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/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurableSet_Ico
|
[] |
[
522,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.Formula.imp_semanticallyEquivalent_not_sup
|
[] |
[
564,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.refl
|
[] |
[
54,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
11
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.prod_comp
|
[] |
[
1148,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1146,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_le_of_unit_norm
|
[
{
"state_after": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ‖↑f x‖ ≤ C * ‖x‖",
"state_before": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\n⊢ ‖f‖ ≤ C",
"tactic": "refine' op_norm_le_bound' f hC fun x hx => _"
},
{
"state_after": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\nH₁ : ‖‖x‖⁻¹ • x‖ = 1\n⊢ ‖↑f x‖ ≤ C * ‖x‖",
"state_before": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ‖↑f x‖ ≤ C * ‖x‖",
"tactic": "have H₁ : ‖‖x‖⁻¹ • x‖ = 1 := by rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hx]"
},
{
"state_after": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\nH₁ : ‖‖x‖⁻¹ • x‖ = 1\nH₂ : ‖↑f (‖x‖⁻¹ • x)‖ ≤ C\n⊢ ‖↑f x‖ ≤ C * ‖x‖",
"state_before": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\nH₁ : ‖‖x‖⁻¹ • x‖ = 1\n⊢ ‖↑f x‖ ≤ C * ‖x‖",
"tactic": "have H₂ := hf _ H₁"
},
{
"state_after": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\nH₁ : ‖‖x‖⁻¹ • x‖ = 1\nH₂ : ‖↑f x‖ / ‖x‖ ≤ C\n⊢ 0 < ‖x‖",
"state_before": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\nH₁ : ‖‖x‖⁻¹ • x‖ = 1\nH₂ : ‖↑f (‖x‖⁻¹ • x)‖ ≤ C\n⊢ ‖↑f x‖ ≤ C * ‖x‖",
"tactic": "rwa [map_smul, norm_smul, norm_inv, norm_norm, ← div_eq_inv_mul, _root_.div_le_iff] at H₂"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\nH₁ : ‖‖x‖⁻¹ • x‖ = 1\nH₂ : ‖↑f x‖ / ‖x‖ ≤ C\n⊢ 0 < ‖x‖",
"tactic": "exact (norm_nonneg x).lt_of_ne' hx"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.381082\n𝕜₂ : Type ?u.381085\n𝕜₃ : Type ?u.381088\nE : Type u_1\nEₗ : Type ?u.381094\nF : Type u_2\nFₗ : Type ?u.381100\nG : Type ?u.381103\nGₗ : Type ?u.381106\n𝓕 : Type ?u.381109\ninst✝¹⁹ : SeminormedAddCommGroup E\ninst✝¹⁸ : SeminormedAddCommGroup Eₗ\ninst✝¹⁷ : SeminormedAddCommGroup F\ninst✝¹⁶ : SeminormedAddCommGroup Fₗ\ninst✝¹⁵ : SeminormedAddCommGroup G\ninst✝¹⁴ : SeminormedAddCommGroup Gₗ\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NontriviallyNormedField 𝕜₂\ninst✝¹¹ : NontriviallyNormedField 𝕜₃\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedSpace 𝕜 Eₗ\ninst✝⁸ : NormedSpace 𝕜₂ F\ninst✝⁷ : NormedSpace 𝕜 Fₗ\ninst✝⁶ : NormedSpace 𝕜₃ G\ninst✝⁵ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝⁴ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\nC : ℝ\nhC : 0 ≤ C\nhf : ∀ (x : E), ‖x‖ = 1 → ‖↑f x‖ ≤ C\nx : E\nhx : ‖x‖ ≠ 0\n⊢ ‖‖x‖⁻¹ • x‖ = 1",
"tactic": "rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hx]"
}
] |
[
268,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.range_le
|
[] |
[
119,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.adj_symm
|
[] |
[
111,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Topology/Bases.lean
|
DenseRange.separableSpace
|
[] |
[
501,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
11
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean
|
PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain
|
[
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\n⊢ I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nh_nzI : I ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "revert h_nzI"
},
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\n⊢ (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) M",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\n⊢ I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),\n Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)\n (fun (M : Ideal A) hgt => _) I"
},
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\n⊢ (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) M",
"tactic": "intro h_nzM"
},
{
"state_after": "case hA_nont\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\n⊢ Nontrivial A\n\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "have hA_nont : Nontrivial A"
},
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case hA_nont\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\n⊢ Nontrivial A\n\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "apply IsDomain.toNontrivial"
},
{
"state_after": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : M = ⊤\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥\n\ncase neg\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "by_cases h_topM : M = ⊤"
},
{
"state_after": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : Ideal.IsPrime M\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥\n\ncase neg\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case neg\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "by_cases h_prM : M.IsPrime"
},
{
"state_after": "case neg.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case neg\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM"
},
{
"state_after": "case neg.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case neg.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "have lt_add : ∀ (z) (_ : z ∉ M), M < M + span A {z} := by\n intro z hz\n refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _\n rw [m_eq]\n exact mem_sup_right (mem_span_singleton_self z)"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case neg.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case neg.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "obtain ⟨Wy, h_Wy_le, h_Wx_ne⟩ := hgt (M + span A {y}) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ Multiset.prod (Multiset.map asIdeal (Wx + Wy)) ≤ M ∧ Multiset.prod (Multiset.map asIdeal (Wx + Wy)) ≠ ⊥",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "use Wx + Wy"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) ≤ M ∧\n Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ Multiset.prod (Multiset.map asIdeal (Wx + Wy)) ≤ M ∧ Multiset.prod (Multiset.map asIdeal (Wx + Wy)) ≠ ⊥",
"tactic": "rw [Multiset.map_add, Multiset.prod_add]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ (M + span A {x}) * (M + span A {y}) ≤ M\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ ¬(Multiset.prod (Multiset.map asIdeal Wx) = ⊥ ∨ Multiset.prod (Multiset.map asIdeal Wy) = ⊥)",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) ≤ M ∧\n Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥",
"tactic": "refine' ⟨le_trans (Submodule.mul_le_mul h_Wx_le h_Wy_le) _, mt Ideal.mul_eq_bot.mp _⟩"
},
{
"state_after": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n ∀ (J : Submodule A A),\n J > ⊤ →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : ⊤ ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ ⊤ ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : M = ⊤\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "rcases h_topM with rfl"
},
{
"state_after": "case pos.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n ∀ (J : Submodule A A),\n J > ⊤ →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : ⊤ ≠ ⊥\np_id : Ideal A\nh_nzp : p_id ≠ ⊥\nh_pp : Ideal.IsPrime p_id\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ ⊤ ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"state_before": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n ∀ (J : Submodule A A),\n J > ⊤ →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : ⊤ ≠ ⊥\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ ⊤ ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by\n apply Ring.not_isField_iff_exists_prime.mp h_fA"
},
{
"state_after": "case pos.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n ∀ (J : Submodule A A),\n J > ⊤ →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : ⊤ ≠ ⊥\np_id : Ideal A\nh_nzp : p_id ≠ ⊥\nh_pp : Ideal.IsPrime p_id\n⊢ Multiset.prod (Multiset.map asIdeal {{ asIdeal := p_id, IsPrime := h_pp }}) ≠ ⊥",
"state_before": "case pos.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n ∀ (J : Submodule A A),\n J > ⊤ →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : ⊤ ≠ ⊥\np_id : Ideal A\nh_nzp : p_id ≠ ⊥\nh_pp : Ideal.IsPrime p_id\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ ⊤ ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n ∀ (J : Submodule A A),\n J > ⊤ →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : ⊤ ≠ ⊥\np_id : Ideal A\nh_nzp : p_id ≠ ⊥\nh_pp : Ideal.IsPrime p_id\n⊢ Multiset.prod (Multiset.map asIdeal {{ asIdeal := p_id, IsPrime := h_pp }}) ≠ ⊥",
"tactic": "rwa [Multiset.map_singleton, Multiset.prod_singleton]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI : Ideal A\nhA_nont : Nontrivial A\nhgt :\n ∀ (J : Submodule A A),\n J > ⊤ →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : ⊤ ≠ ⊥\n⊢ ∃ p, p ≠ ⊥ ∧ Ideal.IsPrime p",
"tactic": "apply Ring.not_isField_iff_exists_prime.mp h_fA"
},
{
"state_after": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : Ideal.IsPrime M\n⊢ Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) ≤ M ∧\n Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) ≠ ⊥",
"state_before": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : Ideal.IsPrime M\n⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥",
"tactic": "use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))"
},
{
"state_after": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : Ideal.IsPrime M\n⊢ { asIdeal := M, IsPrime := h_prM }.asIdeal ≤ M ∧ { asIdeal := M, IsPrime := h_prM }.asIdeal ≠ ⊥",
"state_before": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : Ideal.IsPrime M\n⊢ Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) ≤ M ∧\n Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) ≠ ⊥",
"tactic": "rw [Multiset.map_singleton, Multiset.prod_singleton]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : Ideal.IsPrime M\n⊢ { asIdeal := M, IsPrime := h_prM }.asIdeal ≤ M ∧ { asIdeal := M, IsPrime := h_prM }.asIdeal ≠ ⊥",
"tactic": "exact ⟨le_rfl, h_nzM⟩"
},
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nz : A\nhz : ¬z ∈ M\n⊢ M < M + span A {z}",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\n⊢ ∀ (z : A), ¬z ∈ M → M < M + span A {z}",
"tactic": "intro z hz"
},
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nz : A\nhz : ¬z ∈ M\nm_eq : M = M + span A {z}\n⊢ z ∈ M",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nz : A\nhz : ¬z ∈ M\n⊢ M < M + span A {z}",
"tactic": "refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _"
},
{
"state_after": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nz : A\nhz : ¬z ∈ M\nm_eq : M = M + span A {z}\n⊢ z ∈ M + span A {z}",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nz : A\nhz : ¬z ∈ M\nm_eq : M = M + span A {z}\n⊢ z ∈ M",
"tactic": "rw [m_eq]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nz : A\nhz : ¬z ∈ M\nm_eq : M = M + span A {z}\n⊢ z ∈ M + span A {z}",
"tactic": "exact mem_sup_right (mem_span_singleton_self z)"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ M * (M + span A {y}) + span A {x} * (M + span A {y}) ≤ M",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ (M + span A {x}) * (M + span A {y}) ≤ M",
"tactic": "rw [add_mul]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ span A {x} * (M + span A {y}) ≤ M",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ M * (M + span A {y}) + span A {x} * (M + span A {y}) ≤ M",
"tactic": "apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ span A {x} * M + span A {x} * span A {y} ≤ M",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ span A {x} * (M + span A {y}) ≤ M",
"tactic": "rw [mul_add]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ span A {x} * span A {y} ≤ M",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ span A {x} * M + span A {x} * span A {y} ≤ M",
