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Mathlib/Algebra/GroupPower/Order.lean | pow_le_pow_of_le_one' | [] | [
78,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean | HasStrictFDerivAt.localInverse_continuousAt | [] | [
648,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
646,
1
] |
Mathlib/Topology/Semicontinuous.lean | lowerSemicontinuousOn_sum | [] | [
536,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
533,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.piecewise_empty | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42149\nδ : Type ?u.42152\ninst✝ : MeasurableSpace α\nf g : α →ₛ β\n⊢ Set.piecewise ∅ ↑f ↑g = ↑g",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42149\nδ : Type ?u.42152\ninst✝ : MeasurableSpace α\nf g : α →ₛ β\n⊢ ↑(piecewise ∅ (_ : MeasurableSet ∅) f g) = ↑g",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42149\nδ : Type ?u.42152\ninst✝ : MeasurableSpace α\nf g : α →ₛ β\n⊢ Set.piecewise ∅ ↑f ↑g = ↑g",
"tactic": "convert Set.piecewise_empty f g"
}
] | [
267,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
] |
Mathlib/Analysis/Analytic/Basic.lean | Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero | [
{
"state_after": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nh : (fun y => ↑p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)\ny : E\nc : ℝ\nc_pos : 0 < c\nhc : IsBigOWith c (𝓝 0) (fun y => ↑p fun x => y) fun y => ‖y‖ ^ (n + 1)\n⊢ (↑p fun x => y) = 0",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nh : (fun y => ↑p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)\ny : E\n⊢ (↑p fun x => y) = 0",
"tactic": "obtain ⟨c, c_pos, hc⟩ := h.exists_pos"
},
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nh : (fun y => ↑p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)\ny : E\nc : ℝ\nc_pos : 0 < c\nhc : IsBigOWith c (𝓝 0) (fun y => ↑p fun x => y) fun y => ‖y‖ ^ (n + 1)\nt : Set E\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (n + 1)‖\nt_open : IsOpen t\nz_mem : 0 ∈ t\n⊢ (↑p fun x => y) = 0",
"state_before": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nh : (fun y => ↑p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)\ny : E\nc : ℝ\nc_pos : 0 < c\nhc : IsBigOWith c (𝓝 0) (fun y => ↑p fun x => y) fun y => ‖y‖ ^ (n + 1)\n⊢ (↑p fun x => y) = 0",
"tactic": "obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nh : (fun y => ↑p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)\ny : E\nc : ℝ\nc_pos : 0 < c\nhc : IsBigOWith c (𝓝 0) (fun y => ↑p fun x => y) fun y => ‖y‖ ^ (n + 1)\nt : Set E\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (n + 1)‖\nt_open : IsOpen t\nz_mem : 0 ∈ t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\n⊢ (↑p fun x => y) = 0",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nh : (fun y => ↑p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)\ny : E\nc : ℝ\nc_pos : 0 < c\nhc : IsBigOWith c (𝓝 0) (fun y => ↑p fun x => y) fun y => ‖y‖ ^ (n + 1)\nt : Set E\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (n + 1)‖\nt_open : IsOpen t\nz_mem : 0 ∈ t\n⊢ (↑p fun x => y) = 0",
"tactic": "obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (n + 1)‖\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\n⊢ (↑p fun x => y) = 0",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nh : (fun y => ↑p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)\ny : E\nc : ℝ\nc_pos : 0 < c\nhc : IsBigOWith c (𝓝 0) (fun y => ↑p fun x => y) fun y => ‖y‖ ^ (n + 1)\nt : Set E\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (n + 1)‖\nt_open : IsOpen t\nz_mem : 0 ∈ t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\n⊢ (↑p fun x => y) = 0",
"tactic": "clear h hc z_mem"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.zero\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.zero + 1)‖\n⊢ (↑p fun x => y) = 0\n\ncase intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\n⊢ (↑p fun x => y) = 0",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (n + 1)‖\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\n⊢ (↑p fun x => y) = 0",
"tactic": "cases' n with n"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.zero\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.zero + 1)‖\n⊢ (↑p fun x => y) = 0",
"tactic": "exact norm_eq_zero.mp (by\n simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,\n mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.zero + 1)‖\n⊢ ‖↑p fun x => y‖ = 0",
"tactic": "simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,\n mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : ¬y = 0\n⊢ (↑p fun x => y) = 0",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\n⊢ (↑p fun x => y) = 0",
"tactic": "refine' Or.elim (Classical.em (y = 0))\n (fun hy => by simpa only [hy] using p.map_zero) fun hy => _"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\n⊢ (↑p fun x => y) = 0",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : ¬y = 0\n⊢ (↑p fun x => y) = 0",
"tactic": "replace hy := norm_pos_iff.mpr hy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\n⊢ (↑p fun x => y) = 0",
"tactic": "refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"tactic": "have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"tactic": "obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜\n (lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\nh₁ : ‖k • y‖ < δ\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"tactic": "have h₁ : ‖k • y‖ < δ := by\n rw [norm_smul]\n exact inv_mul_cancel_right₀ hy.ne.symm δ ▸\n mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\nh₁ : ‖k • y‖ < δ\nh₂ : ‖↑p fun x => k • y‖ ≤ ‖k‖ ^ Nat.succ n * (‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)))\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\nh₁ : ‖k • y‖ < δ\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"tactic": "have h₂ :=\n calc\n ‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by\n simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))\n _ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by\n simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]\n ring"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\nh₁ : ‖k • y‖ < δ\nh₂ : ‖↑p fun x => k • y‖ ≤ ‖k‖ ^ Nat.succ n * (‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)))\nh₃ : ‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)) < ε\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\nh₁ : ‖k • y‖ < δ\nh₂ : ‖↑p fun x => k • y‖ ≤ ‖k‖ ^ Nat.succ n * (‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)))\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"tactic": "have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=\n inv_mul_cancel_right₀ h₀.ne.symm ε ▸\n mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\nh₁ : ‖k • y‖ < δ\nh₂ : ‖↑p fun x => k • y‖ ≤ ‖k‖ ^ Nat.succ n * (‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)))\nh₃ : ‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)) < ε\n⊢ ‖↑p fun x => y‖ ≤ 0 + ε",
"tactic": "calc\n ‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by\n simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,\n Finset.card_fin] using\n congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)\n _ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr\n _ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by\n rw [← mul_assoc]\n simp [norm_mul, mul_pow]\n _ ≤ 0 + ε := by\n rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]\n simpa using h₃.le"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : y = 0\n⊢ (↑p fun x => y) = 0",
"tactic": "simpa only [hy] using p.map_zero"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\n⊢ ‖k‖ * ‖y‖ < δ",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\n⊢ ‖k • y‖ < δ",
"tactic": "rw [norm_smul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.1082661\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ny : E\nc : ℝ\nc_pos : 0 < c\nt : Set E\nt_open : IsOpen t\nδ : ℝ\nδ_pos : δ > 0\nδε : Metric.ball 0 δ ⊆ t\nn : ℕ\np : ContinuousMultilinearMap 𝕜 (fun i => E) F\nht : ∀ (x : E), x ∈ t → ‖↑p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖\nhy : 0 < ‖y‖\nε : ℝ\nε_pos : 0 < ε\nh₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)\nk : 𝕜\nk_pos : 0 < ‖k‖\nk_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)\n⊢ ‖k‖ * ‖y‖ < δ",
"tactic": "exact inv_mul_cancel_right₀ hy.ne.symm δ ▸\n mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy"
},
{
"state_after": "no goals",
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Mathlib/Order/Disjoint.lean | Disjoint.of_disjoint_inf_of_le | [] | [
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60
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Mathlib/CategoryTheory/Limits/Cones.lean | CategoryTheory.Limits.Cocones.cocone_iso_of_hom_iso | [
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Mathlib/Combinatorics/Quiver/Cast.lean | Quiver.Hom.cast_eq_cast | [
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Mathlib/Analysis/Calculus/Dslope.lean | differentiableAt_dslope_of_ne | [
