file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
MulSalemSpencer.mul_right₀
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.99156\nα : Type u_1\nβ : Type ?u.99162\n𝕜 : Type ?u.99165\nE : Type ?u.99168\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x => x * a) b * (fun x => x * a) c = (fun x => x * a) d * (fun x => x * a) d\n⊢ (fun x => x * a) b = (fun x => x * a) c",
"state_before": "F : Type ?u.99156\nα : Type u_1\nβ : Type ?u.99162\n𝕜 : Type ?u.99165\nE : Type ?u.99168\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\n⊢ MulSalemSpencer ((fun x => x * a) '' s)",
"tactic": "rintro _ _ _ ⟨b, hb, rfl⟩ ⟨c, hc, rfl⟩ ⟨d, hd, rfl⟩ h"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.99156\nα : Type u_1\nβ : Type ?u.99162\n𝕜 : Type ?u.99165\nE : Type ?u.99168\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : b * c * (a * a) = d * d * (a * a)\n⊢ (fun x => x * a) b = (fun x => x * a) c",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.99156\nα : Type u_1\nβ : Type ?u.99162\n𝕜 : Type ?u.99165\nE : Type ?u.99168\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : (fun x => x * a) b * (fun x => x * a) c = (fun x => x * a) d * (fun x => x * a) d\n⊢ (fun x => x * a) b = (fun x => x * a) c",
"tactic": "rw [mul_mul_mul_comm, mul_mul_mul_comm d] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nF : Type ?u.99156\nα : Type u_1\nβ : Type ?u.99162\n𝕜 : Type ?u.99165\nE : Type ?u.99168\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NoZeroDivisors α\ns : Set α\na : α\nhs : MulSalemSpencer s\nha : a ≠ 0\nb : α\nhb : b ∈ s\nc : α\nhc : c ∈ s\nd : α\nhd : d ∈ s\nh : b * c * (a * a) = d * d * (a * a)\n⊢ (fun x => x * a) b = (fun x => x * a) c",
"tactic": "rw [hs hb hc hd (mul_right_cancel₀ (mul_ne_zero ha ha) h)]"
}
] |
[
248,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Order/Antichain.lean
|
isAntichain_iff_forall_not_lt
|
[] |
[
268,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.interpolate_eq_sum_interpolate_insert_sdiff
|
[
{
"state_after": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\n⊢ ∑ i in s, ↑(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i = ↑(interpolate t v) r",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\n⊢ ↑(interpolate t v) r = ∑ i in s, ↑(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i",
"tactic": "symm"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\n⊢ (sup s fun b => degree (↑(interpolate (insert b (t \\ s)) v) r * Lagrange.basis s v b)) < ↑(card t)\n\ncase refine'_2\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\n⊢ eval (v i) (∑ i in s, ↑(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i) = r i",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\n⊢ ∑ i in s, ↑(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i = ↑(interpolate t v) r",
"tactic": "refine' eq_interpolate_of_eval_eq _ hvt (lt_of_le_of_lt (degree_sum_le _ _) _) fun i hi => _"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\n⊢ ∀ (b : ι), b ∈ s → degree (↑(interpolate (insert b (t \\ s)) v) r) + degree (Lagrange.basis s v b) < ↑(card t)",
"state_before": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\n⊢ (sup s fun b => degree (↑(interpolate (insert b (t \\ s)) v) r * Lagrange.basis s v b)) < ↑(card t)",
"tactic": "simp_rw [Nat.cast_withBot, Finset.sup_lt_iff (WithBot.bot_lt_coe t.card), degree_mul]"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"state_before": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\n⊢ ∀ (b : ι), b ∈ s → degree (↑(interpolate (insert b (t \\ s)) v) r) + degree (Lagrange.basis s v b) < ↑(card t)",
"tactic": "intro i hi"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"state_before": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"tactic": "have hs : 1 ≤ s.card := Nonempty.card_pos ⟨_, hi⟩"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"state_before": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"tactic": "have hst' : s.card ≤ t.card := card_le_of_subset hst"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"state_before": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"tactic": "have H : t.card = 1 + (t.card - s.card) + (s.card - 1) := by\n rw [add_assoc, tsub_add_tsub_cancel hst' hs, ← add_tsub_assoc_of_le (hs.trans hst'),\n Nat.succ_add_sub_one, zero_add]"
},
{
"state_after": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) < ↑(1 + (card t - card s))",
"state_before": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) + degree (Lagrange.basis s v i) < ↑(card t)",
"tactic": "rw [degree_basis (Set.InjOn.mono hst hvt) hi, H, WithBot.coe_add, Nat.cast_withBot,\n WithBot.add_lt_add_iff_right (@WithBot.coe_ne_bot _ (s.card - 1))]"
},
{
"state_after": "case h.e'_4.h.e'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n⊢ 1 + (card t - card s) = card (insert i (t \\ s))",
"state_before": "case refine'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n⊢ degree (↑(interpolate (insert i (t \\ s)) v) r) < ↑(1 + (card t - card s))",
"tactic": "convert degree_interpolate_lt _\n (hvt.mono (coe_subset.mpr (insert_subset.mpr ⟨hst hi, sdiff_subset _ _⟩)))"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_1\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\nH : card t = 1 + (card t - card s) + (card s - 1)\n⊢ 1 + (card t - card s) = card (insert i (t \\ s))",
"tactic": "rw [card_insert_of_not_mem (not_mem_sdiff_of_mem_right hi), card_sdiff hst, add_comm]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs✝ : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ s\nhs : 1 ≤ card s\nhst' : card s ≤ card t\n⊢ card t = 1 + (card t - card s) + (card s - 1)",
"tactic": "rw [add_assoc, tsub_add_tsub_cancel hst' hs, ← add_tsub_assoc_of_le (hs.trans hst'),\n Nat.succ_add_sub_one, zero_add]"
},
{
"state_after": "case refine'_2\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i",
"state_before": "case refine'_2\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\n⊢ eval (v i) (∑ i in s, ↑(interpolate (insert i (t \\ s)) v) r * Lagrange.basis s v i) = r i",
"tactic": "simp_rw [eval_finset_sum, eval_mul]"
},
{
"state_after": "case pos\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i\n\ncase neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i",
"state_before": "case refine'_2\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i",
"tactic": "by_cases hi' : i ∈ s"
},
{
"state_after": "case pos\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\n⊢ ∑ x in Finset.erase s i, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = 0",
"state_before": "case pos\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i",
"tactic": "rw [← add_sum_erase _ _ hi', eval_basis_self (hvt.mono hst) hi',\n eval_interpolate_at_node _\n (hvt.mono (coe_subset.mpr (insert_subset.mpr ⟨hi, sdiff_subset _ _⟩)))\n (mem_insert_self _ _),\n mul_one, add_right_eq_self]"
},
{
"state_after": "case pos\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\nj : ι\nhj : j ∈ Finset.erase s i\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) = 0",
"state_before": "case pos\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\n⊢ ∑ x in Finset.erase s i, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = 0",
"tactic": "refine' sum_eq_zero fun j hj => _"
},
{
"state_after": "case pos.intro\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\nj : ι\nhj : j ∈ Finset.erase s i\nhij : j ≠ i\nright✝ : j ∈ s\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) = 0",
"state_before": "case pos\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\nj : ι\nhj : j ∈ Finset.erase s i\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) = 0",
"tactic": "rcases mem_erase.mp hj with ⟨hij, _⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : i ∈ s\nj : ι\nhj : j ∈ Finset.erase s i\nhij : j ≠ i\nright✝ : j ∈ s\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) = 0",
"tactic": "rw [eval_basis_of_ne hij hi', MulZeroClass.mul_zero]"
},
{
"state_after": "case neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i",
"state_before": "case neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i",
"tactic": "have H : (∑ j in s, eval (v i) (Lagrange.basis s v j)) = 1 := by\n rw [← eval_finset_sum, sum_basis (hvt.mono hst) hs, eval_one]"
},
{
"state_after": "case neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) =\n ∑ x in s, r i * eval (v i) (Lagrange.basis s v x)",
"state_before": "case neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) = r i",
"tactic": "rw [← mul_one (r i), ← H, mul_sum]"
},
{
"state_after": "case neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\nj : ι\nhj : j ∈ s\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) =\n r i * eval (v i) (Lagrange.basis s v j)",
"state_before": "case neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\n⊢ ∑ x in s, eval (v i) (↑(interpolate (insert x (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v x) =\n ∑ x in s, r i * eval (v i) (Lagrange.basis s v x)",
"tactic": "refine' sum_congr rfl fun j hj => _"
},
{
"state_after": "case neg.e_a\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\nj : ι\nhj : j ∈ s\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) = r i",