"tactic": "apply sup_le (show span A {x} * M ≤ M from Ideal.mul_le_left)"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ span A {x} * span A {y} ≤ M",
"tactic": "rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsNoetherianRing A\nh_fA : ¬IsField A\nI M : Ideal A\nhgt :\n ∀ (J : Submodule A A),\n J > M →\n (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J\nh_nzM : M ≠ ⊥\nhA_nont : Nontrivial A\nh_topM : ¬M = ⊤\nh_prM : ¬Ideal.IsPrime M\nx : A\nhx : ¬x ∈ M\ny : A\nhy : ¬y ∈ M\nh_xy : x * y ∈ M\nlt_add : ∀ (z : A), ¬z ∈ M → M < M + span A {z}\nWx : Multiset (PrimeSpectrum A)\nh_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}\nh_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥\nWy : Multiset (PrimeSpectrum A)\nh_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}\nh_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥\n⊢ ¬(Multiset.prod (Multiset.map asIdeal Wx) = ⊥ ∨ Multiset.prod (Multiset.map asIdeal Wy) = ⊥)",
"tactic": "rintro (hx | hy) <;> contradiction"
}
] |
[
102,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
eq_tsub_of_add_eq
|
[] |
[
350,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.toBlock_diagonal_disjoint
|
[
{
"state_after": "case a.mk.h.mk\nl : Type ?u.115337\nm : Type u_1\nn : Type ?u.115343\no : Type ?u.115346\np✝ : Type ?u.115349\nq✝ : Type ?u.115352\nm' : o → Type ?u.115357\nn' : o → Type ?u.115362\np' : o → Type ?u.115367\nR : Type ?u.115370\nS : Type ?u.115373\nα : Type u_2\nβ : Type ?u.115379\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd : m → α\np q : m → Prop\nhpq : Disjoint p q\ni : m\nhi : p i\nj : m\nhj : q j\n⊢ toBlock (diagonal d) p q { val := i, property := hi } { val := j, property := hj } =\n OfNat.ofNat 0 { val := i, property := hi } { val := j, property := hj }",
"state_before": "l : Type ?u.115337\nm : Type u_1\nn : Type ?u.115343\no : Type ?u.115346\np✝ : Type ?u.115349\nq✝ : Type ?u.115352\nm' : o → Type ?u.115357\nn' : o → Type ?u.115362\np' : o → Type ?u.115367\nR : Type ?u.115370\nS : Type ?u.115373\nα : Type u_2\nβ : Type ?u.115379\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd : m → α\np q : m → Prop\nhpq : Disjoint p q\n⊢ toBlock (diagonal d) p q = 0",
"tactic": "ext (⟨i, hi⟩⟨j, hj⟩)"
},
{
"state_after": "case a.mk.h.mk\nl : Type ?u.115337\nm : Type u_1\nn : Type ?u.115343\no : Type ?u.115346\np✝ : Type ?u.115349\nq✝ : Type ?u.115352\nm' : o → Type ?u.115357\nn' : o → Type ?u.115362\np' : o → Type ?u.115367\nR : Type ?u.115370\nS : Type ?u.115373\nα : Type u_2\nβ : Type ?u.115379\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd : m → α\np q : m → Prop\nhpq : Disjoint p q\ni : m\nhi : p i\nj : m\nhj : q j\nthis : i ≠ j\n⊢ toBlock (diagonal d) p q { val := i, property := hi } { val := j, property := hj } =\n OfNat.ofNat 0 { val := i, property := hi } { val := j, property := hj }",
"state_before": "case a.mk.h.mk\nl : Type ?u.115337\nm : Type u_1\nn : Type ?u.115343\no : Type ?u.115346\np✝ : Type ?u.115349\nq✝ : Type ?u.115352\nm' : o → Type ?u.115357\nn' : o → Type ?u.115362\np' : o → Type ?u.115367\nR : Type ?u.115370\nS : Type ?u.115373\nα : Type u_2\nβ : Type ?u.115379\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd : m → α\np q : m → Prop\nhpq : Disjoint p q\ni : m\nhi : p i\nj : m\nhj : q j\n⊢ toBlock (diagonal d) p q { val := i, property := hi } { val := j, property := hj } =\n OfNat.ofNat 0 { val := i, property := hi } { val := j, property := hj }",
"tactic": "have : i ≠ j := fun heq => hpq.le_bot i ⟨hi, heq.symm ▸ hj⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mk.h.mk\nl : Type ?u.115337\nm : Type u_1\nn : Type ?u.115343\no : Type ?u.115346\np✝ : Type ?u.115349\nq✝ : Type ?u.115352\nm' : o → Type ?u.115357\nn' : o → Type ?u.115362\np' : o → Type ?u.115367\nR : Type ?u.115370\nS : Type ?u.115373\nα : Type u_2\nβ : Type ?u.115379\ninst✝² : DecidableEq l\ninst✝¹ : DecidableEq m\ninst✝ : Zero α\nd : m → α\np q : m → Prop\nhpq : Disjoint p q\ni : m\nhi : p i\nj : m\nhj : q j\nthis : i ≠ j\n⊢ toBlock (diagonal d) p q { val := i, property := hi } { val := j, property := hj } =\n OfNat.ofNat 0 { val := i, property := hi } { val := j, property := hj }",
"tactic": "simp [diagonal_apply_ne d this]"
}
] |
[
296,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.pair
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.47877\nσ : Type ?u.47880\ninst✝⁴ : Primcodable α✝\ninst✝³ : Primcodable σ\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable γ\nf : α → β\ng : α → γ\nhf : Primrec f\nhg : Primrec g\nn : ℕ\n⊢ (Nat.casesOn (encode (decode n)) 0 fun n =>\n Nat.succ (Nat.pair (Nat.pred (encode (Option.map f (decode n)))) (Nat.pred (encode (Option.map g (decode n)))))) =\n encode (Option.map (fun a => (f a, g a)) (decode n))",
"tactic": "cases @decode α _ n <;> simp [encodek]"
}
] |
[
371,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Topology/Basic.lean
|
mem_closure_iff_frequently
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\na : α\n⊢ a ∈ interior (sᶜ)ᶜ ↔ ¬a ∈ interior {x | ¬x ∈ s}",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\na : α\n⊢ a ∈ closure s ↔ ∃ᶠ (x : α) in 𝓝 a, x ∈ s",
"tactic": "rw [Filter.Frequently, Filter.Eventually, ← mem_interior_iff_mem_nhds,\n closure_eq_compl_interior_compl]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\na : α\n⊢ a ∈ interior (sᶜ)ᶜ ↔ ¬a ∈ interior {x | ¬x ∈ s}",
"tactic": "rfl"
}
] |
[
1268,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1266,
1
] |
Mathlib/MeasureTheory/Function/Floor.lean
|
Nat.measurable_floor
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.6253\nR : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : LinearOrderedSemiring R\ninst✝⁴ : FloorSemiring R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nf : α → R\nn : R\n⊢ MeasurableSet (floor ⁻¹' {⌊n⌋₊})",
"tactic": "cases' eq_or_ne ⌊n⌋₊ 0 with h h <;> simp_all [h, Nat.preimage_floor_of_ne_zero]"
}
] |
[
77,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
Module.End.eigenspaces_independent
|
[
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\n⊢ CompleteLattice.Independent (eigenspace f)",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\n⊢ CompleteLattice.Independent (eigenspace f)",
"tactic": "let S : @LinearMap K K _ _ (RingHom.id K) (Π₀ μ : K, f.eigenspace μ) V\n (@Dfinsupp.addCommMonoid K (fun μ => f.eigenspace μ) _) _\n (@Dfinsupp.module K _ (fun μ => f.eigenspace μ) _ _ _) _ :=\n @Dfinsupp.lsum K K ℕ _ V _ _ _ _ _ _ _ _ _ fun μ => (f.eigenspace μ).subtype"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\n⊢ ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\n⊢ CompleteLattice.Independent (eigenspace f)",
"tactic": "suffices ∀ l : Π₀ μ, f.eigenspace μ, S l = 0 → l = 0 by\n rw [CompleteLattice.independent_iff_dfinsupp_lsum_injective]\n change Function.Injective S\n rw [← @LinearMap.ker_eq_bot K K (Π₀ μ, f.eigenspace μ) V _ _\n (@Dfinsupp.addCommGroup K (fun μ => f.eigenspace μ) _)]\n rw [eq_bot_iff]\n exact this"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\n⊢ l = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\n⊢ ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0",
"tactic": "intro l hl"
},
{
"state_after": "case empty\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = ∅\n⊢ l = 0\n\ncase insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\n⊢ l = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\n⊢ l = 0",
"tactic": "induction' h_l_support : l.support using Finset.induction with μ₀ l_support' hμ₀ ih generalizing l"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ Function.Injective ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (eigenspace f i))",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ CompleteLattice.Independent (eigenspace f)",
"tactic": "rw [CompleteLattice.independent_iff_dfinsupp_lsum_injective]"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ Function.Injective ↑S",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ Function.Injective ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (eigenspace f i))",
"tactic": "change Function.Injective S"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ LinearMap.ker S = ⊥",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ Function.Injective ↑S",
"tactic": "rw [← @LinearMap.ker_eq_bot K K (Π₀ μ, f.eigenspace μ) V _ _\n (@Dfinsupp.addCommGroup K (fun μ => f.eigenspace μ) _)]"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ LinearMap.ker S ≤ ⊥",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ LinearMap.ker S = ⊥",
"tactic": "rw [eq_bot_iff]"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nthis : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → l = 0\n⊢ LinearMap.ker S ≤ ⊥",
"tactic": "exact this"
},
{
"state_after": "no goals",
"state_before": "case empty\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = ∅\n⊢ l = 0",
"tactic": "exact Dfinsupp.support_eq_empty.1 h_l_support"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\n⊢ l = 0",
"tactic": "let l' := Dfinsupp.mapRange.linearMap\n (fun μ => (μ - μ₀) • @LinearMap.id K (f.eigenspace μ) _ _ _) l"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\n⊢ l = 0",
"tactic": "have h_l_support' : l'.support = l_support' := by\n rw [← Finset.erase_insert hμ₀, ← h_l_support]\n ext a\n have : ¬(a = μ₀ ∨ l a = 0) ↔ ¬a = μ₀ ∧ ¬l a = 0 := not_or\n simp only [Dfinsupp.mapRange.linearMap_apply, Dfinsupp.mapRange_apply,\n Dfinsupp.mem_support_iff, Finset.mem_erase, id.def, LinearMap.id_coe, LinearMap.smul_apply,\n Ne.def, smul_eq_zero, sub_eq_zero, this]"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\n⊢ l = 0",
"tactic": "have l'_eq_0 := ih l' total_l' h_l_support'"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\n⊢ l = 0",
"tactic": "have h_smul_eq_0 : ∀ μ, (μ - μ₀) • l μ = 0 := by\n intro μ\n calc\n (μ - μ₀) • l μ = l' μ := by\n simp only [LinearMap.id_coe, id.def, LinearMap.smul_apply, Dfinsupp.mapRange_apply,\n Dfinsupp.mapRange.linearMap_apply]\n _ = 0 := by rw [l'_eq_0]; rfl"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\n⊢ l = 0",
"tactic": "have h_lμ_eq_0 : ∀ μ : K, μ ≠ μ₀ → l μ = 0 := by\n intro μ hμ\n apply or_iff_not_imp_left.1 (smul_eq_zero.1 (h_smul_eq_0 μ))\n rwa [sub_eq_zero]"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ l = 0",
"tactic": "have h_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => (l μ : V)) = 0 := by\n rw [← Finset.sum_const_zero]\n apply Finset.sum_congr rfl\n intro μ hμ\n rw [Submodule.coe_eq_zero, h_lμ_eq_0]\n rintro rfl\n exact hμ₀ hμ"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\n⊢ l = 0",
"tactic": "have : l μ₀ = 0 := by\n simp only [Dfinsupp.lsum_apply_apply, Dfinsupp.sumAddHom_apply,\n LinearMap.toAddMonoidHom_coe, Dfinsupp.sum, h_l_support, Submodule.subtype_apply,\n Submodule.coe_eq_zero, Finset.sum_insert hμ₀, h_sum_l_support'_eq_0, add_zero] at hl\n exact hl"
},
{
"state_after": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\n⊢ l = 0",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\n⊢ l = 0",
"tactic": "show l = 0"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\n⊢ Dfinsupp.support l' = Finset.erase (Dfinsupp.support l) μ₀",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\n⊢ Dfinsupp.support l' = l_support'",
"tactic": "rw [← Finset.erase_insert hμ₀, ← h_l_support]"
},
{
"state_after": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\na : K\n⊢ a ∈ Dfinsupp.support l' ↔ a ∈ Finset.erase (Dfinsupp.support l) μ₀",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\n⊢ Dfinsupp.support l' = Finset.erase (Dfinsupp.support l) μ₀",
"tactic": "ext a"
},
{
"state_after": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\na : K\nthis : ¬(a = μ₀ ∨ ↑l a = 0) ↔ ¬a = μ₀ ∧ ¬↑l a = 0\n⊢ a ∈ Dfinsupp.support l' ↔ a ∈ Finset.erase (Dfinsupp.support l) μ₀",
