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/Deprecated/Subgroup.lean
|
IsGroupHom.mem_ker
|
[] |
[
339,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt
|
[
{
"state_after": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"state_before": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"tactic": "set f' := fun y => continuousMultilinearCurryFin1 ℝ E F (p y 1)"
},
{
"state_after": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"state_before": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"tactic": "have hder : ∀ y ∈ s, HasFDerivWithinAt f (f' y) s y := fun y hy =>\n (hf.hasFDerivWithinAt le_rfl (subset_insert x s hy)).mono (subset_insert x s)"
},
{
"state_after": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"state_before": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"tactic": "have hcont : ContinuousWithinAt f' s x :=\n (continuousMultilinearCurryFin1 ℝ E F).continuousAt.comp_continuousWithinAt\n ((hf.cont _ le_rfl _ (mem_insert _ _)).mono (subset_insert x s))"
},
{
"state_after": "case hK\n𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\n⊢ ‖f' x‖₊ < K\n\n𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nhK : ‖f' x‖₊ < K\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"state_before": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"tactic": "replace hK : ‖f' x‖₊ < K"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nhK : ‖f' x‖₊ < K\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ LipschitzOnWith K f t",
"tactic": "exact\n hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt\n (eventually_nhdsWithin_iff.2 <| eventually_of_forall hder) hcont K hK"
},
{
"state_after": "no goals",
"state_before": "case hK\n𝕜 : Type ?u.2904330\ninst✝¹⁹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁸ : NormedAddCommGroup D\ninst✝¹⁷ : NormedSpace 𝕜 D\nE✝ : Type uE\ninst✝¹⁶ : NormedAddCommGroup E✝\ninst✝¹⁵ : NormedSpace 𝕜 E✝\nF✝ : Type uF\ninst✝¹⁴ : NormedAddCommGroup F✝\ninst✝¹³ : NormedSpace 𝕜 F✝\nG : Type uG\ninst✝¹² : NormedAddCommGroup G\ninst✝¹¹ : NormedSpace 𝕜 G\nX : Type ?u.2904751\ninst✝¹⁰ : NormedAddCommGroup X\ninst✝⁹ : 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✝\n𝕂 : Type ?u.2908229\ninst✝⁸ : IsROrC 𝕂\nE' : Type ?u.2908235\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : NormedSpace 𝕂 E'\nF' : Type ?u.2908343\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace 𝕂 F'\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\np : E → FormalMultilinearSeries ℝ E F\ns : Set E\nx : E\nhf : HasFTaylorSeriesUpToOn 1 f p (insert x s)\nhs : Convex ℝ s\nK : ℝ≥0\nhK : ‖p x 1‖₊ < K\nf' : E → E →L[ℝ] F := fun y => ↑(continuousMultilinearCurryFin1 ℝ E F) (p y 1)\nhder : ∀ (y : E), y ∈ s → HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\n⊢ ‖f' x‖₊ < K",
"tactic": "simpa only [LinearIsometryEquiv.nnnorm_map]"
}
] |
[
2006,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1992,
1
] |
Mathlib/Data/Finset/Sigma.lean
|
Finset.inf_sigma
|
[] |
[
112,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
ChainComplex.prev
|
[] |
[
119,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Data/Finset/Fold.lean
|
Finset.fold_insert_idem
|
[
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19798\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nhi : IsIdempotent β op\nh : a ∈ s\n⊢ fold op b f (insert a s) = op (f a) (fold op b f s)\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19798\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nhi : IsIdempotent β op\nh : ¬a ∈ s\n⊢ fold op b f (insert a s) = op (f a) (fold op b f s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.19798\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nhi : IsIdempotent β op\n⊢ fold op b f (insert a s) = op (f a) (fold op b f s)",
"tactic": "by_cases h : a ∈ s"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19798\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nhi : IsIdempotent β op\nh : a ∈ s\n⊢ fold op b f (insert a (insert a (erase s a))) = op (f a) (fold op b f (insert a (erase s a)))",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19798\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nhi : IsIdempotent β op\nh : a ∈ s\n⊢ fold op b f (insert a s) = op (f a) (fold op b f s)",
"tactic": "rw [← insert_erase h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19798\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nhi : IsIdempotent β op\nh : a ∈ s\n⊢ fold op b f (insert a (insert a (erase s a))) = op (f a) (fold op b f (insert a (erase s a)))",
"tactic": "simp [← ha.assoc, hi.idempotent]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.19798\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nhi : IsIdempotent β op\nh : ¬a ∈ s\n⊢ fold op b f (insert a s) = op (f a) (fold op b f s)",
"tactic": "apply fold_insert h"
}
] |
[
128,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/CategoryTheory/Sites/Sheaf.lean
|
CategoryTheory.Presheaf.isSeparated_iff_subsingleton
|
[] |
[
187,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.leRecOn_injective
|
[
{
"state_after": "case refl\nm✝ n✝ k : ℕ\nC : ℕ → Sort u\nn m : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\n⊢ Function.Injective (leRecOn (_ : Nat.le n n) fun {k} => next)\n\ncase step\nm✝¹ n✝ k : ℕ\nC : ℕ → Sort u\nn m✝ : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nm : ℕ\nhnm : Nat.le n m\nih : Function.Injective (leRecOn hnm fun {k} => next)\n⊢ Function.Injective (leRecOn (_ : Nat.le n (succ m)) fun {k} => next)",
"state_before": "m✝ n✝ k : ℕ\nC : ℕ → Sort u\nn m : ℕ\nhnm : n ≤ m\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\n⊢ Function.Injective (leRecOn hnm fun {k} => next)",
"tactic": "induction' hnm with m hnm ih"
},
{
"state_after": "case step\nm✝¹ n✝ k : ℕ\nC : ℕ → Sort u\nn m✝ : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nm : ℕ\nhnm : Nat.le n m\nih : Function.Injective (leRecOn hnm fun {k} => next)\nx y : C n\nH : leRecOn (_ : Nat.le n (succ m)) (fun {k} => next) x = leRecOn (_ : Nat.le n (succ m)) (fun {k} => next) y\n⊢ x = y",
"state_before": "case step\nm✝¹ n✝ k : ℕ\nC : ℕ → Sort u\nn m✝ : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nm : ℕ\nhnm : Nat.le n m\nih : Function.Injective (leRecOn hnm fun {k} => next)\n⊢ Function.Injective (leRecOn (_ : Nat.le n (succ m)) fun {k} => next)",
"tactic": "intro x y H"
},
{
"state_after": "case step\nm✝¹ n✝ k : ℕ\nC : ℕ → Sort u\nn m✝ : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nm : ℕ\nhnm : Nat.le n m\nih : Function.Injective (leRecOn hnm fun {k} => next)\nx y : C n\nH : next (leRecOn hnm (fun {k} => next) x) = next (leRecOn hnm (fun {k} => next) y)\n⊢ x = y",
"state_before": "case step\nm✝¹ n✝ k : ℕ\nC : ℕ → Sort u\nn m✝ : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nm : ℕ\nhnm : Nat.le n m\nih : Function.Injective (leRecOn hnm fun {k} => next)\nx y : C n\nH : leRecOn (_ : Nat.le n (succ m)) (fun {k} => next) x = leRecOn (_ : Nat.le n (succ m)) (fun {k} => next) y\n⊢ x = y",
"tactic": "rw [leRecOn_succ hnm, leRecOn_succ hnm] at H"
},
{
"state_after": "no goals",
"state_before": "case step\nm✝¹ n✝ k : ℕ\nC : ℕ → Sort u\nn m✝ : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nm : ℕ\nhnm : Nat.le n m\nih : Function.Injective (leRecOn hnm fun {k} => next)\nx y : C n\nH : next (leRecOn hnm (fun {k} => next) x) = next (leRecOn hnm (fun {k} => next) y)\n⊢ x = y",
"tactic": "exact ih (Hnext _ H)"
},
{
"state_after": "case refl\nm✝ n✝ k : ℕ\nC : ℕ → Sort u\nn m : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nx y : C n\nH : leRecOn (_ : Nat.le n n) (fun {k} => next) x = leRecOn (_ : Nat.le n n) (fun {k} => next) y\n⊢ x = y",
"state_before": "case refl\nm✝ n✝ k : ℕ\nC : ℕ → Sort u\nn m : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\n⊢ Function.Injective (leRecOn (_ : Nat.le n n) fun {k} => next)",
"tactic": "intro x y H"
},
{
"state_after": "no goals",
"state_before": "case refl\nm✝ n✝ k : ℕ\nC : ℕ → Sort u\nn m : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Function.Injective next\nx y : C n\nH : leRecOn (_ : Nat.le n n) (fun {k} => next) x = leRecOn (_ : Nat.le n n) (fun {k} => next) y\n⊢ x = y",
"tactic": "rwa [leRecOn_self, leRecOn_self] at H"
}
] |
[
442,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.sdiff_sdiff_self_left
|
[] |
[
2326,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2325,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.mem_span_insert
|
[] |
[
168,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
coe_basisOfLinearIndependentOfCardEqFinrank
|
[] |
[
1200,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1197,
1
] |
Mathlib/Logic/Nontrivial.lean
|
not_subsingleton
|
[] |
[
127,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.choose_property
|
[] |
[
3716,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3715,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.coe_add
|
[] |
[
304,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsLeast.bddBelow
|
[] |
[
345,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.derivBFamily_eq_derivFamily
|
[] |
[
361,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.assoc_aux_1
|
[] |
[
828,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
824,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.surjOn_logb'
|
[
{
"state_after": "b x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ x ∈ logb b '' Iio 0",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ SurjOn (logb b) (Iio 0) univ",
"tactic": "intro x _"
},
{
"state_after": "b x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ -b ^ x ∈ Iio 0 ∧ logb b (-b ^ x) = x",
"state_before": "b x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ x ∈ logb b '' Iio 0",
"tactic": "use -b ^ x"
},
{
"state_after": "case left\nb x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ -b ^ x ∈ Iio 0\n\ncase right\nb x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ logb b (-b ^ x) = x",
"state_before": "b x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ -b ^ x ∈ Iio 0 ∧ logb b (-b ^ x) = x",
"tactic": "constructor"
},
{
"state_after": "case left\nb x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ 0 < b ^ x",
"state_before": "case left\nb x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ -b ^ x ∈ Iio 0",
"tactic": "simp only [Right.neg_neg_iff, Set.mem_Iio]"
