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/Analysis/Calculus/DiffContOnCl.lean
|
DiffContOnCl.sub_const
|
[] |
[
114,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.mem_closedBall_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.33624\nι : Type ?u.33627\ninst✝ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\nh : 0 ≤ ε\n⊢ x ∈ closedBall x ε",
"tactic": "rwa [mem_closedBall, dist_self]"
}
] |
[
510,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/LinearAlgebra/Trace.lean
|
LinearMap.trace_eq_contract
|
[] |
[
169,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.ofFractionRing_eq
|
[
{
"state_after": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nx✝ : FractionRing K[X]\nx : K[X] × { x // x ∈ K[X]⁰ }\n⊢ IsLocalization.mk' (RatFunc K) x.fst x.snd =\n IsLocalization.mk' (RatFunc K) (↑(RingHom.id K[X]) x.fst)\n { val := ↑(RingHom.id K[X]) ↑x.snd, property := (_ : ↑x.snd ∈ Submonoid.comap (RingHom.id K[X]) K[X]⁰) }",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nx✝ : FractionRing K[X]\nx : K[X] × { x // x ∈ K[X]⁰ }\n⊢ { toFractionRing := Localization.mk x.fst x.snd } =\n ↑(IsLocalization.algEquiv K[X]⁰ (FractionRing K[X]) (RatFunc K)) (Localization.mk x.fst x.snd)",
"tactic": "simp only [IsLocalization.algEquiv_apply, IsLocalization.ringEquivOfRingEquiv_apply,\n Localization.mk_eq_mk'_apply, IsLocalization.map_mk', ofFractionRing_mk',\n RingEquiv.coe_toRingHom, RingEquiv.refl_apply, SetLike.eta]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nx✝ : FractionRing K[X]\nx : K[X] × { x // x ∈ K[X]⁰ }\n⊢ IsLocalization.mk' (RatFunc K) x.fst x.snd =\n IsLocalization.mk' (RatFunc K) (↑(RingHom.id K[X]) x.fst)\n { val := ↑(RingHom.id K[X]) ↑x.snd, property := (_ : ↑x.snd ∈ Submonoid.comap (RingHom.id K[X]) K[X]⁰) }",
"tactic": "simp only [IsFractionRing.mk'_eq_div, RingHom.id_apply, Subtype.coe_eta]"
}
] |
[
1051,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1043,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.eqLocus_same
|
[] |
[
2897,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2896,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.Compacts.coe_top
|
[] |
[
118,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PreservesPullback.iso_inv_snd
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nG : C ⥤ D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ X\nk : W ⟶ Y\ncomm : h ≫ f = k ≫ g\ninst✝² : PreservesLimit (cospan f g) G\ninst✝¹ : HasPullback f g\ninst✝ : HasPullback (G.map f) (G.map g)\n⊢ (iso G f g).inv ≫ G.map pullback.snd = pullback.snd",
"tactic": "simp [PreservesPullback.iso, Iso.inv_comp_eq]"
}
] |
[
141,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Topology/Algebra/MulAction.lean
|
ContinuousOn.smul
|
[] |
[
118,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Control/Basic.lean
|
pure_id'_seq
|
[] |
[
61,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_diagonal
|
[
{
"state_after": "case a.mk.h.mk\nl : Type ?u.136326\nm : Type u_1\nn : Type ?u.136332\no : Type u_2\np : Type ?u.136338\nq : Type ?u.136341\nm' : o → Type ?u.136346\nn' : o → Type ?u.136351\np' : o → Type ?u.136356\nR : Type ?u.136359\nS : Type ?u.136362\nα : Type u_3\nβ : Type ?u.136368\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : DecidableEq m\nd : o → m → α\ni : m\nk : o\nj : m\nk' : o\n⊢ blockDiagonal (fun k => diagonal (d k)) (i, k) (j, k') = diagonal (fun ik => d ik.snd ik.fst) (i, k) (j, k')",
"state_before": "l : Type ?u.136326\nm : Type u_1\nn : Type ?u.136332\no : Type u_2\np : Type ?u.136338\nq : Type ?u.136341\nm' : o → Type ?u.136346\nn' : o → Type ?u.136351\np' : o → Type ?u.136356\nR : Type ?u.136359\nS : Type ?u.136362\nα : Type u_3\nβ : Type ?u.136368\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : DecidableEq m\nd : o → m → α\n⊢ (blockDiagonal fun k => diagonal (d k)) = diagonal fun ik => d ik.snd ik.fst",