"state_before": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\na : K\n⊢ a ∈ Dfinsupp.support l' ↔ a ∈ Finset.erase (Dfinsupp.support l) μ₀",
"tactic": "have : ¬(a = μ₀ ∨ l a = 0) ↔ ¬a = μ₀ ∧ ¬l a = 0 := not_or"
},
{
"state_after": "no goals",
"state_before": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\na : K\nthis : ¬(a = μ₀ ∨ ↑l a = 0) ↔ ¬a = μ₀ ∧ ¬↑l a = 0\n⊢ a ∈ Dfinsupp.support l' ↔ a ∈ Finset.erase (Dfinsupp.support l) μ₀",
"tactic": "simp only [Dfinsupp.mapRange.linearMap_apply, Dfinsupp.mapRange_apply,\n Dfinsupp.mem_support_iff, Finset.mem_erase, id.def, LinearMap.id_coe, LinearMap.smul_apply,\n Ne.def, smul_eq_zero, sub_eq_zero, this]"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\n⊢ ↑S l' = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\n⊢ ↑S l' = 0",
"tactic": "let g := f - algebraMap K (End K V) μ₀"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑S l' = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\n⊢ ↑S l' = 0",
"tactic": "let a : Π₀ _ : K, V := Dfinsupp.mapRange.linearMap (fun μ => (f.eigenspace μ).subtype) l"
},
{
"state_after": "case calc_1\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑S l' = ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) l\n\ncase calc_2\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) l =\n ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp g (Submodule.subtype (eigenspace f μ))) l\n\ncase calc_3\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp g (Submodule.subtype (eigenspace f μ))) l =\n ↑(↑(Dfinsupp.lsum ℕ) fun x => g) a\n\ncase calc_4\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑(↑(Dfinsupp.lsum ℕ) fun x => g) a = ↑g (↑(↑(Dfinsupp.lsum ℕ) fun x => LinearMap.id) a)\n\ncase calc_5\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑g (↑(↑(Dfinsupp.lsum ℕ) fun x => LinearMap.id) a) = ↑g (↑S l)",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑S l' = 0",
"tactic": "calc\n S l' =\n Dfinsupp.lsum ℕ (fun μ => (f.eigenspace μ).subtype.comp ((μ - μ₀) • LinearMap.id)) l := ?_\n _ = Dfinsupp.lsum ℕ (fun μ => g.comp (f.eigenspace μ).subtype) l := ?_\n _ = Dfinsupp.lsum ℕ (fun _ => g) a := ?_\n _ = g (Dfinsupp.lsum ℕ (fun _ => (LinearMap.id : V →ₗ[K] V)) a) := ?_\n _ = g (S l) := ?_\n _ = 0 := by rw [hl, g.map_zero]"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑g (↑S l) = 0",
"tactic": "rw [hl, g.map_zero]"
},
{
"state_after": "no goals",
"state_before": "case calc_1\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑S l' = ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) l",
"tactic": "exact Dfinsupp.sum_mapRange_index.linearMap"
},
{
"state_after": "case calc_2.e_a.h.e_6.h\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ (fun μ => LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) = fun μ =>\n LinearMap.comp g (Submodule.subtype (eigenspace f μ))",
"state_before": "case calc_2\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) l =\n ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp g (Submodule.subtype (eigenspace f μ))) l",
"tactic": "congr"
},
{
"state_after": "case calc_2.e_a.h.e_6.h.h.h\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\nμ : K\nv : { x // x ∈ eigenspace f μ }\n⊢ ↑(LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) v =\n ↑(LinearMap.comp g (Submodule.subtype (eigenspace f μ))) v",
"state_before": "case calc_2.e_a.h.e_6.h\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ (fun μ => LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) = fun μ =>\n LinearMap.comp g (Submodule.subtype (eigenspace f μ))",
"tactic": "ext (μ v)"
},
{
"state_after": "no goals",
"state_before": "case calc_2.e_a.h.e_6.h.h.h\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\nμ : K\nv : { x // x ∈ eigenspace f μ }\n⊢ ↑(LinearMap.comp (Submodule.subtype (eigenspace f μ)) ((μ - μ₀) • LinearMap.id)) v =\n ↑(LinearMap.comp g (Submodule.subtype (eigenspace f μ))) v",
"tactic": "simp only [LinearMap.coe_comp, Function.comp_apply, LinearMap.smul_apply, LinearMap.id_coe,\n id.def, sub_smul, Submodule.subtype_apply, Submodule.coe_sub, Submodule.coe_smul_of_tower,\n LinearMap.sub_apply, mem_eigenspace_iff.1 v.prop, algebraMap_end_apply]"
},
{
"state_after": "no goals",
"state_before": "case calc_3\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑(↑(Dfinsupp.lsum ℕ) fun μ => LinearMap.comp g (Submodule.subtype (eigenspace f μ))) l =\n ↑(↑(Dfinsupp.lsum ℕ) fun x => g) a",
"tactic": "rw [Dfinsupp.sum_mapRange_index.linearMap]"
},
{
"state_after": "no goals",
"state_before": "case calc_4\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑(↑(Dfinsupp.lsum ℕ) fun x => g) a = ↑g (↑(↑(Dfinsupp.lsum ℕ) fun x => LinearMap.id) a)",
"tactic": "simp only [Dfinsupp.sumAddHom_apply, LinearMap.id_coe, LinearMap.map_dfinsupp_sum, id.def,\n LinearMap.toAddMonoidHom_coe, Dfinsupp.lsum_apply_apply]"
},
{
"state_after": "no goals",
"state_before": "case calc_5\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ng : End K V := f - ↑(algebraMap K (End K V)) μ₀\na : Π₀ (x : K), V := ↑(Dfinsupp.mapRange.linearMap fun μ => Submodule.subtype (eigenspace f μ)) l\n⊢ ↑g (↑(↑(Dfinsupp.lsum ℕ) fun x => LinearMap.id) a) = ↑g (↑S l)",
"tactic": "simp only [Dfinsupp.sum_mapRange_index.linearMap, LinearMap.id_comp]"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nμ : K\n⊢ (μ - μ₀) • ↑l μ = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\n⊢ ∀ (μ : K), (μ - μ₀) • ↑l μ = 0",
"tactic": "intro μ"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nμ : K\n⊢ (μ - μ₀) • ↑l μ = 0",
"tactic": "calc\n (μ - μ₀) • l μ = l' μ := by\n simp only [LinearMap.id_coe, id.def, LinearMap.smul_apply, Dfinsupp.mapRange_apply,\n Dfinsupp.mapRange.linearMap_apply]\n _ = 0 := by rw [l'_eq_0]; rfl"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nμ : K\n⊢ (μ - μ₀) • ↑l μ = ↑l' μ",
"tactic": "simp only [LinearMap.id_coe, id.def, LinearMap.smul_apply, Dfinsupp.mapRange_apply,\n Dfinsupp.mapRange.linearMap_apply]"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nμ : K\n⊢ ↑0 μ = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nμ : K\n⊢ ↑l' μ = 0",
"tactic": "rw [l'_eq_0]"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nμ : K\n⊢ ↑0 μ = 0",
"tactic": "rfl"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nμ : K\nhμ : μ ≠ μ₀\n⊢ ↑l μ = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\n⊢ ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0",
"tactic": "intro μ hμ"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nμ : K\nhμ : μ ≠ μ₀\n⊢ ¬μ - μ₀ = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nμ : K\nhμ : μ ≠ μ₀\n⊢ ↑l μ = 0",
"tactic": "apply or_iff_not_imp_left.1 (smul_eq_zero.1 (h_smul_eq_0 μ))"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nμ : K\nhμ : μ ≠ μ₀\n⊢ ¬μ - μ₀ = 0",
"tactic": "rwa [sub_eq_zero]"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ (Finset.sum l_support' fun μ => ↑(↑l μ)) = Finset.sum ?m.327909 fun _x => 0\n\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ Type ?u.327905\n\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ Finset ?m.327908",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0",
"tactic": "rw [← Finset.sum_const_zero]"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ ∀ (x : K), x ∈ l_support' → ↑(↑l x) = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ (Finset.sum l_support' fun μ => ↑(↑l μ)) = Finset.sum ?m.327909 fun _x => 0\n\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ Type ?u.327905\n\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ Finset ?m.327908",
"tactic": "apply Finset.sum_congr rfl"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nμ : K\nhμ : μ ∈ l_support'\n⊢ ↑(↑l μ) = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\n⊢ ∀ (x : K), x ∈ l_support' → ↑(↑l x) = 0",
"tactic": "intro μ hμ"
},
{
"state_after": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nμ : K\nhμ : μ ∈ l_support'\n⊢ μ ≠ μ₀",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nμ : K\nhμ : μ ∈ l_support'\n⊢ ↑(↑l μ) = 0",
"tactic": "rw [Submodule.coe_eq_zero, h_lμ_eq_0]"
},
{
"state_after": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nl_support' : Finset K\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nμ : K\nhμ : μ ∈ l_support'\nhμ₀ : ¬μ ∈ l_support'\nh_l_support : Dfinsupp.support l = insert μ l_support'\nl' : Π₀ (i : K), { x // x ∈ eigenspace f i } := ↑(Dfinsupp.mapRange.linearMap fun μ_1 => (μ_1 - μ) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ_1 : K), (μ_1 - μ) • ↑l μ_1 = 0\nh_lμ_eq_0 : ∀ (μ_1 : K), μ_1 ≠ μ → ↑l μ_1 = 0\n⊢ False",
"state_before": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nμ : K\nhμ : μ ∈ l_support'\n⊢ μ ≠ μ₀",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nl_support' : Finset K\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nμ : K\nhμ : μ ∈ l_support'\nhμ₀ : ¬μ ∈ l_support'\nh_l_support : Dfinsupp.support l = insert μ l_support'\nl' : Π₀ (i : K), { x // x ∈ eigenspace f i } := ↑(Dfinsupp.mapRange.linearMap fun μ_1 => (μ_1 - μ) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ_1 : K), (μ_1 - μ) • ↑l μ_1 = 0\nh_lμ_eq_0 : ∀ (μ_1 : K), μ_1 ≠ μ → ↑l μ_1 = 0\n⊢ False",
"tactic": "exact hμ₀ hμ"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nhl : ↑l μ₀ = 0\n⊢ ↑l μ₀ = 0",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\n⊢ ↑l μ₀ = 0",
"tactic": "simp only [Dfinsupp.lsum_apply_apply, Dfinsupp.sumAddHom_apply,\n LinearMap.toAddMonoidHom_coe, Dfinsupp.sum, h_l_support, Submodule.subtype_apply,\n Submodule.coe_eq_zero, Finset.sum_insert hμ₀, h_sum_l_support'_eq_0, add_zero] at hl"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nhl : ↑l μ₀ = 0\n⊢ ↑l μ₀ = 0",
"tactic": "exact hl"
},
{
"state_after": "case insert.h.a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\nμ : K\n⊢ ↑(↑l μ) = ↑(↑0 μ)",
"state_before": "case insert\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\n⊢ l = 0",
"tactic": "ext μ"
},
{
"state_after": "case pos\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\nμ : K\nh_cases : μ = μ₀\n⊢ ↑(↑l μ) = ↑(↑0 μ)\n\ncase neg\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\nμ : K\nh_cases : ¬μ = μ₀\n⊢ ↑(↑l μ) = ↑(↑0 μ)",
"state_before": "case insert.h.a\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\nμ : K\n⊢ ↑(↑l μ) = ↑(↑0 μ)",
"tactic": "by_cases h_cases : μ = μ₀"
},
{
"state_after": "no goals",
"state_before": "case pos\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\nμ : K\nh_cases : μ = μ₀\n⊢ ↑(↑l μ) = ↑(↑0 μ)",
"tactic": "rwa [h_cases, SetLike.coe_eq_coe, Dfinsupp.coe_zero, Pi.zero_apply]"
},
{
"state_after": "no goals",
"state_before": "case neg\nK R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End K V\nS : (Π₀ (μ : K), { x // x ∈ eigenspace f μ }) →ₗ[K] V := ↑(Dfinsupp.lsum ℕ) fun μ => Submodule.subtype (eigenspace f μ)\nl✝ : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl✝ : ↑S l✝ = 0\nx✝ : Finset K\nh_l_support✝ : Dfinsupp.support l✝ = x✝\nμ₀ : K\nl_support' : Finset K\nhμ₀ : ¬μ₀ ∈ l_support'\nih : ∀ (l : Π₀ (μ : K), { x // x ∈ eigenspace f μ }), ↑S l = 0 → Dfinsupp.support l = l_support' → l = 0\nl : Π₀ (μ : K), { x // x ∈ eigenspace f μ }\nhl : ↑S l = 0\nh_l_support : Dfinsupp.support l = insert μ₀ l_support'\nl' : (fun x => Π₀ (i : K), { x // x ∈ eigenspace f i }) l :=\n ↑(Dfinsupp.mapRange.linearMap fun μ => (μ - μ₀) • LinearMap.id) l\nh_l_support' : Dfinsupp.support l' = l_support'\ntotal_l' : ↑S l' = 0\nl'_eq_0 : l' = 0\nh_smul_eq_0 : ∀ (μ : K), (μ - μ₀) • ↑l μ = 0\nh_lμ_eq_0 : ∀ (μ : K), μ ≠ μ₀ → ↑l μ = 0\nh_sum_l_support'_eq_0 : (Finset.sum l_support' fun μ => ↑(↑l μ)) = 0\nthis : ↑l μ₀ = 0\nμ : K\nh_cases : ¬μ = μ₀\n⊢ ↑(↑l μ) = ↑(↑0 μ)",
"tactic": "exact congr_arg _ (h_lμ_eq_0 μ h_cases)"
}
] |
[
236,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Tactic/NormNum/Core.lean
|
Mathlib.Meta.NormNum.IsNat.to_eq