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] |
Mathlib/Analysis/NormedSpace/PiLp.lean | PiLp.infty_equiv_isometry | [
{
"state_after": "no goals",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.215698\n𝕜' : Type ?u.215701\nι : Type u_2\nα : ι → Type ?u.215709\nβ : ι → Type u_1\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : PiLp ⊤ β\n⊢ edist (↑(PiLp.equiv ⊤ β) x) (↑(PiLp.equiv ⊤ β) y) ≤ edist x y",
"tactic": "simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_equiv ∞ β x y"
},
{
"state_after": "no goals",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.215698\n𝕜' : Type ?u.215701\nι : Type u_2\nα : ι → Type ?u.215709\nβ : ι → Type u_1\ninst✝² : Fintype ι\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : PiLp ⊤ β\n⊢ edist x y ≤ edist (↑(PiLp.equiv ⊤ β) x) (↑(PiLp.equiv ⊤ β) y)",
"tactic": "simpa only [ENNReal.div_top, ENNReal.zero_toReal, NNReal.rpow_zero, ENNReal.coe_one,\n one_mul] using antilipschitzWith_equiv ∞ β x y"
}
] | [
531,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
526,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.lift_mk | [] | [
286,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
] |
Mathlib/Algebra/DirectSum/Module.lean | DirectSum.component.lof_self | [] | [
225,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
] |
Mathlib/Data/Part.lean | Part.inv_mem_inv | [
{
"state_after": "α : Type u_1\nβ : Type ?u.58569\nγ : Type ?u.58572\ninst✝ : Inv α\na : Part α\nma : α\nha : ma ∈ a\n⊢ ∃ a_1, a_1 ∈ a ∧ a_1⁻¹ = ma⁻¹",
"state_before": "α : Type u_1\nβ : Type ?u.58569\nγ : Type ?u.58572\ninst✝ : Inv α\na : Part α\nma : α\nha : ma ∈ a\n⊢ ma⁻¹ ∈ a⁻¹",
"tactic": "simp [inv_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.58569\nγ : Type ?u.58572\ninst✝ : Inv α\na : Part α\nma : α\nha : ma ∈ a\n⊢ ∃ a_1, a_1 ∈ a ∧ a_1⁻¹ = ma⁻¹",
"tactic": "aesop"
}
] | [
746,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
745,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.norm_fst | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CstarRing A\na : 𝓜(𝕜, A)\n⊢ ‖a.fst‖ = ‖a‖",
"tactic": "simp only [norm_def, toProdHom_apply, Prod.norm_def, norm_fst_eq_snd, max_eq_right le_rfl]"
}
] | [
643,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
642,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | Real.contDiffAt_arcsin_iff | [] | [
133,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean | BoxIntegral.Box.coe_subset_coe | [] | [
170,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
170,
1
] |
Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | AlgebraicTopology.DoldKan.PInfty_idem | [
{
"state_after": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f (PInfty ≫ PInfty) n = HomologicalComplex.Hom.f PInfty n",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ PInfty ≫ PInfty = PInfty",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f (PInfty ≫ PInfty) n = HomologicalComplex.Hom.f PInfty n",
"tactic": "exact PInfty_f_idem n"
}
] | [
114,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean | Ideal.span_singleton_eq_span_singleton | [
{
"state_after": "α✝ : Type u\nβ : Type v\na b : α✝\ninst✝² : CommSemiring α✝\nI : Ideal α✝\nα : Type u\ninst✝¹ : CommRing α\ninst✝ : IsDomain α\nx y : α\n⊢ span {y} ≤ span {x} ∧ span {x} ≤ span {y} ↔ x ∣ y ∧ y ∣ x",
"state_before": "α✝ : Type u\nβ : Type v\na b : α✝\ninst✝² : CommSemiring α✝\nI : Ideal α✝\nα : Type u\ninst✝¹ : CommRing α\ninst✝ : IsDomain α\nx y : α\n⊢ span {x} = span {y} ↔ Associated x y",
"tactic": "rw [← dvd_dvd_iff_associated, le_antisymm_iff, and_comm]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\na b : α✝\ninst✝² : CommSemiring α✝\nI : Ideal α✝\nα : Type u\ninst✝¹ : CommRing α\ninst✝ : IsDomain α\nx y : α\n⊢ span {y} ≤ span {x} ∧ span {x} ≤ span {y} ↔ x ∣ y ∧ y ∣ x",
"tactic": "apply and_congr <;> rw [span_singleton_le_span_singleton]"
}
] | [
514,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
511,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.mul_invOfUnit | [] | [
1933,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1931,
1
] |
Mathlib/Topology/Order/Hom/Basic.lean | ContinuousOrderHom.coe_comp | [] | [
166,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsBigO.smul | [
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.566716\nE : Type ?u.566719\nF : Type ?u.566722\nG : Type ?u.566725\nE' : Type u_4\nF' : Type u_5\nG' : Type ?u.566734\nE'' : Type ?u.566737\nF'' : Type ?u.566740\nG'' : Type ?u.566743\nR : Type ?u.566746\nR' : Type ?u.566749\n𝕜 : Type u_2\n𝕜' : Type u_3\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝¹ : NormedSpace 𝕜 E'\ninst✝ : NormedSpace 𝕜' F'\nk₁ : α → 𝕜\nk₂ : α → 𝕜'\nh₁ : k₁ =O[l] k₂\nh₂ : f' =O[l] g'\nx✝ : α\n⊢ ‖k₂ x✝‖ * ‖g' x✝‖ = ‖k₂ x✝ • g' x✝‖",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.566716\nE : Type ?u.566719\nF : Type ?u.566722\nG : Type ?u.566725\nE' : Type u_4\nF' : Type u_5\nG' : Type ?u.566734\nE'' : Type ?u.566737\nF'' : Type ?u.566740\nG'' : Type ?u.566743\nR : Type ?u.566746\nR' : Type ?u.566749\n𝕜 : Type u_2\n𝕜' : Type u_3\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝¹ : NormedSpace 𝕜 E'\ninst✝ : NormedSpace 𝕜' F'\nk₁ : α → 𝕜\nk₂ : α → 𝕜'\nh₁ : k₁ =O[l] k₂\nh₂ : f' =O[l] g'\n⊢ ∀ (x : α), ‖k₂ x‖ * ‖g' x‖ = ‖k₂ x • g' x‖",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.566716\nE : Type ?u.566719\nF : Type ?u.566722\nG : Type ?u.566725\nE' : Type u_4\nF' : Type u_5\nG' : Type ?u.566734\nE'' : Type ?u.566737\nF'' : Type ?u.566740\nG'' : Type ?u.566743\nR : Type ?u.566746\nR' : Type ?u.566749\n𝕜 : Type u_2\n𝕜' : Type u_3\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝¹ : NormedSpace 𝕜 E'\ninst✝ : NormedSpace 𝕜' F'\nk₁ : α → 𝕜\nk₂ : α → 𝕜'\nh₁ : k₁ =O[l] k₂\nh₂ : f' =O[l] g'\nx✝ : α\n⊢ ‖k₂ x✝‖ * ‖g' x✝‖ = ‖k₂ x✝ • g' x✝‖",
"tactic": "simp only [norm_smul]"
}
] | [
1759,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1756,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.image_const_sub_Ici | [
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ici b = Iic (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => a - x) '' Ici b = Iic (a - b)",
"tactic": "have := image_comp (fun x => a + x) fun x => -x"
},
{
"state_after": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ici b = Iic (a - b)",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), ((fun x => a + x) ∘ fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ici b = Iic (a - b)",
"tactic": "dsimp [Function.comp] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\nthis : ∀ (a_1 : Set α), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1)\n⊢ (fun x => a - x) '' Ici b = Iic (a - b)",
"tactic": "simp [sub_eq_add_neg, this, add_comm]"
}
] | [
323,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
321,
1
] |
Mathlib/Order/UpperLower/LocallyFinite.lean | Set.Finite.upperClosure | [
{
"state_after": "α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderTop α\nhs : Set.Finite s\n⊢ Set.Finite (⋃ (a : α) (_ : a ∈ s), Ici a)",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderTop α\nhs : Set.Finite s\n⊢ Set.Finite ↑(upperClosure s)",
"tactic": "rw [coe_upperClosure]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderTop α\nhs : Set.Finite s\n⊢ Set.Finite (⋃ (a : α) (_ : a ∈ s), Ici a)",
"tactic": "exact hs.biUnion fun _ _ => finite_Ici _"
}
] | [
28,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
25,
11
] |
Mathlib/Data/Prod/PProd.lean | PProd.forall' | [] | [
40,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | MeasureTheory.measure_inter_null_of_null_left | [] | [
348,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
347,
1
] |
Mathlib/Data/MvPolynomial/Monad.lean | MvPolynomial.aeval_eq_bind₁ | [] | [
110,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Data/List/Rotate.lean | List.isRotated_reverse_comm_iff | [
{
"state_after": "case mpr\nα : Type u\nl l' : List α\nh : l ~r reverse l'\n⊢ reverse l ~r l'",
"state_before": "case mpr\nα : Type u\nl l' : List α\n⊢ l ~r reverse l' → reverse l ~r l'",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u\nl l' : List α\nh : l ~r reverse l'\n⊢ reverse l ~r l'",
"tactic": "simpa using h.reverse"
}
] | [
526,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
523,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.two_nsmul_toReal_eq_two_mul_add_two_pi | [
{
"state_after": "θ : Angle\n⊢ toReal (2 • ↑(toReal θ)) = 2 * toReal θ + 2 * π ↔ toReal θ ≤ -π / 2",
"state_before": "θ : Angle\n⊢ toReal (2 • θ) = 2 * toReal θ + 2 * π ↔ toReal θ ≤ -π / 2",
"tactic": "nth_rw 1 [← coe_toReal θ]"
},
{
"state_after": "θ : Angle\n⊢ -3 * π < 2 * toReal θ ∧ 2 * toReal θ ≤ -π ↔ toReal θ ≤ -π / 2",
"state_before": "θ : Angle\n⊢ toReal (2 • ↑(toReal θ)) = 2 * toReal θ + 2 * π ↔ toReal θ ≤ -π / 2",
"tactic": "rw [← coe_nsmul, two_nsmul, ← two_mul, toReal_coe_eq_self_add_two_pi_iff, Set.mem_Ioc]"
},
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ -3 * π < 2 * toReal θ ∧ 2 * toReal θ ≤ -π ↔ toReal θ ≤ -π / 2",