"tactic": "ext (⟨i, k⟩⟨j, k'⟩)"
},
{
"state_after": "case a.mk.h.mk\nl : Type ?u.136326\nm : Type u_1\nn : Type ?u.136332\no : Type u_2\np : Type ?u.136338\nq : Type ?u.136341\nm' : o → Type ?u.136346\nn' : o → Type ?u.136351\np' : o → Type ?u.136356\nR : Type ?u.136359\nS : Type ?u.136362\nα : Type u_3\nβ : Type ?u.136368\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : DecidableEq m\nd : o → m → α\ni : m\nk : o\nj : m\nk' : o\n⊢ (if k = k' ∧ i = j then d k i else 0) = if i = j ∧ k = k' then d k i else 0",
"state_before": "case a.mk.h.mk\nl : Type ?u.136326\nm : Type u_1\nn : Type ?u.136332\no : Type u_2\np : Type ?u.136338\nq : Type ?u.136341\nm' : o → Type ?u.136346\nn' : o → Type ?u.136351\np' : o → Type ?u.136356\nR : Type ?u.136359\nS : Type ?u.136362\nα : Type u_3\nβ : Type ?u.136368\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : DecidableEq m\nd : o → m → α\ni : m\nk : o\nj : m\nk' : o\n⊢ blockDiagonal (fun k => diagonal (d k)) (i, k) (j, k') = diagonal (fun ik => d ik.snd ik.fst) (i, k) (j, k')",
"tactic": "simp only [blockDiagonal_apply, diagonal_apply, Prod.mk.inj_iff, ← ite_and]"
},
{
"state_after": "case a.mk.h.mk.e_c\nl : Type ?u.136326\nm : Type u_1\nn : Type ?u.136332\no : Type u_2\np : Type ?u.136338\nq : Type ?u.136341\nm' : o → Type ?u.136346\nn' : o → Type ?u.136351\np' : o → Type ?u.136356\nR : Type ?u.136359\nS : Type ?u.136362\nα : Type u_3\nβ : Type ?u.136368\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : DecidableEq m\nd : o → m → α\ni : m\nk : o\nj : m\nk' : o\n⊢ (k = k' ∧ i = j) = (i = j ∧ k = k')",
"state_before": "case a.mk.h.mk\nl : Type ?u.136326\nm : Type u_1\nn : Type ?u.136332\no : Type u_2\np : Type ?u.136338\nq : Type ?u.136341\nm' : o → Type ?u.136346\nn' : o → Type ?u.136351\np' : o → Type ?u.136356\nR : Type ?u.136359\nS : Type ?u.136362\nα : Type u_3\nβ : Type ?u.136368\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : DecidableEq m\nd : o → m → α\ni : m\nk : o\nj : m\nk' : o\n⊢ (if k = k' ∧ i = j then d k i else 0) = if i = j ∧ k = k' then d k i else 0",
"tactic": "congr 1"
},
{
"state_after": "no goals",
"state_before": "case a.mk.h.mk.e_c\nl : Type ?u.136326\nm : Type u_1\nn : Type ?u.136332\no : Type u_2\np : Type ?u.136338\nq : Type ?u.136341\nm' : o → Type ?u.136346\nn' : o → Type ?u.136351\np' : o → Type ?u.136356\nR : Type ?u.136359\nS : Type ?u.136362\nα : Type u_3\nβ : Type ?u.136368\ninst✝³ : DecidableEq o\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : DecidableEq m\nd : o → m → α\ni : m\nk : o\nj : m\nk' : o\n⊢ (k = k' ∧ i = j) = (i = j ∧ k = k')",
"tactic": "rw [and_comm]"
}
] |
[
410,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_le_of_ae_le_trim
|
[] |
[
4401,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4399,
1
] |
Mathlib/NumberTheory/Bernoulli.lean
|
bernoulli'_eq_bernoulli
|
[
{
"state_after": "no goals",
"state_before": "A : Type ?u.631794\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ bernoulli' n = (-1) ^ n * bernoulli n",
"tactic": "simp [bernoulli, ← mul_assoc, ← sq, ← pow_mul, mul_comm n 2, pow_mul]"
}
] |
[
209,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_reverse
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\n⊢ formPerm l * formPerm (reverse l) = 1",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\n⊢ formPerm (reverse l) = (formPerm l)⁻¹",
"tactic": "rw [eq_comm, inv_eq_iff_mul_eq_one]"
},
{
"state_after": "case H\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\n⊢ ↑(formPerm l * formPerm (reverse l)) x = ↑1 x",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\n⊢ formPerm l * formPerm (reverse l) = 1",