|
[] |
[
49,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
RingHom.eq_of_eqOn_set_top
|
[] |
[
1241,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1240,
1
] |
Mathlib/Data/PFun.lean
|
PFun.mem_dom
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.8010\nδ : Type ?u.8013\nε : Type ?u.8016\nι : Type ?u.8019\nf : α →. β\nx : α\n⊢ x ∈ Dom f ↔ ∃ y, y ∈ f x",
"tactic": "simp [Dom, Part.dom_iff_mem]"
}
] |
[
83,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Nat/Bits.lean
|
Nat.div2_bits_eq_tail
|
[
{
"state_after": "case z\nn : ℕ\n⊢ bits (div2 0) = List.tail (bits 0)\n\ncase f\nn✝ : ℕ\nb : Bool\nn : ℕ\nh : n = 0 → b = true\na✝ : bits (div2 n) = List.tail (bits n)\n⊢ bits (div2 (bit b n)) = List.tail (bits (bit b n))",
"state_before": "n✝ n : ℕ\n⊢ bits (div2 n) = List.tail (bits n)",
"tactic": "induction' n using Nat.binaryRec' with b n h _"
},
{
"state_after": "no goals",
"state_before": "case f\nn✝ : ℕ\nb : Bool\nn : ℕ\nh : n = 0 → b = true\na✝ : bits (div2 n) = List.tail (bits n)\n⊢ bits (div2 (bit b n)) = List.tail (bits (bit b n))",
"tactic": "simp [div2_bit, bits_append_bit _ _ h]"
},
{
"state_after": "no goals",
"state_before": "case z\nn : ℕ\n⊢ bits (div2 0) = List.tail (bits 0)",
"tactic": "simp"
}
] |
[
249,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.IsFundamentalSequence.monotone
|
[
{
"state_after": "case inl\nα : Type ?u.69712\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\nhf : IsFundamentalSequence a o f\ni j : Ordinal\nhi : i < o\nhj : j < o\nhij✝ : i ≤ j\nhij : i < j\n⊢ f i hi ≤ f j hj\n\ncase inr\nα : Type ?u.69712\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\nhf : IsFundamentalSequence a o f\ni : Ordinal\nhi hj : i < o\nhij : i ≤ i\n⊢ f i hi ≤ f i hj",
"state_before": "α : Type ?u.69712\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\nhf : IsFundamentalSequence a o f\ni j : Ordinal\nhi : i < o\nhj : j < o\nhij : i ≤ j\n⊢ f i hi ≤ f j hj",
"tactic": "rcases lt_or_eq_of_le hij with (hij | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.69712\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\nhf : IsFundamentalSequence a o f\ni j : Ordinal\nhi : i < o\nhj : j < o\nhij✝ : i ≤ j\nhij : i < j\n⊢ f i hi ≤ f j hj",
"tactic": "exact (hf.2.1 hi hj hij).le"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type ?u.69712\nr : α → α → Prop\na o : Ordinal\nf : (b : Ordinal) → b < o → Ordinal\nhf : IsFundamentalSequence a o f\ni : Ordinal\nhi hj : i < o\nhij : i ≤ i\n⊢ f i hi ≤ f i hj",
"tactic": "rfl"
}
] |
[
598,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
594,
11
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.cmp_swap
|
[
{
"state_after": "case pos.pos\nα : Type ?u.468320\na✝¹ a✝ : PosNum\n⊢ Ordering.swap (cmp (pos a✝¹) (pos a✝)) = cmp (pos a✝) (pos a✝¹)",
"state_before": "α : Type ?u.468320\nm n : Num\n⊢ Ordering.swap (cmp m n) = cmp n m",
"tactic": "cases m <;> cases n <;> try { unfold cmp } <;> try { rfl }"
},
{
"state_after": "no goals",
"state_before": "case pos.pos\nα : Type ?u.468320\na✝¹ a✝ : PosNum\n⊢ Ordering.swap (cmp (pos a✝¹) (pos a✝)) = cmp (pos a✝) (pos a✝¹)",
"tactic": "apply PosNum.cmp_swap"
}
] |
[
855,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
854,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.coe_toAddSubgroup
|
[] |
[
527,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.ext
|
[
{
"state_after": "case mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : E →ₗ.[R] F\nf_dom : Submodule R E\nf : { x // x ∈ f_dom } →ₗ[R] F\nh : { domain := f_dom, toFun := f }.domain = g.domain\nh' :\n ∀ ⦃x : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄ ⦃y : { x // x ∈ g.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑g y\n⊢ { domain := f_dom, toFun := f } = g",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf g : E →ₗ.[R] F\nh : f.domain = g.domain\nh' : ∀ ⦃x : { x // x ∈ f.domain }⦄ ⦃y : { x // x ∈ g.domain }⦄, ↑x = ↑y → ↑f x = ↑g y\n⊢ f = g",
"tactic": "rcases f with ⟨f_dom, f⟩"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf_dom : Submodule R E\nf : { x // x ∈ f_dom } →ₗ[R] F\ng_dom : Submodule R E\ng : { x // x ∈ g_dom } →ₗ[R] F\nh : { domain := f_dom, toFun := f }.domain = { domain := g_dom, toFun := g }.domain\nh' :\n ∀ ⦃x : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄ ⦃y : { x // x ∈ { domain := g_dom, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑{ domain := g_dom, toFun := g } y\n⊢ { domain := f_dom, toFun := f } = { domain := g_dom, toFun := g }",
"state_before": "case mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : E →ₗ.[R] F\nf_dom : Submodule R E\nf : { x // x ∈ f_dom } →ₗ[R] F\nh : { domain := f_dom, toFun := f }.domain = g.domain\nh' :\n ∀ ⦃x : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄ ⦃y : { x // x ∈ g.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑g y\n⊢ { domain := f_dom, toFun := f } = g",
"tactic": "rcases g with ⟨g_dom, g⟩"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf_dom : Submodule R E\nf g : { x // x ∈ f_dom } →ₗ[R] F\nh' :\n ∀ ⦃x : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄ ⦃y : { x // x ∈ { domain := f_dom, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑{ domain := f_dom, toFun := g } y\n⊢ { domain := f_dom, toFun := f } = { domain := f_dom, toFun := g }",
"state_before": "case mk.mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf_dom : Submodule R E\nf : { x // x ∈ f_dom } →ₗ[R] F\ng_dom : Submodule R E\ng : { x // x ∈ g_dom } →ₗ[R] F\nh : { domain := f_dom, toFun := f }.domain = { domain := g_dom, toFun := g }.domain\nh' :\n ∀ ⦃x : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄ ⦃y : { x // x ∈ { domain := g_dom, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑{ domain := g_dom, toFun := g } y\n⊢ { domain := f_dom, toFun := f } = { domain := g_dom, toFun := g }",
"tactic": "obtain rfl : f_dom = g_dom := h"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf_dom : Submodule R E\nf : { x // x ∈ f_dom } →ₗ[R] F\nh' :\n ∀ ⦃x y : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑{ domain := f_dom, toFun := f } y\n⊢ { domain := f_dom, toFun := f } = { domain := f_dom, toFun := f }",
"state_before": "case mk.mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf_dom : Submodule R E\nf g : { x // x ∈ f_dom } →ₗ[R] F\nh' :\n ∀ ⦃x : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄ ⦃y : { x // x ∈ { domain := f_dom, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑{ domain := f_dom, toFun := g } y\n⊢ { domain := f_dom, toFun := f } = { domain := f_dom, toFun := g }",
"tactic": "obtain rfl : f = g := LinearMap.ext fun x => h' rfl"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.20860\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nf_dom : Submodule R E\nf : { x // x ∈ f_dom } →ₗ[R] F\nh' :\n ∀ ⦃x y : { x // x ∈ { domain := f_dom, toFun := f }.domain }⦄,\n ↑x = ↑y → ↑{ domain := f_dom, toFun := f } x = ↑{ domain := f_dom, toFun := f } y\n⊢ { domain := f_dom, toFun := f } = { domain := f_dom, toFun := f }",
"tactic": "rfl"
}
] |
[
76,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.eq_of_coeff_monomial_ne_zero
|
[] |
[
173,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.of_isBilimit
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPullback b.fst b.snd 0 0",
"tactic": "convert IsPullback.of_is_product' h.isLimit HasZeroObject.zeroIsTerminal"
}
] |
[
562,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
1
] |
src/lean/Init/Data/Nat/Div.lean
|
Nat.mod_zero
|
[] |
[
109,
27
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
105,
9
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_subtype_map_embedding
|
[
{
"state_after": "ι : Type ?u.382203\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\ns : Finset { x // p x }\nf : { x // p x } → β\ng : α → β\nh : ∀ (x : { x // p x }), x ∈ s → g ↑x = f x\n⊢ ∏ x in s, g (↑(Function.Embedding.subtype fun x => p x) x) = ∏ x in s, f x",
"state_before": "ι : Type ?u.382203\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\ns : Finset { x // p x }\nf : { x // p x } → β\ng : α → β\nh : ∀ (x : { x // p x }), x ∈ s → g ↑x = f x\n⊢ ∏ x in map (Function.Embedding.subtype fun x => p x) s, g x = ∏ x in s, f x",
"tactic": "rw [Finset.prod_map]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.382203\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\ns : Finset { x // p x }\nf : { x // p x } → β\ng : α → β\nh : ∀ (x : { x // p x }), x ∈ s → g ↑x = f x\n⊢ ∏ x in s, g (↑(Function.Embedding.subtype fun x => p x) x) = ∏ x in s, f x",
"tactic": "exact Finset.prod_congr rfl h"
}
] |
[
895,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_sin_mul_cos₁
|
[
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, sin x * cos x) = (sin b ^ 2 - sin a ^ 2) / 2",
"tactic": "simpa using integral_sin_pow_mul_cos_pow_odd 1 0"
}
] |
[
769,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.nil_toFinset
|
[] |
[
815,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
814,
1
] |
Std/Tactic/Ext.lean
|
Unit.ext
|
[] |
[
206,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
206,
11
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.prod_eq_inf
|
[
{
"state_after": "case a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ (f₁ ⨯ f₂) ≤ f₁ ⊓ f₂\n\ncase a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ f₁ ⊓ f₂ ≤ (f₁ ⨯ f₂)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ (f₁ ⨯ f₂) = f₁ ⊓ f₂",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ f₁ ⊓ f₂ ≤ (f₁ ⨯ f₂)",
"state_before": "case a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ (f₁ ⨯ f₂) ≤ f₁ ⊓ f₂\n\ncase a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ f₁ ⊓ f₂ ≤ (f₁ ⨯ f₂)",
"tactic": ". refine' le_inf _ _ _ (Limits.prod.fst.le) (Limits.prod.snd.le)"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ f₁ ⊓ f₂ ≤ (f₁ ⨯ f₂)",
"tactic": ". apply leOfHom\n exact prod.lift (inf_le_left _ _).hom (inf_le_right _ _).hom"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ (f₁ ⨯ f₂) ≤ f₁ ⊓ f₂",
"tactic": "refine' le_inf _ _ _ (Limits.prod.fst.le) (Limits.prod.snd.le)"
},
{
"state_after": "case a.h\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ f₁ ⊓ f₂ ⟶ f₁ ⨯ f₂",
"state_before": "case a\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ f₁ ⊓ f₂ ≤ (f₁ ⨯ f₂)",
"tactic": "apply leOfHom"
},
{
"state_after": "no goals",
"state_before": "case a.h\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasPullbacks C\nA : C\nf₁ f₂ : Subobject A\ninst✝ : HasBinaryProduct f₁ f₂\n⊢ f₁ ⊓ f₂ ⟶ f₁ ⨯ f₂",
"tactic": "exact prod.lift (inf_le_left _ _).hom (inf_le_right _ _).hom"
}
] |
[
480,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.cauchy₃
|
[] |
[
179,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.prod_assoc_symm_preimage
|
[
{
"state_after": "case h\nα✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Set α\nt : Set β\nu : Set γ\nx✝ : α × β × γ\n⊢ x✝ ∈ ↑(prodAssoc α β γ).symm ⁻¹' (s ×ˢ t) ×ˢ u ↔ x✝ ∈ s ×ˢ t ×ˢ u",
"state_before": "α✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Set α\nt : Set β\nu : Set γ\n⊢ ↑(prodAssoc α β γ).symm ⁻¹' (s ×ˢ t) ×ˢ u = s ×ˢ t ×ˢ u",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Sort u\nβ✝ : Sort v\nγ✝ : Sort w\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Set α\nt : Set β\nu : Set γ\nx✝ : α × β × γ\n⊢ x✝ ∈ ↑(prodAssoc α β γ).symm ⁻¹' (s ×ˢ t) ×ˢ u ↔ x✝ ∈ s ×ˢ t ×ˢ u",
"tactic": "simp [and_assoc]"
}
] |
[
150,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Algebra/Regular/SMul.lean
|
IsSMulRegular.mul_iff
|
[
{
"state_after": "R : Type u_1\nS : Type ?u.29294\nM : Type u_2\na b : R\ns : S\ninst✝² : CommSemigroup R\ninst✝¹ : SMul R M\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M (a * b) ↔ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a)",
