (i + 1) l",
"tactic": "rw [concat_eq_append, append_assoc, singleton_append, get_drop_eq_drop, take_append_drop]"
}
] |
[
154,
94
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
151,
1
] |
Mathlib/Data/MvPolynomial/Monad.lean
|
MvPolynomial.eval₂Hom_C_eq_bind₁
|
[] |
[
115,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_fst_neBot
|
[] |
[
2385,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2383,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.monoidHom_ext
|
[] |
[
536,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/UV.lean
|
UV.disjoint_of_mem_compression_of_not_mem
|
[
{
"state_after": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nh : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\nha : ¬a ∈ s\n⊢ Disjoint v a",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nh : a ∈ 𝓒 u v s\nha : ¬a ∈ s\n⊢ Disjoint v a",
"tactic": "rw [mem_compression] at h"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : ¬a ∈ s\nh : a ∈ s ∧ compress u v a ∈ s\n⊢ Disjoint v a\n\ncase inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nhba : compress u v b = a\n⊢ Disjoint v a",
"state_before": "α : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nh : a ∈ s ∧ compress u v a ∈ s ∨ ¬a ∈ s ∧ ∃ b, b ∈ s ∧ compress u v b = a\nha : ¬a ∈ s\n⊢ Disjoint v a",
"tactic": "obtain h | ⟨-, b, hb, hba⟩ := h"
},
{
"state_after": "case inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nhba : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \\ v else b) = a\n⊢ Disjoint v a",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nhba : compress u v b = a\n⊢ Disjoint v a",
"tactic": "unfold compress at hba"
},
{
"state_after": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ Disjoint v a\n\ncase inr.intro.intro.intro.inr\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : ¬(Disjoint u b ∧ v ≤ b)\nhba : b = a\n⊢ Disjoint v a",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nhba : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \\ v else b) = a\n⊢ Disjoint v a",
"tactic": "split_ifs at hba"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b : α\nha : ¬a ∈ s\nh : a ∈ s ∧ compress u v a ∈ s\n⊢ Disjoint v a",
"tactic": "cases ha h.1"
},
{
"state_after": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ Disjoint v ((b ⊔ u) \\ v)",
"state_before": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ Disjoint v a",
"tactic": "rw [← hba]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.inl\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ Disjoint v ((b ⊔ u) \\ v)",
"tactic": "exact disjoint_sdiff_self_right"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.inr\nα : Type u_1\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : DecidableRel Disjoint\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : Finset α\nu v a b✝ : α\nha : ¬a ∈ s\nb : α\nhb : b ∈ s\nh✝ : ¬(Disjoint u b ∧ v ≤ b)\nhba : b = a\n⊢ Disjoint v a",
"tactic": "cases ne_of_mem_of_not_mem hb ha hba"
}
] |
[
244,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.modEq_zero_iff_dvd
|
[
{
"state_after": "no goals",
"state_before": "m n a b c d : ℕ\n⊢ a ≡ 0 [MOD n] ↔ n ∣ a",
"tactic": "rw [ModEq, zero_mod, dvd_iff_mod_eq_zero]"
}
] |
[
79,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Finset/Sort.lean
|
Finset.orderEmbOfFin_unique
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\n⊢ Set.range f = Set.range ↑(orderEmbOfFin s h)",
"state_before": "α : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\n⊢ f = ↑(orderEmbOfFin s h)",
"tactic": "apply Fin.strictMono_unique hmono (s.orderEmbOfFin h).strictMono"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\n⊢ image f univ = s",
"state_before": "α : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\n⊢ Set.range f = Set.range ↑(orderEmbOfFin s h)",
"tactic": "rw [range_orderEmbOfFin, ← Set.image_univ, ← coe_univ, ← coe_image, coe_inj]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\nx : α\nhx : x ∈ image f univ\n⊢ x ∈ s\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\n⊢ card s ≤ card (image f univ)",
"state_before": "α : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\n⊢ image f univ = s",
"tactic": "refine' eq_of_subset_of_card_le (fun x hx => _) _"
},
{
"state_after": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\nx : Fin k\nleft✝ : x ∈ univ\nhx : f x ∈ image f univ\n⊢ f x ∈ s",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\nx : α\nhx : x ∈ image f univ\n⊢ x ∈ s",
"tactic": "rcases mem_image.1 hx with ⟨x, _, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nα : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\nx : Fin k\nleft✝ : x ∈ univ\nhx : f x ∈ image f univ\n⊢ f x ∈ s",
"tactic": "exact hfs x"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.51338\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : card s = k\nf : Fin k → α\nhfs : ∀ (x : Fin k), f x ∈ s\nhmono : StrictMono f\n⊢ card s ≤ card (image f univ)",
"tactic": "rw [h, card_image_of_injective _ hmono.injective, card_univ, Fintype.card_fin]"
}
] |
[
223,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/LinearAlgebra/Vandermonde.lean
|
Matrix.vandermonde_transpose_mul_vandermonde
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nv : Fin n → R\ni j : Fin n\n⊢ ((vandermonde v)ᵀ ⬝ vandermonde v) i j = ∑ k : Fin n, v k ^ (↑i + ↑j)",
"tactic": "simp only [vandermonde_apply, Matrix.mul_apply, Matrix.transpose_apply, pow_add]"
}
] |
[
75,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
HasStrictFDerivAt.prodMap
|
[] |
[
342,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
11
] |
Mathlib/Data/Setoid/Partition.lean
|
IndexedPartition.mem_iff_index_eq
|
[] |
[
383,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/Logic/Embedding/Basic.lean
|
Function.Embedding.coe_subtype
|
[] |
[
236,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Analysis/NormedSpace/CompactOperator.lean
|
isCompactOperator_iff_image_closedBall_subset_compact
|
[] |
[
181,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.one_imK
|
[] |
[
202,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
9
] |
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