},
{
"state_after": "no goals",
"state_before": "case left\nb x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ 0 < b ^ x",
"tactic": "apply rpow_pos_of_pos b_pos"
},
{
"state_after": "no goals",
"state_before": "case right\nb x✝ y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\nx : ℝ\na✝ : x ∈ univ\n⊢ logb b (-b ^ x) = x",
"tactic": "rw [logb_neg_eq_logb, logb_rpow b_pos b_ne_one]"
}
] |
[
133,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
SeminormFamily.basisSets_intersect
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU : U ∈ basisSets p\nhV : V ∈ basisSets p\n⊢ ∃ z, z ∈ basisSets p ∧ z ⊆ U ∩ V",
"tactic": "classical\n rcases p.basisSets_iff.mp hU with ⟨s, r₁, hr₁, hU⟩\n rcases p.basisSets_iff.mp hV with ⟨t, r₂, hr₂, hV⟩\n use ((s ∪ t).sup p).ball 0 (min r₁ r₂)\n refine' ⟨p.basisSets_mem (s ∪ t) (lt_min_iff.mpr ⟨hr₁, hr₂⟩), _⟩\n rw [hU, hV, ball_finset_sup_eq_iInter _ _ _ (lt_min_iff.mpr ⟨hr₁, hr₂⟩),\n ball_finset_sup_eq_iInter _ _ _ hr₁, ball_finset_sup_eq_iInter _ _ _ hr₂]\n exact\n Set.subset_inter\n (Set.iInter₂_mono' fun i hi =>\n ⟨i, Finset.subset_union_left _ _ hi, ball_mono <| min_le_left _ _⟩)\n (Set.iInter₂_mono' fun i hi =>\n ⟨i, Finset.subset_union_right _ _ hi, ball_mono <| min_le_right _ _⟩)"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\n⊢ ∃ z, z ∈ basisSets p ∧ z ⊆ U ∩ V",
"state_before": "𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU : U ∈ basisSets p\nhV : V ∈ basisSets p\n⊢ ∃ z, z ∈ basisSets p ∧ z ⊆ U ∩ V",
"tactic": "rcases p.basisSets_iff.mp hU with ⟨s, r₁, hr₁, hU⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ ∃ z, z ∈ basisSets p ∧ z ⊆ U ∩ V",
"state_before": "case intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\n⊢ ∃ z, z ∈ basisSets p ∧ z ⊆ U ∩ V",
"tactic": "rcases p.basisSets_iff.mp hV with ⟨t, r₂, hr₂, hV⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ ball (Finset.sup (s ∪ t) p) 0 (min r₁ r₂) ∈ basisSets p ∧ ball (Finset.sup (s ∪ t) p) 0 (min r₁ r₂) ⊆ U ∩ V",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ ∃ z, z ∈ basisSets p ∧ z ⊆ U ∩ V",
"tactic": "use ((s ∪ t).sup p).ball 0 (min r₁ r₂)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ ball (Finset.sup (s ∪ t) p) 0 (min r₁ r₂) ⊆ U ∩ V",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ ball (Finset.sup (s ∪ t) p) 0 (min r₁ r₂) ∈ basisSets p ∧ ball (Finset.sup (s ∪ t) p) 0 (min r₁ r₂) ⊆ U ∩ V",
"tactic": "refine' ⟨p.basisSets_mem (s ∪ t) (lt_min_iff.mpr ⟨hr₁, hr₂⟩), _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ (⋂ (i : ι) (_ : i ∈ s ∪ t), ball (p i) 0 (min r₁ r₂)) ⊆\n (⋂ (i : ι) (_ : i ∈ s), ball (p i) 0 r₁) ∩ ⋂ (i : ι) (_ : i ∈ t), ball (p i) 0 r₂",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ ball (Finset.sup (s ∪ t) p) 0 (min r₁ r₂) ⊆ U ∩ V",
"tactic": "rw [hU, hV, ball_finset_sup_eq_iInter _ _ _ (lt_min_iff.mpr ⟨hr₁, hr₂⟩),\n ball_finset_sup_eq_iInter _ _ _ hr₁, ball_finset_sup_eq_iInter _ _ _ hr₂]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_2\n𝕜₂ : Type ?u.22222\n𝕝 : Type ?u.22225\n𝕝₂ : Type ?u.22228\nE : Type u_1\nF : Type ?u.22234\nG : Type ?u.22237\nι : Type u_3\nι' : Type ?u.22243\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nU V : Set E\nhU✝ : U ∈ basisSets p\nhV✝ : V ∈ basisSets p\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = ball (Finset.sup s p) 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = ball (Finset.sup t p) 0 r₂\n⊢ (⋂ (i : ι) (_ : i ∈ s ∪ t), ball (p i) 0 (min r₁ r₂)) ⊆\n (⋂ (i : ι) (_ : i ∈ s), ball (p i) 0 r₁) ∩ ⋂ (i : ι) (_ : i ∈ t), ball (p i) 0 r₂",
"tactic": "exact\n Set.subset_inter\n (Set.iInter₂_mono' fun i hi =>\n ⟨i, Finset.subset_union_left _ _ hi, ball_mono <| min_le_left _ _⟩)\n (Set.iInter₂_mono' fun i hi =>\n ⟨i, Finset.subset_union_right _ _ hi, ball_mono <| min_le_right _ _⟩)"
}
] |
[
115,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/GroupTheory/GroupAction/Embedding.lean
|
Function.Embedding.smul_def
|
[] |
[
37,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMaxOn.max
|
[] |
[
627,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
Submodule.finrank_lt
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\ns : Submodule K V\nh : s < ⊤\n⊢ finrank K { x // x ∈ s } < finrank K { x // x ∈ s } + finrank K (V ⧸ s)",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\ns : Submodule K V\nh : s < ⊤\n⊢ finrank K { x // x ∈ s } < finrank K V",
"tactic": "rw [← s.finrank_quotient_add_finrank, add_comm]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\ns : Submodule K V\nh : s < ⊤\n⊢ finrank K { x // x ∈ s } < finrank K { x // x ∈ s } + finrank K (V ⧸ s)",
"tactic": "exact Nat.lt_add_of_pos_right (finrank_pos_iff.mpr (Quotient.nontrivial_of_lt_top _ h))"
}
] |
[
772,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.copy_eq
|
[] |
[
161,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.IsSubordinate.disjUnion
|
[
{
"state_after": "case refine'_1\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₁.boxes\n⊢ ↑Box.Icc J ⊆\n closedBall (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J) ↑(r (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J))\n\ncase refine'_2\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₂.boxes\n⊢ ↑Box.Icc J ⊆\n closedBall (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J) ↑(r (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J))",
"state_before": "ι : Type u_1\nI J : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\n⊢ IsSubordinate (TaggedPrepartition.disjUnion π₁ π₂ h) r",
"tactic": "refine' fun J hJ => (Finset.mem_union.1 hJ).elim (fun hJ => _) fun hJ => _"
},
{
"state_after": "case refine'_1\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₁.boxes\n⊢ ↑Box.Icc J ⊆ closedBall (tag π₁ J) ↑(r (tag π₁ J))",
"state_before": "case refine'_1\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₁.boxes\n⊢ ↑Box.Icc J ⊆\n closedBall (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J) ↑(r (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J))",
"tactic": "rw [disjUnion_tag_of_mem_left _ hJ]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₁.boxes\n⊢ ↑Box.Icc J ⊆ closedBall (tag π₁ J) ↑(r (tag π₁ J))",
"tactic": "exact h₁ _ hJ"
},
{
"state_after": "case refine'_2\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₂.boxes\n⊢ ↑Box.Icc J ⊆ closedBall (tag π₂ J) ↑(r (tag π₂ J))",
"state_before": "case refine'_2\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₂.boxes\n⊢ ↑Box.Icc J ⊆\n closedBall (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J) ↑(r (tag (TaggedPrepartition.disjUnion π₁ π₂ h) J))",
"tactic": "rw [disjUnion_tag_of_mem_right _ hJ]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type u_1\nI J✝ : Box ι\nπ π₁ π₂ : TaggedPrepartition I\nx : ι → ℝ\nr r₁ r₂ : (ι → ℝ) → ↑(Ioi 0)\ninst✝ : Fintype ι\nh₁ : IsSubordinate π₁ r\nh₂ : IsSubordinate π₂ r\nh : Disjoint (iUnion π₁) (iUnion π₂)\nJ : Box ι\nhJ✝ : J ∈ TaggedPrepartition.disjUnion π₁ π₂ h\nhJ : J ∈ π₂.boxes\n⊢ ↑Box.Icc J ⊆ closedBall (tag π₂ J) ↑(r (tag π₂ J))",
"tactic": "exact h₂ _ hJ"
}
] |
[
389,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
src/lean/Init/Data/AC.lean
|
Lean.Data.AC.Context.eval_norm
|
[
{
"state_after": "α : Sort u_1\nctx : Context α\ne : Expr\n⊢ evalList α ctx\n (if ContextInformation.isIdem ctx = true then\n mergeIdem\n (if ContextInformation.isComm ctx = true then sort (removeNeutrals ctx (Expr.toList e))\n else removeNeutrals ctx (Expr.toList e))\n else\n if ContextInformation.isComm ctx = true then sort (removeNeutrals ctx (Expr.toList e))\n else removeNeutrals ctx (Expr.toList e)) =\n eval α ctx e",
"state_before": "α : Sort u_1\nctx : Context α\ne : Expr\n⊢ evalList α ctx (norm ctx e) = eval α ctx e",
"tactic": "simp [norm]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nctx : Context α\ne : Expr\n⊢ evalList α ctx\n (if ContextInformation.isIdem ctx = true then\n mergeIdem\n (if ContextInformation.isComm ctx = true then sort (removeNeutrals ctx (Expr.toList e))\n else removeNeutrals ctx (Expr.toList e))\n else\n if ContextInformation.isComm ctx = true then sort (removeNeutrals ctx (Expr.toList e))\n else removeNeutrals ctx (Expr.toList e)) =\n eval α ctx e",
"tactic": "cases h₁ : ContextInformation.isIdem ctx <;> cases h₂ : ContextInformation.isComm ctx <;>\nsimp_all [evalList_removeNeutrals, eval_toList, toList_nonEmpty, evalList_mergeIdem, evalList_sort]"
}
] |
[
315,
102
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
312,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.finset_sup_apply_lt
|
[
{
"state_after": "case intro\nR : Type ?u.528629\nR' : Type ?u.528632\n𝕜 : Type u_1\n𝕜₂ : Type ?u.528638\n𝕜₃ : Type ?u.528641\n𝕝 : Type ?u.528644\nE : Type u_2\nE₂ : Type ?u.528650\nE₃ : Type ?u.528653\nF : Type ?u.528656\nG : Type ?u.528659\nι : Type u_3\ninst✝¹⁸ : SeminormedRing 𝕜\ninst✝¹⁷ : SeminormedRing 𝕜₂\ninst✝¹⁶ : SeminormedRing 𝕜₃\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹⁵ : RingHomIsometric σ₁₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁴ : RingHomIsometric σ₂₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹³ : RingHomIsometric σ₁₃\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : AddCommGroup E₂\ninst✝¹⁰ : AddCommGroup E₃\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : AddCommGroup G\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜₂ E₂\ninst✝⁵ : Module 𝕜₃ E₃\ninst✝⁴ : Module 𝕜 F\ninst✝³ : Module 𝕜 G\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\na : ℝ≥0\nha : 0 < ↑a\nh : ∀ (i : ι), i ∈ s → ↑(p i) x < ↑a\n⊢ ↑(Finset.sup s p) x < ↑a",
"state_before": "R : Type ?u.528629\nR' : Type ?u.528632\n𝕜 : Type u_1\n𝕜₂ : Type ?u.528638\n𝕜₃ : Type ?u.528641\n𝕝 : Type ?u.528644\nE : Type u_2\nE₂ : Type ?u.528650\nE₃ : Type ?u.528653\nF : Type ?u.528656\nG : Type ?u.528659\nι : Type u_3\ninst✝¹⁸ : SeminormedRing 𝕜\ninst✝¹⁷ : SeminormedRing 𝕜₂\ninst✝¹⁶ : SeminormedRing 𝕜₃\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹⁵ : RingHomIsometric σ₁₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁴ : RingHomIsometric σ₂₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹³ : RingHomIsometric σ₁₃\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : AddCommGroup E₂\ninst✝¹⁰ : AddCommGroup E₃\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : AddCommGroup G\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜₂ E₂\ninst✝⁵ : Module 𝕜₃ E₃\ninst✝⁴ : Module 𝕜 F\ninst✝³ : Module 𝕜 G\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\na : ℝ\nha : 0 < a\nh : ∀ (i : ι), i ∈ s → ↑(p i) x < a\n⊢ ↑(Finset.sup s p) x < a",