"state_before": "R : Type u_1\nS : Type ?u.29294\nM : Type u_2\na b : R\ns : S\ninst✝² : CommSemigroup R\ninst✝¹ : SMul R M\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b",
"tactic": "rw [← mul_and_mul_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.29294\nM : Type u_2\na b : R\ns : S\ninst✝² : CommSemigroup R\ninst✝¹ : SMul R M\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M (a * b) ↔ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a)",
"tactic": "exact ⟨fun ab => ⟨ab, by rwa [mul_comm]⟩, fun rab => rab.1⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.29294\nM : Type u_2\na b : R\ns : S\ninst✝² : CommSemigroup R\ninst✝¹ : SMul R M\ninst✝ : IsScalarTower R R M\nab : IsSMulRegular M (a * b)\n⊢ IsSMulRegular M (b * a)",
"tactic": "rwa [mul_comm]"
}
] |
[
226,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
Algebra.FinitePresentation.mvPolynomial
|
[
{
"state_after": "case intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type u_2\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ FinitePresentation R (MvPolynomial ι R)",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type u_2\ninst✝ : Finite ι\n⊢ FinitePresentation R (MvPolynomial ι R)",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type u_2\ninst✝ : Finite ι\nval✝ : Fintype ι\neqv : MvPolynomial (Fin (Fintype.card ι)) R ≃ₐ[R] MvPolynomial ι R :=\n AlgEquiv.symm (MvPolynomial.renameEquiv R (Fintype.equivFin ι))\n⊢ FinitePresentation R (MvPolynomial ι R)",
"state_before": "case intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type u_2\ninst✝ : Finite ι\nval✝ : Fintype ι\n⊢ FinitePresentation R (MvPolynomial ι R)",
"tactic": "let eqv := (MvPolynomial.renameEquiv R <| Fintype.equivFin ι).symm"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type u_2\ninst✝ : Finite ι\nval✝ : Fintype ι\neqv : MvPolynomial (Fin (Fintype.card ι)) R ≃ₐ[R] MvPolynomial ι R :=\n AlgEquiv.symm (MvPolynomial.renameEquiv R (Fintype.equivFin ι))\n⊢ FinitePresentation R (MvPolynomial ι R)",
"tactic": "exact\n ⟨Fintype.card ι, eqv, eqv.surjective,\n ((RingHom.injective_iff_ker_eq_bot _).1 eqv.injective).symm ▸ Submodule.fg_bot⟩"
}
] |
[
112,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
11
] |
Mathlib/Data/Prod/TProd.lean
|
List.TProd.fst_mk
|
[] |
[
71,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.coprime.gcd_mul
|
[] |
[
409,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
404,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
NonemptyInterval.snd_pow
|
[] |
[
267,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.comp_continuousAt_iff'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.71605\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\nx : α\n⊢ range ↑h ∈ 𝓝 (↑h x)",
"tactic": "simp"
}
] |
[
426,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Order/Ideal.lean
|
Order.Ideal.sup_mem
|
[] |
[
348,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
AffineBasis.coord_apply_centroid
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nι' : Type ?u.181001\nk : Type u_1\nV : Type u_3\nP : Type u_4\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : DivisionRing k\ninst✝¹ : Module k V\ninst✝ : CharZero k\nb : AffineBasis ι k P\ns : Finset ι\ni : ι\nhi : i ∈ s\n⊢ ↑(coord b i) (Finset.centroid k s ↑b) = (↑(Finset.card s))⁻¹",
"tactic": "rw [Finset.centroid,\n b.coord_apply_combination_of_mem hi (s.sum_centroidWeights_eq_one_of_nonempty _ ⟨i, hi⟩),\n Finset.centroidWeights, Function.const_apply]"
}
] |
[
317,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Data/Polynomial/Module.lean
|
PolynomialModule.smul_single_apply
|
[
{
"state_after": "case h_add\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\n⊢ ↑((p + q) • ↑(single R i) m) n = if i ≤ n then coeff (p + q) (n - i) • m else 0\n\ncase h_monomial\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\n⊢ ↑(↑(monomial n✝) a✝ • ↑(single R i) m) n = if i ≤ n then coeff (↑(monomial n✝) a✝) (n - i) • m else 0",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nf : R[X]\nm : M\nn : ℕ\n⊢ ↑(f • ↑(single R i) m) n = if i ≤ n then coeff f (n - i) • m else 0",
"tactic": "induction' f using Polynomial.induction_on' with p q hp hq"
},
{
"state_after": "case h_add\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\n⊢ ((if i ≤ n then coeff p (n - i) • m else 0) + if i ≤ n then coeff q (n - i) • m else 0) =\n if i ≤ n then coeff p (n - i) • m + coeff q (n - i) • m else 0",
"state_before": "case h_add\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\n⊢ ↑((p + q) • ↑(single R i) m) n = if i ≤ n then coeff (p + q) (n - i) • m else 0",
"tactic": "rw [add_smul, Finsupp.add_apply, hp, hq, coeff_add, add_smul]"
},
{
"state_after": "case h_add.inl\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\nh✝ : i ≤ n\n⊢ coeff p (n - i) • m + coeff q (n - i) • m = coeff p (n - i) • m + coeff q (n - i) • m\n\ncase h_add.inr\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\nh✝ : ¬i ≤ n\n⊢ 0 + 0 = 0",
"state_before": "case h_add\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\n⊢ ((if i ≤ n then coeff p (n - i) • m else 0) + if i ≤ n then coeff q (n - i) • m else 0) =\n if i ≤ n then coeff p (n - i) • m + coeff q (n - i) • m else 0",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case h_add.inl\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\nh✝ : i ≤ n\n⊢ coeff p (n - i) • m + coeff q (n - i) • m = coeff p (n - i) • m + coeff q (n - i) • m\n\ncase h_add.inr\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn : ℕ\np q : R[X]\nhp : ↑(p • ↑(single R i) m) n = if i ≤ n then coeff p (n - i) • m else 0\nhq : ↑(q • ↑(single R i) m) n = if i ≤ n then coeff q (n - i) • m else 0\nh✝ : ¬i ≤ n\n⊢ 0 + 0 = 0",
"tactic": "exacts [rfl, zero_add 0]"
},
{
"state_after": "case h_monomial\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\n⊢ (if n✝ + i = n then a✝ • m else 0) = if i ≤ n then if n✝ = n - i then a✝ • m else 0 else 0",
"state_before": "case h_monomial\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\n⊢ ↑(↑(monomial n✝) a✝ • ↑(single R i) m) n = if i ≤ n then coeff (↑(monomial n✝) a✝) (n - i) • m else 0",
"tactic": "rw [monomial_smul_single, single_apply, coeff_monomial, ite_smul, zero_smul]"
},
{
"state_after": "case pos\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : i ≤ n\n⊢ (if n✝ + i = n then a✝ • m else 0) = if i ≤ n then if n✝ = n - i then a✝ • m else 0 else 0\n\ncase neg\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\n⊢ (if n✝ + i = n then a✝ • m else 0) = if i ≤ n then if n✝ = n - i then a✝ • m else 0 else 0",
"state_before": "case h_monomial\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\n⊢ (if n✝ + i = n then a✝ • m else 0) = if i ≤ n then if n✝ = n - i then a✝ • m else 0 else 0",
"tactic": "by_cases h : i ≤ n"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : i ≤ n\n⊢ (if n✝ + i = n then a✝ • m else 0) = if i ≤ n then if n✝ = n - i then a✝ • m else 0 else 0",
"tactic": "simp_rw [eq_tsub_iff_add_eq_of_le h, if_pos h]"
},
{
"state_after": "case neg\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\n⊢ n✝ + i = n → a✝ • m = 0",
"state_before": "case neg\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\n⊢ (if n✝ + i = n then a✝ • m else 0) = if i ≤ n then if n✝ = n - i then a✝ • m else 0 else 0",
"tactic": "rw [if_neg h, ite_eq_right_iff]"
},
{
"state_after": "case neg\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\ne : n✝ + i = n\n⊢ a✝ • m = 0",
"state_before": "case neg\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\n⊢ n✝ + i = n → a✝ • m = 0",
"tactic": "intro e"
},
{
"state_after": "case neg.h\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\ne : n✝ + i = n\n⊢ False",
"state_before": "case neg\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\ne : n✝ + i = n\n⊢ a✝ • m = 0",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case neg.h\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nI : Ideal R\nS : Type ?u.264988\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ni : ℕ\nm : M\nn n✝ : ℕ\na✝ : R\nh : ¬i ≤ n\ne : n✝ + i = n\n⊢ False",
"tactic": "linarith"
}
] |
[
177,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.continuous_map
|
[] |
[
593,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Analysis/Calculus/ExtendDeriv.lean
|
has_deriv_at_interval_left_endpoint_of_tendsto_deriv
|
[
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "obtain ⟨b, ab : a < b, sab : Ioc a b ⊆ s⟩ := mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 hs"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "let t := Ioo a b"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "have ts : t ⊆ s := Subset.trans Ioo_subset_Ioc_self sab"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "have t_diff : DifferentiableOn ℝ f t := f_diff.mono ts"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "have t_conv : Convex ℝ t := convex_Ioo a b"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "have t_open : IsOpen t := isOpen_Ioo"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "have t_closure : closure t = Icc a b := closure_Ioo ab.ne"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\nt_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "have t_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e)) := by\n simp only [deriv_fderiv.symm]\n exact Tendsto.comp\n (isBoundedBilinearMap_smulRight : IsBoundedBilinearMap ℝ _).continuous_right.continuousAt\n (tendsto_nhdsWithin_mono_left Ioo_subset_Ioi_self f_lim')"
},
{
"state_after": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\nt_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))\nthis : HasDerivWithinAt f e (Icc a b) a\n⊢ HasDerivWithinAt f e (Ici a) a",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\nt_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "have : HasDerivWithinAt f e (Icc a b) a := by\n rw [hasDerivWithinAt_iff_hasFDerivWithinAt, ← t_closure]\n exact has_fderiv_at_boundary_of_tendsto_fderiv t_diff t_conv t_open t_cont t_diff'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\nt_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))\nthis : HasDerivWithinAt f e (Icc a b) a\n⊢ HasDerivWithinAt f e (Ici a) a",
"tactic": "exact this.nhdsWithin (Icc_mem_nhdsWithin_Ici <| left_mem_Ico.2 ab)"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\n⊢ ∀ (y : ℝ), y ∈ Icc a b → ContinuousWithinAt f t y",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\n⊢ ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y",
"tactic": "rw [t_closure]"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\n⊢ ContinuousWithinAt f t y",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\n⊢ ∀ (y : ℝ), y ∈ Icc a b → ContinuousWithinAt f t y",
"tactic": "intro y hy"
},
{
"state_after": "case pos\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : y = a\n⊢ ContinuousWithinAt f t y\n\ncase neg\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : ¬y = a\n⊢ ContinuousWithinAt f t y",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\n⊢ ContinuousWithinAt f t y",
"tactic": "by_cases h : y = a"
},
{
"state_after": "case pos\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : y = a\n⊢ ContinuousWithinAt f t a",
"state_before": "case pos\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : y = a\n⊢ ContinuousWithinAt f t y",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : y = a\n⊢ ContinuousWithinAt f t a",
"tactic": "exact f_lim.mono ts"
},
{
"state_after": "case neg\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : ¬y = a\nthis : y ∈ s\n⊢ ContinuousWithinAt f t y",
"state_before": "case neg\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : ¬y = a\n⊢ ContinuousWithinAt f t y",
"tactic": "have : y ∈ s := sab ⟨lt_of_le_of_ne hy.1 (Ne.symm h), hy.2⟩"
},
{
"state_after": "no goals",
"state_before": "case neg\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\ny : ℝ\nhy : y ∈ Icc a b\nh : ¬y = a\nthis : y ∈ s\n⊢ ContinuousWithinAt f t y",