"tactic": "refine'\n ⟨fun h => by linarith, fun h =>\n ⟨by linarith [pi_pos, neg_pi_lt_toReal θ], (le_div_iff' (zero_lt_two' ℝ)).1 h⟩⟩"
},
{
"state_after": "no goals",
"state_before": "θ : Angle\nh : -3 * π < 2 * toReal θ ∧ 2 * toReal θ ≤ -π\n⊢ toReal θ ≤ -π / 2",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "θ : Angle\nh : toReal θ ≤ -π / 2\n⊢ -3 * π < 2 * toReal θ",
"tactic": "linarith [pi_pos, neg_pi_lt_toReal θ]"
}
] | [
718,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
712,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.comap_sSup | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.255463\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns✝ : Set α\nt : Set β\ns : Set (Filter β)\nm : α → β\n⊢ comap m (sSup s) = ⨆ (f : Filter β) (_ : f ∈ s), comap m f",
"tactic": "simp only [sSup_eq_iSup, comap_iSup, eq_self_iff_true]"
}
] | [
2240,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2239,
1
] |
Mathlib/Topology/PartitionOfUnity.lean | PartitionOfUnity.locallyFinite | [] | [
144,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
11
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Iio_injective | [] | [
1120,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1119,
1
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.card_support_eq_zero | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.20446\nγ : Type ?u.20449\nι : Type ?u.20452\nM : Type u_2\nM' : Type ?u.20458\nN : Type ?u.20461\nP : Type ?u.20464\nG : Type ?u.20467\nH : Type ?u.20470\nR : Type ?u.20473\nS : Type ?u.20476\ninst✝ : Zero M\nf : α →₀ M\n⊢ card f.support = 0 ↔ f = 0",
"tactic": "simp"
}
] | [
229,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
229,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean | Unitization.inl_zero | [] | [
258,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Mathlib/Algebra/Module/LinearMap.lean | IsLinearMap.map_neg | [] | [
724,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
723,
1
] |
Mathlib/Data/Finset/Fold.lean | Finset.fold_cons | [
{
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{
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51,
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Mathlib/GroupTheory/Complement.lean | Subgroup.smul_toFun | [] | [
466,
99
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461,
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Mathlib/Data/MvPolynomial/Variables.lean | MvPolynomial.degreeOf_mul_X_ne | [
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{
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{
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{
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{
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},
{
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}
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563,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
554,
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Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | BoxIntegral.IntegrationParams.henstock_le_mcShane | [
{
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282,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
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Mathlib/Data/PFun.lean | PFun.preimage_inter | [] | [
447,
27
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446,
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Mathlib/Algebra/Homology/ImageToKernel.lean | image_le_kernel | [
{
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43,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/List/Basic.lean | List.getLast_cons_cons | [] | [
732,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
730,
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Mathlib/RingTheory/PowerBasis.lean | PowerBasis.finiteDimensional | [] | [
85,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
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Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub | [
{
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] | [
1621,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1617,
1
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Mathlib/Analysis/InnerProductSpace/Calculus.lean | contDiff_inner | [] | [
64,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
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Mathlib/Topology/Covering.lean | IsCoveringMap.continuous | [] | [
165,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
11
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Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | HomogeneousLocalization.NumDenSameDeg.deg_add | [] | [
194,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
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Mathlib/Topology/Instances/Real.lean | Function.Periodic.compact_of_continuous | [
{
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"tactic": "rw [← hp.image_uIcc hc 0]"
},
{
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}
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207,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
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Mathlib/GroupTheory/Subgroup/ZPowers.lean | Int.mem_zmultiples_iff | [
{
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"tactic": "rw [mul_comm, eq_comm, ← smul_eq_mul]"
}
] | [
162,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
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Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | MeasureTheory.Measure.prod_sum | [
{
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"tactic": "refine' prod_eq fun s t hs ht => _"
},
{
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"tactic": "simp_rw [sum_apply _ (hs.prod ht), sum_apply _ ht, prod_prod, ENNReal.tsum_mul_left]"
}
] | [
574,
87
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571,
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Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.FinMeasAdditive.zero | [
{
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"tactic": "simp"
}
] | [
106,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
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Mathlib/Algebra/Ring/Semiconj.lean | SemiconjBy.sub_right | [
{
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}
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94,
61
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92,
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Mathlib/MeasureTheory/Function/Jacobian.lean | MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2 | [
{
"state_after": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 ((∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑0 * ↑↑μ s))\n⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"tactic": "have :\n Tendsto (fun ε : ℝ≥0 => (∫⁻ x in s, ENNReal.ofReal (|(f' x).det|) ∂μ) + 2 * ε * μ s) (𝓝[>] 0)\n (𝓝 ((∫⁻ x in s, ENNReal.ofReal (|(f' x).det|) ∂μ) + 2 * (0 : ℝ≥0) * μ s)) := by\n apply Tendsto.mono_left _ nhdsWithin_le_nhds\n refine' tendsto_const_nhds.add _\n refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)\n exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)"
},
{
"state_after": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\n⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 ((∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑0 * ↑↑μ s))\n⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"tactic": "simp only [add_zero, MulZeroClass.zero_mul, MulZeroClass.mul_zero, ENNReal.coe_zero] at this"
},
{
"state_after": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[Ioi 0] 0,\n ↑↑μ (f '' s) ≤ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑c * ↑↑μ s",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\n⊢ ↑↑μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ",
"tactic": "apply ge_of_tendsto this"
},
{
"state_after": "case h\nE : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\n⊢ ∀ (a : ℝ≥0),\n a ∈ Ioi 0 →\n ↑↑μ (f '' s) ≤ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑a * ↑↑μ s",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[Ioi 0] 0,\n ↑↑μ (f '' s) ≤ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑c * ↑↑μ s",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nE : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\nε : ℝ≥0\nεpos : 0 < ε\n⊢ ↑↑μ (f '' s) ≤ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s",
"state_before": "case h\nE : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\n⊢ ∀ (a : ℝ≥0),\n a ∈ Ioi 0 →\n ↑↑μ (f '' s) ≤ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑a * ↑↑μ s",
"tactic": "rintro ε (εpos : 0 < ε)"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nthis :\n Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ))\nε : ℝ≥0\nεpos : 0 < ε\n⊢ ↑↑μ (f '' s) ≤ (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s",
"tactic": "exact addHaar_image_le_lintegral_abs_det_fderiv_aux1 μ hs hf' εpos"
},
{