"state_before": "case neg\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\nj : ι\nhj : j ∈ s\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) * eval (v i) (Lagrange.basis s v j) =\n r i * eval (v i) (Lagrange.basis s v j)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case neg.e_a\nF : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j✝ : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\nH : ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1\nj : ι\nhj : j ∈ s\n⊢ eval (v i) (↑(interpolate (insert j (t \\ s)) v) r) = r i",
"tactic": "exact\n eval_interpolate_at_node _ (hvt.mono (insert_subset.mpr ⟨hst hj, sdiff_subset _ _⟩))\n (mem_insert.mpr (Or.inr (mem_sdiff.mpr ⟨hi, hi'⟩)))"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni✝ j : ι\nv r r' : ι → F\nhvt : Set.InjOn v ↑t\nhs : Finset.Nonempty s\nhst : s ⊆ t\ni : ι\nhi : i ∈ t\nhi' : ¬i ∈ s\n⊢ ∑ j in s, eval (v i) (Lagrange.basis s v j) = 1",
"tactic": "rw [← eval_finset_sum, sum_basis (hvt.mono hst) hs, eval_one]"
}
] |
[
454,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
iUnion_Iic_eq_Iic_iSup
|
[] |
[
249,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
Units.nnnorm_pos
|
[] |
[
447,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.one_def
|
[] |
[
463,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
|
strictConcaveOn_sin_Icc
|
[
{
"state_after": "x : ℝ\nhx : x ∈ interior (Icc 0 π)\n⊢ (deriv^[2]) sin x < 0",
"state_before": "⊢ StrictConcaveOn ℝ (Icc 0 π) sin",
"tactic": "apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_sin fun x hx => ?_"
},
{
"state_after": "x : ℝ\nhx : x ∈ Ioo 0 π\n⊢ (deriv^[2]) sin x < 0",
"state_before": "x : ℝ\nhx : x ∈ interior (Icc 0 π)\n⊢ (deriv^[2]) sin x < 0",
"tactic": "rw [interior_Icc] at hx"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ∈ Ioo 0 π\n⊢ (deriv^[2]) sin x < 0",
"tactic": "simp [sin_pos_of_mem_Ioo hx]"
}
] |
[
173,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Combinatorics/Colex.lean
|
Nat.sum_two_pow_lt
|
[
{
"state_after": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\n⊢ ∑ x in range k, Nat.pow 2 x < 2 ^ k",
"state_before": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\n⊢ Finset.sum A (Nat.pow 2) < 2 ^ k",
"tactic": "apply lt_of_le_of_lt (sum_le_sum_of_subset fun t => mem_range.2 ∘ h₁)"
},
{
"state_after": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\nz : (∑ i in range k, (1 + 1) ^ i) * 1 + 1 = (1 + 1) ^ k\n⊢ ∑ x in range k, Nat.pow 2 x < 2 ^ k",
"state_before": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\n⊢ ∑ x in range k, Nat.pow 2 x < 2 ^ k",
"tactic": "have z := geom_sum_mul_add 1 k"
},
{
"state_after": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\nz : ∑ i in range k, 2 ^ i + 1 = 2 ^ k\n⊢ ∑ x in range k, Nat.pow 2 x < 2 ^ k",
"state_before": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\nz : (∑ i in range k, (1 + 1) ^ i) * 1 + 1 = (1 + 1) ^ k\n⊢ ∑ x in range k, Nat.pow 2 x < 2 ^ k",
"tactic": "rw [mul_one, one_add_one_eq_two] at z"
},
{
"state_after": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\nz : ∑ i in range k, 2 ^ i + 1 = 2 ^ k\n⊢ ∑ x in range k, Nat.pow 2 x < ∑ i in range k, 2 ^ i + 1",
"state_before": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\nz : ∑ i in range k, 2 ^ i + 1 = 2 ^ k\n⊢ ∑ x in range k, Nat.pow 2 x < 2 ^ k",
"tactic": "rw [← z]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.638\nk : ℕ\nA : Finset ℕ\nh₁ : ∀ {x : ℕ}, x ∈ A → x < k\nz : ∑ i in range k, 2 ^ i + 1 = 2 ^ k\n⊢ ∑ x in range k, Nat.pow 2 x < ∑ i in range k, 2 ^ i + 1",
"tactic": "apply Nat.lt_succ_self"
}
] |
[
111,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_rotate'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.8273\nβ : Type ?u.8276\nG : Type u_1\ninst✝ : CommSemigroup G\na b c : G\n⊢ a * (b * c) = b * (c * a)",
"tactic": "simp only [mul_left_comm, mul_comm]"
}
] |
[
122,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.symm_toLocalEquiv
|
[] |
[
337,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
Homeomorph.symm_comp_toContinuousMap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.69498\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α ≃ₜ β\ng : β ≃ₜ γ\n⊢ ContinuousMap.comp (toContinuousMap (Homeomorph.symm f)) (toContinuousMap f) = ContinuousMap.id α",
"tactic": "rw [← coe_trans, self_trans_symm, coe_refl]"
}
] |
[
492,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
491,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_pi_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.396731\nι : Type ?u.396734\ninst✝³ : PseudoMetricSpace α\nπ : β → Type ?u.396742\ninst✝² : Fintype β\ninst✝¹ : (b : β) → PseudoMetricSpace (π b)\ninst✝ : Nonempty β\na b : α\n⊢ (dist (fun x => a) fun x => b) = dist a b",
"tactic": "simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b)"
}
] |
[
2029,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2028,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_eq_sum
|
[
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.348297\nδ : Type ?u.348300\nι : Type ?u.348303\nR : Type ?u.348306\nR' : Type ?u.348309\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : Countable β\ninst✝ : MeasurableSingletonClass β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(map f μ) s = ↑↑(sum fun b => ↑↑μ (f ⁻¹' {b}) • dirac b) s",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.348297\nδ : Type ?u.348300\nι : Type ?u.348303\nR : Type ?u.348306\nR' : Type ?u.348309\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝¹ : Countable β\ninst✝ : MeasurableSingletonClass β\nμ : Measure α\nf : α → β\nhf : Measurable f\n⊢ map f μ = sum fun b => ↑↑μ (f ⁻¹' {b}) • dirac b",
"tactic": "ext1 s hs"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.348297\nδ : Type ?u.348300\nι : Type ?u.348303\nR : Type ?u.348306\nR' : Type ?u.348309\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : Countable β\ninst✝ : MeasurableSingletonClass β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\nthis : ∀ (y : β), y ∈ s → MeasurableSet (f ⁻¹' {y})\n⊢ ↑↑(map f μ) s = ↑↑(sum fun b => ↑↑μ (f ⁻¹' {b}) • dirac b) s",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.348297\nδ : Type ?u.348300\nι : Type ?u.348303\nR : Type ?u.348306\nR' : Type ?u.348309\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : Countable β\ninst✝ : MeasurableSingletonClass β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(map f μ) s = ↑↑(sum fun b => ↑↑μ (f ⁻¹' {b}) • dirac b) s",
"tactic": "have : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y}) := fun y _ => hf (measurableSet_singleton _)"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.348297\nδ : Type ?u.348300\nι : Type ?u.348303\nR : Type ?u.348306\nR' : Type ?u.348309\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : Countable β\ninst✝ : MeasurableSingletonClass β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\nthis : ∀ (y : β), y ∈ s → MeasurableSet (f ⁻¹' {y})\n⊢ ↑↑(map f μ) s = ↑↑(sum fun b => ↑↑μ (f ⁻¹' {b}) • dirac b) s",
"tactic": "simp [← tsum_measure_preimage_singleton (to_countable s) this, *,\n tsum_subtype s fun b => μ (f ⁻¹' {b}), ← indicator_mul_right s fun b => μ (f ⁻¹' {b})]"
}
] |
[
2138,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2133,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelEmbedding.map_rel_iff
|
[] |
[
276,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.iUnion_filter_not
|
[
{
"state_after": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ (⋃ (J : Box ι) (_ : J ∈ filter π fun J => ¬p J), ↑J) =\n (⋃ (J : Box ι) (_ : J ∈ π), ↑J) \\ ⋃ (J : Box ι) (_ : J ∈ filter π p), ↑J",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ Prepartition.iUnion (filter π fun J => ¬p J) = Prepartition.iUnion π \\ Prepartition.iUnion (filter π p)",
"tactic": "simp only [Prepartition.iUnion]"
},
{
"state_after": "case h.e'_2.h.e'_3.h.pq.a.a\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\nx✝ : Box ι\n⊢ (x✝ ∈ filter π fun J => ¬p J) ↔ x✝ ∈ ↑π.boxes \\ ↑(filter π p).boxes\n\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ PairwiseDisjoint (↑π.boxes ∪ ↑(filter π p).boxes) Box.toSet",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ (⋃ (J : Box ι) (_ : J ∈ filter π fun J => ¬p J), ↑J) =\n (⋃ (J : Box ι) (_ : J ∈ π), ↑J) \\ ⋃ (J : Box ι) (_ : J ∈ filter π p), ↑J",
"tactic": "convert (@Set.biUnion_diff_biUnion_eq _ (Box ι) π.boxes (π.filter p).boxes (↑) _).symm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_3.h.pq.a.a\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\nx✝ : Box ι\n⊢ (x✝ ∈ filter π fun J => ¬p J) ↔ x✝ ∈ ↑π.boxes \\ ↑(filter π p).boxes",
"tactic": "simp (config := { contextual := true })"
},
{
"state_after": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ Set.Pairwise (↑π.boxes ∪ ↑(filter π p).boxes) (Disjoint on Box.toSet)",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ PairwiseDisjoint (↑π.boxes ∪ ↑(filter π p).boxes) Box.toSet",
"tactic": "rw [Set.PairwiseDisjoint]"
},
{
"state_after": "case h.e'_2\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ ↑π.boxes ∪ ↑(filter π p).boxes = ↑π.boxes",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ Set.Pairwise (↑π.boxes ∪ ↑(filter π p).boxes) (Disjoint on Box.toSet)",