"tactic": "lift a to ℝ≥0 using ha.le"
},
{
"state_after": "case intro\nR : Type ?u.528629\nR' : Type ?u.528632\n𝕜 : Type u_1\n𝕜₂ : Type ?u.528638\n𝕜₃ : Type ?u.528641\n𝕝 : Type ?u.528644\nE : Type u_2\nE₂ : Type ?u.528650\nE₃ : Type ?u.528653\nF : Type ?u.528656\nG : Type ?u.528659\nι : Type u_3\ninst✝¹⁸ : SeminormedRing 𝕜\ninst✝¹⁷ : SeminormedRing 𝕜₂\ninst✝¹⁶ : SeminormedRing 𝕜₃\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹⁵ : RingHomIsometric σ₁₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁴ : RingHomIsometric σ₂₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹³ : RingHomIsometric σ₁₃\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : AddCommGroup E₂\ninst✝¹⁰ : AddCommGroup E₃\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : AddCommGroup G\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜₂ E₂\ninst✝⁵ : Module 𝕜₃ E₃\ninst✝⁴ : Module 𝕜 F\ninst✝³ : Module 𝕜 G\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\na : ℝ≥0\nha : 0 < ↑a\nh : ∀ (i : ι), i ∈ s → ↑(p i) x < ↑a\n⊢ ∀ (b : ι), b ∈ s → { val := ↑(p b) x, property := (_ : 0 ≤ ↑(p b) x) } < a\n\ncase intro\nR : Type ?u.528629\nR' : Type ?u.528632\n𝕜 : Type u_1\n𝕜₂ : Type ?u.528638\n𝕜₃ : Type ?u.528641\n𝕝 : Type ?u.528644\nE : Type u_2\nE₂ : Type ?u.528650\nE₃ : Type ?u.528653\nF : Type ?u.528656\nG : Type ?u.528659\nι : Type u_3\ninst✝¹⁸ : SeminormedRing 𝕜\ninst✝¹⁷ : SeminormedRing 𝕜₂\ninst✝¹⁶ : SeminormedRing 𝕜₃\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹⁵ : RingHomIsometric σ₁₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁴ : RingHomIsometric σ₂₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹³ : RingHomIsometric σ₁₃\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : AddCommGroup E₂\ninst✝¹⁰ : AddCommGroup E₃\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : AddCommGroup G\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜₂ E₂\ninst✝⁵ : Module 𝕜₃ E₃\ninst✝⁴ : Module 𝕜 F\ninst✝³ : Module 𝕜 G\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\na : ℝ≥0\nha : 0 < ↑a\nh : ∀ (i : ι), i ∈ s → ↑(p i) x < ↑a\n⊢ ⊥ < a",
"state_before": "case intro\nR : Type ?u.528629\nR' : Type ?u.528632\n𝕜 : Type u_1\n𝕜₂ : Type ?u.528638\n𝕜₃ : Type ?u.528641\n𝕝 : Type ?u.528644\nE : Type u_2\nE₂ : Type ?u.528650\nE₃ : Type ?u.528653\nF : Type ?u.528656\nG : Type ?u.528659\nι : Type u_3\ninst✝¹⁸ : SeminormedRing 𝕜\ninst✝¹⁷ : SeminormedRing 𝕜₂\ninst✝¹⁶ : SeminormedRing 𝕜₃\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹⁵ : RingHomIsometric σ₁₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁴ : RingHomIsometric σ₂₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹³ : RingHomIsometric σ₁₃\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : AddCommGroup E₂\ninst✝¹⁰ : AddCommGroup E₃\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : AddCommGroup G\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜₂ E₂\ninst✝⁵ : Module 𝕜₃ E₃\ninst✝⁴ : Module 𝕜 F\ninst✝³ : Module 𝕜 G\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\na : ℝ≥0\nha : 0 < ↑a\nh : ∀ (i : ι), i ∈ s → ↑(p i) x < ↑a\n⊢ ↑(Finset.sup s p) x < ↑a",
"tactic": "rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type ?u.528629\nR' : Type ?u.528632\n𝕜 : Type u_1\n𝕜₂ : Type ?u.528638\n𝕜₃ : Type ?u.528641\n𝕝 : Type ?u.528644\nE : Type u_2\nE₂ : Type ?u.528650\nE₃ : Type ?u.528653\nF : Type ?u.528656\nG : Type ?u.528659\nι : Type u_3\ninst✝¹⁸ : SeminormedRing 𝕜\ninst✝¹⁷ : SeminormedRing 𝕜₂\ninst✝¹⁶ : SeminormedRing 𝕜₃\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹⁵ : RingHomIsometric σ₁₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁴ : RingHomIsometric σ₂₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹³ : RingHomIsometric σ₁₃\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : AddCommGroup E₂\ninst✝¹⁰ : AddCommGroup E₃\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : AddCommGroup G\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜₂ E₂\ninst✝⁵ : Module 𝕜₃ E₃\ninst✝⁴ : Module 𝕜 F\ninst✝³ : Module 𝕜 G\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\na : ℝ≥0\nha : 0 < ↑a\nh : ∀ (i : ι), i ∈ s → ↑(p i) x < ↑a\n⊢ ∀ (b : ι), b ∈ s → { val := ↑(p b) x, property := (_ : 0 ≤ ↑(p b) x) } < a",
"tactic": "exact h"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type ?u.528629\nR' : Type ?u.528632\n𝕜 : Type u_1\n𝕜₂ : Type ?u.528638\n𝕜₃ : Type ?u.528641\n𝕝 : Type ?u.528644\nE : Type u_2\nE₂ : Type ?u.528650\nE₃ : Type ?u.528653\nF : Type ?u.528656\nG : Type ?u.528659\nι : Type u_3\ninst✝¹⁸ : SeminormedRing 𝕜\ninst✝¹⁷ : SeminormedRing 𝕜₂\ninst✝¹⁶ : SeminormedRing 𝕜₃\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹⁵ : RingHomIsometric σ₁₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ninst✝¹⁴ : RingHomIsometric σ₂₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹³ : RingHomIsometric σ₁₃\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : AddCommGroup E₂\ninst✝¹⁰ : AddCommGroup E₃\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : AddCommGroup G\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜₂ E₂\ninst✝⁵ : Module 𝕜₃ E₃\ninst✝⁴ : Module 𝕜 F\ninst✝³ : Module 𝕜 G\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\np : ι → Seminorm 𝕜 E\ns : Finset ι\nx : E\na : ℝ≥0\nha : 0 < ↑a\nh : ∀ (i : ι), i ∈ s → ↑(p i) x < ↑a\n⊢ ⊥ < a",
"tactic": "exact NNReal.coe_pos.mpr ha"
}
] |
[
422,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.Nonempty.of_disjSups_right
|
[
{
"state_after": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ (∃ x a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = x) → ∃ x, x ∈ t",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ Finset.Nonempty (s ○ t) → Finset.Nonempty t",
"tactic": "simp_rw [Finset.Nonempty, mem_disjSups]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ (∃ x a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = x) → ∃ x, x ∈ t",
"tactic": "exact fun ⟨_, _, _, b, hb, _⟩ => ⟨b, hb⟩"
}
] |
[
486,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.deleteEdges_sdiff_eq_of_le
|
[
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.161200\n𝕜 : Type ?u.161203\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\nH : SimpleGraph V\nh : H ≤ G\nv w : V\n⊢ Adj (deleteEdges G (edgeSet G \\ edgeSet H)) v w ↔ Adj H v w",
"state_before": "ι : Sort ?u.161200\n𝕜 : Type ?u.161203\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nH : SimpleGraph V\nh : H ≤ G\n⊢ deleteEdges G (edgeSet G \\ edgeSet H) = H",
"tactic": "ext (v w)"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a\nι : Sort ?u.161200\n𝕜 : Type ?u.161203\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\nH : SimpleGraph V\nh : H ≤ G\nv w : V\n⊢ Adj (deleteEdges G (edgeSet G \\ edgeSet H)) v w ↔ Adj H v w",
"tactic": "constructor <;> simp (config := { contextual := true }) [@h v w]"
}
] |
[
1170,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1167,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.const_def
|
[] |
[
40,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Order/Compare.lean
|
Ordering.Compares.eq_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6425\ninst✝ : Preorder α\na b : α\nh✝ : Compares eq a b\nh : eq = lt\n⊢ a < b",
"tactic": "injection h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6425\ninst✝ : Preorder α\na b : α\nh✝ : Compares gt a b\nh : gt = lt\n⊢ a < b",
"tactic": "injection h"
}
] |
[
88,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Fintype.exists_card_fiber_le_of_card_le_mul
|
[] |
[
454,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/CategoryTheory/Preadditive/Projective.lean
|
CategoryTheory.Adjunction.map_projective
|
[
{
"state_after": "case intro\nC : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE✝ X✝ : D\nf : F.obj P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nf' : P ⟶ G.obj E✝\nhf' : f' ≫ G.map g = adj.unit.app P ≫ G.map f\n⊢ ∃ f', f' ≫ g = f",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE✝ X✝ : D\nf : F.obj P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\n⊢ ∃ f', f' ≫ g = f",
"tactic": "rcases hP.factors (adj.unit.app P ≫ G.map f) (G.map g) with ⟨f', hf'⟩"
},
{
"state_after": "case intro\nC : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE✝ X✝ : D\nf : F.obj P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nf' : P ⟶ G.obj E✝\nhf' : f' ≫ G.map g = adj.unit.app P ≫ G.map f\n⊢ (F.map f' ≫ adj.counit.app E✝) ≫ g = f",
"state_before": "case intro\nC : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE✝ X✝ : D\nf : F.obj P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nf' : P ⟶ G.obj E✝\nhf' : f' ≫ G.map g = adj.unit.app P ≫ G.map f\n⊢ ∃ f', f' ≫ g = f",
"tactic": "use F.map f' ≫ adj.counit.app _"
},
{
"state_after": "case intro\nC : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE✝ X✝ : D\nf : F.obj P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nf' : P ⟶ G.obj E✝\nhf' : f' ≫ G.map g = adj.unit.app P ≫ G.map f\n⊢ F.map (adj.unit.app P ≫ G.map f) ≫ adj.counit.app X✝ = f",
"state_before": "case intro\nC : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE✝ X✝ : D\nf : F.obj P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nf' : P ⟶ G.obj E✝\nhf' : f' ≫ G.map g = adj.unit.app P ≫ G.map f\n⊢ (F.map f' ≫ adj.counit.app E✝) ≫ g = f",
"tactic": "rw [Category.assoc, ← Adjunction.counit_naturality, ← Category.assoc, ← F.map_comp, hf']"
},
{
"state_after": "no goals",
"state_before": "case intro\nC : Type u\ninst✝² : Category C\nD : Type u'\ninst✝¹ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝ : Functor.PreservesEpimorphisms G\nP : C\nhP : Projective P\nE✝ X✝ : D\nf : F.obj P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nf' : P ⟶ G.obj E✝\nhf' : f' ≫ G.map g = adj.unit.app P ≫ G.map f\n⊢ F.map (adj.unit.app P ≫ G.map f) ≫ adj.counit.app X✝ = f",
"tactic": "simp"
}
] |
[
214,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.map₂'_mk''
|
[] |
[
762,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.length_pos_of_prod_ne_one
|
[
{