"state_after": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s) (𝓝 0)\n (𝓝 ((∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑0 * ↑↑μ s))",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s)\n (𝓝[Ioi 0] 0) (𝓝 ((∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑0 * ↑↑μ s))",
"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds"
},
{
"state_after": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ Tendsto (fun ε => 2 * ↑ε * ↑↑μ s) (𝓝 0) (𝓝 (2 * ↑0 * ↑↑μ s))",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ Tendsto (fun ε => (∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑ε * ↑↑μ s) (𝓝 0)\n (𝓝 ((∫⁻ (x : E) in s, ENNReal.ofReal (abs (ContinuousLinearMap.det (f' x))) ∂μ) + 2 * ↑0 * ↑↑μ s))",
"tactic": "refine' tendsto_const_nhds.add _"
},
{
"state_after": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ Tendsto (fun ε => 2 * ↑ε) (𝓝 0) (𝓝 (2 * ↑0))",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ Tendsto (fun ε => 2 * ↑ε * ↑↑μ s) (𝓝 0) (𝓝 (2 * ↑0 * ↑↑μ s))",
"tactic": "refine' ENNReal.Tendsto.mul_const _ (Or.inr h's)"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type ?u.785438\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nh's : ↑↑μ s ≠ ⊤\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\n⊢ Tendsto (fun ε => 2 * ↑ε) (𝓝 0) (𝓝 (2 * ↑0))",
"tactic": "exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top)"
}
] | [
903,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
888,
1
] |
Mathlib/Data/PNat/Factors.lean | PNat.factorMultiset_pow | [
{
"state_after": "n : ℕ+\nm : ℕ\nu : PrimeMultiset := factorMultiset n\n⊢ factorMultiset (Pow.pow n m) = m • factorMultiset n",
"state_before": "n : ℕ+\nm : ℕ\n⊢ factorMultiset (Pow.pow n m) = m • factorMultiset n",
"tactic": "let u := factorMultiset n"
},
{
"state_after": "n : ℕ+\nm : ℕ\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (Pow.pow n m) = m • factorMultiset n",
"state_before": "n : ℕ+\nm : ℕ\nu : PrimeMultiset := factorMultiset n\n⊢ factorMultiset (Pow.pow n m) = m • factorMultiset n",
"tactic": "have : n = u.prod := (prod_factorMultiset n).symm"
},
{
"state_after": "n : ℕ+\nm : ℕ\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (PrimeMultiset.prod (m • u)) = m • factorMultiset (PrimeMultiset.prod u)",
"state_before": "n : ℕ+\nm : ℕ\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (Pow.pow n m) = m • factorMultiset n",
"tactic": "rw [this, ← PrimeMultiset.prod_smul]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ+\nm : ℕ\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n⊢ factorMultiset (PrimeMultiset.prod (m • u)) = m • factorMultiset (PrimeMultiset.prod u)",
"tactic": "repeat' rw [PrimeMultiset.factorMultiset_prod]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ+\nm : ℕ\nu : PrimeMultiset := factorMultiset n\nthis : n = PrimeMultiset.prod u\n⊢ m • u = m • factorMultiset (PrimeMultiset.prod u)",
"tactic": "rw [PrimeMultiset.factorMultiset_prod]"
}
] | [
314,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/Topology/Homotopy/Basic.lean | ContinuousMap.Homotopy.congr_arg | [] | [
203,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
1
] |
Mathlib/Topology/Constructions.lean | Continuous.fst | [] | [
332,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.map_iInf_comap | [
{
"state_after": "α : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\n⊢ ↑(⨅ (i : ι), ↑(map f) (↑(comap f) (m i))) s ≤ ↑(↑(map f) (⨅ (i : ι), ↑(comap f) (m i))) s",
"state_before": "α : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\n⊢ ↑(map f) (⨅ (i : ι), ↑(comap f) (m i)) = ⨅ (i : ι), ↑(map f) (↑(comap f) (m i))",
"tactic": "refine' (map_iInf_le _ _).antisymm fun s => _"
},
{
"state_after": "α : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\n⊢ ∀ (i : ℕ → Set α),\n f ⁻¹' s ⊆ iUnion i →\n (⨅ (t : ℕ → Set β) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ (i : ι), ↑(m i) (f '' (f ⁻¹' t n))) ≤\n ∑' (n : ℕ), ⨅ (i_2 : ι), ↑(m i_2) (f '' i n)",
"state_before": "α : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\n⊢ ↑(⨅ (i : ι), ↑(map f) (↑(comap f) (m i))) s ≤ ↑(↑(map f) (⨅ (i : ι), ↑(comap f) (m i))) s",
"tactic": "simp only [map_apply, comap_apply, iInf_apply, le_iInf_iff]"
},
{
"state_after": "case refine'_1\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ s ⊆ ⋃ (n : ℕ), f '' t n ∪ range fᶜ\n\ncase refine'_2\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ (∑' (n : ℕ), ⨅ (i : ι), ↑(m i) (f '' (f ⁻¹' (fun n => f '' t n ∪ range fᶜ) n))) ≤\n ∑' (n : ℕ), ⨅ (i : ι), ↑(m i) (f '' t n)",
"state_before": "α : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\n⊢ ∀ (i : ℕ → Set α),\n f ⁻¹' s ⊆ iUnion i →\n (⨅ (t : ℕ → Set β) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ (i : ι), ↑(m i) (f '' (f ⁻¹' t n))) ≤\n ∑' (n : ℕ), ⨅ (i_2 : ι), ↑(m i_2) (f '' i n)",
"tactic": "refine' fun t ht => iInf_le_of_le (fun n => f '' t n ∪ range fᶜ) (iInf_le_of_le _ _)"
},
{
"state_after": "case refine'_1\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ f '' (f ⁻¹' s) ⊆ f '' ⋃ (i : ℕ), t i",
"state_before": "case refine'_1\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ s ⊆ ⋃ (n : ℕ), f '' t n ∪ range fᶜ",
"tactic": "rw [← iUnion_union, Set.union_comm, ← inter_subset, ← image_iUnion, ←\n image_preimage_eq_inter_range]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ f '' (f ⁻¹' s) ⊆ f '' ⋃ (i : ℕ), t i",
"tactic": "exact image_subset _ ht"
},
{
"state_after": "case refine'_2\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\nn : ℕ\ni : ι\n⊢ f '' (f ⁻¹' (fun n => f '' t n ∪ range fᶜ) n) ⊆ f '' t n",
"state_before": "case refine'_2\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ (∑' (n : ℕ), ⨅ (i : ι), ↑(m i) (f '' (f ⁻¹' (fun n => f '' t n ∪ range fᶜ) n))) ≤\n ∑' (n : ℕ), ⨅ (i : ι), ↑(m i) (f '' t n)",
"tactic": "refine' ENNReal.tsum_le_tsum fun n => iInf_mono fun i => (m i).mono _"
},
{
"state_after": "case refine'_2\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\nn : ℕ\ni : ι\n⊢ f ⁻¹' (f '' t n) ⊆ f ⁻¹' (f '' t n)",
"state_before": "case refine'_2\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\nn : ℕ\ni : ι\n⊢ f '' (f ⁻¹' (fun n => f '' t n ∪ range fᶜ) n) ⊆ f '' t n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_3\nι : Sort u_1\nβ : Type u_2\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\nn : ℕ\ni : ι\n⊢ f ⁻¹' (f '' t n) ⊆ f ⁻¹' (f '' t n)",
"tactic": "exact subset_refl _"
}
] | [
1253,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1243,
1
] |
Mathlib/Order/Ideal.lean | Order.Ideal.coe_ssubset_coe | [] | [
159,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/Data/Nat/Multiplicity.lean | Nat.Prime.multiplicity_choose_prime_pow_add_multiplicity | [
{
"state_after": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\nhdisj :\n Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i ∣ k) (Ico 1 (succ n)))\n⊢ multiplicity p (choose (p ^ n) k) + multiplicity p k ≤ ↑n",
"state_before": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\n⊢ multiplicity p (choose (p ^ n) k) + multiplicity p k ≤ ↑n",
"tactic": "have hdisj :\n Disjoint ((Ico 1 n.succ).filter fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i)\n ((Ico 1 n.succ).filter fun i => p ^ i ∣ k) := by\n simp (config := { contextual := true }) [disjoint_right, *, dvd_iff_mod_eq_zero,\n Nat.mod_lt _ (pow_pos hp.pos _)]"
},
{
"state_after": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\nhdisj :\n Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i ∣ k) (Ico 1 (succ n)))\n⊢ card (filter (fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k) (Ico 1 (succ n))) ≤ n",
"state_before": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\nhdisj :\n Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i ∣ k) (Ico 1 (succ n)))\n⊢ multiplicity p (choose (p ^ n) k) + multiplicity p k ≤ ↑n",
"tactic": "rw [multiplicity_choose hp hkn (lt_succ_self _),\n multiplicity_eq_card_pow_dvd (ne_of_gt hp.one_lt) hk0.bot_lt\n (lt_succ_of_le (log_mono_right hkn)),\n ← Nat.cast_add, PartENat.coe_le_coe, log_pow hp.one_lt, ← card_disjoint_union hdisj,\n filter_union_right]"
},
{
"state_after": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\nhdisj :\n Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i ∣ k) (Ico 1 (succ n)))\nfilter_le_Ico :\n card (filter (fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k) (Ico 1 (succ n))) ≤ card (Ico 1 (succ n))\n⊢ card (filter (fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k) (Ico 1 (succ n))) ≤ n",
"state_before": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\nhdisj :\n Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i ∣ k) (Ico 1 (succ n)))\n⊢ card (filter (fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k) (Ico 1 (succ n))) ≤ n",
"tactic": "have filter_le_Ico := (Ico 1 n.succ).card_filter_le\n fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k"
},
{
"state_after": "no goals",
"state_before": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\nhdisj :\n Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i ∣ k) (Ico 1 (succ n)))\nfilter_le_Ico :\n card (filter (fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k) (Ico 1 (succ n))) ≤ card (Ico 1 (succ n))\n⊢ card (filter (fun x => p ^ x ≤ k % p ^ x + (p ^ n - k) % p ^ x ∨ p ^ x ∣ k) (Ico 1 (succ n))) ≤ n",