"tactic": "ext x"
},
{
"state_after": "case H\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x",
"state_before": "case H\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\n⊢ ↑(formPerm l * formPerm (reverse l)) x = ↑1 x",
"tactic": "rw [mul_apply, one_apply]"
},
{
"state_after": "case H.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : x ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x\n\ncase H.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x",
"state_before": "case H\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x",
"tactic": "cases' Classical.em (x ∈ l) with hx hx"
},
{
"state_after": "case H.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x",
"state_before": "case H.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : x ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x\n\ncase H.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x",
"tactic": ". obtain ⟨k, hk, rfl⟩ := nthLe_of_mem ((mem_reverse _ _).mpr hx)\n have h1 : l.length - 1 - k < l.length := by\n rw [Nat.sub_sub, add_comm]\n exact Nat.sub_lt_self (Nat.succ_pos _) (Nat.succ_le_of_lt (by simpa using hk))\n have h2 : length l - 1 - (k + 1) % length (reverse l) < length l := by\n rw [Nat.sub_sub, length_reverse];\n exact Nat.sub_lt_self (by rw [add_comm]; exact Nat.succ_pos _)\n (by rw [add_comm]; exact Nat.succ_le_of_lt (Nat.mod_lt _ (length_pos_of_mem hx)))\n rw [formPerm_apply_nthLe l.reverse (nodup_reverse.mpr h), nthLe_reverse' _ _ _ h1,\n nthLe_reverse' _ _ _ h2, formPerm_apply_nthLe _ h]\n congr\n rw [length_reverse] at *\n cases' lt_or_eq_of_le (Nat.succ_le_of_lt hk) with h h\n . rw [Nat.mod_eq_of_lt h, ← Nat.sub_add_comm, Nat.succ_sub_succ_eq_sub,\n Nat.mod_eq_of_lt h1]\n exact (Nat.le_sub_iff_add_le (length_pos_of_mem hx)).2 (Nat.succ_le_of_lt h)\n . rw [← h]; simp"
},
{
"state_after": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk",
"state_before": "case H.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : x ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x",
"tactic": "obtain ⟨k, hk, rfl⟩ := nthLe_of_mem ((mem_reverse _ _).mpr hx)"
},
{
"state_after": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk",
"state_before": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk",
"tactic": "have h1 : l.length - 1 - k < l.length := by\n rw [Nat.sub_sub, add_comm]\n exact Nat.sub_lt_self (Nat.succ_pos _) (Nat.succ_le_of_lt (by simpa using hk))"
},
{
"state_after": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length (reverse l) < length l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk",
"state_before": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk",
"tactic": "have h2 : length l - 1 - (k + 1) % length (reverse l) < length l := by\n rw [Nat.sub_sub, length_reverse];\n exact Nat.sub_lt_self (by rw [add_comm]; exact Nat.succ_pos _)\n (by rw [add_comm]; exact Nat.succ_le_of_lt (Nat.mod_lt _ (length_pos_of_mem hx)))"
},
{
"state_after": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length (reverse l) < length l\n⊢ nthLe l ((length l - 1 - (k + 1) % length (reverse l) + 1) % length l)\n (_ : (length l - 1 - (k + 1) % length (reverse l) + 1) % length l < length l) =\n nthLe l (length l - 1 - k) h1",
"state_before": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length (reverse l) < length l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk",
"tactic": "rw [formPerm_apply_nthLe l.reverse (nodup_reverse.mpr h), nthLe_reverse' _ _ _ h1,\n nthLe_reverse' _ _ _ h2, formPerm_apply_nthLe _ h]"
},
{
"state_after": "case H.inl.intro.intro.e_n\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length (reverse l) < length l\n⊢ (length l - 1 - (k + 1) % length (reverse l) + 1) % length l = length l - 1 - k",