"tactic": "exact (f_diff.continuousOn y this).mono ts"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\n⊢ Tendsto (fun x => smulRight 1 (deriv f x)) (𝓝[Ioo a b] a) (𝓝 (smulRight 1 e))",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\n⊢ Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))",
"tactic": "simp only [deriv_fderiv.symm]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\n⊢ Tendsto (fun x => smulRight 1 (deriv f x)) (𝓝[Ioo a b] a) (𝓝 (smulRight 1 e))",
"tactic": "exact Tendsto.comp\n (isBoundedBilinearMap_smulRight : IsBoundedBilinearMap ℝ _).continuous_right.continuousAt\n (tendsto_nhdsWithin_mono_left Ioo_subset_Ioi_self f_lim')"
},
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\nt_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))\n⊢ HasFDerivWithinAt f (smulRight 1 e) (closure t) a",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\nt_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))\n⊢ HasDerivWithinAt f e (Icc a b) a",
"tactic": "rw [hasDerivWithinAt_iff_hasFDerivWithinAt, ← t_closure]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.104384\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\ne : E\na : ℝ\nf : ℝ → E\nf_diff : DifferentiableOn ℝ f s\nf_lim : ContinuousWithinAt f s a\nhs : s ∈ 𝓝[Ioi a] a\nf_lim' : Tendsto (fun x => deriv f x) (𝓝[Ioi a] a) (𝓝 e)\nb : ℝ\nab : a < b\nsab : Ioc a b ⊆ s\nt : Set ℝ := Ioo a b\nts : t ⊆ s\nt_diff : DifferentiableOn ℝ f t\nt_conv : Convex ℝ t\nt_open : IsOpen t\nt_closure : closure t = Icc a b\nt_cont : ∀ (y : ℝ), y ∈ closure t → ContinuousWithinAt f t y\nt_diff' : Tendsto (fun x => fderiv ℝ f x) (𝓝[t] a) (𝓝 (smulRight 1 e))\n⊢ HasFDerivWithinAt f (smulRight 1 e) (closure t) a",
"tactic": "exact has_fderiv_at_boundary_of_tendsto_fderiv t_diff t_conv t_open t_cont t_diff'"
}
] |
[
143,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.toLin_finTwoProd
|
[] |
[
826,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_eq_real_sin_iff_eq_or_add_eq_pi
|
[
{
"state_after": "case h\nψ x✝ : ℝ\n⊢ sin ↑x✝ = Real.sin ψ ↔ ↑x✝ = ↑ψ ∨ ↑x✝ + ↑ψ = ↑π",
"state_before": "θ : Angle\nψ : ℝ\n⊢ sin θ = Real.sin ψ ↔ θ = ↑ψ ∨ θ + ↑ψ = ↑π",
"tactic": "induction θ using Real.Angle.induction_on"
},
{
"state_after": "no goals",
"state_before": "case h\nψ x✝ : ℝ\n⊢ sin ↑x✝ = Real.sin ψ ↔ ↑x✝ = ↑ψ ∨ ↑x✝ + ↑ψ = ↑π",
"tactic": "exact sin_eq_iff_coe_eq_or_add_eq_pi"
}
] |
[
349,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.eq_empty_of_forall_not_mem
|
[] |
[
585,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
LinearMap.comp_id
|
[] |
[
554,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isCodetecting_iff_isCoseparating
|
[] |
[
198,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Topology/Connected.lean
|
isPreconnected_iff_preconnectedSpace
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.72105\nπ : ι → Type ?u.72110\ninst✝ : TopologicalSpace α\ns✝ t u v s : Set α\nh : PreconnectedSpace ↑s\n⊢ IsPreconnected s",
"tactic": "simpa using isPreconnected_univ.image ((↑) : s → α) continuous_subtype_val.continuousOn"
}
] |
[
873,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
1
] |
Mathlib/Topology/Order/Basic.lean
|
IsGLB.isGLB_of_tendsto
|
[] |
[
2059,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2056,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
norm_algebraMap'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.480957\nβ : Type ?u.480960\nγ : Type ?u.480963\nι : Type ?u.480966\n𝕜 : Type u_2\n𝕜' : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\nx : 𝕜\n⊢ ‖↑(algebraMap 𝕜 𝕜') x‖ = ‖x‖",
"tactic": "rw [norm_algebraMap, norm_one, mul_one]"
}
] |
[
512,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Data/Rel.lean
|
Rel.dom_mono
|
[] |
[
74,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean
|
Nat.gcd_add_self_right
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ gcd m (n + m) = gcd m (n + 1 * m)",
"tactic": "rw [one_mul]"
}
] |
[
77,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_zero_right
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ choose n 0 = 1",
"tactic": "cases n <;> rfl"
}
] |
[
56,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Order/Basic.lean
|
PUnit.le
|
[] |
[
1376,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1375,
11
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.apriori_edist_iterate_efixedPoint_le
|
[] |
[
141,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/LinearAlgebra/Dfinsupp.lean
|
Submodule.mem_iSup_iff_exists_dfinsupp'
|
[
{
"state_after": "ι : Type u_3\nR : Type u_1\nS : Type ?u.436512\nM : ι → Type ?u.436517\nN : Type u_2\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Submodule R N\ninst✝ : (i : ι) → (x : { x // x ∈ p i }) → Decidable (x ≠ 0)\nx : N\n⊢ (∃ f, ↑(↑(lsum ℕ) fun i => Submodule.subtype (p i)) f = x) ↔ ∃ f, (sum f fun i xi => ↑xi) = x",
"state_before": "ι : Type u_3\nR : Type u_1\nS : Type ?u.436512\nM : ι → Type ?u.436517\nN : Type u_2\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Submodule R N\ninst✝ : (i : ι) → (x : { x // x ∈ p i }) → Decidable (x ≠ 0)\nx : N\n⊢ x ∈ iSup p ↔ ∃ f, (sum f fun i xi => ↑xi) = x",
"tactic": "rw [mem_iSup_iff_exists_dfinsupp]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nR : Type u_1\nS : Type ?u.436512\nM : ι → Type ?u.436517\nN : Type u_2\ndec_ι : DecidableEq ι\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\np : ι → Submodule R N\ninst✝ : (i : ι) → (x : { x // x ∈ p i }) → Decidable (x ≠ 0)\nx : N\n⊢ (∃ f, ↑(↑(lsum ℕ) fun i => Submodule.subtype (p i)) f = x) ↔ ∃ f, (sum f fun i xi => ↑xi) = x",
"tactic": "simp_rw [Dfinsupp.lsum_apply_apply, Dfinsupp.sumAddHom_apply,\n LinearMap.toAddMonoidHom_coe, coeSubtype]"
}
] |
[
366,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisCoinsertion.u_sup_l
|
[] |
[
805,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.disjoint_or_nonempty_inter
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.203129\nγ : Type ?u.203132\ninst✝ : DecidableEq α\ns✝ s₁ s₂ t✝ t₁ t₂ u v : Finset α\na b : α\ns t : Finset α\n⊢ _root_.Disjoint s t ∨ ¬_root_.Disjoint s t",
"state_before": "α : Type u_1\nβ : Type ?u.203129\nγ : Type ?u.203132\ninst✝ : DecidableEq α\ns✝ s₁ s₂ t✝ t₁ t₂ u v : Finset α\na b : α\ns t : Finset α\n⊢ _root_.Disjoint s t ∨ Finset.Nonempty (s ∩ t)",
"tactic": "rw [← not_disjoint_iff_nonempty_inter]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.203129\nγ : Type ?u.203132\ninst✝ : DecidableEq α\ns✝ s₁ s₂ t✝ t₁ t₂ u v : Finset α\na b : α\ns t : Finset α\n⊢ _root_.Disjoint s t ∨ ¬_root_.Disjoint s t",
"tactic": "exact em _"
}
] |
[
1835,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1833,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bfamilyOfFamily'_typein
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.271133\nγ : Type ?u.271136\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nι : Type u_1\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\nf : ι → α\ni : ι\n⊢ bfamilyOfFamily' r f (typein r i) (_ : typein r i < type r) = f i",
"tactic": "simp only [bfamilyOfFamily', enum_typein]"
}
] |
[
1144,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1142,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_dvd
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nh : m ∣ n\n⊢ fib m ∣ fib n",
"tactic": "rwa [gcd_eq_left_iff_dvd, ← fib_gcd, gcd_eq_left_iff_dvd.mp]"
}
] |
[
289,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.diam_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.293762\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns t : Set α\nd : ℝ≥0∞\n⊢ diam (insert x s) ≤ d ↔ max (⨆ (y : α) (_ : y ∈ s), edist x y) (diam s) ≤ d",
"tactic": "simp only [diam_le_iff, ball_insert_iff, edist_self, edist_comm x, max_le_iff, iSup_le_iff,\n zero_le, true_and_iff, forall_and, and_self_iff, ← and_assoc]"
}
] |
[
926,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
923,
1
] |
Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
|
GeneralizedContinuedFraction.squashSeq_succ_n_tail_eq_squashSeq_tail_n
|
[
{
"state_after": "case none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = none\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n\n\ncase some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"tactic": "cases' s_succ_succ_nth_eq : s.get? (n + 2) with gp_succ_succ_n"
},
{
"state_after": "case some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"state_before": "case none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = none\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n\n\ncase some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"tactic": "case none =>\n cases s_succ_nth_eq : s.get? (n + 1) <;>\n simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = none\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"tactic": "cases s_succ_nth_eq : s.get? (n + 1) <;>\n simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq]"
},
{
"state_after": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\n⊢ ∃ gp_succ_n, Stream'.Seq.get? s (n + 1) = some gp_succ_n\n\ncase intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"tactic": "obtain ⟨gp_succ_n, s_succ_nth_eq⟩ : ∃ gp_succ_n, s.get? (n + 1) = some gp_succ_n"
},
{
"state_after": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"state_before": "K : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\n⊢ ∃ gp_succ_n, Stream'.Seq.get? s (n + 1) = some gp_succ_n\n\ncase intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"tactic": "exact s.ge_stable (n + 1).le_succ s_succ_succ_nth_eq"
},
{
"state_after": "case intro.h\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"state_before": "case intro\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\n⊢ Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n",
"tactic": "ext1 m"
},
{
"state_after": "case intro.h.inl\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_eq_n : m = n\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m\n\ncase intro.h.inr\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"state_before": "case intro.h\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "cases' Decidable.em (m = n) with m_eq_n m_ne_n"
},
{
"state_after": "no goals",
"state_before": "case intro.h.inl\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_eq_n : m = n\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "simp [*, squashSeq]"
},
{
"state_after": "case intro.h.inr\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"state_before": "case intro.h.inr\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "have : s.tail.get? m = s.get? (m + 1) := s.get?_tail m"
},
{
"state_after": "case intro.h.inr.none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = none\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m\n\ncase intro.h.inr.some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\nval✝ : Pair K\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"state_before": "case intro.h.inr\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "cases s_succ_mth_eq : s.get? (m + 1)"
},
{
"state_after": "case intro.h.inr.none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = none\nx✝ : Stream'.Seq.get? (Stream'.Seq.tail s) m = none\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m\n\ncase intro.h.inr.some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\nval✝ : Pair K\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝\nx✝ : Stream'.Seq.get? (Stream'.Seq.tail s) m = some val✝\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"state_before": "case intro.h.inr.none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = none\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m\n\ncase intro.h.inr.some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\nval✝ : Pair K\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "all_goals have _ := this.trans s_succ_mth_eq"