"tactic": "rwa [card_Ico 1 n.succ] at filter_le_Ico"
},
{
"state_after": "no goals",
"state_before": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\n⊢ Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 (succ n)))\n (filter (fun i => p ^ i ∣ k) (Ico 1 (succ n)))",
"tactic": "simp (config := { contextual := true }) [disjoint_right, *, dvd_iff_mod_eq_zero,\n Nat.mod_lt _ (pow_pos hp.pos _)]"
},
{
"state_after": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\n⊢ multiplicity p (p ^ n) ≤ multiplicity p (choose (p ^ n) k) + multiplicity p k",
"state_before": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\n⊢ ↑n ≤ multiplicity p (choose (p ^ n) k) + multiplicity p k",
"tactic": "rw [← hp.multiplicity_pow_self]"
},
{
"state_after": "no goals",
"state_before": "p n k : ℕ\nhp : Prime p\nhkn : k ≤ p ^ n\nhk0 : k ≠ 0\n⊢ multiplicity p (p ^ n) ≤ multiplicity p (choose (p ^ n) k) + multiplicity p k",
"tactic": "exact multiplicity_le_multiplicity_choose_add hp _ _"
}
] | [
240,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
223,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | UniqueFactorizationMonoid.multiplicative_of_coprime | [
{
"state_after": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\n⊢ f (a * b) = f a * f b",
"state_before": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\n⊢ f (a * b) = f a * f b",
"tactic": "letI := Classical.decEq α"
},
{
"state_after": "case pos\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : a = 0\n⊢ f (a * b) = f a * f b\n\ncase neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\n⊢ f (a * b) = f a * f b",
"state_before": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\n⊢ f (a * b) = f a * f b",
"tactic": "by_cases ha0 : a = 0"
},
{
"state_after": "case pos\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : b = 0\n⊢ f (a * b) = f a * f b\n\ncase neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\n⊢ f (a * b) = f a * f b",
"state_before": "case neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\n⊢ f (a * b) = f a * f b",
"tactic": "by_cases hb0 : b = 0"
},
{
"state_after": "case pos\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : f 1 = 0\n⊢ f (a * b) = f a * f b\n\ncase neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\n⊢ f (a * b) = f a * f b",
"state_before": "case neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\n⊢ f (a * b) = f a * f b",
"tactic": "by_cases hf1 : f 1 = 0"
},
{
"state_after": "case neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis : Nontrivial α\n⊢ f (a * b) = f a * f b",
"state_before": "case neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\n⊢ f (a * b) = f a * f b",
"tactic": "haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩"
},
{
"state_after": "case neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ f (a * b) = f a * f b",
"state_before": "case neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis : Nontrivial α\n⊢ f (a * b) = f a * f b",
"tactic": "letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid"
},
{
"state_after": "case neg.refine'_1\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α), p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) → Prime p\n\ncase neg.refine'_2\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α),\n p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ∀ (q : α), q ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) → p ∣ q → p = q",
"state_before": "case neg\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))",
"tactic": "refine' multiplicative_prime_power _ _ _ _ _ @h1 @hpr @hcp"
},
{
"state_after": "case neg.refine'_1\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α), p ∈ normalizedFactors a ∨ p ∈ normalizedFactors b → Prime p\n\ncase neg.refine'_2\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α),\n p ∈ normalizedFactors a ∨ p ∈ normalizedFactors b →\n ∀ (q : α), q ∈ normalizedFactors a ∨ q ∈ normalizedFactors b → p ∣ q → p = q",
"state_before": "case neg.refine'_1\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α), p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) → Prime p\n\ncase neg.refine'_2\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α),\n p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ∀ (q : α), q ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) → p ∣ q → p = q",
"tactic": "all_goals simp only [Multiset.mem_toFinset, Finset.mem_union]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : a = 0\n⊢ f (a * b) = f a * f b",
"tactic": "rw [ha0, MulZeroClass.zero_mul, h0, MulZeroClass.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : b = 0\n⊢ f (a * b) = f a * f b",
"tactic": "rw [hb0, MulZeroClass.mul_zero, h0, MulZeroClass.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : f 1 = 0\n⊢ f (a * b) = f a * f b",
"tactic": "calc\n f (a * b) = f (a * b * 1) := by rw [mul_one]\n _ = 0 := by simp only [h1 isUnit_one, hf1, MulZeroClass.mul_zero]\n _ = f a * f (b * 1) := by simp only [h1 isUnit_one, hf1, MulZeroClass.mul_zero]\n _ = f a * f b := by rw [mul_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : f 1 = 0\n⊢ f (a * b) = f (a * b * 1)",
"tactic": "rw [mul_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : f 1 = 0\n⊢ f (a * b * 1) = 0",
"tactic": "simp only [h1 isUnit_one, hf1, MulZeroClass.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : f 1 = 0\n⊢ 0 = f a * f (b * 1)",
"tactic": "simp only [h1 isUnit_one, hf1, MulZeroClass.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : f 1 = 0\n⊢ f a * f (b * 1) = f a * f b",
"tactic": "rw [mul_one]"
},
{
"state_after": "case intro\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\n⊢ f (a * b) = f a * f b",
"state_before": "α : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\n⊢ f (a * b) = f a * f b",
"tactic": "obtain ⟨ua, a_eq⟩ := normalizedFactors_prod ha0"
},
{
"state_after": "case intro.intro\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f (a * b) = f a * f b",
"state_before": "case intro\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\n⊢ f (a * b) = f a * f b",
"tactic": "obtain ⟨ub, b_eq⟩ := normalizedFactors_prod hb0"
},
{
"state_after": "case intro.intro\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f (Multiset.prod (normalizedFactors a) * Multiset.prod (normalizedFactors b)) * f ↑ub * f ↑ua =\n f (Multiset.prod (normalizedFactors a)) * f (Multiset.prod (normalizedFactors b)) * f ↑ub * f ↑ua",
"state_before": "case intro.intro\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f (a * b) = f a * f b",
"tactic": "rw [← a_eq, ← b_eq, mul_right_comm (Multiset.prod (normalizedFactors a)) ua\n (Multiset.prod (normalizedFactors b) * ub), h1 ua.isUnit, h1 ub.isUnit, h1 ua.isUnit, ←\n mul_assoc, h1 ub.isUnit, mul_right_comm _ (f ua), ← mul_assoc]"
},
{
"state_after": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f (Multiset.prod (normalizedFactors a) * Multiset.prod (normalizedFactors b)) =\n f (Multiset.prod (normalizedFactors a)) * f (Multiset.prod (normalizedFactors b))",
"state_before": "case intro.intro\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f (Multiset.prod (normalizedFactors a) * Multiset.prod (normalizedFactors b)) * f ↑ub * f ↑ua =\n f (Multiset.prod (normalizedFactors a)) * f (Multiset.prod (normalizedFactors b)) * f ↑ub * f ↑ua",
"tactic": "congr"
},
{
"state_after": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f\n (∏ x in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors a) * id x ^ Multiset.count x (normalizedFactors b)) =\n f\n (∏ x in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors a)) *\n f\n (∏ x in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors b))\n\ncase intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors b) → id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors a) → id x ^ Multiset.count x (normalizedFactors a) = 1",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f (Multiset.prod (normalizedFactors a) * Multiset.prod (normalizedFactors b)) =\n f (Multiset.prod (normalizedFactors a)) * f (Multiset.prod (normalizedFactors b))",
"tactic": "rw [← (normalizedFactors a).map_id, ← (normalizedFactors b).map_id,\n Finset.prod_multiset_map_count, Finset.prod_multiset_map_count,\n Finset.prod_subset (Finset.subset_union_left _ (normalizedFactors b).toFinset),\n Finset.prod_subset (Finset.subset_union_right _ (normalizedFactors b).toFinset), ←\n Finset.prod_mul_distrib]"
},
{
"state_after": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors b) → id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors a) → id x ^ Multiset.count x (normalizedFactors a) = 1",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ f\n (∏ x in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors a) * id x ^ Multiset.count x (normalizedFactors b)) =\n f\n (∏ x in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors a)) *\n f\n (∏ x in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors b))\n\ncase intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors b) → id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors a) → id x ^ Multiset.count x (normalizedFactors a) = 1",