"state_after": "case nil\nι : Type ?u.58476\nα : Type ?u.58479\nM : Type u_1\nN : Type ?u.58485\nP : Type ?u.58488\nM₀ : Type ?u.58491\nG : Type ?u.58494\nR : Type ?u.58497\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na : M\nh : prod [] ≠ 1\n⊢ 0 < length []\n\ncase cons\nι : Type ?u.58476\nα : Type ?u.58479\nM : Type u_1\nN : Type ?u.58485\nP : Type ?u.58488\nM₀ : Type ?u.58491\nG : Type ?u.58494\nR : Type ?u.58497\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na head✝ : M\ntail✝ : List M\nh : prod (head✝ :: tail✝) ≠ 1\n⊢ 0 < length (head✝ :: tail✝)",
"state_before": "ι : Type ?u.58476\nα : Type ?u.58479\nM : Type u_1\nN : Type ?u.58485\nP : Type ?u.58488\nM₀ : Type ?u.58491\nG : Type ?u.58494\nR : Type ?u.58497\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na : M\nL : List M\nh : prod L ≠ 1\n⊢ 0 < length L",
"tactic": "cases L"
},
{
"state_after": "case nil\nι : Type ?u.58476\nα : Type ?u.58479\nM : Type u_1\nN : Type ?u.58485\nP : Type ?u.58488\nM₀ : Type ?u.58491\nG : Type ?u.58494\nR : Type ?u.58497\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na : M\nh : ¬0 < length []\n⊢ ¬prod [] ≠ 1",
"state_before": "case nil\nι : Type ?u.58476\nα : Type ?u.58479\nM : Type u_1\nN : Type ?u.58485\nP : Type ?u.58488\nM₀ : Type ?u.58491\nG : Type ?u.58494\nR : Type ?u.58497\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na : M\nh : prod [] ≠ 1\n⊢ 0 < length []",
"tactic": "contrapose h"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.58476\nα : Type ?u.58479\nM : Type u_1\nN : Type ?u.58485\nP : Type ?u.58488\nM₀ : Type ?u.58491\nG : Type ?u.58494\nR : Type ?u.58497\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na : M\nh : ¬0 < length []\n⊢ ¬prod [] ≠ 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.58476\nα : Type ?u.58479\nM : Type u_1\nN : Type ?u.58485\nP : Type ?u.58488\nM₀ : Type ?u.58491\nG : Type ?u.58494\nR : Type ?u.58497\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl l₁ l₂ : List M\na head✝ : M\ntail✝ : List M\nh : prod (head✝ :: tail✝) ≠ 1\n⊢ 0 < length (head✝ :: tail✝)",
"tactic": "simp"
}
] |
[
193,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Algebra/Category/ModuleCat/Basic.lean
|
ModuleCat.coe_comp
|
[] |
[
241,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Init/Logic.lean
|
and_true_iff
|
[] |
[
142,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
AlgHom.Finite.finiteType
|
[] |
[
287,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.bodd_add
|
[
{
"state_after": "case negSucc.negSucc\nm n : ℕ\n⊢ Nat.bodd (Nat.succ (m + n) + 1) = Nat.bodd (m + 1 + (n + 1))",
"state_before": "m n : ℤ\n⊢ bodd (m + n) = xor (bodd m) (bodd n)",
"tactic": "cases' m with m m <;>\ncases' n with n n <;>\nsimp only [ofNat_eq_coe, ofNat_add_negSucc, negSucc_add_ofNat,\n negSucc_add_negSucc, bodd_subNatNat] <;>\nsimp only [negSucc_coe, bodd_neg, bodd_coe, ←Nat.bodd_add, Bool.xor_comm, ←Nat.cast_add]"
},
{
"state_after": "no goals",
"state_before": "case negSucc.negSucc\nm n : ℕ\n⊢ Nat.bodd (Nat.succ (m + n) + 1) = Nat.bodd (m + 1 + (n + 1))",
"tactic": "rw [←Nat.succ_add, add_assoc]"
}
] |
[
83,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.isPell_norm
|
[
{
"state_after": "d x y : ℤ\n⊢ x * x - y * (d * y) = 1 ↔ x * x + -(y * (d * y)) = 1",
"state_before": "d x y : ℤ\n⊢ IsPell { re := x, im := y } ↔ { re := x, im := y } * star { re := x, im := y } = 1",
"tactic": "simp [Zsqrtd.ext, IsPell, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "d x y : ℤ\n⊢ x * x - y * (d * y) = 1 ↔ x * x + -(y * (d * y)) = 1",
"tactic": "ring_nf"
}
] |
[
73,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
algebraMap.coe_mul
|
[] |
[
164,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.FinStronglyMeasurable.aefinStronglyMeasurable
|
[] |
[
999,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.MapsTo.iterate_restrict
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nn : ℕ\nx : ↑s\n⊢ (restrict f s s h^[n]) x = restrict (f^[n]) s s (_ : MapsTo (f^[n]) s s) x",
"state_before": "α : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nn : ℕ\n⊢ restrict f s s h^[n] = restrict (f^[n]) s s (_ : MapsTo (f^[n]) s s)",
"tactic": "funext x"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nn : ℕ\nx : ↑s\n⊢ ↑((restrict f s s h^[n]) x) = (f^[n]) ↑x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nn : ℕ\nx : ↑s\n⊢ (restrict f s s h^[n]) x = restrict (f^[n]) s s (_ : MapsTo (f^[n]) s s) x",
"tactic": "rw [Subtype.ext_iff, MapsTo.val_restrict_apply]"
},
{
"state_after": "case h.zero\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nx✝ x : ↑s\n⊢ ↑((restrict f s s h^[Nat.zero]) x) = (f^[Nat.zero]) ↑x\n\ncase h.succ\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nx✝ : ↑s\nn : ℕ\nihn : ∀ (x : ↑s), ↑((restrict f s s h^[n]) x) = (f^[n]) ↑x\nx : ↑s\n⊢ ↑((restrict f s s h^[Nat.succ n]) x) = (f^[Nat.succ n]) ↑x",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nn : ℕ\nx : ↑s\n⊢ ↑((restrict f s s h^[n]) x) = (f^[n]) ↑x",
"tactic": "induction' n with n ihn generalizing x"
},
{
"state_after": "no goals",
"state_before": "case h.zero\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nx✝ x : ↑s\n⊢ ↑((restrict f s s h^[Nat.zero]) x) = (f^[Nat.zero]) ↑x",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.succ\nα : Type u_1\nβ : Type ?u.18768\nγ : Type ?u.18771\nι : Sort ?u.18774\nπ : α → Type ?u.18779\ns✝ s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf✝ f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nf : α → α\ns : Set α\nh : MapsTo f s s\nx✝ : ↑s\nn : ℕ\nihn : ∀ (x : ↑s), ↑((restrict f s s h^[n]) x) = (f^[n]) ↑x\nx : ↑s\n⊢ ↑((restrict f s s h^[Nat.succ n]) x) = (f^[Nat.succ n]) ↑x",
"tactic": "simp [Nat.iterate, ihn]"
}
] |
[
441,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
1
] |
Mathlib/Topology/Homotopy/HomotopyGroup.lean
|
homotopic_genLoopZeroEquiv_symm_iff
|
[
{
"state_after": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx a b : X\n⊢ Homotopic (↑genLoopZeroEquiv.symm a).toContinuousMap (↑genLoopZeroEquiv.symm b).toContinuousMap ↔ Joined a b",
"state_before": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx a b : X\n⊢ GenLoop.Homotopic (↑genLoopZeroEquiv.symm a) (↑genLoopZeroEquiv.symm b) ↔ Joined a b",
"tactic": "rw [GenLoop.Homotopic, Cube.boundary_zero, homotopicRel_empty]"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx a b : X\n⊢ Homotopic (↑genLoopZeroEquiv.symm a).toContinuousMap (↑genLoopZeroEquiv.symm b).toContinuousMap ↔ Joined a b",
"tactic": "exact homotopic_const_iff"
}
] |
[
235,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
Subalgebra.starClosure_eq_adjoin
|
[] |
[
488,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.neg_of_int
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.1055224\n⊢ ↑(-0) = -↑0",
"tactic": "rw [Int.cast_neg, Int.cast_zero]"
}
] |
[
1532,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1529,
1
] |
Mathlib/Logic/Function/Iterate.lean
|
Function.iterate_one
|
[] |
[
79,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Order/Cover.lean
|
densely_ordered_iff_forall_not_covby
|
[] |
[
247,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
OrthonormalBasis.coe_toHilbertBasis
|
[] |
[
589,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Limits.biprod.total
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : Preadditive C\nJ : Type\ninst✝¹ : Fintype J\nX Y : C\ninst✝ : HasBinaryBiproduct X Y\n⊢ fst ≫ inl + snd ≫ inr = 𝟙 (X ⊞ Y)",
"tactic": "ext <;> simp [add_comp]"
}
] |
[
451,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
450,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
deriv_mul_const_field'
|
[] |
[
246,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.Integrable.simpleFunc_mul
|
[
{
"state_after": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ Integrable (↑(SimpleFunc.piecewise s hs (SimpleFunc.const β c) (SimpleFunc.const β 0)) * f)",
"state_before": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\n⊢ Integrable (↑g * f)",
"tactic": "refine'\n SimpleFunc.induction (fun c s hs => _)\n (fun g₁ g₂ _ h_int₁ h_int₂ =>\n (h_int₁.add h_int₂).congr (by rw [SimpleFunc.coe_add, add_mul]))\n g"
},
{
"state_after": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ Integrable (Set.indicator s (const β c) * f)",
"state_before": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ Integrable (↑(SimpleFunc.piecewise s hs (SimpleFunc.const β c) (SimpleFunc.const β 0)) * f)",
"tactic": "simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator]"
},
{
"state_after": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nthis : Set.indicator s (const β c) * f = Set.indicator s (c • f)\n⊢ IntegrableOn (c • f) s",
"state_before": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nthis : Set.indicator s (const β c) * f = Set.indicator s (c • f)\n⊢ Integrable (Set.indicator s (const β c) * f)",
"tactic": "rw [this, integrable_indicator_iff hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nthis : Set.indicator s (const β c) * f = Set.indicator s (c • f)\n⊢ IntegrableOn (c • f) s",
"tactic": "exact (hf.smul c).integrableOn"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\ng₁ g₂ : SimpleFunc β ℝ\nx✝ : Disjoint (support ↑g₁) (support ↑g₂)\nh_int₁ : Integrable (↑g₁ * f)\nh_int₂ : Integrable (↑g₂ * f)\n⊢ ↑g₁ * f + ↑g₂ * f =ᵐ[μ] ↑(g₁ + g₂) * f",
"tactic": "rw [SimpleFunc.coe_add, add_mul]"
},
{
"state_after": "case h\nα : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nx : β\n⊢ (Set.indicator s (const β c) * f) x = Set.indicator s (c • f) x",
"state_before": "α : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ Set.indicator s (const β c) * f = Set.indicator s (c • f)",
"tactic": "ext1 x"
},
{
"state_after": "case pos\nα : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nx : β\nhx : x ∈ s\n⊢ (Set.indicator s (const β c) * f) x = Set.indicator s (c • f) x\n\ncase neg\nα : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nx : β\nhx : ¬x ∈ s\n⊢ (Set.indicator s (const β c) * f) x = Set.indicator s (c • f) x",
"state_before": "case h\nα : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nx : β\n⊢ (Set.indicator s (const β c) * f) x = Set.indicator s (c • f) x",
"tactic": "by_cases hx : x ∈ s"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nx : β\nhx : x ∈ s\n⊢ (Set.indicator s (const β c) * f) x = Set.indicator s (c • f) x",