"tactic": "convert π.pairwiseDisjoint"
},
{
"state_after": "case h.e'_2\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ {x | x ∈ π.boxes ∧ p x} ⊆ ↑π.boxes",
"state_before": "case h.e'_2\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ ↑π.boxes ∪ ↑(filter π p).boxes = ↑π.boxes",
"tactic": "rw [Set.union_eq_left_iff_subset, filter_boxes, coe_filter]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nι : Type u_1\nI J J₁ J₂ : Box ι\nπ✝ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ : Prepartition I\np : Box ι → Prop\n⊢ {x | x ∈ π.boxes ∧ p x} ⊆ ↑π.boxes",
"tactic": "exact fun _ ⟨h, _⟩ => h"
}
] |
[
629,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Topology/Constructions.lean
|
isClosed_set_pi
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.257106\nδ : Type ?u.257109\nε : Type ?u.257112\nζ : Type ?u.257115\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.257126\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ni : Set ι\ns : (a : ι) → Set (π a)\nhs : ∀ (a : ι), a ∈ i → IsClosed (s a)\n⊢ IsClosed (⋂ (a : ι) (_ : a ∈ i), eval a ⁻¹' s a)",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.257106\nδ : Type ?u.257109\nε : Type ?u.257112\nζ : Type ?u.257115\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.257126\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ni : Set ι\ns : (a : ι) → Set (π a)\nhs : ∀ (a : ι), a ∈ i → IsClosed (s a)\n⊢ IsClosed (Set.pi i s)",
"tactic": "rw [pi_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.257106\nδ : Type ?u.257109\nε : Type ?u.257112\nζ : Type ?u.257115\nι : Type u_1\nπ : ι → Type u_2\nκ : Type ?u.257126\ninst✝¹ : TopologicalSpace α\ninst✝ : (i : ι) → TopologicalSpace (π i)\nf : α → (i : ι) → π i\ni : Set ι\ns : (a : ι) → Set (π a)\nhs : ∀ (a : ι), a ∈ i → IsClosed (s a)\n⊢ IsClosed (⋂ (a : ι) (_ : a ∈ i), eval a ⁻¹' s a)",
"tactic": "exact isClosed_biInter fun a ha => (hs _ ha).preimage (continuous_apply _)"
}
] |
[
1335,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1333,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.optionEquivSumPUnit_symm_inl
|
[] |
[
440,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
439,
1
] |
Mathlib/RepresentationTheory/Basic.lean
|
Representation.ofModule_asAlgebraHom_apply_apply
|
[
{
"state_after": "case hM\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\n⊢ ∀ (g : G),\n ↑(↑(asAlgebraHom (ofModule M)) (↑(of k G) g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (↑(of k G) g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\n\ncase hadd\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\n⊢ ∀ (f g : MonoidAlgebra k G),\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m) →\n ↑(↑(asAlgebraHom (ofModule M)) g) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m) →\n ↑(↑(asAlgebraHom (ofModule M)) (f + g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((f + g) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\n\ncase hsmul\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\n⊢ ∀ (r : k) (f : MonoidAlgebra k G),\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m) →\n ↑(↑(asAlgebraHom (ofModule M)) (r • f)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((r • f) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"state_before": "k : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\n⊢ ↑(↑(asAlgebraHom (ofModule M)) r) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (r • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"tactic": "apply MonoidAlgebra.induction_on r"
},
{
"state_after": "case hM\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\ng : G\n⊢ ↑(↑(asAlgebraHom (ofModule M)) (↑(of k G) g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (↑(of k G) g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"state_before": "case hM\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\n⊢ ∀ (g : G),\n ↑(↑(asAlgebraHom (ofModule M)) (↑(of k G) g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (↑(of k G) g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"tactic": "intro g"
},
{
"state_after": "no goals",
"state_before": "case hM\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\ng : G\n⊢ ↑(↑(asAlgebraHom (ofModule M)) (↑(of k G) g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (↑(of k G) g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"tactic": "simp only [one_smul, MonoidAlgebra.lift_symm_apply, MonoidAlgebra.of_apply,\n Representation.asAlgebraHom_single, Representation.ofModule, AddEquiv.apply_eq_iff_eq,\n RestrictScalars.lsmul_apply_apply]"
},
{
"state_after": "case hadd\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\nf g : MonoidAlgebra k G\nfw :\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\ngw :\n ↑(↑(asAlgebraHom (ofModule M)) g) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\n⊢ ↑(↑(asAlgebraHom (ofModule M)) (f + g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((f + g) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"state_before": "case hadd\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\n⊢ ∀ (f g : MonoidAlgebra k G),\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m) →\n ↑(↑(asAlgebraHom (ofModule M)) g) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m) →\n ↑(↑(asAlgebraHom (ofModule M)) (f + g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((f + g) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"tactic": "intro f g fw gw"
},
{
"state_after": "no goals",
"state_before": "case hadd\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\nf g : MonoidAlgebra k G\nfw :\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\ngw :\n ↑(↑(asAlgebraHom (ofModule M)) g) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (g • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\n⊢ ↑(↑(asAlgebraHom (ofModule M)) (f + g)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((f + g) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"tactic": "simp only [fw, gw, map_add, add_smul, LinearMap.add_apply]"
},
{
"state_after": "case hsmul\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr✝ : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\nr : k\nf : MonoidAlgebra k G\nw :\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\n⊢ ↑(↑(asAlgebraHom (ofModule M)) (r • f)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((r • f) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"state_before": "case hsmul\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\n⊢ ∀ (r : k) (f : MonoidAlgebra k G),\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m) →\n ↑(↑(asAlgebraHom (ofModule M)) (r • f)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((r • f) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"tactic": "intro r f w"
},
{
"state_after": "no goals",
"state_before": "case hsmul\nk : Type u_1\nG : Type u_2\nV : Type ?u.309923\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : Module (MonoidAlgebra k G) M\nr✝ : MonoidAlgebra k G\nm : RestrictScalars k (MonoidAlgebra k G) M\nr : k\nf : MonoidAlgebra k G\nw :\n ↑(↑(asAlgebraHom (ofModule M)) f) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n (f • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)\n⊢ ↑(↑(asAlgebraHom (ofModule M)) (r • f)) m =\n ↑(AddEquiv.symm (RestrictScalars.addEquiv k (MonoidAlgebra k G) ((fun x => M) m)))\n ((r • f) • ↑(RestrictScalars.addEquiv k (MonoidAlgebra k G) M) m)",
"tactic": "simp only [w, AlgHom.map_smul, LinearMap.smul_apply,\n RestrictScalars.addEquiv_symm_map_smul_smul]"
}
] |
[
221,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.opNorm_le_bound
|
[] |
[
283,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.coe_strictlyPositive
|
[] |
[
618,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.derivFamily_succ
|
[] |
[
173,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.SimplicialObject.δ_comp_σ_of_gt'
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\nhi : i = 0\n⊢ False",
"tactic": "simp only [Fin.not_lt_zero, hi] at H"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n⊢ X.map (SimplexCategory.σ j).op ≫ X.map (SimplexCategory.δ i).op =\n X.map (SimplexCategory.δ (Fin.pred i (_ : i = 0 → False))).op ≫\n X.map (SimplexCategory.σ (Fin.castLT j (_ : ↑j < n + 1))).op",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n⊢ σ X j ≫ δ X i = δ X (Fin.pred i (_ : i = 0 → False)) ≫ σ X (Fin.castLT j (_ : ↑j < n + 1))",
"tactic": "dsimp [δ, σ]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n⊢ X.map (SimplexCategory.σ j).op ≫ X.map (SimplexCategory.δ i).op =\n X.map (SimplexCategory.δ (Fin.pred i (_ : i = 0 → False))).op ≫\n X.map (SimplexCategory.σ (Fin.castLT j (_ : ↑j < n + 1))).op",
"tactic": "simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_gt' H]"
}
] |
[
197,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/LinearAlgebra/Matrix/AbsoluteValue.lean
|
Matrix.det_sum_smul_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : Nontrivial R\ninst✝² : LinearOrderedCommRing S\nn : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nι : Type u_1\ns : Finset ι\nc : ι → R\nA : ι → Matrix n n R\nabv : AbsoluteValue R S\nx : S\nhx : ∀ (k : ι) (i j : n), ↑abv (A k i j) ≤ x\ny : S\nhy : ∀ (k : ι), ↑abv (c k) ≤ y\n⊢ ↑abv (det (∑ k in s, c k • A k)) ≤ Nat.factorial (Fintype.card n) • (card s • y * x) ^ Fintype.card n",