"state_before": "case H.inl.intro.intro\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length (reverse l) < length l\n⊢ nthLe l ((length l - 1 - (k + 1) % length (reverse l) + 1) % length l)\n (_ : (length l - 1 - (k + 1) % length (reverse l) + 1) % length l < length l) =\n nthLe l (length l - 1 - k) h1",
"tactic": "congr"
},
{
"state_after": "case H.inl.intro.intro.e_n\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"state_before": "case H.inl.intro.intro.e_n\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length (reverse l) < length l\n⊢ (length l - 1 - (k + 1) % length (reverse l) + 1) % length l = length l - 1 - k",
"tactic": "rw [length_reverse] at *"
},
{
"state_after": "case H.inl.intro.intro.e_n.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k < length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k\n\ncase H.inl.intro.intro.e_n.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k = length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"state_before": "case H.inl.intro.intro.e_n\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"tactic": "cases' lt_or_eq_of_le (Nat.succ_le_of_lt hk) with h h"
},
{
"state_after": "case H.inl.intro.intro.e_n.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k = length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"state_before": "case H.inl.intro.intro.e_n.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k < length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k\n\ncase H.inl.intro.intro.e_n.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k = length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"tactic": ". rw [Nat.mod_eq_of_lt h, ← Nat.sub_add_comm, Nat.succ_sub_succ_eq_sub,\n Nat.mod_eq_of_lt h1]\n exact (Nat.le_sub_iff_add_le (length_pos_of_mem hx)).2 (Nat.succ_le_of_lt h)"
},
{
"state_after": "no goals",
"state_before": "case H.inl.intro.intro.e_n.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k = length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"tactic": ". rw [← h]; simp"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\n⊢ length l - (k + 1) < length l",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\n⊢ length l - 1 - k < length l",
"tactic": "rw [Nat.sub_sub, add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\n⊢ length l - (k + 1) < length l",
"tactic": "exact Nat.sub_lt_self (Nat.succ_pos _) (Nat.succ_le_of_lt (by simpa using hk))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\n⊢ k < length l",
"tactic": "simpa using hk"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ length l - (1 + (k + 1) % length l) < length l",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ length l - 1 - (k + 1) % length (reverse l) < length l",
"tactic": "rw [Nat.sub_sub, length_reverse]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ length l - (1 + (k + 1) % length l) < length l",
"tactic": "exact Nat.sub_lt_self (by rw [add_comm]; exact Nat.succ_pos _)\n (by rw [add_comm]; exact Nat.succ_le_of_lt (Nat.mod_lt _ (length_pos_of_mem hx)))"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ 0 < (k + 1) % length l + 1",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ 0 < 1 + (k + 1) % length l",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ 0 < (k + 1) % length l + 1",
"tactic": "exact Nat.succ_pos _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ (k + 1) % length l + 1 ≤ length l",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ 1 + (k + 1) % length l ≤ length l",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh : Nodup l\nk : ℕ\nhk : k < length (reverse l)\nhx : nthLe (reverse l) k hk ∈ l\nh1 : length l - 1 - k < length l\n⊢ (k + 1) % length l + 1 ≤ length l",