"tactic": "simp only [hx, Pi.mul_apply, Set.indicator_of_mem, Pi.smul_apply, Algebra.id.smul_eq_mul,\n ← Function.const_def]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.5134274\nβ : Type u_1\nE : Type ?u.5134280\nF : Type ?u.5134283\ninst✝¹ : MeasurableSpace α\nι : Type ?u.5134289\ninst✝ : NormedAddCommGroup E\nf : β → ℝ\nm m0 : MeasurableSpace β\nμ : Measure β\ng : SimpleFunc β ℝ\nhf : Integrable f\nc : ℝ\ns : Set β\nhs : MeasurableSet s\nx : β\nhx : ¬x ∈ s\n⊢ (Set.indicator s (const β c) * f) x = Set.indicator s (c • f) x",
"tactic": "simp only [hx, Pi.mul_apply, Set.indicator_of_not_mem, not_false_iff, MulZeroClass.zero_mul]"
}
] |
[
1339,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1323,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.lintegral_tendsto_of_monotone_of_nat
|
[] |
[
366,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
9
] |
Mathlib/Algebra/Group/Commute.lean
|
inv_mul_cancel_comm
|
[] |
[
424,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Algebra/Free.lean
|
FreeSemigroup.length_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nf : α → β\nx✝ : FreeSemigroup α\nx : α\ny : FreeSemigroup α\nhx : length (↑(map f) (of x)) = length (of x)\nhy : length (↑(map f) y) = length y\n⊢ length (↑(map f) (of x * y)) = length (of x * y)",
"tactic": "simp only [map_mul, length_mul, *]"
}
] |
[
575,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.perm_inv_on_of_perm_on_finset
|
[
{
"state_after": "α : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\n⊢ ↑f⁻¹ y ∈ s",
"state_before": "α : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\n⊢ ↑f⁻¹ y ∈ s",
"tactic": "have h0 : ∀ y ∈ s, ∃ (x : _)(hx : x ∈ s), y = (fun i (_ : i ∈ s) => f i) x hx :=\n Finset.surj_on_of_inj_on_of_card_le (fun x hx => (fun i _ => f i) x hx) (fun a ha => h a ha)\n (fun a₁ a₂ ha₁ ha₂ heq => (Equiv.apply_eq_iff_eq f).mp heq) rfl.ge"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\ny2 : α\nhy2 : y2 ∈ s\nheq : y = (fun i x => ↑f i) y2 hy2\n⊢ ↑f⁻¹ y ∈ s",
"state_before": "α : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\n⊢ ↑f⁻¹ y ∈ s",
"tactic": "obtain ⟨y2, hy2, heq⟩ := h0 y hy"
},
{
"state_after": "case h.e'_4\nα : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\ny2 : α\nhy2 : y2 ∈ s\nheq : y = (fun i x => ↑f i) y2 hy2\n⊢ ↑f⁻¹ y = y2",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\ny2 : α\nhy2 : y2 ∈ s\nheq : y = (fun i x => ↑f i) y2 hy2\n⊢ ↑f⁻¹ y ∈ s",
"tactic": "convert hy2"
},
{
"state_after": "case h.e'_4\nα : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\ny2 : α\nhy2 : y2 ∈ s\nheq : y = (fun i x => ↑f i) y2 hy2\n⊢ ↑f⁻¹ ((fun i x => ↑f i) y2 hy2) = y2",
"state_before": "case h.e'_4\nα : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\ny2 : α\nhy2 : y2 ∈ s\nheq : y = (fun i x => ↑f i) y2 hy2\n⊢ ↑f⁻¹ y = y2",
"tactic": "rw [heq]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u\nβ : Type v\ns : Finset α\nf : Perm α\nh : ∀ (x : α), x ∈ s → ↑f x ∈ s\ny : α\nhy : y ∈ s\nh0 : ∀ (y : α), y ∈ s → ∃ x hx, y = (fun i x => ↑f i) x hx\ny2 : α\nhy2 : y2 ∈ s\nheq : y = (fun i x => ↑f i) y2 hy2\n⊢ ↑f⁻¹ ((fun i x => ↑f i) y2 hy2) = y2",
"tactic": "simp only [inv_apply_self]"
}
] |
[
70,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasStrictFDerivAt_exp_zero_of_radius_pos
|
[
{
"state_after": "case h.e'_10\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < FormalMultilinearSeries.radius (expSeries 𝕂 𝔸)\n⊢ 1 = ↑(continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)",
"state_before": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < FormalMultilinearSeries.radius (expSeries 𝕂 𝔸)\n⊢ HasStrictFDerivAt (exp 𝕂) 1 0",
"tactic": "convert (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt"
},
{
"state_after": "case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < FormalMultilinearSeries.radius (expSeries 𝕂 𝔸)\nx : 𝔸\n⊢ ↑1 x = ↑(↑(continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)) x",
"state_before": "case h.e'_10\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < FormalMultilinearSeries.radius (expSeries 𝕂 𝔸)\n⊢ 1 = ↑(continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)",
"tactic": "ext x"
},
{
"state_after": "case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < FormalMultilinearSeries.radius (expSeries 𝕂 𝔸)\nx : 𝔸\n⊢ x = ↑(expSeries 𝕂 𝔸 1) fun x_1 => x",
"state_before": "case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < FormalMultilinearSeries.radius (expSeries 𝕂 𝔸)\nx : 𝔸\n⊢ ↑1 x = ↑(↑(continuousMultilinearCurryFin1 𝕂 𝔸 𝔸) (expSeries 𝕂 𝔸 1)) x",
"tactic": "change x = expSeries 𝕂 𝔸 1 fun _ => x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < FormalMultilinearSeries.radius (expSeries 𝕂 𝔸)\nx : 𝔸\n⊢ x = ↑(expSeries 𝕂 𝔸 1) fun x_1 => x",
"tactic": "simp [expSeries_apply_eq]"
}
] |
[
75,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigOWith_abs_left
|
[] |
[
758,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.updateColumn_subsingleton
|
[
{
"state_after": "case a.h\nl : Type ?u.1221200\nm : Type u_2\nn : Type u_1\no : Type ?u.1221209\nm' : o → Type ?u.1221214\nn' : o → Type ?u.1221219\nR : Type u_3\nS : Type ?u.1221225\nα : Type v\nβ : Type w\nγ : Type ?u.1221232\nM : Matrix m n α\ni✝ : m\nj : n\nb✝ : n → α\nc : m → α\ninst✝ : Subsingleton n\nA : Matrix m n R\ni : n\nb : m → R\nx : m\ny : n\n⊢ updateColumn A i b x y = submatrix (col b) id (Function.const n ()) x y",
"state_before": "l : Type ?u.1221200\nm : Type u_2\nn : Type u_1\no : Type ?u.1221209\nm' : o → Type ?u.1221214\nn' : o → Type ?u.1221219\nR : Type u_3\nS : Type ?u.1221225\nα : Type v\nβ : Type w\nγ : Type ?u.1221232\nM : Matrix m n α\ni✝ : m\nj : n\nb✝ : n → α\nc : m → α\ninst✝ : Subsingleton n\nA : Matrix m n R\ni : n\nb : m → R\n⊢ updateColumn A i b = submatrix (col b) id (Function.const n ())",
"tactic": "ext x y"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.1221200\nm : Type u_2\nn : Type u_1\no : Type ?u.1221209\nm' : o → Type ?u.1221214\nn' : o → Type ?u.1221219\nR : Type u_3\nS : Type ?u.1221225\nα : Type v\nβ : Type w\nγ : Type ?u.1221232\nM : Matrix m n α\ni✝ : m\nj : n\nb✝ : n → α\nc : m → α\ninst✝ : Subsingleton n\nA : Matrix m n R\ni : n\nb : m → R\nx : m\ny : n\n⊢ updateColumn A i b x y = submatrix (col b) id (Function.const n ()) x y",
"tactic": "simp [updateColumn_apply, Subsingleton.elim i y]"
}
] |
[
2791,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2788,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
MvPolynomial.isWeightedHomogeneous_of_total_degree_zero
|
[
{
"state_after": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\n⊢ ↑(weightedDegree' w) d = ⊥",
"state_before": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\n⊢ IsWeightedHomogeneous w p ⊥",
"tactic": "intro d hd"
},
{
"state_after": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\nh : weightedTotalDegree' w p = ↑(weightedTotalDegree w p)\n⊢ ↑(weightedDegree' w) d = ⊥",
"state_before": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\n⊢ ↑(weightedDegree' w) d = ⊥",
"tactic": "have h := weightedTotalDegree_coe w p (MvPolynomial.ne_zero_iff.mpr ⟨d, hd⟩)"
},
{
"state_after": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\nh : (sup (support p) fun s => ↑(↑(weightedDegree' w) s)) = ↑⊥\n⊢ ↑(weightedDegree' w) d = ⊥",
"state_before": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\nh : weightedTotalDegree' w p = ↑(weightedTotalDegree w p)\n⊢ ↑(weightedDegree' w) d = ⊥",
"tactic": "simp only [weightedTotalDegree', hp] at h"
},
{
"state_after": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\nh : (sup (support p) fun s => ↑(↑(weightedDegree' w) s)) = ↑⊥\n⊢ ↑(↑(weightedDegree' w) d) ≤ sup (support p) fun s => ↑(↑(weightedDegree' w) s)",
"state_before": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\nh : (sup (support p) fun s => ↑(↑(weightedDegree' w) s)) = ↑⊥\n⊢ ↑(weightedDegree' w) d = ⊥",
"tactic": "rw [eq_bot_iff, ← WithBot.coe_le_coe, ← h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nM : Type u_1\ninst✝³ : CommSemiring R\nσ : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : weightedTotalDegree w p = ⊥\nd : σ →₀ ℕ\nhd : coeff d p ≠ 0\nh : (sup (support p) fun s => ↑(↑(weightedDegree' w) s)) = ↑⊥\n⊢ ↑(↑(weightedDegree' w) d) ≤ sup (support p) fun s => ↑(↑(weightedDegree' w) s)",
"tactic": "apply Finset.le_sup (mem_support_iff.mpr hd)"
}
] |
[
213,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_eq_compl_iUnion_compl
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.69319\nβ : Type u_1\nγ : Type ?u.69325\nι : Sort u_2\nι' : Sort ?u.69331\nι₂ : Sort ?u.69334\nκ : ι → Sort ?u.69339\nκ₁ : ι → Sort ?u.69344\nκ₂ : ι → Sort ?u.69349\nκ' : ι' → Sort ?u.69354\ns : ι → Set β\n⊢ (⋂ (i : ι), s i) = (⋃ (i : ι), s iᶜ)ᶜ",
"tactic": "simp only [compl_iUnion, compl_compl]"
}
] |
[
518,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearEquiv.prod_apply
|
[] |
[
787,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
786,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.associator_inv_apply
|
[] |
[
255,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.coe_sub
|
[] |
[
582,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
581,
1
] |
Mathlib/Logic/Basic.lean
|
of_not_not
|
[] |
[
251,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
Ring.inverse_star
|
[
{
"state_after": "case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\na : R\nha : IsUnit a\n⊢ inverse (star a) = star (inverse a)\n\ncase neg\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\na : R\nha : ¬IsUnit a\n⊢ inverse (star a) = star (inverse a)",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\na : R\n⊢ inverse (star a) = star (inverse a)",
"tactic": "by_cases ha : IsUnit a"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\na : R\nha : ¬IsUnit a\n⊢ inverse (star a) = star (inverse a)",
"tactic": "rw [Ring.inverse_non_unit _ ha, Ring.inverse_non_unit _ (mt isUnit_star.mp ha), star_zero]"
},
{
"state_after": "case pos.intro\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nu : Rˣ\n⊢ inverse (star ↑u) = star (inverse ↑u)",
"state_before": "case pos\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\na : R\nha : IsUnit a\n⊢ inverse (star a) = star (inverse a)",
"tactic": "obtain ⟨u, rfl⟩ := ha"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : StarRing R\nu : Rˣ\n⊢ inverse (star ↑u) = star (inverse ↑u)",
"tactic": "rw [Ring.inverse_unit, ← Units.coe_star, Ring.inverse_unit, ← Units.coe_star_inv]"