"tactic": "simpa only [smul_mul_assoc] using\n det_sum_le s fun k i j =>\n calc\n abv (c k * A k i j) = abv (c k) * abv (A k i j) := abv.map_mul _ _\n _ ≤ y * x := mul_le_mul (hy k) (hx k i j) (abv.nonneg _) ((abv.nonneg _).trans (hy k))"
}
] |
[
79,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sup_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.102338\nβ₂ : Type ?u.102341\nγ : Type ?u.102344\nι : Sort u_1\nι' : Sort ?u.102350\nκ : ι → Sort ?u.102355\nκ' : ι' → Sort ?u.102360\ninst✝¹ : CompleteLattice α\nf✝ g s t : ι → α\na✝ b : α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ (a ⊔ ⨆ (x : ι), f x) = ⨆ (x : ι), a ⊔ f x",
"tactic": "rw [iSup_sup_eq, iSup_const]"
}
] |
[
1258,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1257,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.le_prod_of_submultiplicative_on_pred
|
[
{
"state_after": "case inl\nι : Type u_3\nα : Type ?u.6427\nβ : Type ?u.6430\nM : Type u_2\nN : Type u_1\nG : Type ?u.6439\nk : Type ?u.6442\nR : Type ?u.6445\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_one : f 1 = 1\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\nhs : ∀ (i : ι), i ∈ ∅ → p (g i)\n⊢ f (∏ i in ∅, g i) ≤ ∏ i in ∅, f (g i)\n\ncase inr\nι : Type u_3\nα : Type ?u.6427\nβ : Type ?u.6430\nM : Type u_2\nN : Type u_1\nG : Type ?u.6439\nk : Type ?u.6442\nR : Type ?u.6445\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_one : f 1 = 1\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs : ∀ (i : ι), i ∈ s → p (g i)\nhs_nonempty : Finset.Nonempty s\n⊢ f (∏ i in s, g i) ≤ ∏ i in s, f (g i)",
"state_before": "ι : Type u_3\nα : Type ?u.6427\nβ : Type ?u.6430\nM : Type u_2\nN : Type u_1\nG : Type ?u.6439\nk : Type ?u.6442\nR : Type ?u.6445\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_one : f 1 = 1\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs : ∀ (i : ι), i ∈ s → p (g i)\n⊢ f (∏ i in s, g i) ≤ ∏ i in s, f (g i)",
"tactic": "rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u_3\nα : Type ?u.6427\nβ : Type ?u.6430\nM : Type u_2\nN : Type u_1\nG : Type ?u.6439\nk : Type ?u.6442\nR : Type ?u.6445\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_one : f 1 = 1\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\nhs : ∀ (i : ι), i ∈ ∅ → p (g i)\n⊢ f (∏ i in ∅, g i) ≤ ∏ i in ∅, f (g i)",
"tactic": "simp [h_one]"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type u_3\nα : Type ?u.6427\nβ : Type ?u.6430\nM : Type u_2\nN : Type u_1\nG : Type ?u.6439\nk : Type ?u.6442\nR : Type ?u.6445\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf : M → N\np : M → Prop\nh_one : f 1 = 1\nh_mul : ∀ (x y : M), p x → p y → f (x * y) ≤ f x * f y\nhp_mul : ∀ (x y : M), p x → p y → p (x * y)\ng : ι → M\ns : Finset ι\nhs : ∀ (i : ι), i ∈ s → p (g i)\nhs_nonempty : Finset.Nonempty s\n⊢ f (∏ i in s, g i) ≤ ∏ i in s, f (g i)",
"tactic": "exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs"
}
] |
[
83,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.blsub_lt_ord_lift
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.39723\nr : α → α → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nc : Ordinal\nho : Cardinal.lift (card o) < cof c\nhf : ∀ (i : Ordinal) (hi : i < o), f i hi < c\nh : blsub o f = c\n⊢ cof c ≤ Cardinal.lift (card o)",
"tactic": "simpa [← iSup_ord, hf, h] using cof_blsub_le_lift.{u, v} f"
}
] |
[
446,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_transpose
|
[
{
"state_after": "case a.h\nl : Type u_2\nm : Type u_4\nn : Type u_1\no : Type u_5\np : Type ?u.12383\nq : Type ?u.12386\nm' : o → Type ?u.12391\nn' : o → Type ?u.12396\np' : o → Type ?u.12401\nR : Type ?u.12404\nS : Type ?u.12407\nα : Type u_3\nβ : Type ?u.12413\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\ni : l ⊕ m\nj : n ⊕ o\n⊢ (fromBlocks A B C D)ᵀ i j = fromBlocks Aᵀ Cᵀ Bᵀ Dᵀ i j",
"state_before": "l : Type u_2\nm : Type u_4\nn : Type u_1\no : Type u_5\np : Type ?u.12383\nq : Type ?u.12386\nm' : o → Type ?u.12391\nn' : o → Type ?u.12396\np' : o → Type ?u.12401\nR : Type ?u.12404\nS : Type ?u.12407\nα : Type u_3\nβ : Type ?u.12413\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\n⊢ (fromBlocks A B C D)ᵀ = fromBlocks Aᵀ Cᵀ Bᵀ Dᵀ",
"tactic": "ext i j"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type u_2\nm : Type u_4\nn : Type u_1\no : Type u_5\np : Type ?u.12383\nq : Type ?u.12386\nm' : o → Type ?u.12391\nn' : o → Type ?u.12396\np' : o → Type ?u.12401\nR : Type ?u.12404\nS : Type ?u.12407\nα : Type u_3\nβ : Type ?u.12413\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\ni : l ⊕ m\nj : n ⊕ o\n⊢ (fromBlocks A B C D)ᵀ i j = fromBlocks Aᵀ Cᵀ Bᵀ Dᵀ i j",
"tactic": "rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> simp [fromBlocks]"
}
] |
[
155,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
Submodule.norm_subtypeL_le
|
[] |
[
1299,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1298,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
BoundedContinuousFunction.NNReal.coe_ennreal_comp_measurable
|
[] |
[
320,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degrees_pow
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.41650\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\n⊢ 0 ≤ 0 • degrees p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.41650\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\n⊢ degrees (p ^ 0) ≤ 0 • degrees p",
"tactic": "rw [pow_zero, degrees_one]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.41650\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\n⊢ 0 ≤ 0 • degrees p",
"tactic": "exact Multiset.zero_le _"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.41650\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\nn : ℕ\n⊢ degrees (p * p ^ n) ≤ degrees p + n • degrees p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.41650\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\nn : ℕ\n⊢ degrees (p ^ (n + 1)) ≤ (n + 1) • degrees p",
"tactic": "rw [pow_succ, add_smul, add_comm, one_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.41650\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\nn : ℕ\n⊢ degrees (p * p ^ n) ≤ degrees p + n • degrees p",
"tactic": "exact le_trans (degrees_mul _ _) (add_le_add_left (degrees_pow _ n) _)"
}
] |
[
186,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.monomorphisms.infer_property
|
[] |
[
420,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.not_iff
|
[] |
[
477,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
11
] |
Std/Data/List/Lemmas.lean
|
List.cons_diff_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl₂ : List α\nh : a ∈ l₂\nl₁ : List α\n⊢ List.diff (a :: l₁) l₂ = List.diff l₁ (List.erase l₂ a)",
"tactic": "rw [cons_diff, if_pos h]"
}
] |
[
1513,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1512,
1
] |
Mathlib/Order/Basic.lean
|
Prod.mk_lt_mk
|
[] |
[
1248,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1247,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieSubmodule.ucs_zero
|
[] |
[
364,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.neg_apply
|
[] |
[
693,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
692,
1
] |
Mathlib/CategoryTheory/Subobject/Limits.lean
|
CategoryTheory.Limits.imageSubobject_zero
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nX Y Z : C\nf : X ⟶ Y\ninst✝² : HasImage f\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nA B : C\n⊢ (imageSubobjectIso 0 ≪≫ imageZero ≪≫ botCoeIsoZero.symm).hom ≫ arrow ⊥ = arrow (imageSubobject 0)",
"tactic": "simp"
}
] |
[
379,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.image_source_inter_eq
|
[] |
[
282,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.coe_neg_one
|
[
{
"state_after": "case zero\nm : ℕ\n⊢ ↑(-1) = zero\n\ncase succ\nm n✝ : ℕ\n⊢ ↑(-1) = Nat.succ n✝",
"state_before": "n m : ℕ\n⊢ ↑(-1) = n",
"tactic": "cases n"
},
{
"state_after": "case succ\nm n✝ : ℕ\n⊢ n✝ + 1 < Nat.succ n✝ + 1",
"state_before": "case succ\nm n✝ : ℕ\n⊢ ↑(-1) = Nat.succ n✝",
"tactic": "rw [Fin.coe_neg, Fin.val_one, Nat.succ_sub_one, Nat.mod_eq_of_lt]"
},
{
"state_after": "no goals",
"state_before": "case succ\nm n✝ : ℕ\n⊢ n✝ + 1 < Nat.succ n✝ + 1",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : ℕ\n⊢ ↑(-1) = zero",
"tactic": "simp"
}
] |
[
1946,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1942,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
DifferentiableWithinAt.cpow
|
[] |
[
129,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.foldl_eq
|
[] |
[
522,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.sum_apply_eq_zero
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.337219\nγ : Type ?u.337222\nδ : Type ?u.337225\nι : Type u_1\nR : Type ?u.337231\nR' : Type ?u.337234\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\nh : ∀ (i : ι), ↑↑(μ i) s = 0\n⊢ ↑↑(sum μ) s ≤ 0",
"state_before": "α : Type u_2\nβ : Type ?u.337219\nγ : Type ?u.337222\nδ : Type ?u.337225\nι : Type u_1\nR : Type ?u.337231\nR' : Type ?u.337234\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\n⊢ ↑↑(sum μ) s = 0 ↔ ∀ (i : ι), ↑↑(μ i) s = 0",