},
{
"state_after": "case intro.h.inr.some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\nval✝ : Pair K\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝\nx✝ : Stream'.Seq.get? (Stream'.Seq.tail s) m = some val✝\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"state_before": "case intro.h.inr.some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\nval✝ : Pair K\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "have _ := this.trans s_succ_mth_eq"
},
{
"state_after": "no goals",
"state_before": "case intro.h.inr.none\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = none\nx✝ : Stream'.Seq.get? (Stream'.Seq.tail s) m = none\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "simp only [*, squashSeq, Stream'.Seq.get?_tail, Stream'.Seq.get?_zipWith,\n Option.map₂_none_right]"
},
{
"state_after": "no goals",
"state_before": "case intro.h.inr.some\nK : Type u_1\nn : ℕ\ng : GeneralizedContinuedFraction K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\ngp_succ_succ_n : Pair K\ns_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n\ngp_succ_n : Pair K\ns_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n\nm : ℕ\nm_ne_n : ¬m = n\nthis : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1)\nval✝ : Pair K\ns_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val✝\nx✝ : Stream'.Seq.get? (Stream'.Seq.tail s) m = some val✝\n⊢ Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m",
"tactic": "simp [*, squashSeq]"
}
] |
[
155,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
exists_hasDerivAt_eq_slope
|
[
{
"state_after": "case intro.intro\nE : Type ?u.317721\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.317817\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : (b - a) * f' c = (f b - f a) * 1\n⊢ ∃ c, c ∈ Ioo a b ∧ f' c = (f b - f a) / (b - a)",
"state_before": "E : Type ?u.317721\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.317817\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\n⊢ ∃ c, c ∈ Ioo a b ∧ f' c = (f b - f a) / (b - a)",
"tactic": "obtain ⟨c, cmem, hc⟩ : ∃ c ∈ Ioo a b, (b - a) * f' c = (f b - f a) * 1 :=\n exists_ratio_hasDerivAt_eq_ratio_slope f f' hab hfc hff' id 1 continuousOn_id\n fun x _ => hasDerivAt_id x"
},
{
"state_after": "case intro.intro\nE : Type ?u.317721\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.317817\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : (b - a) * f' c = (f b - f a) * 1\n⊢ f' c = (f b - f a) / (b - a)",
"state_before": "case intro.intro\nE : Type ?u.317721\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.317817\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : (b - a) * f' c = (f b - f a) * 1\n⊢ ∃ c, c ∈ Ioo a b ∧ f' c = (f b - f a) / (b - a)",
"tactic": "use c, cmem"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nE : Type ?u.317721\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.317817\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : (b - a) * f' c = (f b - f a) * 1\n⊢ f' c = (f b - f a) / (b - a)",
"tactic": "rwa [mul_one, mul_comm, ← eq_div_iff (sub_ne_zero.2 hab.ne')] at hc"
}
] |
[
758,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/Algebra/Order/Monoid/MinMax.lean
|
mul_lt_mul_iff_of_le_of_le
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.6644\ninst✝⁵ : LinearOrder α\ninst✝⁴ : Mul α\ninst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\nha : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nh : a₁ < a₂ ∨ b₁ < b₂\n⊢ a₁ * b₁ < a₂ * b₂",
"state_before": "α : Type u_1\nβ : Type ?u.6644\ninst✝⁵ : LinearOrder α\ninst✝⁴ : Mul α\ninst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\nha : a₁ ≤ a₂\nhb : b₁ ≤ b₂\n⊢ a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂",
"tactic": "refine' ⟨lt_or_lt_of_mul_lt_mul, fun h => _⟩"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.6644\ninst✝⁵ : LinearOrder α\ninst✝⁴ : Mul α\ninst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\nha : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nha' : a₁ < a₂\n⊢ a₁ * b₁ < a₂ * b₂\n\ncase inr\nα : Type u_1\nβ : Type ?u.6644\ninst✝⁵ : LinearOrder α\ninst✝⁴ : Mul α\ninst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\nha : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nhb' : b₁ < b₂\n⊢ a₁ * b₁ < a₂ * b₂",
"state_before": "α : Type u_1\nβ : Type ?u.6644\ninst✝⁵ : LinearOrder α\ninst✝⁴ : Mul α\ninst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\nha : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nh : a₁ < a₂ ∨ b₁ < b₂\n⊢ a₁ * b₁ < a₂ * b₂",
"tactic": "cases' h with ha' hb'"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.6644\ninst✝⁵ : LinearOrder α\ninst✝⁴ : Mul α\ninst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\nha : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nha' : a₁ < a₂\n⊢ a₁ * b₁ < a₂ * b₂",
"tactic": "exact mul_lt_mul_of_lt_of_le ha' hb"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.6644\ninst✝⁵ : LinearOrder α\ninst✝⁴ : Mul α\ninst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na₁ a₂ b₁ b₂ : α\nha : a₁ ≤ a₂\nhb : b₁ ≤ b₂\nhb' : b₁ < b₂\n⊢ a₁ * b₁ < a₂ * b₂",
"tactic": "exact mul_lt_mul_of_le_of_lt ha hb'"
}
] |
[
127,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean
|
NonUnitalAlgHom.coe_lmul_eq_mul
|
[] |
[
135,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.IntegrableOn.continuousOn_mul_of_subset
|
[
{
"state_after": "case intro\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : IntegrableOn g' A\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\n⊢ IntegrableOn (fun x => g x * g' x) A",
"state_before": "X : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : IntegrableOn g' A\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\n⊢ IntegrableOn (fun x => g x * g' x) A",
"tactic": "rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\n⊢ Memℒp (fun x => g x * g' x) 1",
"state_before": "case intro\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : IntegrableOn g' A\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\n⊢ IntegrableOn (fun x => g x * g' x) A",
"tactic": "rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg'⊢"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\nthis : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖\n⊢ Memℒp (fun x => g x * g' x) 1",
"state_before": "case intro\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\n⊢ Memℒp (fun x => g x * g' x) 1",
"tactic": "have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by\n filter_upwards [ae_restrict_mem hA]with x hx\n refine' (norm_mul_le _ _).trans _\n apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "case intro\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\nthis : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖\n⊢ Memℒp (fun x => g x * g' x) 1",
"tactic": "exact\n Memℒp.of_le_mul hg' (((hg.mono hAK).aestronglyMeasurable hA).mul hg'.aestronglyMeasurable) this"
},
{
"state_after": "case h\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖",
"state_before": "X : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\n⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖",
"tactic": "filter_upwards [ae_restrict_mem hA]with x hx"
},
{
"state_after": "case h\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖",
"state_before": "case h\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖",
"tactic": "refine' (norm_mul_le _ _).trans _"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Type u_1\nY : Type ?u.2480793\nE : Type ?u.2480796\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : ContinuousOn g K\nhg' : Memℒp g' 1\nhK : IsCompact K\nhA : MeasurableSet A\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖",
"tactic": "apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)"
}
] |
[
414,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aevalTower_X
|
[] |
[
398,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.digits_zero_zero
|
[] |
[
95,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.flip_add
|
[] |
[
815,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
814,
1
] |
Mathlib/Data/Real/Pointwise.lean
|
Real.iSup_mul_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nα : Type ?u.16214\ninst✝ : LinearOrderedField α\nr : ℝ\nha : 0 ≤ r\nf : ι → ℝ\n⊢ (⨆ (i : ι), f i) * r = ⨆ (i : ι), f i * r",
"tactic": "simp only [Real.mul_iSup_of_nonneg ha, mul_comm]"
}
] |
[
139,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
finite_cover_nhds_interior
|
[] |
[
806,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
802,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sInter_prod_sInter_subset
|
[] |
[
1825,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1823,
1
] |
Mathlib/Analysis/Convex/Exposed.lean
|
IsExposed.eq_inter_halfspace'
|
[
{
"state_after": "case intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderedRing 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nl✝ : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\nA : Set E\nl : E →L[𝕜] 𝕜\nhAB : IsExposed 𝕜 A {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\nhB : Set.Nonempty {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\n⊢ ∃ l_1 a, {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x} = {x | x ∈ A ∧ a ≤ ↑l_1 x}",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderedRing 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nl : E →L[𝕜] 𝕜\nA✝ B✝ C : Set E\nX : Finset E\nx : E\nA B : Set E\nhAB : IsExposed 𝕜 A B\nhB : Set.Nonempty B\n⊢ ∃ l a, B = {x | x ∈ A ∧ a ≤ ↑l x}",
"tactic": "obtain ⟨l, rfl⟩ := hAB hB"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderedRing 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nl✝ : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\nA : Set E\nl : E →L[𝕜] 𝕜\nhAB : IsExposed 𝕜 A {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\nw : E\nhw : w ∈ {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\n⊢ ∃ l_1 a, {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x} = {x | x ∈ A ∧ a ≤ ↑l_1 x}",
"state_before": "case intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderedRing 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nl✝ : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\nA : Set E\nl : E →L[𝕜] 𝕜\nhAB : IsExposed 𝕜 A {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\nhB : Set.Nonempty {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\n⊢ ∃ l_1 a, {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x} = {x | x ∈ A ∧ a ≤ ↑l_1 x}",
"tactic": "obtain ⟨w, hw⟩ := hB"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderedRing 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nl✝ : E →L[𝕜] 𝕜\nA✝ B C : Set E\nX : Finset E\nx : E\nA : Set E\nl : E →L[𝕜] 𝕜\nhAB : IsExposed 𝕜 A {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\nw : E\nhw : w ∈ {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x}\n⊢ ∃ l_1 a, {x | x ∈ A ∧ ∀ (y : E), y ∈ A → ↑l y ≤ ↑l x} = {x | x ∈ A ∧ a ≤ ↑l_1 x}",
"tactic": "exact ⟨l, l w, Subset.antisymm (fun x hx => ⟨hx.1, hx.2 w hw.1⟩) fun x hx =>\n ⟨hx.1, fun y hy => (hw.2 y hy).trans hx.2⟩⟩"
}
] |
[
124,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.closedBall_zero_eq_preimage_closedBall
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1026847\nR' : Type ?u.1026850\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1026856\n𝕜₃ : Type ?u.1026859\n𝕝 : Type ?u.1026862\nE : Type u_1\nE₂ : Type ?u.1026868\nE₃ : Type ?u.1026871\nF : Type ?u.1026874\nG : Type ?u.1026877\nι : Type ?u.1026880\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜 E\nr : ℝ\n⊢ closedBall p 0 r = ↑p ⁻¹' Metric.closedBall 0 r",
"tactic": "rw [closedBall_zero_eq, preimage_metric_closedBall]"
}
] |
[
824,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
822,
1
] |
Mathlib/Data/Nat/Squarefree.lean
|
Nat.squarefree_and_prime_pow_iff_prime