}
] |
[
571,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.Prepartition.iUnion_biUnionTagged
|
[] |
[
152,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Analysis/Convex/StrictConvexSpace.lean
|
not_sameRay_iff_abs_lt_norm_sub
|
[] |
[
232,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
addSalemSpencer_sphere
|
[
{
"state_after": "case inl\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\n⊢ AddSalemSpencer (sphere x 0)\n\ncase inr\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ AddSalemSpencer (sphere x r)",
"state_before": "F : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\nr : ℝ\n⊢ AddSalemSpencer (sphere x r)",
"tactic": "obtain rfl | hr := eq_or_ne r 0"
},
{
"state_after": "case inl\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\n⊢ AddSalemSpencer {x}",
"state_before": "case inl\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\n⊢ AddSalemSpencer (sphere x 0)",
"tactic": "rw [sphere_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\n⊢ AddSalemSpencer {x}",
"tactic": "exact addSalemSpencer_singleton _"
},
{
"state_after": "case h.e'_3\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ sphere x r = frontier (closedBall x r)",
"state_before": "case inr\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ AddSalemSpencer (sphere x r)",
"tactic": "convert addSalemSpencer_frontier isClosed_ball (strictConvex_closedBall ℝ x r)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nF : Type ?u.119922\nα : Type ?u.119925\nβ : Type ?u.119928\n𝕜 : Type ?u.119931\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ sphere x r = frontier (closedBall x r)",
"tactic": "exact (frontier_closedBall _ hr).symm"
}
] |
[
305,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/RingTheory/WittVector/IsPoly.lean
|
WittVector.IsPoly.ext
|
[
{
"state_after": "case mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nhf : IsPoly p f\nhg : IsPoly p g\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x",
"tactic": "obtain ⟨φ, hf⟩ := hf"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x",
"state_before": "case mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nhg : IsPoly p g\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x",
"tactic": "obtain ⟨ψ, hg⟩ := hg"
},
{
"state_after": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\n⊢ f x✝ = g x✝",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x",
"tactic": "intros"
},
{
"state_after": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn : ℕ\n⊢ coeff (f x✝) n = coeff (g x✝) n",
"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\n⊢ f x✝ = g x✝",
"tactic": "ext n"
},
{
"state_after": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn : ℕ\n⊢ ∀ (n : ℕ), ↑(bind₁ φ) (wittPolynomial p ℤ n) = ↑(bind₁ ψ) (wittPolynomial p ℤ n)",
"state_before": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn : ℕ\n⊢ coeff (f x✝) n = coeff (g x✝) n",
"tactic": "rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq p φ ψ]"
},
{
"state_after": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\n⊢ ↑(bind₁ φ) (wittPolynomial p ℤ k) = ↑(bind₁ ψ) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn : ℕ\n⊢ ∀ (n : ℕ), ↑(bind₁ φ) (wittPolynomial p ℤ n) = ↑(bind₁ ψ) (wittPolynomial p ℤ n)",
"tactic": "intro k"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\n⊢ ∀ (x : ℕ → ℤ),\n ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"state_before": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\n⊢ ↑(bind₁ φ) (wittPolynomial p ℤ k) = ↑(bind₁ ψ) (wittPolynomial p ℤ k)",
"tactic": "apply MvPolynomial.funext"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\n⊢ ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\n⊢ ∀ (x : ℕ → ℤ),\n ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"tactic": "intro x"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\n⊢ ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"tactic": "simp only [hom_bind₁]"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh : ↑(ghostComponent k) (f (mk p fun i => { down := x i })) = ↑(ghostComponent k) (g (mk p fun i => { down := x i }))\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x) = ↑(ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"tactic": "specialize h (ULift ℤ) (mk p fun i => ⟨x i⟩) k"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh : ↑(ghostComponent k) (f (mk p fun i => { down := x i })) = ↑(ghostComponent k) (g (mk p fun i => { down := x i }))\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"tactic": "simp only [ghostComponent_apply, aeval_eq_eval₂Hom] at h"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i))\n (wittPolynomial p ℤ k)) =\n ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k))",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"tactic": "apply (ULift.ringEquiv.symm : ℤ ≃+* _).injective"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i))\n (wittPolynomial p ℤ k)) =\n ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k))",
"tactic": "simp only [← RingEquiv.coe_toRingHom, map_eval₂Hom]"
},
{
"state_after": "case h.e'_2\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n\ncase h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k)",
"tactic": "convert h using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n\ncase h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)",
"tactic": "all_goals\n simp only [hf, hg, MvPolynomial.eval, map_eval₂Hom]\n apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl\n ext1\n apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl\n simp only [coeff_mk]; rfl"
},
{
"state_after": "case h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (eval₂Hom (RingHom.id ℤ) x) C)) fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i))\n (wittPolynomial p ℤ k) =\n ↑(aeval fun n => ↑(aeval (mk p fun i => { down := x i }).coeff) (ψ n)) (wittPolynomial p ℤ k)",
"state_before": "case h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)",
"tactic": "simp only [hf, hg, MvPolynomial.eval, map_eval₂Hom]"
},
{
"state_after": "p : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ (fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i)) =\n fun n => ↑(aeval (mk p fun i => { down := x i }).coeff) (ψ n)",
"state_before": "case h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (eval₂Hom (RingHom.id ℤ) x) C)) fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i))\n (wittPolynomial p ℤ k) =\n ↑(aeval fun n => ↑(aeval (mk p fun i => { down := x i }).coeff) (ψ n)) (wittPolynomial p ℤ k)",
"tactic": "apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl"
},
{
"state_after": "case h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ x✝) =\n ↑(aeval (mk p fun i => { down := x i }).coeff) (ψ x✝)",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\n⊢ (fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i)) =\n fun n => ↑(aeval (mk p fun i => { down := x i }).coeff) (ψ n)",
"tactic": "ext1"
},
{
"state_after": "p : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ (fun i => ↑↑(RingEquiv.symm ULift.ringEquiv) (x i)) = (mk p fun i => { down := x i }).coeff",
"state_before": "case h\np : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ x✝) =\n ↑(aeval (mk p fun i => { down := x i }).coeff) (ψ x✝)",
"tactic": "apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl"
},
{
"state_after": "p : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ (fun i => ↑↑(RingEquiv.symm ULift.ringEquiv) (x i)) = fun i => { down := x i }",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ (fun i => ↑↑(RingEquiv.symm ULift.ringEquiv) (x i)) = (mk p fun i => { down := x i }).coeff",
"tactic": "simp only [coeff_mk]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.168788\nidx : Type ?u.168791\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => ↑(aeval x.coeff) (φ n)\nψ : ℕ → MvPolynomial ℕ ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => ↑(aeval x.coeff) (ψ n)\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ : 𝕎 R✝\nn k : ℕ\nx : ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ (fun i => ↑↑(RingEquiv.symm ULift.ringEquiv) (x i)) = fun i => { down := x i }",
"tactic": "rfl"
}
] |
[
269,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_left_surjective
|
[] |
[
617,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Topology/Order/Basic.lean
|
mem_nhds_iff_exists_Ioo_subset'
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\n⊢ s ∈ 𝓝 a → ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s\n\ncase mpr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\n⊢ (∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s) → s ∈ 𝓝 a",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\n⊢ s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\nh : s ∈ 𝓝 a\n⊢ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\n⊢ s ∈ 𝓝 a → ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"tactic": "intro h"
},
{
"state_after": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu✝ : ∃ u, a < u\nh : s ∈ 𝓝 a\nu : α\nau : a < u\nhu : Ico a u ⊆ s\n⊢ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\nh : s ∈ 𝓝 a\n⊢ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"tactic": "rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl✝ : ∃ l, l < a\nhu✝ : ∃ u, a < u\nh : s ∈ 𝓝 a\nu : α\nau : a < u\nhu : Ico a u ⊆ s\nl : α\nla : l < a\nhl : Ioc l a ⊆ s\n⊢ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"state_before": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu✝ : ∃ u, a < u\nh : s ∈ 𝓝 a\nu : α\nau : a < u\nhu : Ico a u ⊆ s\n⊢ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"tactic": "rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl✝ : ∃ l, l < a\nhu✝ : ∃ u, a < u\nh : s ∈ 𝓝 a\nu : α\nau : a < u\nhu : Ico a u ⊆ s\nl : α\nla : l < a\nhl : Ioc l a ⊆ s\n⊢ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s",
"tactic": "exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\nl u : α\nha : a ∈ Ioo l u\nh : Ioo l u ⊆ s\n⊢ s ∈ 𝓝 a",
"state_before": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\n⊢ (∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s) → s ∈ 𝓝 a",
"tactic": "rintro ⟨l, u, ha, h⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhl : ∃ l, l < a\nhu : ∃ u, a < u\nl u : α\nha : a ∈ Ioo l u\nh : Ioo l u ⊆ s\n⊢ s ∈ 𝓝 a",
"tactic": "apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h"
}