"tactic": "refine'\n ⟨fun h i => nonpos_iff_eq_zero.1 <| h ▸ le_iff'.1 (le_sum μ i) _, fun h =>\n nonpos_iff_eq_zero.1 _⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.337219\nγ : Type ?u.337222\nδ : Type ?u.337225\nι : Type u_1\nR : Type ?u.337231\nR' : Type ?u.337234\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\nh : ∀ (i : ι), ↑↑(μ i) s = 0\nt : Set α\nhst : s ⊆ t\nhtm : MeasurableSet t\nht : ∀ (i : ι), ↑↑(μ i) t = ↑↑(μ i) s\n⊢ ↑↑(sum μ) s ≤ 0",
"state_before": "α : Type u_2\nβ : Type ?u.337219\nγ : Type ?u.337222\nδ : Type ?u.337225\nι : Type u_1\nR : Type ?u.337231\nR' : Type ?u.337234\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\nh : ∀ (i : ι), ↑↑(μ i) s = 0\n⊢ ↑↑(sum μ) s ≤ 0",
"tactic": "rcases exists_measurable_superset_forall_eq μ s with ⟨t, hst, htm, ht⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_2\nβ : Type ?u.337219\nγ : Type ?u.337222\nδ : Type ?u.337225\nι : Type u_1\nR : Type ?u.337231\nR' : Type ?u.337234\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\nh : ∀ (i : ι), ↑↑(μ i) s = 0\nt : Set α\nhst : s ⊆ t\nhtm : MeasurableSet t\nht : ∀ (i : ι), ↑↑(μ i) t = ↑↑(μ i) s\n⊢ ↑↑(sum μ) s ≤ 0",
"tactic": "calc\n sum μ s ≤ sum μ t := measure_mono hst\n _ = 0 := by simp [*]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.337219\nγ : Type ?u.337222\nδ : Type ?u.337225\nι : Type u_1\nR : Type ?u.337231\nR' : Type ?u.337234\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\nh : ∀ (i : ι), ↑↑(μ i) s = 0\nt : Set α\nhst : s ⊆ t\nhtm : MeasurableSet t\nht : ∀ (i : ι), ↑↑(μ i) t = ↑↑(μ i) s\n⊢ ↑↑(sum μ) t = 0",
"tactic": "simp [*]"
}
] |
[
2051,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2043,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finSumFinEquiv_symm_apply_castAdd
|
[] |
[
344,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Mathlib/Order/Filter/FilterProduct.lean
|
Filter.Germ.const_inf
|
[] |
[
99,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_preimage_mul_right
|
[
{
"state_after": "ι : Type ?u.2210287\ninst✝ : Fintype ι\na : ℝ\nh : a ≠ 0\ns : Set ℝ\n⊢ ↑↑(ofReal (abs a⁻¹) • volume) s = ofReal (abs a⁻¹) * ↑↑volume s",
"state_before": "ι : Type ?u.2210287\ninst✝ : Fintype ι\na : ℝ\nh : a ≠ 0\ns : Set ℝ\n⊢ ↑↑(Measure.map (fun x => x * a) volume) s = ofReal (abs a⁻¹) * ↑↑volume s",
"tactic": "rw [map_volume_mul_right h]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2210287\ninst✝ : Fintype ι\na : ℝ\nh : a ≠ 0\ns : Set ℝ\n⊢ ↑↑(ofReal (abs a⁻¹) • volume) s = ofReal (abs a⁻¹) * ↑↑volume s",
"tactic": "rfl"
}
] |
[
343,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.liftPropWithinAt_self_target
|
[] |
[
236,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
CauchySeq.mul
|
[] |
[
442,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Data/Part.lean
|
Part.inv_some
|
[] |
[
752,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
751,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean
|
uniqueDiffWithinAt_congr
|
[] |
[
295,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/GroupTheory/NielsenSchreier.lean
|
IsFreeGroupoid.path_nonempty_of_hom
|
[
{
"state_after": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\n⊢ Nonempty (Path (IsFreeGroupoid.symgen a) (IsFreeGroupoid.symgen b))",
"state_before": "G : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\n⊢ Nonempty (a ⟶ b) → Nonempty (Path (IsFreeGroupoid.symgen a) (IsFreeGroupoid.symgen b))",
"tactic": "rintro ⟨p⟩"
},
{
"state_after": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"state_before": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\n⊢ Nonempty (Path (IsFreeGroupoid.symgen a) (IsFreeGroupoid.symgen b))",
"tactic": "rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ←\n mul_inv_eq_one]"
},
{
"state_after": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"state_before": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"tactic": "let X := FreeGroup (WeaklyConnectedComponent <| Generators G)"
},
{
"state_after": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"state_before": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"tactic": "let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)"
},
{
"state_after": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"state_before": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"tactic": "let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Type u) := SingleObj.differenceFunctor f"
},
{
"state_after": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\n⊢ F.map p = ((Functor.const G).obj ()).map p",
"state_before": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\n⊢ FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1",
"tactic": "change (F.map p) = ((@CategoryTheory.Functor.const G _ _ (SingleObj.category X)).obj ()).map p"
},
{
"state_after": "case intro.h.e_5.h.e_self\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\n⊢ F = (Functor.const G).obj ()",
"state_before": "case intro\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\n⊢ F.map p = ((Functor.const G).obj ()).map p",
"tactic": "congr"
},
{
"state_after": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ F.map (of e✝) = ((Functor.const G).obj ()).map (of e✝)",
"state_before": "case intro.h.e_5.h.e_self\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\n⊢ F = (Functor.const G).obj ()",
"tactic": "ext"
},
{
"state_after": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ f b✝ =\n f\n (let_fun this := a✝;\n this)",
"state_before": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ F.map (of e✝) = ((Functor.const G).obj ()).map (of e✝)",
"tactic": "rw [Functor.const_obj_map, id_as_one, differenceFunctor_map, @mul_inv_eq_one _ _ (f _)]"
},
{
"state_after": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ WeaklyConnectedComponent.mk b✝ =\n WeaklyConnectedComponent.mk\n (let_fun this := a✝;\n this)",
"state_before": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ f b✝ =\n f\n (let_fun this := a✝;\n this)",
"tactic": "apply congr_arg FreeGroup.of"
},
{
"state_after": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ Nonempty\n (Path b✝\n (let_fun this := a✝;\n this))",
"state_before": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ WeaklyConnectedComponent.mk b✝ =\n WeaklyConnectedComponent.mk\n (let_fun this := a✝;\n this)",
"tactic": "apply (WeaklyConnectedComponent.eq _ _).mpr"
},
{
"state_after": "no goals",
"state_before": "case intro.h.e_5.h.e_self.h\nG : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ Nonempty\n (Path b✝\n (let_fun this := a✝;\n this))",
"tactic": "exact ⟨Hom.toPath (Sum.inr (by assumption))⟩"
},
{
"state_after": "no goals",
"state_before": "G : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\np : a ⟶ b\nX : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))\nf : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\nF : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f\na✝ b✝ : Generators G\ne✝ : a✝ ⟶ b✝\n⊢ (let_fun this := a✝;\n this) ⟶\n b✝",
"tactic": "assumption"
}
] |
[
291,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.ssubset_univ_iff
|
[] |
[
142,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.minFac_le
|
[] |
[
348,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/Order/Max.lean
|
isMin_toDual_iff
|
[] |
[
243,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.inter_eq_left_iff_subset
|
[] |
[
1794,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1793,
1
] |
Mathlib/RingTheory/Int/Basic.lean
|
Int.normUnit_eq
|
[] |
[
102,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_union_Ioc_eq_Icc
|
[] |
[
1587,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1584,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.succ_succAbove_zero
|
[
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\ninst✝ : NeZero n\ni : Fin n\n⊢ ↑castSucc 0 < succ i",
"tactic": "simp only [castSucc_zero, succ_pos]"
}
] |
[
2212,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2211,
1
] |
Mathlib/Data/Set/Intervals/WithBotTop.lean
|
WithBot.image_coe_Ioo
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\n⊢ some '' Ioo a b = Ioo ↑a ↑b",
"tactic": "rw [← preimage_coe_Ioo, image_preimage_eq_inter_range, range_coe,\n inter_eq_self_of_subset_left (Subset.trans Ioo_subset_Ioi_self <| Ioi_subset_Ioi bot_le)]"
}
] |
[
235,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
concaveOn_id
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type ?u.14565\nF : Type ?u.14568\nα : Type ?u.14571\nβ : Type u_1\nι : Type ?u.14577\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : OrderedAddCommMonoid α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : SMul 𝕜 α\ninst✝ : SMul 𝕜 β\ns✝ : Set E\nf : E → β\ng : β → α\ns : Set β\nhs : Convex 𝕜 s\nx✝ : β\na✝⁵ : x✝ ∈ s\ny✝ : β\na✝⁴ : y✝ ∈ s\na✝³ b✝ : 𝕜\na✝² : 0 ≤ a✝³\na✝¹ : 0 ≤ b✝\na✝ : a✝³ + b✝ = 1\n⊢ a✝³ • _root_.id x✝ + b✝ • _root_.id y✝ ≤ _root_.id (a✝³ • x✝ + b✝ • y✝)",