"tactic": "simp_rw [id.def, ← pow_add, this]"
},
{
"state_after": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ normalizedFactors b → id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ normalizedFactors a → id x ^ Multiset.count x (normalizedFactors a) = 1",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors b) → id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors a) → id x ^ Multiset.count x (normalizedFactors a) = 1",
"tactic": "all_goals simp only [Multiset.mem_toFinset]"
},
{
"state_after": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ normalizedFactors a → id x ^ Multiset.count x (normalizedFactors a) = 1",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ Multiset.toFinset (normalizedFactors a) → id x ^ Multiset.count x (normalizedFactors a) = 1",
"tactic": "simp only [Multiset.mem_toFinset]"
},
{
"state_after": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\np : α\na✝ : p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b)\nhpb : ¬p ∈ normalizedFactors b\n⊢ id p ^ Multiset.count p (normalizedFactors b) = 1",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ normalizedFactors b → id x ^ Multiset.count x (normalizedFactors b) = 1",
"tactic": "intro p _ hpb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\np : α\na✝ : p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b)\nhpb : ¬p ∈ normalizedFactors b\n⊢ id p ^ Multiset.count p (normalizedFactors b) = 1",
"tactic": "simp [hpb]"
},
{
"state_after": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\np : α\na✝ : p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b)\nhpa : ¬p ∈ normalizedFactors a\n⊢ id p ^ Multiset.count p (normalizedFactors a) = 1",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\n⊢ ∀ (x : α),\n x ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ¬x ∈ normalizedFactors a → id x ^ Multiset.count x (normalizedFactors a) = 1",
"tactic": "intro p _ hpa"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.e_a.e_a\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝² : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝¹ : Nontrivial α\nthis✝ : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (∏ p in Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : αˣ\na_eq : Multiset.prod (normalizedFactors a) * ↑ua = a\nub : αˣ\nb_eq : Multiset.prod (normalizedFactors b) * ↑ub = b\np : α\na✝ : p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b)\nhpa : ¬p ∈ normalizedFactors a\n⊢ id p ^ Multiset.count p (normalizedFactors a) = 1",
"tactic": "simp [hpa]"
},
{
"state_after": "case neg.refine'_2\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α),\n p ∈ normalizedFactors a ∨ p ∈ normalizedFactors b →\n ∀ (q : α), q ∈ normalizedFactors a ∨ q ∈ normalizedFactors b → p ∣ q → p = q",
"state_before": "case neg.refine'_2\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α),\n p ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) →\n ∀ (q : α), q ∈ Multiset.toFinset (normalizedFactors a) ∪ Multiset.toFinset (normalizedFactors b) → p ∣ q → p = q",
"tactic": "simp only [Multiset.mem_toFinset, Finset.mem_union]"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α), p ∈ normalizedFactors a ∨ p ∈ normalizedFactors b → Prime p",
"tactic": "rintro p (hpa | hpb) <;> apply prime_of_normalized_factor <;> assumption"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nα : Type u_2\nR : Type ?u.1916855\ninst✝⁴ : CancelCommMonoidWithZero R\ninst✝³ : UniqueFactorizationMonoid R\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_1\ninst✝ : CancelCommMonoidWithZero β\nf : α → β\na b : α\nh0 : f 0 = 0\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → f (x * y) = f x * f y\nthis✝¹ : DecidableEq α := Classical.decEq α\nha0 : ¬a = 0\nhb0 : ¬b = 0\nhf1 : ¬f 1 = 0\nthis✝ : Nontrivial α\nthis : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid\n⊢ ∀ (p : α),\n p ∈ normalizedFactors a ∨ p ∈ normalizedFactors b →\n ∀ (q : α), q ∈ normalizedFactors a ∨ q ∈ normalizedFactors b → p ∣ q → p = q",
"tactic": "rintro p (hp | hp) q (hq | hq) hdvd <;>\n rw [← normalize_normalized_factor _ hp, ← normalize_normalized_factor _ hq] <;>\n exact\n normalize_eq_normalize hdvd\n ((prime_of_normalized_factor _ hp).irreducible.dvd_symm\n (prime_of_normalized_factor _ hq).irreducible hdvd)"
}
] | [
1189,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1139,
1
] |
Mathlib/Order/JordanHolder.lean | CompositionSeries.lt_top_of_mem_eraseTop | [] | [
441,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
439,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.lt_or_eq_of_le | [] | [
389,
51
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
388,
11
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean | AddMonoidAlgebra.lift_symm_apply | [] | [
1985,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1983,
1
] |
Mathlib/Data/Polynomial/Lifts.lean | Polynomial.base_mul_mem_lifts | [
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nr : R\nhp : ∃ x, ↑(mapRingHom f) x = p\n⊢ ∃ x, ↑(mapRingHom f) x = ↑C (↑f r) * p",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nr : R\nhp : p ∈ lifts f\n⊢ ↑C (↑f r) * p ∈ lifts f",
"tactic": "simp only [lifts, RingHom.mem_rangeS] at hp⊢"
},
{
"state_after": "case intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nr : R\np₁ : R[X]\n⊢ ∃ x, ↑(mapRingHom f) x = ↑C (↑f r) * ↑(mapRingHom f) p₁",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\np : S[X]\nr : R\nhp : ∃ x, ↑(mapRingHom f) x = p\n⊢ ∃ x, ↑(mapRingHom f) x = ↑C (↑f r) * p",
"tactic": "obtain ⟨p₁, rfl⟩ := hp"
},
{
"state_after": "case intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nr : R\np₁ : R[X]\n⊢ ↑(mapRingHom f) (↑C r * p₁) = ↑C (↑f r) * ↑(mapRingHom f) p₁",
"state_before": "case intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nr : R\np₁ : R[X]\n⊢ ∃ x, ↑(mapRingHom f) x = ↑C (↑f r) * ↑(mapRingHom f) p₁",
"tactic": "use C r * p₁"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nr : R\np₁ : R[X]\n⊢ ↑(mapRingHom f) (↑C r * p₁) = ↑C (↑f r) * ↑(mapRingHom f) p₁",
"tactic": "simp only [coe_mapRingHom, map_C, map_mul]"
}
] | [
119,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean | Dfinsupp.subset_mapRange_neLocus | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nN : α → Type u_1\ninst✝⁵ : DecidableEq α\nM : α → Type u_2\nP : α → Type ?u.12530\ninst✝⁴ : (a : α) → Zero (N a)\ninst✝³ : (a : α) → Zero (M a)\ninst✝² : (a : α) → Zero (P a)\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → DecidableEq (M a)\nf g : Π₀ (a : α), N a\nF : (a : α) → N a → M a\nF0 : ∀ (a : α), F a 0 = 0\na : α\n⊢ a ∈ neLocus (mapRange F F0 f) (mapRange F F0 g) → a ∈ neLocus f g",
"tactic": "simpa only [mem_neLocus, mapRange_apply, not_imp_not] using congr_arg (F a)"
}
] | [
94,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero | [
{
"state_after": "𝕜 : Type ?u.3103380\nE : Type ?u.3103383\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ 2 = 0 ∨ ↑re (inner x y) = 0 ↔ inner x y = 0",
"state_before": "𝕜 : Type ?u.3103380\nE : Type ?u.3103383\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ inner x y = 0",
"tactic": "rw [@norm_add_mul_self ℝ, add_right_cancel_iff, add_right_eq_self, mul_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.3103380\nE : Type ?u.3103383\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\n⊢ 2 = 0 ∨ ↑re (inner x y) = 0 ↔ inner x y = 0",
"tactic": "norm_num"
}
] | [
1463,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1460,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasSum_le_inj | [
{
"state_after": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\n⊢ a₁ ≤ a₂",
"state_before": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum f a₁\nhg : HasSum g a₂\n⊢ a₁ ≤ a₂",
"tactic": "rw [← hasSum_extend_zero he] at hf"
},
{
"state_after": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\nc : κ\n⊢ extend e f 0 c ≤ g c",
"state_before": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\n⊢ a₁ ≤ a₂",
"tactic": "refine hasSum_le (fun c => ?_) hf hg"
},
{
"state_after": "case inl.intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\ni : ι\n⊢ extend e f 0 (e i) ≤ g (e i)\n\ncase inr\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh✝ : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\nc : κ\nh : ¬c ∈ Set.range e\n⊢ extend e f 0 c ≤ g c",
"state_before": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\nc : κ\n⊢ extend e f 0 c ≤ g c",
"tactic": "obtain ⟨i, rfl⟩ | h := em (c ∈ Set.range e)"
},
{
"state_after": "case inl.intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\ni : ι\n⊢ f i ≤ g (e i)",
"state_before": "case inl.intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\ni : ι\n⊢ extend e f 0 (e i) ≤ g (e i)",
"tactic": "rw [he.extend_apply]"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\ni : ι\n⊢ f i ≤ g (e i)",