"tactic": "exact Nat.succ_le_of_lt (Nat.mod_lt _ (length_pos_of_mem hx))"
},
{
"state_after": "case H.inl.intro.intro.e_n.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k < length l\n⊢ Nat.succ k ≤ length l - 1",
"state_before": "case H.inl.intro.intro.e_n.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k < length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"tactic": "rw [Nat.mod_eq_of_lt h, ← Nat.sub_add_comm, Nat.succ_sub_succ_eq_sub,\n Nat.mod_eq_of_lt h1]"
},
{
"state_after": "no goals",
"state_before": "case H.inl.intro.intro.e_n.inl\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k < length l\n⊢ Nat.succ k ≤ length l - 1",
"tactic": "exact (Nat.le_sub_iff_add_le (length_pos_of_mem hx)).2 (Nat.succ_le_of_lt h)"
},
{
"state_after": "case H.inl.intro.intro.e_n.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k = length l\n⊢ (Nat.succ k - 1 - (k + 1) % Nat.succ k + 1) % Nat.succ k = Nat.succ k - 1 - k",
"state_before": "case H.inl.intro.intro.e_n.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k = length l\n⊢ (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "case H.inl.intro.intro.e_n.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx : α\nl : List α\nh✝ : Nodup l\nk : ℕ\nhk✝ : k < length (reverse l)\nhk : k < length l\nhx : nthLe (reverse l) k hk✝ ∈ l\nh1 : length l - 1 - k < length l\nh2 : length l - 1 - (k + 1) % length l < length l\nh : Nat.succ k = length l\n⊢ (Nat.succ k - 1 - (k + 1) % Nat.succ k + 1) % Nat.succ k = Nat.succ k - 1 - k",
"tactic": "simp"
},
{
"state_after": "case H.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ¬x ∈ reverse l",
"state_before": "case H.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ↑(formPerm l) (↑(formPerm (reverse l)) x) = x",
"tactic": "rw [formPerm_apply_of_not_mem x l.reverse, formPerm_apply_of_not_mem _ _ hx]"
},
{
"state_after": "no goals",
"state_before": "case H.inr\nα : Type u_1\nβ : Type ?u.774227\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ : α\nl : List α\nh : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ¬x ∈ reverse l",
"tactic": "simpa using hx"
}
] |
[
325,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
src/lean/Init/SimpLemmas.lean
|
heq_eq_eq
|
[] |
[
143,
113
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
143,
9
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
|
ContinuousLinearEquiv.comp_fderiv
|
[
{
"state_after": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.150633\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\nx : G\n⊢ fderivWithin 𝕜 (↑iso ∘ f) univ x = comp (↑iso) (fderivWithin 𝕜 f univ x)",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.150633\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\nx : G\n⊢ fderiv 𝕜 (↑iso ∘ f) x = comp (↑iso) (fderiv 𝕜 f x)",
"tactic": "rw [← fderivWithin_univ, ← fderivWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_1\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_2\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.150633\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\niso : E ≃L[𝕜] F\nf : G → E\nx : G\n⊢ fderivWithin 𝕜 (↑iso ∘ f) univ x = comp (↑iso) (fderivWithin 𝕜 f univ x)",
"tactic": "exact iso.comp_fderivWithin uniqueDiffWithinAt_univ"
}
] |
[
171,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.infDist_le_infDist_of_subset
|
[] |
[
516,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackConeOfRightIso_π_app_none
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : IsIso g\n⊢ (pullbackConeOfRightIso f g).π.app none = f",
"tactic": "simp"
}
] |
[
1705,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1705,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Complex.deriv_cos
|
[] |
[