"state_before": "𝕜 : Type u_2\nE : Type ?u.14565\nF : Type ?u.14568\nα : Type ?u.14571\nβ : Type u_1\nι : Type ?u.14577\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : OrderedAddCommMonoid α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : SMul 𝕜 α\ninst✝ : SMul 𝕜 β\ns✝ : Set E\nf : E → β\ng : β → α\ns : Set β\nhs : Convex 𝕜 s\n⊢ ∀ ⦃x : β⦄,\n x ∈ s →\n ∀ ⦃y : β⦄,\n y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • _root_.id x + b • _root_.id y ≤ _root_.id (a • x + b • y)",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type ?u.14565\nF : Type ?u.14568\nα : Type ?u.14571\nβ : Type u_1\nι : Type ?u.14577\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : OrderedAddCommMonoid α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : SMul 𝕜 α\ninst✝ : SMul 𝕜 β\ns✝ : Set E\nf : E → β\ng : β → α\ns : Set β\nhs : Convex 𝕜 s\nx✝ : β\na✝⁵ : x✝ ∈ s\ny✝ : β\na✝⁴ : y✝ ∈ s\na✝³ b✝ : 𝕜\na✝² : 0 ≤ a✝³\na✝¹ : 0 ≤ b✝\na✝ : a✝³ + b✝ = 1\n⊢ a✝³ • _root_.id x✝ + b✝ • _root_.id y✝ ≤ _root_.id (a✝³ • x✝ + b✝ • y✝)",
"tactic": "rfl"
}
] |
[
103,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.StableUnderBaseChange.snd
|
[] |
[
211,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.pi_apply
|
[] |
[
1223,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1222,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
nnnorm_pow_le_mul_norm
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.791224\n𝕜 : Type ?u.791227\nα : Type ?u.791230\nι : Type ?u.791233\nκ : Type ?u.791236\nE : Type u_1\nF : Type ?u.791242\nG : Type ?u.791245\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nn : ℕ\na : E\n⊢ ‖a ^ n‖₊ ≤ ↑n * ‖a‖₊",
"tactic": "simpa only [← NNReal.coe_le_coe, NNReal.coe_mul, NNReal.coe_nat_cast] using\n norm_pow_le_mul_norm n a"
}
] |
[
1558,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1556,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.coe_im
|
[] |
[
293,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/CategoryTheory/LiftingProperties/Basic.lean
|
CategoryTheory.HasLiftingProperty.op
|
[
{
"state_after": "C : Type u_1\ninst✝ : Category C\nA B B' X Y Y' : C\ni : A ⟶ B\ni' : B ⟶ B'\np : X ⟶ Y\np' : Y ⟶ Y'\nh : HasLiftingProperty i p\nf : Y.op ⟶ B.op\ng : X.op ⟶ A.op\nsq : CommSq f p.op i.op g\n⊢ CommSq.HasLift (_ : CommSq g.unop i p f.unop)",
"state_before": "C : Type u_1\ninst✝ : Category C\nA B B' X Y Y' : C\ni : A ⟶ B\ni' : B ⟶ B'\np : X ⟶ Y\np' : Y ⟶ Y'\nh : HasLiftingProperty i p\nf : Y.op ⟶ B.op\ng : X.op ⟶ A.op\nsq : CommSq f p.op i.op g\n⊢ CommSq.HasLift sq",
"tactic": "simp only [CommSq.HasLift.iff_unop, Quiver.Hom.unop_op]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\nA B B' X Y Y' : C\ni : A ⟶ B\ni' : B ⟶ B'\np : X ⟶ Y\np' : Y ⟶ Y'\nh : HasLiftingProperty i p\nf : Y.op ⟶ B.op\ng : X.op ⟶ A.op\nsq : CommSq f p.op i.op g\n⊢ CommSq.HasLift (_ : CommSq g.unop i p f.unop)",
"tactic": "infer_instance"
}
] |
[
61,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Topology/UniformSpace/Compact.lean
|
Continuous.tendstoUniformly
|
[] |
[
262,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Computability/Encoding.lean
|
Computability.FinEncoding.card_le_aleph0
|
[] |
[
261,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieHom.mem_idealRange
|
[
{
"state_after": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nx : L\n⊢ ↑f x ∈ LieIdeal.map f ⊤",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nx : L\n⊢ ↑f x ∈ idealRange f",
"tactic": "rw [idealRange_eq_map]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nx : L\n⊢ ↑f x ∈ LieIdeal.map f ⊤",
"tactic": "exact LieIdeal.mem_map (LieSubmodule.mem_top x)"
}
] |
[
963,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
961,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
CategoryTheory.Limits.pullbackDiagonalMapIdIso_inv_snd_snd
|
[
{
"state_after": "C : Type u_2\ninst✝⁵ : Category C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ :\n HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) (_ : (f ≫ i) ≫ 𝟙 S = f ≫ i) (_ : (g ≫ i) ≫ 𝟙 S = g ≫ i))\n⊢ snd ≫ snd = (pullbackDiagonalMapIdIso f g i).hom ≫ snd",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ :\n HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) (_ : (f ≫ i) ≫ 𝟙 S = f ≫ i) (_ : (g ≫ i) ≫ 𝟙 S = g ≫ i))\n⊢ (pullbackDiagonalMapIdIso f g i).inv ≫ snd ≫ snd = snd",
"tactic": "rw [Iso.inv_comp_eq]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\nX Y Z : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf : X ⟶ T\ng : Y ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback (f ≫ i) (g ≫ i)\ninst✝ :\n HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) (_ : (f ≫ i) ≫ 𝟙 S = f ≫ i) (_ : (g ≫ i) ≫ 𝟙 S = g ≫ i))\n⊢ snd ≫ snd = (pullbackDiagonalMapIdIso f g i).hom ≫ snd",
"tactic": "simp"
}
] |
[
274,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Functor.map_hom_inv
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ F.map f ≫ F.map (inv f) = 𝟙 (F.obj X)",
"tactic": "simp"
}
] |
[
628,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_filter_lt_of_lt_right
|
[] |
[
356,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Data/List/Duplicate.lean
|
List.duplicate_cons_iff_of_ne
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\nx y : α\nhne : x ≠ y\n⊢ x ∈+ y :: l ↔ x ∈+ l",
"tactic": "simp [duplicate_cons_iff, hne.symm]"
}
] |
[
107,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.mem_localTrivAt_target
|
[] |
[
728,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
726,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.ndinter_eq_zero_iff_disjoint
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ s t : Multiset α\n⊢ ndinter s t ⊆ 0 ↔ Disjoint s t",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ s t : Multiset α\n⊢ ndinter s t = 0 ↔ Disjoint s t",
"tactic": "rw [← subset_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ s t : Multiset α\n⊢ ndinter s t ⊆ 0 ↔ Disjoint s t",
"tactic": "simp [subset_iff, Disjoint]"
}
] |
[
282,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
LinearIsometry.toLinearIsometryEquiv_apply
|
[] |
[
86,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.insert.valid
|
[
{
"state_after": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : IsTotal α fun x x_1 => x ≤ x_1\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordnode α\nh : Valid t\n⊢ Valid (insertWith (fun x_1 => x) x t)",
"state_before": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : IsTotal α fun x x_1 => x ≤ x_1\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordnode α\nh : Valid t\n⊢ Valid (Ordnode.insert x t)",
"tactic": "rw [insert_eq_insertWith]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : IsTotal α fun x x_1 => x ≤ x_1\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordnode α\nh : Valid t\n⊢ Valid (insertWith (fun x_1 => x) x t)",
"tactic": "exact insertWith.valid _ _ (fun _ _ => ⟨le_rfl, le_rfl⟩) h"
}
] |
[
1549,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1547,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.translationNumber_lt_of_forall_lt_add
|
[
{
"state_after": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ τ f < z",
"state_before": "f g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\n⊢ τ f < z",
"tactic": "obtain ⟨x, -, hx⟩ : ∃ x ∈ Icc (0 : ℝ) 1, ∀ y ∈ Icc (0 : ℝ) 1, f y - y ≤ f x - x :=\n isCompact_Icc.exists_forall_ge (nonempty_Icc.2 zero_le_one)\n (hf.sub continuous_id).continuousOn"
},
{
"state_after": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ τ f ≤ ↑f x - x",
"state_before": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ τ f < z",
"tactic": "refine' lt_of_le_of_lt _ (sub_lt_iff_lt_add'.2 <| hz x)"
},
{
"state_after": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ ∀ (x_1 : ℝ), ↑f x_1 ≤ x_1 + (↑f x - x)",
"state_before": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ τ f ≤ ↑f x - x",
"tactic": "apply translationNumber_le_of_le_add"
},
{
"state_after": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ ∀ (x_1 : ℝ), ↑f x_1 - x_1 ≤ ↑f x - x",
"state_before": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ ∀ (x_1 : ℝ), ↑f x_1 ≤ x_1 + (↑f x - x)",
"tactic": "simp only [← sub_le_iff_le_add']"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), ↑f x < x + z\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f y - y ≤ ↑f x - x\n⊢ ∀ (x_1 : ℝ), ↑f x_1 - x_1 ≤ ↑f x - x",
"tactic": "exact f.forall_map_sub_of_Icc (fun a => a ≤ f x - x) hx"
}
] |
[
915,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
907,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.restrict_mono
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.817681\nδ : Type ?u.817684\ninst✝² : MeasurableSpace α\nK : Type ?u.817690\ninst✝¹ : Zero β\ninst✝ : Preorder β\ns : Set α\nf g : α →ₛ β\nH : f ≤ g\nhs : MeasurableSet s\nx : α\n⊢ ↑(restrict f s) x ≤ ↑(restrict g s) x",
"tactic": "simp only [coe_restrict _ hs, indicator_le_indicator (H x)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.817681\nδ : Type ?u.817684\ninst✝² : MeasurableSpace α\nK : Type ?u.817690\ninst✝¹ : Zero β\ninst✝ : Preorder β\ns : Set α\nf g : α →ₛ β\nH : f ≤ g\nhs : ¬MeasurableSet s\n⊢ restrict f s ≤ restrict g s",
"tactic": "simp only [restrict_of_not_measurable hs, le_refl]"
}
] |
[
815,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
811,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
affineIndependent_of_ne_of_mem_of_mem_of_not_mem