|
[
{
"state_after": "n : ℕ\n⊢ Squarefree n ∧ IsPrimePow n → Prime n",
"state_before": "n : ℕ\n⊢ Squarefree n ∧ IsPrimePow n ↔ Prime n",
"tactic": "refine' ⟨_, fun hn => ⟨hn.squarefree, hn.isPrimePow⟩⟩"
},
{
"state_after": "n : ℕ\n⊢ (Squarefree n ∧ ∃ p k, Prime p ∧ 0 < k ∧ p ^ k = n) → Prime n",
"state_before": "n : ℕ\n⊢ Squarefree n ∧ IsPrimePow n → Prime n",
"tactic": "rw [isPrimePow_nat_iff]"
},
{
"state_after": "case intro.intro.intro.intro.intro\np k : ℕ\nhp : Prime p\nhk : 0 < k\nh : Squarefree (p ^ k)\n⊢ Prime (p ^ k)",
"state_before": "n : ℕ\n⊢ (Squarefree n ∧ ∃ p k, Prime p ∧ 0 < k ∧ p ^ k = n) → Prime n",
"tactic": "rintro ⟨h, p, k, hp, hk, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\np k : ℕ\nhp : Prime p\nhk : 0 < k\nh : Squarefree p ∧ k = 1\n⊢ Prime (p ^ k)",
"state_before": "case intro.intro.intro.intro.intro\np k : ℕ\nhp : Prime p\nhk : 0 < k\nh : Squarefree (p ^ k)\n⊢ Prime (p ^ k)",
"tactic": "rw [squarefree_pow_iff hp.ne_one hk.ne'] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\np k : ℕ\nhp : Prime p\nhk : 0 < k\nh : Squarefree p ∧ k = 1\n⊢ Prime (p ^ k)",
"tactic": "rwa [h.2, pow_one]"
}
] |
[
103,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.isIso_of_stalkFunctor_map_iso
|
[
{
"state_after": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\n⊢ IsIso ((Sheaf.forget C X).map f)",
"state_before": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\n⊢ IsIso f",
"tactic": "suffices IsIso ((Sheaf.forget C X).map f) by exact isIso_of_fully_faithful (Sheaf.forget C X) f"
},
{
"state_after": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\n⊢ ∀ (U : (Opens ↑X)ᵒᵖ), IsIso (f.val.app U)",
"state_before": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\n⊢ IsIso ((Sheaf.forget C X).map f)",
"tactic": "suffices ∀ U : (Opens X)ᵒᵖ, IsIso (f.1.app U) by\n exact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this"
},
{
"state_after": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\nU : (Opens ↑X)ᵒᵖ\n⊢ IsIso (f.val.app U)",
"state_before": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\n⊢ ∀ (U : (Opens ↑X)ᵒᵖ), IsIso (f.val.app U)",
"tactic": "intro U"
},
{
"state_after": "case h\nC : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\nX✝ : Opens ↑X\n⊢ IsIso (f.val.app X✝.op)",
"state_before": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\nU : (Opens ↑X)ᵒᵖ\n⊢ IsIso (f.val.app U)",
"tactic": "induction U"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\nX✝ : Opens ↑X\n⊢ IsIso (f.val.app X✝.op)",
"tactic": "apply app_isIso_of_stalkFunctor_map_iso"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\nthis : IsIso ((Sheaf.forget C X).map f)\n⊢ IsIso f",
"tactic": "exact isIso_of_fully_faithful (Sheaf.forget C X) f"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁷ : Category C\ninst✝⁶ : HasColimits C\nX Y Z : TopCat\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : PreservesFilteredColimits (forget C)\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (forget C)\ninst✝¹ : ReflectsIsomorphisms (forget C)\nF G : Sheaf C X\nf : F ⟶ G\ninst✝ : ∀ (x : ↑X), IsIso ((stalkFunctor C x).map f.val)\nthis : ∀ (U : (Opens ↑X)ᵒᵖ), IsIso (f.val.app U)\n⊢ IsIso ((Sheaf.forget C X).map f)",
"tactic": "exact @NatIso.isIso_of_isIso_app _ _ _ _ F.1 G.1 f.1 this"
}
] |
[
625,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
eq_orthogonalProjection_of_mem_of_inner_eq_zero
|
[] |
[
498,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
496,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.default_eq
|
[] |
[
43,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.coe_mul
|
[] |
[
551,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
550,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsGLB.inter_Iic_of_mem
|
[] |
[
480,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.IsPath.dropUntil
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\np : Walk G v w\nhc : IsPath p\nh : u ∈ support p\n⊢ IsPath (append (?m.256082 hc h) (dropUntil p u h))",
"tactic": "rwa [← take_spec _ h] at hc"
}
] |
[
1199,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1197,
11
] |
Mathlib/Order/Filter/ENNReal.lean
|
ENNReal.limsup_add_le
|
[] |
[
87,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
div_self_mul_self'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.26175\nM₀ : Type ?u.26178\nG₀ : Type u_1\nM₀' : Type ?u.26184\nG₀' : Type ?u.26187\nF : Type ?u.26190\nF' : Type ?u.26193\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\n⊢ a / (a * a) = a⁻¹⁻¹ * a⁻¹ * a⁻¹",
"tactic": "simp [mul_inv_rev]"
}
] |
[
390,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_eq_of_eq_inv_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.52331\nβ : Type ?u.52334\nG : Type u_1\ninst✝ : Group G\na b c d : G\nh : b = a⁻¹ * c\n⊢ a * b = c",
"tactic": "rw [h, mul_inv_cancel_left]"
}
] |
[
658,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/MeasureTheory/Measure/Content.lean
|
MeasureTheory.Content.outerMeasure_preimage
|
[
{
"state_after": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nf : G ≃ₜ G\nh : ∀ ⦃K : Compacts G⦄, (fun s => ↑(toFun μ s)) (Compacts.map ↑f (_ : Continuous ↑f) K) = (fun s => ↑(toFun μ s)) K\nA : Set G\n⊢ ∀ (s : Set G) (hs : IsOpen s),\n innerContent μ { carrier := ↑f.toEquiv ⁻¹' s, is_open' := (_ : IsOpen (↑f.toEquiv ⁻¹' s)) } =\n innerContent μ { carrier := s, is_open' := hs }",
"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nf : G ≃ₜ G\nh : ∀ ⦃K : Compacts G⦄, (fun s => ↑(toFun μ s)) (Compacts.map ↑f (_ : Continuous ↑f) K) = (fun s => ↑(toFun μ s)) K\nA : Set G\n⊢ ↑(Content.outerMeasure μ) (↑f ⁻¹' A) = ↑(Content.outerMeasure μ) A",
"tactic": "refine' inducedOuterMeasure_preimage _ μ.innerContent_iUnion_nat μ.innerContent_mono _\n (fun _ => f.isOpen_preimage) _"
},
{
"state_after": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nf : G ≃ₜ G\nh : ∀ ⦃K : Compacts G⦄, (fun s => ↑(toFun μ s)) (Compacts.map ↑f (_ : Continuous ↑f) K) = (fun s => ↑(toFun μ s)) K\nA s : Set G\nhs : IsOpen s\n⊢ innerContent μ { carrier := ↑f.toEquiv ⁻¹' s, is_open' := (_ : IsOpen (↑f.toEquiv ⁻¹' s)) } =\n innerContent μ { carrier := s, is_open' := hs }",
"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nf : G ≃ₜ G\nh : ∀ ⦃K : Compacts G⦄, (fun s => ↑(toFun μ s)) (Compacts.map ↑f (_ : Continuous ↑f) K) = (fun s => ↑(toFun μ s)) K\nA : Set G\n⊢ ∀ (s : Set G) (hs : IsOpen s),\n innerContent μ { carrier := ↑f.toEquiv ⁻¹' s, is_open' := (_ : IsOpen (↑f.toEquiv ⁻¹' s)) } =\n innerContent μ { carrier := s, is_open' := hs }",
"tactic": "intro s hs"
},
{
"state_after": "no goals",
"state_before": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : T2Space G\nf : G ≃ₜ G\nh : ∀ ⦃K : Compacts G⦄, (fun s => ↑(toFun μ s)) (Compacts.map ↑f (_ : Continuous ↑f) K) = (fun s => ↑(toFun μ s)) K\nA s : Set G\nhs : IsOpen s\n⊢ innerContent μ { carrier := ↑f.toEquiv ⁻¹' s, is_open' := (_ : IsOpen (↑f.toEquiv ⁻¹' s)) } =\n innerContent μ { carrier := s, is_open' := hs }",
"tactic": "convert μ.innerContent_comap f h ⟨s, hs⟩"
}
] |
[
308,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.iSup_join
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.103075\nι✝ : Sort x\nf✝ g : Filter α\ns t : Set α\nι : Sort w\nf : ι → Filter (Filter α)\nx : Set α\n⊢ (x ∈ ⨆ (x : ι), join (f x)) ↔ x ∈ join (⨆ (x : ι), f x)",
"tactic": "simp only [mem_iSup, mem_join]"
}
] |
[
865,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/RingTheory/NonZeroDivisors.lean
|
mem_nonZeroDivisors_of_ne_zero
|
[] |
[
123,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Order/CompleteBooleanAlgebra.lean
|
biInf_sup_biInf
|
[] |
[
225,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Algebra/Opposites.lean
|
MulOpposite.unop_comp_op
|
[] |
[
118,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.isPartition_single_iff
|
[] |
[
310,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.compl_Ici
|
[] |
[
1066,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1065,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
|
Complex.hasSum_sin
|
[
{
"state_after": "case h.e'_5\nz : ℂ\n⊢ (fun n => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) = fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I",
"state_before": "z : ℂ\n⊢ HasSum (fun n => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) (sin z)",
"tactic": "convert Complex.hasSum_sin' z using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nz : ℂ\n⊢ (fun n => (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)!) = fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I",
"tactic": "simp_rw [mul_pow, pow_succ', pow_mul, Complex.I_sq, ← mul_assoc, mul_div_assoc, div_right_comm,\n div_self Complex.I_ne_zero, mul_comm _ ((-1 : ℂ) ^ _), mul_one_div, mul_div_assoc, mul_assoc]"
}
] |
[
81,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.dvd_gcd
|
[
{
"state_after": "no goals",
"state_before": "k m n : Nat\n⊢ k ∣ m → k ∣ n → k ∣ gcd m n",
"tactic": "induction m, n using gcd.induction with intro km kn\n| H0 n => rw [gcd_zero_left]; exact kn\n| H1 n m _ IH => rw [gcd_rec]; exact IH ((dvd_mod_iff km).2 kn) km"
},
{
"state_after": "case H0\nk n : Nat\nkm : k ∣ 0\nkn : k ∣ n\n⊢ k ∣ n",
"state_before": "case H0\nk n : Nat\nkm : k ∣ 0\nkn : k ∣ n\n⊢ k ∣ gcd 0 n",
"tactic": "rw [gcd_zero_left]"
},
{
"state_after": "no goals",
"state_before": "case H0\nk n : Nat\nkm : k ∣ 0\nkn : k ∣ n\n⊢ k ∣ n",
"tactic": "exact kn"
},
{
"state_after": "case H1\nk n m : Nat\na✝ : 0 < n\nIH : k ∣ m % n → k ∣ n → k ∣ gcd (m % n) n\nkm : k ∣ n\nkn : k ∣ m\n⊢ k ∣ gcd (m % n) n",
"state_before": "case H1\nk n m : Nat\na✝ : 0 < n\nIH : k ∣ m % n → k ∣ n → k ∣ gcd (m % n) n\nkm : k ∣ n\nkn : k ∣ m\n⊢ k ∣ gcd n m",
"tactic": "rw [gcd_rec]"
},
{
"state_after": "no goals",
"state_before": "case H1\nk n m : Nat\na✝ : 0 < n\nIH : k ∣ m % n → k ∣ n → k ∣ gcd (m % n) n\nkm : k ∣ n\nkn : k ∣ m\n⊢ k ∣ gcd (m % n) n",
"tactic": "exact IH ((dvd_mod_iff km).2 kn) km"
}
] |
[
52,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
49,
1
] |
Mathlib/GroupTheory/GroupAction/Prod.lean
|
Prod.smul_mk
|
[] |
[
61,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_sin_pow_three
|
[
{
"state_after": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, sin x ^ (2 * 1 + 1) * cos x ^ 0) = ∫ (u : ℝ) in cos b..cos a, u ^ 0 * (1 - u ^ 2) ^ 1\n⊢ (∫ (x : ℝ) in a..b, sin x ^ 3) = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3",
"state_before": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, sin x ^ 3) = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3",
"tactic": "have := @integral_sin_pow_odd_mul_cos_pow a b 1 0"
},
{
"state_after": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, sin x ^ 3) = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3\n⊢ (∫ (x : ℝ) in a..b, sin x ^ 3) = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3",
"state_before": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, sin x ^ (2 * 1 + 1) * cos x ^ 0) = ∫ (u : ℝ) in cos b..cos a, u ^ 0 * (1 - u ^ 2) ^ 1\n⊢ (∫ (x : ℝ) in a..b, sin x ^ 3) = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3",
"tactic": "norm_num at this"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nn : ℕ\nthis : (∫ (x : ℝ) in a..b, sin x ^ 3) = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3\n⊢ (∫ (x : ℝ) in a..b, sin x ^ 3) = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3",
"tactic": "exact this"
}
] |
[
822,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.smul_iUnion₂
|
[] |
[
264,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/CategoryTheory/Action.lean
|
CategoryTheory.ActionCategory.coe_back
|
[] |
[
88,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.angle_lt_pi_of_not_collinear
|
[] |
[
477,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
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