"tactic": "exact h _"
},
{
"state_after": "case inr\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh✝ : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\nc : κ\nh : ¬c ∈ Set.range e\n⊢ OfNat.ofNat 0 c ≤ g c",
"state_before": "case inr\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh✝ : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\nc : κ\nh : ¬c ∈ Set.range e\n⊢ extend e f 0 c ≤ g c",
"tactic": "rw [extend_apply' _ _ _ h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝² : OrderedAddCommMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf g✝ : ι → α\na a₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ (c : κ), ¬c ∈ Set.range e → 0 ≤ g c\nh✝ : ∀ (i : ι), f i ≤ g (e i)\nhf : HasSum (extend e f 0) a₁\nhg : HasSum g a₂\nc : κ\nh : ¬c ∈ Set.range e\n⊢ OfNat.ofNat 0 c ≤ g c",
"tactic": "exact hs _ h"
}
] | [
72,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
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Mathlib/Algebra/Invertible.lean | Commute.invOf_left | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : Monoid α\na b : α\ninst✝ : Invertible b\nh : Commute b a\n⊢ ⅟b * a = ⅟b * (a * b * ⅟b)",
"tactic": "simp [mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : Monoid α\na b : α\ninst✝ : Invertible b\nh : Commute b a\n⊢ ⅟b * (a * b * ⅟b) = ⅟b * (b * a * ⅟b)",
"tactic": "rw [h.eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : Monoid α\na b : α\ninst✝ : Invertible b\nh : Commute b a\n⊢ ⅟b * (b * a * ⅟b) = a * ⅟b",
"tactic": "simp [mul_assoc]"
}
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336,
39
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331,
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Mathlib/FieldTheory/SeparableDegree.lean | Polynomial.HasSeparableContraction.dvd_degree | [] | [
90,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
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Mathlib/Data/List/Sublists.lean | List.sublistsLenAux_zero | [
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nl : List α\nf : List α → β\nr : List β\n⊢ sublistsLenAux 0 l f r = f [] :: r",
"tactic": "cases l <;> rfl"
}
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259,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
258,
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Mathlib/GroupTheory/Submonoid/Operations.lean | Submonoid.mem_prod | [] | [
860,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
858,
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Mathlib/Combinatorics/SimpleGraph/Matching.lean | SimpleGraph.Subgraph.IsMatching.even_card | [
{
"state_after": "no goals",
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"tactic": "classical\nrw [isMatching_iff_forall_degree] at h\nuse M.coe.edgeFinset.card\nrw [← two_mul, ← M.coe.sum_degrees_eq_twice_card_edges]\nconvert_to _ = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v using 3\nsimp [h, Finset.card_univ]"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\ninst✝ : Fintype ↑M.verts\nh✝ : IsMatching M\nh : ∀ (v : V), v ∈ M.verts → degree M v = 1\n⊢ Even (Finset.card (Set.toFinset M.verts))",
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"tactic": "rw [isMatching_iff_forall_degree] at h"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\ninst✝ : Fintype ↑M.verts\nh✝ : IsMatching M\nh : ∀ (v : V), v ∈ M.verts → degree M v = 1\n⊢ Finset.card (Set.toFinset M.verts) =\n Finset.card (edgeFinset (Subgraph.coe M)) + Finset.card (edgeFinset (Subgraph.coe M))",
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"tactic": "use M.coe.edgeFinset.card"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\ninst✝ : Fintype ↑M.verts\nh✝ : IsMatching M\nh : ∀ (v : V), v ∈ M.verts → degree M v = 1\n⊢ Finset.card (Set.toFinset M.verts) = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v",
"state_before": "V : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\ninst✝ : Fintype ↑M.verts\nh✝ : IsMatching M\nh : ∀ (v : V), v ∈ M.verts → degree M v = 1\n⊢ Finset.card (Set.toFinset M.verts) =\n Finset.card (edgeFinset (Subgraph.coe M)) + Finset.card (edgeFinset (Subgraph.coe M))",
"tactic": "rw [← two_mul, ← M.coe.sum_degrees_eq_twice_card_edges]"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\nM✝ M : Subgraph G\ninst✝ : Fintype ↑M.verts\nh✝ : IsMatching M\nh : ∀ (v : V), v ∈ M.verts → degree M v = 1\n⊢ Finset.card (Set.toFinset M.verts) = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v",
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"tactic": "convert_to _ = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v using 3"
},
{
"state_after": "no goals",
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"tactic": "simp [h, Finset.card_univ]"
}
] | [
114,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
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Mathlib/Topology/Constructions.lean | comap_sigmaMk_nhds | [] | [
1511,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1510,
1
] |
Mathlib/Topology/Instances/Rat.lean | Rat.uniformContinuous_neg | [
{
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"tactic": "rw [dist_comm] at h"
},
{
"state_after": "no goals",
"state_before": "ε : ℝ\nε0 : ε > 0\na✝ b✝ : ℚ\nh : dist b✝ a✝ < ε\n⊢ dist (-a✝) (-b✝) < ε",
"tactic": "simpa only [dist_eq, cast_neg, neg_sub_neg] using h"
}
] | [
100,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
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Mathlib/LinearAlgebra/Dual.lean | Submodule.dualCoannihilator_bot | [] | [
866,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/CategoryTheory/Abelian/Homology.lean | homology.condition_ι | [
{
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"tactic": "dsimp [ι]"
},
{
"state_after": "no goals",
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"tactic": "simp"
}
] | [
162,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
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Mathlib/Data/Set/Sigma.lean | Set.mk_preimage_sigma | [] | [
183,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
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Mathlib/Analysis/Normed/Group/Hom.lean | NormedAddGroupHom.range_comp_incl_top | [
{
"state_after": "V : Type ?u.494024\nW : Type ?u.494027\nV₁ : Type u_2\nV₂ : Type u_1\nV₃ : Type ?u.494036\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\n⊢ AddMonoidHom.range (toAddMonoidHom f) = range f",
"state_before": "V : Type ?u.494024\nW : Type ?u.494027\nV₁ : Type u_2\nV₂ : Type u_1\nV₃ : Type ?u.494036\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\n⊢ range (NormedAddGroupHom.comp f (incl ⊤)) = range f",
"tactic": "simp [comp_range, incl_range, ← AddMonoidHom.range_eq_map]"
},
{
"state_after": "no goals",
"state_before": "V : Type ?u.494024\nW : Type ?u.494027\nV₁ : Type u_2\nV₂ : Type u_1\nV₃ : Type ?u.494036\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\n⊢ AddMonoidHom.range (toAddMonoidHom f) = range f",
"tactic": "rfl"
}
] | [
814,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
813,
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Mathlib/Topology/Sets/Compacts.lean | TopologicalSpace.NonemptyCompacts.nonempty | [] | [
233,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
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Mathlib/Order/Antichain.lean | IsAntichain.eq | [] | [
64,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.Parallel.trans | [
{
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"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ s₃ : AffineSubspace k P\nh₁₂ : s₁ ∥ s₂\nh₂₃ : s₂ ∥ s₃\n⊢ s₁ ∥ s₃",
"tactic": "rcases h₁₂ with ⟨v₁₂, rfl⟩"
},
{
"state_after": "case intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ : AffineSubspace k P\nv₁₂ v₂₃ : V\n⊢ s₁ ∥ map (↑(constVAdd k P v₂₃)) (map (↑(constVAdd k P v₁₂)) s₁)",
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"tactic": "rcases h₂₃ with ⟨v₂₃, rfl⟩"
},
{
"state_after": "case intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ : AffineSubspace k P\nv₁₂ v₂₃ : V\n⊢ map (↑(constVAdd k P v₂₃)) (map (↑(constVAdd k P v₁₂)) s₁) = map (↑(constVAdd k P (v₂₃ + v₁₂))) s₁",
"state_before": "case intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ : AffineSubspace k P\nv₁₂ v₂₃ : V\n⊢ s₁ ∥ map (↑(constVAdd k P v₂₃)) (map (↑(constVAdd k P v₁₂)) s₁)",
"tactic": "refine' ⟨v₂₃ + v₁₂, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ : AffineSubspace k P\nv₁₂ v₂₃ : V\n⊢ map (↑(constVAdd k P v₂₃)) (map (↑(constVAdd k P v₁₂)) s₁) = map (↑(constVAdd k P (v₂₃ + v₁₂))) s₁",
"tactic": "rw [map_map, ← coe_trans_to_affineMap, ← constVAdd_add]"
}
] | [
1744,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1739,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.disjoint_left | [] | [
915,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
912,
1
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Mathlib/Algebra/BigOperators/Finprod.lean | finprod_mem_of_eqOn_one | [
{
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"tactic": "rw [← finprod_mem_one s]"