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\n⊢ AffineIndependent k ![p₁, p₂, p₃]",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\n⊢ AffineIndependent k ![p₁, p₂, p₃]",
"tactic": "have ha : AffineIndependent k fun x : { x : Fin 3 // x ≠ 2 } => ![p₁, p₂, p₃] x := by\n rw [← affineIndependent_equiv (finSuccAboveEquiv (2 : Fin 3)).toEquiv]\n convert affineIndependent_of_ne k hp₁p₂\n ext x\n fin_cases x <;> rfl"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\n⊢ ¬Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\n⊢ AffineIndependent k ![p₁, p₂, p₃]",
"tactic": "refine' ha.affineIndependent_of_not_mem_span _"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\n⊢ False",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\n⊢ ¬Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})",
"tactic": "intro h"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\n⊢ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2}) ≤ s",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\n⊢ False",
"tactic": "refine' hp₃ ((AffineSubspace.le_def' _ s).1 _ p₃ h)"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\n⊢ ∀ (x : Fin 3), x ∈ {x | x ≠ 2} → Matrix.vecCons p₁ ![p₂, p₃] x ∈ ↑s",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\n⊢ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2}) ≤ s",
"tactic": "simp_rw [affineSpan_le, Set.image_subset_iff, Set.subset_def, Set.mem_preimage]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\nx : Fin 3\n⊢ x ∈ {x | x ≠ 2} → Matrix.vecCons p₁ ![p₂, p₃] x ∈ ↑s",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\n⊢ ∀ (x : Fin 3), x ∈ {x | x ≠ 2} → Matrix.vecCons p₁ ![p₂, p₃] x ∈ ↑s",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nha : AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x\nh : Matrix.vecCons p₁ ![p₂, p₃] 2 ∈ affineSpan k (![p₁, p₂, p₃] '' {x | x ≠ 2})\nx : Fin 3\n⊢ x ∈ {x | x ≠ 2} → Matrix.vecCons p₁ ![p₂, p₃] x ∈ ↑s",
"tactic": "fin_cases x <;> simp [hp₁, hp₂]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\n⊢ AffineIndependent k ((fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x) ∘ ↑(finSuccAboveEquiv 2).toEquiv)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\n⊢ AffineIndependent k fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x",
"tactic": "rw [← affineIndependent_equiv (finSuccAboveEquiv (2 : Fin 3)).toEquiv]"
},
{
"state_after": "case h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\n⊢ (fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x) ∘ ↑(finSuccAboveEquiv 2).toEquiv = ![p₁, p₂]",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\n⊢ AffineIndependent k ((fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x) ∘ ↑(finSuccAboveEquiv 2).toEquiv)",
"tactic": "convert affineIndependent_of_ne k hp₁p₂"
},
{
"state_after": "case h.e'_9.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nx : Fin 2\n⊢ ((fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x) ∘ ↑(finSuccAboveEquiv 2).toEquiv) x = Matrix.vecCons p₁ ![p₂] x",
"state_before": "case h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\n⊢ (fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x) ∘ ↑(finSuccAboveEquiv 2).toEquiv = ![p₁, p₂]",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_9.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.444767\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : ¬p₃ ∈ s\nx : Fin 2\n⊢ ((fun x => Matrix.vecCons p₁ ![p₂, p₃] ↑x) ∘ ↑(finSuccAboveEquiv 2).toEquiv) x = Matrix.vecCons p₁ ![p₂] x",
"tactic": "fin_cases x <;> rfl"
}
] |
[
693,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
680,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
AddSubgroup.le_pointwise_smul_iff₀
|
[] |
[
557,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.liminf_le_of_le
|
[] |
[
475,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Algebra/Module/Submodule/Pointwise.lean
|
Submodule.neg_sup
|
[] |
[
123,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.rightLim
|
[] |
[
144,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
11
] |
Mathlib/NumberTheory/LSeries.lean
|
Nat.ArithmeticFunction.LSeriesSummable_zero
|
[
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ LSeriesSummable 0 z",
"tactic": "simp [LSeriesSummable, summable_zero]"
}
] |
[
61,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.floor_logb_nat_cast
|
[
{
"state_after": "case inl\nb✝ x y : ℝ\nb : ℕ\nhb : 1 < b\nhr : 0 ≤ 0\n⊢ ⌊logb (↑b) 0⌋ = Int.log b 0\n\ncase inr\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"state_before": "b✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr : 0 ≤ r\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"tactic": "obtain rfl | hr := hr.eq_or_lt"
},
{
"state_after": "case inr\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"state_before": "case inr\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"tactic": "have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb"
},
{
"state_after": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ⌊logb (↑b) r⌋ ≤ Int.log b r\n\ncase inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ Int.log b r ≤ ⌊logb (↑b) r⌋",
"state_before": "case inr\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"tactic": "apply le_antisymm"
},
{
"state_after": "no goals",
"state_before": "case inl\nb✝ x y : ℝ\nb : ℕ\nhb : 1 < b\nhr : 0 ≤ 0\n⊢ ⌊logb (↑b) 0⌋ = Int.log b 0",
"tactic": "rw [logb_zero, Int.log_zero_right, Int.floor_zero]"
},
{
"state_after": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ↑b ^ ↑⌊logb (↑b) r⌋ ≤ r",
"state_before": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ⌊logb (↑b) r⌋ ≤ Int.log b r",
"tactic": "rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_int_cast b]"
},
{
"state_after": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ↑b ^ ↑⌊logb (↑b) r⌋ ≤ ↑b ^ logb (↑b) r",
"state_before": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ↑b ^ ↑⌊logb (↑b) r⌋ ≤ r",
"tactic": "refine' le_of_le_of_eq _ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr)"
},
{
"state_after": "no goals",
"state_before": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ↑b ^ ↑⌊logb (↑b) r⌋ ≤ ↑b ^ logb (↑b) r",
"tactic": "exact rpow_le_rpow_of_exponent_le hb1'.le (Int.floor_le _)"
},
{
"state_after": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ↑b ^ Int.log b r ≤ r",
"state_before": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ Int.log b r ≤ ⌊logb (↑b) r⌋",
"tactic": "rw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_int_cast]"
},
{
"state_after": "no goals",
"state_before": "case inr.a\nb✝ x y : ℝ\nb : ℕ\nr : ℝ\nhb : 1 < b\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb1' : 1 < ↑b\n⊢ ↑b ^ Int.log b r ≤ r",
"tactic": "exact Int.zpow_log_le_self hb hr"
}
] |
[
366,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.real_toNNReal
|
[] |
[
1753,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1751,
1
] |
Std/Data/Int/DivMod.lean
|
Int.natAbs_div_le_natAbs
|
[
{
"state_after": "a b : Int\nn : Nat\n⊢ natAbs (a / ↑n) ≤ natAbs a",
"state_before": "a b : Int\nn : Nat\n⊢ natAbs (a / -↑n) ≤ natAbs a",
"tactic": "rw [Int.ediv_neg, natAbs_neg]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nn : Nat\n⊢ natAbs (a / ↑n) ≤ natAbs a",
"tactic": "apply aux"
}
] |
[
573,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
565,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.map_smul
|
[] |
[
178,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
11
] |
Mathlib/Data/Nat/Interval.lean
|
Nat.card_fintypeIoo
|
[
{
"state_after": "no goals",
"state_before": "a b c : ℕ\n⊢ Fintype.card ↑(Set.Ioo a b) = b - a - 1",
"tactic": "rw [Fintype.card_ofFinset, card_Ioo]"
}
] |
[
145,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.coe_ne_zero
|
[] |
[
80,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
stronglyMeasurable_of_stronglyMeasurable_union_cover
|
[
{
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"tactic": "refine' ⟨f, fun y => _⟩"
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"tactic": "rw [dif_neg hy]"
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"state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : y ∈ s\n⊢ (fun n => ↑(f n) y) = fun n => ↑(StronglyMeasurable.approx hc n) { val := y, property := hy }",
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"tactic": "convert hc.tendsto_approx ⟨y, hy⟩ using 1"
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{
"state_after": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : y ∈ s\nn : ℕ\n⊢ ↑(f n) y = ↑(StronglyMeasurable.approx hc n) { val := y, property := hy }",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : y ∈ s\n⊢ (fun n => ↑(f n) y) = fun n => ↑(StronglyMeasurable.approx hc n) { val := y, property := hy }",
"tactic": "ext1 n"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : y ∈ s\nn : ℕ\n⊢ ↑(f n) y = ↑(StronglyMeasurable.approx hc n) { val := y, property := hy }",
"tactic": "simp only [dif_pos hy, SimpleFunc.apply_mk]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\nA : y ∈ t\n⊢ Tendsto (fun n => ↑(f n) y) atTop (𝓝 (f✝ y))",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\n⊢ Tendsto (fun n => ↑(f n) y) atTop (𝓝 (f✝ y))",