] |
[
1318,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1310,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
|
Measurable.measurePreserving
|
[] |
[
55,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
11
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.bodd_two
|
[] |
[
40,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Data/Finsupp/NeLocus.lean
|
Finsupp.neLocus_self_add_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type ?u.35751\nN : Type u_2\nP : Type ?u.35757\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq N\ninst✝ : AddGroup N\nf f₁ f₂ g g₁ g₂ : α →₀ N\n⊢ neLocus (f + g) f = g.support",
"tactic": "rw [neLocus_comm, neLocus_self_add_right]"
}
] |
[
168,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Order/Basic.lean
|
max_rec'
|
[] |
[
953,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
952,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.add_smul_mem
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.204606\nβ : Type ?u.204609\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : x + y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ x + t • y ∈ s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.204606\nβ : Type ?u.204609\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : x + y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ x + t • y ∈ s",
"tactic": "have h : x + t • y = (1 - t) • x + t • (x + y) := by\n rw [smul_add, ← add_assoc, ← add_smul, sub_add_cancel, one_smul]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.204606\nβ : Type ?u.204609\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : x + y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ (1 - t) • x + t • (x + y) ∈ s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.204606\nβ : Type ?u.204609\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : x + y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ x + t • y ∈ s",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.204606\nβ : Type ?u.204609\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : x + y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ (1 - t) • x + t • (x + y) ∈ s",
"tactic": "exact hs hx hy (sub_nonneg_of_le ht.2) ht.1 (sub_add_cancel _ _)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.204606\nβ : Type ?u.204609\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t✝ : Set E\nhs : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : x + y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ x + t • y = (1 - t) • x + t • (x + y)",
"tactic": "rw [smul_add, ← add_assoc, ← add_smul, sub_add_cancel, one_smul]"
}
] |
[
467,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/ModelTheory/Types.lean
|
FirstOrder.Language.Theory.CompleteType.isMaximal
|
[] |
[
82,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/LinearAlgebra/UnitaryGroup.lean
|
Matrix.mem_orthogonalGroup_iff
|
[
{
"state_after": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nα : Type v\ninst✝² : CommRing α\ninst✝¹ : StarRing α\nA✝ : Matrix n n α\nβ : Type v\ninst✝ : CommRing β\nA : Matrix n n β\nhA : A * star A = 1\n⊢ star A * A = 1",
"state_before": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nα : Type v\ninst✝² : CommRing α\ninst✝¹ : StarRing α\nA✝ : Matrix n n α\nβ : Type v\ninst✝ : CommRing β\nA : Matrix n n β\n⊢ A ∈ orthogonalGroup n β ↔ A * star A = 1",
"tactic": "refine' ⟨And.right, fun hA => ⟨_, hA⟩⟩"
},
{
"state_after": "no goals",
"state_before": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nα : Type v\ninst✝² : CommRing α\ninst✝¹ : StarRing α\nA✝ : Matrix n n α\nβ : Type v\ninst✝ : CommRing β\nA : Matrix n n β\nhA : A * star A = 1\n⊢ star A * A = 1",
"tactic": "simpa only [mul_eq_mul, mul_eq_one_comm] using hA"
}
] |
[
222,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.ofReal_bit1
|
[
{
"state_after": "no goals",
"state_before": "r : ℝ\n⊢ (↑(bit1 r)).re = (bit1 ↑r).re ∧ (↑(bit1 r)).im = (bit1 ↑r).im",
"tactic": "simp [bit1]"
}
] |
[
237,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_eq_bot
|
[] |
[
529,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.frequently_lt_of_lt_limsSup
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.185642\nγ : Type ?u.185645\nι : Type ?u.185648\nf : Filter α\ninst✝ : ConditionallyCompleteLinearOrder α\na : α\nhf : autoParam (IsCobounded (fun x x_1 => x ≤ x_1) f) _auto✝\nh : ¬∃ᶠ (n : α) in f, a < n\n⊢ limsSup f ≤ a",
"state_before": "α : Type u_1\nβ : Type ?u.185642\nγ : Type ?u.185645\nι : Type ?u.185648\nf : Filter α\ninst✝ : ConditionallyCompleteLinearOrder α\na : α\nhf : autoParam (IsCobounded (fun x x_1 => x ≤ x_1) f) _auto✝\nh : a < limsSup f\n⊢ ∃ᶠ (n : α) in f, a < n",
"tactic": "contrapose! h"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.185642\nγ : Type ?u.185645\nι : Type ?u.185648\nf : Filter α\ninst✝ : ConditionallyCompleteLinearOrder α\na : α\nhf : autoParam (IsCobounded (fun x x_1 => x ≤ x_1) f) _auto✝\nh : ∀ᶠ (x : α) in f, x ≤ a\n⊢ limsSup f ≤ a",
"state_before": "α : Type u_1\nβ : Type ?u.185642\nγ : Type ?u.185645\nι : Type ?u.185648\nf : Filter α\ninst✝ : ConditionallyCompleteLinearOrder α\na : α\nhf : autoParam (IsCobounded (fun x x_1 => x ≤ x_1) f) _auto✝\nh : ¬∃ᶠ (n : α) in f, a < n\n⊢ limsSup f ≤ a",
"tactic": "simp only [not_frequently, not_lt] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.185642\nγ : Type ?u.185645\nι : Type ?u.185648\nf : Filter α\ninst✝ : ConditionallyCompleteLinearOrder α\na : α\nhf : autoParam (IsCobounded (fun x x_1 => x ≤ x_1) f) _auto✝\nh : ∀ᶠ (x : α) in f, x ≤ a\n⊢ limsSup f ≤ a",
"tactic": "exact limsSup_le_of_le hf h"
}
] |
[
1109,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1104,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.FiniteAtFilter.mono
|
[] |
[
4134,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4133,
11
] |
Std/Data/String/Lemmas.lean
|
String.ext_iff
|
[] |
[
18,
90
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
18,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprod.map_swap
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ A B X Y : C\nf : A ⟶ B\ng : X ⟶ Y\ninst✝ : HasColimitsOfShape (Discrete WalkingPair) C\n⊢ map (𝟙 X) f ≫ map g (𝟙 B) = map g (𝟙 A) ≫ map (𝟙 Y) f",
"tactic": "simp"
}
] |
[
892,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
890,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.sInf_caratheodory
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"tactic": "rw [OuterMeasure.sInf_eq_boundedBy_sInfGen]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\n⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) (t ∩ s) + OuterMeasure.sInfGen (toOuterMeasure '' m) (t \\ s) ≤\n OuterMeasure.sInfGen (toOuterMeasure '' m) t",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"tactic": "refine' OuterMeasure.boundedBy_caratheodory fun t => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\n⊢ ∀ (x : Measure α),\n x ∈ m →\n ∀ (i : Set α),\n t ⊆ i →\n MeasurableSet i →\n ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑x i",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\n⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) (t ∩ s) + OuterMeasure.sInfGen (toOuterMeasure '' m) (t \\ s) ≤\n OuterMeasure.sInfGen (toOuterMeasure '' m) t",
"tactic": "simp only [OuterMeasure.sInfGen, le_iInf_iff, ball_image_iff,\n measure_eq_iInf t]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\n⊢ ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑μ u",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\n⊢ ∀ (x : Measure α),\n x ∈ m →\n ∀ (i : Set α),\n t ⊆ i →\n MeasurableSet i →\n ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑x i",
"tactic": "intro μ hμ u htu _hu"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\nhm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ ↑↑μ t\n⊢ ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑μ u",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\n⊢ ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑μ u",
"tactic": "have hm : ∀ {s t}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t := by\n intro s t hst\n rw [OuterMeasure.sInfGen_def]\n refine' iInf_le_of_le μ.toOuterMeasure (iInf_le_of_le (mem_image_of_mem _ hμ) _)\n refine' measure_mono hst"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\nhm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ ↑↑μ t\n⊢ ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑μ (u ∩ s) + ↑↑μ (u \\ s)",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\nhm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ ↑↑μ t\n⊢ ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑μ u",
"tactic": "rw [← measure_inter_add_diff u hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\nhm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ ↑↑μ t\n⊢ ((⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t ∩ s)) +\n ⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ (t \\ s)) ≤\n ↑↑μ (u ∩ s) + ↑↑μ (u \\ s)",
"tactic": "refine' add_le_add (hm <| inter_subset_inter_left _ htu) (hm <| diff_subset_diff_left htu)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝¹ s' t✝¹ : Set α\nm : Set (Measure α)\ns✝ : Set α\nhs : MeasurableSet s✝\nt✝ : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t✝ ⊆ u\n_hu : MeasurableSet u\ns t : Set α\nhst : s ⊆ t\n⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ ↑↑μ t",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\n⊢ ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ ↑↑μ t",
"tactic": "intro s t hst"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝¹ s' t✝¹ : Set α\nm : Set (Measure α)\ns✝ : Set α\nhs : MeasurableSet s✝\nt✝ : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t✝ ⊆ u\n_hu : MeasurableSet u\ns t : Set α\nhst : s ⊆ t\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ s) ≤ ↑↑μ t",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝¹ s' t✝¹ : Set α\nm : Set (Measure α)\ns✝ : Set α\nhs : MeasurableSet s✝\nt✝ : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t✝ ⊆ u\n_hu : MeasurableSet u\ns t : Set α\nhst : s ⊆ t\n⊢ OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ ↑↑μ t",