},
{
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"tactic": "exact finprod_mem_congr rfl hf"
}
] | [
654,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
652,
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Mathlib/Data/Nat/Basic.lean | Nat.add_mod_eq_add_mod_right | [
{
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"tactic": "rw [← mod_add_mod, ← mod_add_mod k, H]"
}
] | [
758,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
756,
1
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Mathlib/Algebra/Category/GroupCat/Colimits.lean | AddCommGroupCat.Colimits.quot_add | [] | [
193,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
189,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.range_le_bot_iff | [
{
"state_after": "R : Type u_1\nR₁ : Type ?u.1255386\nR₂ : Type u_2\nR₃ : Type ?u.1255392\nR₄ : Type ?u.1255395\nS : Type ?u.1255398\nK : Type ?u.1255401\nK₂ : Type ?u.1255404\nM : Type u_3\nM' : Type ?u.1255410\nM₁ : Type ?u.1255413\nM₂ : Type u_4\nM₃ : Type ?u.1255419\nM₄ : Type ?u.1255422\nN : Type ?u.1255425\nN₂ : Type ?u.1255428\nι : Type ?u.1255431\nV : Type ?u.1255434\nV₂ : Type ?u.1255437\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝¹ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type ?u.1255847\nsc : SemilinearMapClass F τ₁₂ M M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\n⊢ comap f ⊥ = ⊤ ↔ f = 0",
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"tactic": "rw [range_le_iff_comap]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1255386\nR₂ : Type u_2\nR₃ : Type ?u.1255392\nR₄ : Type ?u.1255395\nS : Type ?u.1255398\nK : Type ?u.1255401\nK₂ : Type ?u.1255404\nM : Type u_3\nM' : Type ?u.1255410\nM₁ : Type ?u.1255413\nM₂ : Type u_4\nM₃ : Type ?u.1255419\nM₄ : Type ?u.1255422\nN : Type ?u.1255425\nN₂ : Type ?u.1255428\nι : Type ?u.1255431\nV : Type ?u.1255434\nV₂ : Type ?u.1255437\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : Module R M\ninst✝³ : Module R₂ M₂\ninst✝² : Module R₃ M₃\nσ₂₁ : R₂ →+* R\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝¹ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type ?u.1255847\nsc : SemilinearMapClass F τ₁₂ M M₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\n⊢ comap f ⊥ = ⊤ ↔ f = 0",
"tactic": "exact ker_eq_top"
}
] | [
1416,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1415,
1
] |
Mathlib/Topology/Semicontinuous.lean | upperSemicontinuous_iff_isOpen_preimage | [] | [
786,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
783,
1
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Mathlib/Algebra/Module/Submodule/Basic.lean | Submodule.coe_set_mk | [] | [
88,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean | summable_one_div_pow_of_le | [
{
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"tactic": "refine'\n summable_of_nonneg_of_le (fun a => one_div_nonneg.mpr (pow_nonneg (zero_le_one.trans hm.le) _))\n (fun a => _)\n (summable_geometric_of_lt_1 (one_div_nonneg.mpr (zero_le_one.trans hm.le))\n ((one_div_lt (zero_lt_one.trans hm) zero_lt_one).mpr (one_div_one.le.trans_lt hm)))"
},
{
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"tactic": "rw [div_pow, one_pow]"
},
{
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"state_before": "α : Type ?u.462750\nβ : Type ?u.462753\nι : Type ?u.462756\nm : ℝ\nf : ℕ → ℕ\nhm : 1 < m\nfi : ∀ (i : ℕ), i ≤ f i\na : ℕ\n⊢ 1 / m ^ f a ≤ 1 / m ^ a",
"tactic": "refine' (one_div_le_one_div _ _).mpr (pow_le_pow hm.le (fi a)) <;>\n exact pow_pos (zero_lt_one.trans hm) _"
}
] | [
445,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
436,
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Mathlib/Data/Real/Irrational.lean | irrational_add_rat_iff | [] | [
548,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
547,
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Mathlib/Algebra/Opposites.lean | MulOpposite.unop_inj | [] | [
182,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
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Mathlib/Topology/ContinuousFunction/Basic.lean | ContinuousMap.coe_comp | [] | [
241,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
240,
1
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src/lean/Init/Data/Array/Basic.lean | Array.size_pop | [
{
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"state_before": "α : Type u_1\na : Array α\n⊢ size (pop a) = size a - 1",
"tactic": "match a with\n| ⟨[]⟩ => rfl\n| ⟨a::as⟩ => simp [pop, Nat.succ_sub_succ_eq_sub, size]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : Array α\n⊢ size (pop { data := List.nil }) = size { data := List.nil } - 1",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na✝ : Array α\na : α\nas : List α\n⊢ size (pop { data := a :: as }) = size { data := a :: as } - 1",
"tactic": "simp [pop, Nat.succ_sub_succ_eq_sub, size]"
}
] | [
630,
58
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
627,
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Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_range_zero | [
{
"state_after": "no goals",
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"tactic": "rw [range_zero, prod_empty]"
}
] | [
1261,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1261,
1
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Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | SimpleGraph.sdiff_compl_neighborFinset_inter_eq | [
{
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"tactic": "ext"
},
{
"state_after": "case a\nV : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\nh : Adj G v w\na✝ : V\n⊢ ¬Adj G v a✝ → ¬Adj G w a✝ → ¬(a✝ = w ∨ a✝ = v)",
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"tactic": "simp only [and_imp, mem_union, mem_sdiff, mem_compl, and_iff_left_iff_imp, mem_neighborFinset,\n mem_inter, mem_singleton]"
},
{
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"tactic": "rintro hnv hnw (rfl | rfl)"
},
{
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"tactic": "exact hnv h"
},
{
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"state_before": "case a.inr\nV : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nw a✝ : V\nhnw : ¬Adj G w a✝\nh : Adj G a✝ w\nhnv : ¬Adj G a✝ a✝\n⊢ False",
"tactic": "apply hnw"
},
{
"state_after": "no goals",
"state_before": "case a.inr\nV : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nw a✝ : V\nhnw : ¬Adj G w a✝\nh : Adj G a✝ w\nhnv : ¬Adj G a✝ a✝\n⊢ Adj G w a✝",
"tactic": "rwa [adj_comm]"
}
] | [
134,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/Data/List/Basic.lean | List.length_le_length_insertNth | [
{
"state_after": "case inl\nι : Type ?u.123956\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : n ≤ length l\n⊢ length l ≤ length (insertNth n x l)\n\ncase inr\nι : Type ?u.123956\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : length l < n\n⊢ length l ≤ length (insertNth n x l)",
"state_before": "ι : Type ?u.123956\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\n⊢ length l ≤ length (insertNth n x l)",
"tactic": "cases' le_or_lt n l.length with hn hn"
},
{
"state_after": "case inl\nι : Type ?u.123956\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : n ≤ length l\n⊢ length l ≤ length l + 1",
"state_before": "case inl\nι : Type ?u.123956\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : n ≤ length l\n⊢ length l ≤ length (insertNth n x l)",
"tactic": "rw [length_insertNth _ _ hn]"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.123956\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : n ≤ length l\n⊢ length l ≤ length l + 1",
"tactic": "exact (Nat.lt_succ_self _).le"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.123956\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\nhn : length l < n\n⊢ length l ≤ length (insertNth n x l)",
"tactic": "rw [insertNth_of_length_lt _ _ _ hn]"
}
] | [
1685,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1680,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_subtype'' | [] | [
1216,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1215,
1
] |
Mathlib/CategoryTheory/Functor/Const.lean | CategoryTheory.Functor.const.opObjUnop_hom_app | [] | [
76,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Topology/Sheaves/PUnit.lean | TopCat.Presheaf.isSheaf_on_pUnit_iff_isTerminal | [] | [
62,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
] |
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | Matrix.SpecialLinearGroup.coe_matrix_coe | [] | [
254,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
252,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean | Trivialization.coordChangeL_apply' | [
{
"state_after": "no goals",
"state_before": "R : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ ↑(coordChangeL R e e' b) y = (↑e' (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, y))).snd",
"tactic": "rw [e.coordChangeL_apply e' hb, e.mk_symm hb.1]"
}
] | [
359,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
356,
1
] |
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