"tactic": "have A : y ∈ t := by simpa [hy] using h (mem_univ y)"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\nA : y ∈ t\n⊢ (fun n => ↑(f n) y) = fun n => ↑(StronglyMeasurable.approx hd n) { val := y, property := A }",
"state_before": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\nA : y ∈ t\n⊢ Tendsto (fun n => ↑(f n) y) atTop (𝓝 (f✝ y))",
"tactic": "convert hd.tendsto_approx ⟨y, A⟩ using 1"
},
{
"state_after": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\nA : y ∈ t\nn : ℕ\n⊢ ↑(f n) y = ↑(StronglyMeasurable.approx hd n) { val := y, property := A }",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\nA : y ∈ t\n⊢ (fun n => ↑(f n) y) = fun n => ↑(StronglyMeasurable.approx hd n) { val := y, property := A }",
"tactic": "ext1 n"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\nA : y ∈ t\nn : ℕ\n⊢ ↑(f n) y = ↑(StronglyMeasurable.approx hd n) { val := y, property := A }",
"tactic": "simp only [dif_neg hy, SimpleFunc.apply_mk]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.197809\nι : Type ?u.197812\ninst✝¹ : Countable ι\nf✝¹ g : α → β\nm : MeasurableSpace α\ninst✝ : TopologicalSpace β\nf✝ : α → β\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : univ ⊆ s ∪ t\nhc : StronglyMeasurable fun a => f✝ ↑a\nhd : StronglyMeasurable fun a => f✝ ↑a\nf : ℕ → α →ₛ β :=\n fun n =>\n {\n toFun := fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) },\n measurableSet_fiber' :=\n (_ :\n ∀ (x : β),\n MeasurableSet\n ((fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) }) ⁻¹'\n {x})),\n finite_range' :=\n (_ :\n Set.Finite\n (range fun x =>\n if hx : x ∈ s then ↑(StronglyMeasurable.approx hc n) { val := x, property := hx }\n else ↑(StronglyMeasurable.approx hd n) { val := x, property := (_ : x ∈ t) })) }\ny : α\nhy : ¬y ∈ s\n⊢ y ∈ t",
"tactic": "simpa [hy] using h (mem_univ y)"
}
] |
[
799,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
tendsto_nhds_of_tendsto_nhdsWithin
|
[] |
[
390,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Filter.tendsto_iff_ultrafilter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.52285\nf✝ : Filter α\ns : Set α\na : α\nf : α → β\nl₁ : Filter α\nl₂ : Filter β\n⊢ Tendsto f l₁ l₂ ↔ ∀ (g : Ultrafilter α), ↑g ≤ l₁ → Tendsto f (↑g) l₂",
"tactic": "simpa only [tendsto_iff_comap] using le_iff_ultrafilter"
}
] |
[
454,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Combinatorics/Quiver/SingleObj.lean
|
Quiver.SingleObj.toPrefunctor_symm_comp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : SingleObj α ⥤q SingleObj β\ng : SingleObj β ⥤q SingleObj γ\n⊢ ↑toPrefunctor.symm (f ⋙q g) = ↑toPrefunctor.symm g ∘ ↑toPrefunctor.symm f",
"tactic": "simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]"
}
] |
[
118,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.mem_range_embedding_iff'
|
[
{
"state_after": "case mp\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\n⊢ j ∈ Set.range ↑(embedding c i) → i = index c j\n\ncase mpr\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\n⊢ i = index c j → j ∈ Set.range ↑(embedding c i)",
"state_before": "n : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\n⊢ j ∈ Set.range ↑(embedding c i) ↔ i = index c j",
"tactic": "constructor"
},
{
"state_after": "case mp\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\n⊢ ¬i = index c j → ¬j ∈ Set.range ↑(embedding c i)",
"state_before": "case mp\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\n⊢ j ∈ Set.range ↑(embedding c i) → i = index c j",
"tactic": "rw [← not_imp_not]"
},
{
"state_after": "case mp\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\nh : ¬i = index c j\n⊢ ¬j ∈ Set.range ↑(embedding c i)",
"state_before": "case mp\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\n⊢ ¬i = index c j → ¬j ∈ Set.range ↑(embedding c i)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\nh : ¬i = index c j\n⊢ ¬j ∈ Set.range ↑(embedding c i)",
"tactic": "exact Set.disjoint_right.1 (c.disjoint_range h) (c.mem_range_embedding j)"
},
{
"state_after": "case mpr\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\nh : i = index c j\n⊢ j ∈ Set.range ↑(embedding c i)",
"state_before": "case mpr\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\n⊢ i = index c j → j ∈ Set.range ↑(embedding c i)",
"tactic": "intro h"
},
{
"state_after": "case mpr\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\nh : i = index c j\n⊢ j ∈ Set.range ↑(embedding c (index c j))",
"state_before": "case mpr\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\nh : i = index c j\n⊢ j ∈ Set.range ↑(embedding c i)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nn : ℕ\nc : Composition n\nj : Fin n\ni : Fin (length c)\nh : i = index c j\n⊢ j ∈ Set.range ↑(embedding c (index c j))",
"tactic": "exact c.mem_range_embedding j"
}
] |
[
431,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
Commute.zpow_zpow_self
|
[] |
[
514,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/LinearAlgebra/ProjectiveSpace/Subspace.lean
|
Projectivization.Subspace.mem_add
|
[] |
[
73,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.map_toPoly
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type u_1\nF : Type ?u.679043\nK : Type ?u.679046\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nφ : R →+* S\n⊢ toPoly (map φ P) = Polynomial.map φ (toPoly P)",
"tactic": "simp only [map, toPoly, map_C, map_X, Polynomial.map_add, Polynomial.map_mul, Polynomial.map_pow]"
}
] |
[
462,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.eq_of_ne_mem
|
[] |
[
385,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_pos_iff
|
[
{
"state_after": "x y : ℝ\nhx : 0 < x\n⊢ log 1 < log x ↔ 1 < x",
"state_before": "x y : ℝ\nhx : 0 < x\n⊢ 0 < log x ↔ 1 < x",
"tactic": "rw [← log_one]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : 0 < x\n⊢ log 1 < log x ↔ 1 < x",
"tactic": "exact log_lt_log_iff zero_lt_one hx"
}
] |
[
172,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/MeasureTheory/Function/L2Space.lean
|
MeasureTheory.L2.snorm_inner_lt_top
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ snorm (fun x => inner (↑↑f x) (↑↑g x)) 1 μ < ⊤",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\n⊢ snorm (fun x => inner (↑↑f x) (↑↑g x)) 1 μ < ⊤",
"tactic": "have h : ∀ x, ‖⟪f x, g x⟫‖ ≤ ‖‖f x‖ ^ (2 : ℝ) + ‖g x‖ ^ (2 : ℝ)‖ := by\n intro x\n rw [← @Nat.cast_two ℝ, Real.rpow_nat_cast, Real.rpow_nat_cast]\n calc\n ‖⟪f x, g x⟫‖ ≤ ‖f x‖ * ‖g x‖ := norm_inner_le_norm _ _\n _ ≤ 2 * ‖f x‖ * ‖g x‖ :=\n (mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_nonneg _) one_le_two)\n (norm_nonneg _))\n _ ≤ ‖‖f x‖ ^ 2 + ‖g x‖ ^ 2‖ := (two_mul_le_add_sq _ _).trans (le_abs_self _)"
},
{
"state_after": "case refine'_1\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ\n\ncase refine'_2\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ AEStronglyMeasurable (fun a => ‖↑↑g a‖ ^ 2) μ\n\ncase refine'_3\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ snorm (fun a => ‖↑↑f a‖ ^ 2) 1 μ + snorm (fun a => ‖↑↑g a‖ ^ 2) 1 μ < ⊤",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ snorm (fun x => inner (↑↑f x) (↑↑g x)) 1 μ < ⊤",
"tactic": "refine' (snorm_mono_ae (ae_of_all _ h)).trans_lt ((snorm_add_le _ _ le_rfl).trans_lt _)"
},
{
"state_after": "case refine'_3\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ snorm (fun a => ‖↑↑f a‖ ^ 2) 1 μ < ⊤ ∧ snorm (fun a => ‖↑↑g a‖ ^ 2) 1 μ < ⊤",
"state_before": "case refine'_3\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ snorm (fun a => ‖↑↑f a‖ ^ 2) 1 μ + snorm (fun a => ‖↑↑g a‖ ^ 2) 1 μ < ⊤",
"tactic": "rw [ENNReal.add_lt_top]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ snorm (fun a => ‖↑↑f a‖ ^ 2) 1 μ < ⊤ ∧ snorm (fun a => ‖↑↑g a‖ ^ 2) 1 μ < ⊤",
"tactic": "exact ⟨snorm_rpow_two_norm_lt_top f, snorm_rpow_two_norm_lt_top g⟩"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nx : α\n⊢ ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\n⊢ ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖",
"tactic": "intro x"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nx : α\n⊢ ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nx : α\n⊢ ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖",
"tactic": "rw [← @Nat.cast_two ℝ, Real.rpow_nat_cast, Real.rpow_nat_cast]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nx : α\n⊢ ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖",
"tactic": "calc\n ‖⟪f x, g x⟫‖ ≤ ‖f x‖ * ‖g x‖ := norm_inner_le_norm _ _\n _ ≤ 2 * ‖f x‖ * ‖g x‖ :=\n (mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_nonneg _) one_le_two)\n (norm_nonneg _))\n _ ≤ ‖‖f x‖ ^ 2 + ‖g x‖ ^ 2‖ := (two_mul_le_add_sq _ _).trans (le_abs_self _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ",
"tactic": "exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nE : Type u_2\nF : Type ?u.44217\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf g : { x // x ∈ Lp E 2 }\nh : ∀ (x : α), ‖inner (↑↑f x) (↑↑g x)‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖\n⊢ AEStronglyMeasurable (fun a => ‖↑↑g a‖ ^ 2) μ",
"tactic": "exact ((Lp.aestronglyMeasurable g).norm.aemeasurable.pow_const _).aestronglyMeasurable"
}
] |
[
146,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
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