"tactic": "rw [OuterMeasure.sInfGen_def]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝¹ s' t✝¹ : Set α\nm : Set (Measure α)\ns✝ : Set α\nhs : MeasurableSet s✝\nt✝ : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t✝ ⊆ u\n_hu : MeasurableSet u\ns t : Set α\nhst : s ⊆ t\n⊢ ↑↑μ s ≤ ↑↑μ t",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝¹ s' t✝¹ : Set α\nm : Set (Measure α)\ns✝ : Set α\nhs : MeasurableSet s✝\nt✝ : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t✝ ⊆ u\n_hu : MeasurableSet u\ns t : Set α\nhst : s ⊆ t\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ toOuterMeasure '' m), ↑μ s) ≤ ↑↑μ t",
"tactic": "refine' iInf_le_of_le μ.toOuterMeasure (iInf_le_of_le (mem_image_of_mem _ hμ) _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.147381\nγ : Type ?u.147384\nδ : Type ?u.147387\nι : Type ?u.147390\nR : Type ?u.147393\nR' : Type ?u.147396\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝¹ s' t✝¹ : Set α\nm : Set (Measure α)\ns✝ : Set α\nhs : MeasurableSet s✝\nt✝ : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t✝ ⊆ u\n_hu : MeasurableSet u\ns t : Set α\nhst : s ⊆ t\n⊢ ↑↑μ s ≤ ↑↑μ t",
"tactic": "refine' measure_mono hst"
}
] |
[
1016,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1003,
1
] |
Mathlib/Algebra/BigOperators/Intervals.lean
|
Finset.sum_range_by_parts
|
[
{
"state_after": "case pos\nR : Type u_2\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhn : n = 0\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i\n\ncase neg\nR : Type u_2\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhn : ¬n = 0\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i",
"tactic": "by_cases hn : n = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_2\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhn : n = 0\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i",
"tactic": "simp [hn]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_2\nM : Type u_1\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ℕ → R\ng : ℕ → M\nm n : ℕ\nhn : ¬n = 0\n⊢ ∑ i in range n, f i • g i =\n f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i",
"tactic": "rw [range_eq_Ico, sum_Ico_by_parts f g (Nat.pos_of_ne_zero hn), sum_range_zero, smul_zero,\n sub_zero, range_eq_Ico]"
}
] |
[
319,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_mem_inter_mulSupport
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.206039\nι : Type ?u.206042\nG : Type ?u.206045\nM : Type u_2\nN : Type ?u.206051\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\ns : Set α\n⊢ (∏ᶠ (i : α) (_ : i ∈ s ∩ mulSupport f), f i) = ∏ᶠ (i : α) (_ : i ∈ s), f i",
"tactic": "rw [finprod_mem_def, finprod_mem_def, mulIndicator_inter_mulSupport]"
}
] |
[
551,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.left_mem_Icc
|
[] |
[
118,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Data/List/Basic.lean
|
List.drop_take
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.217889\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nn : ℕ\nx✝ : List α\n⊢ drop 0 (take (0 + n) x✝) = take n (drop 0 x✝)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.217889\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nm n : ℕ\n⊢ drop (m + 1) (take (m + 1 + n) []) = take n (drop (m + 1) [])",
"tactic": "simp"
},
{
"state_after": "ι : Type ?u.217889\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nm n : ℕ\nhead✝ : α\nl : List α\nh : m + 1 + n = m + n + 1\n⊢ drop (m + 1) (take (m + 1 + n) (head✝ :: l)) = take n (drop (m + 1) (head✝ :: l))",
"state_before": "ι : Type ?u.217889\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nm n : ℕ\nhead✝ : α\nl : List α\n⊢ drop (m + 1) (take (m + 1 + n) (head✝ :: l)) = take n (drop (m + 1) (head✝ :: l))",
"tactic": "have h : m + 1 + n = m + n + 1 := by ac_rfl"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.217889\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nm n : ℕ\nhead✝ : α\nl : List α\nh : m + 1 + n = m + n + 1\n⊢ drop (m + 1) (take (m + 1 + n) (head✝ :: l)) = take n (drop (m + 1) (head✝ :: l))",
"tactic": "simpa [take_cons, h] using drop_take m n l"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.217889\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nm n : ℕ\nhead✝ : α\nl : List α\n⊢ m + 1 + n = m + n + 1",
"tactic": "ac_rfl"
}
] |
[
2262,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2257,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
kstar_mul_le_kstar
|
[] |
[
211,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Topology/Spectral/Hom.lean
|
SpectralMap.cancel_right
|
[] |
[
219,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.continuousAt_iff_continuousAt_comp_left
|
[
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.102343\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\nx : γ\nh : f ⁻¹' e.source ∈ 𝓝 x\nhx : f x ∈ e.source\n⊢ ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.102343\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\nx : γ\nh : f ⁻¹' e.source ∈ 𝓝 x\n⊢ ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x",
"tactic": "have hx : f x ∈ e.source := (mem_of_mem_nhds h : _)"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.102343\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\nx : γ\nh : f ⁻¹' e.source ∈ 𝓝 x\nhx : f x ∈ e.source\nh' : f ⁻¹' e.source ∈ 𝓝[univ] x\n⊢ ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.102343\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\nx : γ\nh : f ⁻¹' e.source ∈ 𝓝 x\nhx : f x ∈ e.source\n⊢ ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x",
"tactic": "have h' : f ⁻¹' e.source ∈ 𝓝[univ] x := by rwa [nhdsWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.102343\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\nx : γ\nh : f ⁻¹' e.source ∈ 𝓝 x\nhx : f x ∈ e.source\nh' : f ⁻¹' e.source ∈ 𝓝[univ] x\n⊢ ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x",
"tactic": "rw [← continuousWithinAt_univ, ← continuousWithinAt_univ,\n e.continuousWithinAt_iff_continuousWithinAt_comp_left hx h']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.102343\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\nf : γ → α\nx : γ\nh : f ⁻¹' e.source ∈ 𝓝 x\nhx : f x ∈ e.source\n⊢ f ⁻¹' e.source ∈ 𝓝[univ] x",
"tactic": "rwa [nhdsWithin_univ]"
}
] |
[
1187,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1182,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.ptendsto_of_ptendsto'
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nf : α →. β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (s : Set β), s ∈ l₂ → PFun.preimage f s ∈ l₁) → ∀ (s : Set β), s ∈ l₂ → PFun.core f s ∈ l₁",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α →. β\nl₁ : Filter α\nl₂ : Filter β\n⊢ PTendsto' f l₁ l₂ → PTendsto f l₁ l₂",
"tactic": "rw [ptendsto_def, ptendsto'_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α →. β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (s : Set β), s ∈ l₂ → PFun.preimage f s ∈ l₁) → ∀ (s : Set β), s ∈ l₂ → PFun.core f s ∈ l₁",
"tactic": "exact fun h s sl₂ => mem_of_superset (h s sl₂) (PFun.preimage_subset_core _ _)"
}
] |
[
278,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inv.lean
|
Differentiable.inv
|
[] |
[
164,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/RingTheory/Derivation/Basic.lean
|
Derivation.leibniz_invOf
|
[] |
[
410,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/MeasureTheory/Function/L2Space.lean
|
integral_inner
|
[] |
[
100,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isRat_inv_neg
|
[
{
"state_after": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn d : ℕ\ninv✝ : Invertible ↑d\n⊢ IsRat (↑(Int.negOfNat (Nat.succ n)) * ⅟↑d)⁻¹ (Int.negOfNat d) (Nat.succ n)",
"state_before": "α : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\na : α\nn d : ℕ\n⊢ IsRat a (Int.negOfNat (Nat.succ n)) d → IsRat a⁻¹ (Int.negOfNat d) (Nat.succ n)",
"tactic": "rintro ⟨_, rfl⟩"
},
{
"state_after": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn d : ℕ\ninv✝ : Invertible ↑d\n⊢ IsRat (↑(-Int.ofNat (Nat.succ n)) * ⅟↑d)⁻¹ (-Int.ofNat d) (Nat.succ n)",
"state_before": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn d : ℕ\ninv✝ : Invertible ↑d\n⊢ IsRat (↑(Int.negOfNat (Nat.succ n)) * ⅟↑d)⁻¹ (Int.negOfNat d) (Nat.succ n)",
"tactic": "simp only [Int.negOfNat_eq]"
},
{
"state_after": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn d : ℕ\ninv✝ : Invertible ↑d\nthis : Invertible ↑(Nat.succ n)\n⊢ IsRat (↑(-Int.ofNat (Nat.succ n)) * ⅟↑d)⁻¹ (-Int.ofNat d) (Nat.succ n)",
"state_before": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn d : ℕ\ninv✝ : Invertible ↑d\n⊢ IsRat (↑(-Int.ofNat (Nat.succ n)) * ⅟↑d)⁻¹ (-Int.ofNat d) (Nat.succ n)",
"tactic": "have := invertibleOfNonzero (α := α) (Nat.cast_ne_zero.2 (Nat.succ_ne_zero n))"
},
{
"state_after": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn✝ d : ℕ\ninv✝ : Invertible ↑d\nn : ℕ\nthis : Invertible ↑n\n⊢ IsRat (↑(-Int.ofNat n) * ⅟↑d)⁻¹ (-Int.ofNat d) n",
"state_before": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn d : ℕ\ninv✝ : Invertible ↑d\nthis : Invertible ↑(Nat.succ n)\n⊢ IsRat (↑(-Int.ofNat (Nat.succ n)) * ⅟↑d)⁻¹ (-Int.ofNat d) (Nat.succ n)",
"tactic": "generalize Nat.succ n = n at *"
},
{
"state_after": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn✝ d : ℕ\ninv✝ : Invertible ↑d\nn : ℕ\nthis : Invertible ↑n\n⊢ (↑(-Int.ofNat n) * ⅟↑d)⁻¹ = ↑(-Int.ofNat d) * ⅟↑n",
"state_before": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn✝ d : ℕ\ninv✝ : Invertible ↑d\nn : ℕ\nthis : Invertible ↑n\n⊢ IsRat (↑(-Int.ofNat n) * ⅟↑d)⁻¹ (-Int.ofNat d) n",
"tactic": "use this"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_1\ninst✝¹ : DivisionRing α\ninst✝ : CharZero α\nn✝ d : ℕ\ninv✝ : Invertible ↑d\nn : ℕ\nthis : Invertible ↑n\n⊢ (↑(-Int.ofNat n) * ⅟↑d)⁻¹ = ↑(-Int.ofNat d) * ⅟↑n",
"tactic": "simp only [Int.ofNat_eq_coe, Int.cast_neg,\nInt.cast_ofNat, invOf_eq_inv, inv_neg, neg_mul, mul_inv_rev, inv_inv]"
}
] |
[
510,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.snd_dual
|
[] |
[
118,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Topology/Order.lean
|
isOpen_iff_continuous_mem
|
[] |
[
927,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
926,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.land'_zero
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ land' n 0 = 0",
"tactic": "simp [land']"
}
] |
[
203,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/LinearAlgebra/AffineSpace/MidpointZero.lean
|
homothety_invOf_two
|
[] |
[
40,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.