96,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.coe_lt_coe
|
[] |
[
120,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.linMulLinSelfPosDef
|
[] |
[
959,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
957,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.card_biUnion_le_card_mul
|
[] |
[
232,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
dist_le_of_le_geometric_of_tendsto
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ dist (f n) a ≤ C * r ^ n / (1 - r)",
"state_before": "α : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ dist (f n) a ≤ C * r ^ n / (1 - r)",
"tactic": "have := aux_hasSum_of_le_geometric hr hu"
},
{
"state_after": "case h.e'_4\nα : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ C * r ^ n / (1 - r) = ∑' (m : ℕ), C * r ^ (n + m)",
"state_before": "α : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ dist (f n) a ≤ C * r ^ n / (1 - r)",
"tactic": "convert dist_le_tsum_of_dist_le_of_tendsto _ hu ⟨_, this⟩ ha n"
},
{
"state_after": "case h.e'_4\nα : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ C / (1 - r) * r ^ n = ∑' (m : ℕ), r ^ n * (C * r ^ m)",
"state_before": "case h.e'_4\nα : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ C * r ^ n / (1 - r) = ∑' (m : ℕ), C * r ^ (n + m)",
"tactic": "simp only [pow_add, mul_left_comm C, mul_div_right_comm]"
},
{
"state_after": "case h.e'_4\nα : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ r ^ n * (C / (1 - r)) = ∑' (m : ℕ), r ^ n * (C * r ^ m)",
"state_before": "case h.e'_4\nα : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ C / (1 - r) * r ^ n = ∑' (m : ℕ), r ^ n * (C * r ^ m)",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\nβ : Type ?u.451019\nι : Type ?u.451022\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nhr : r < 1\nf : ℕ → α\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n => C * r ^ n) (C / (1 - r))\n⊢ r ^ n * (C / (1 - r)) = ∑' (m : ℕ), r ^ n * (C * r ^ m)",
"tactic": "exact (this.mul_left _).tsum_eq.symm"
}
] |
[
403,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_bihimp_sup
|
[] |
[
322,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Data/List/Zip.lean
|
List.zip_nil_right
|
[] |
[
65,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
monotone_snd
|
[] |
[
1153,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1153,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.tan_nonneg_of_nonneg_of_le_pi_div_two
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nh0x : 0 ≤ x\nhxp✝ : x ≤ π / 2\nh✝ : 0 < x\nhxp : x = π / 2\n⊢ 0 ≤ tan x",
"tactic": "simp [hxp, tan_eq_sin_div_cos]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh0x : 0 ≤ x\nhxp : x ≤ π / 2\nhx0 : 0 = x\nx✝ : x < π / 2 ∨ x = π / 2\n⊢ 0 ≤ tan x",
"tactic": "simp [hx0.symm]"
}
] |
[
948,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
944,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
RingEquiv.mapMatrix_trans
|
[] |
[
1577,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1575,
1
] |
Mathlib/Order/Interval.lean
|
Interval.coe_inf
|
[
{
"state_after": "case none\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nt : Interval α\n⊢ ↑⊥ = ↑⊥ ∩ ↑t",
"state_before": "case none\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nt : Interval α\n⊢ ↑(none ⊓ t) = ↑none ∩ ↑t",
"tactic": "rw [WithBot.none_eq_bot, bot_inf_eq]"
},
{
"state_after": "no goals",
"state_before": "case none\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nt : Interval α\n⊢ ↑⊥ = ↑⊥ ∩ ↑t",
"tactic": "exact (empty_inter _).symm"
},
{
"state_after": "case some.none\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : NonemptyInterval α\n⊢ ↑⊥ = ↑(some s) ∩ ↑⊥",
"state_before": "case some.none\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : NonemptyInterval α\n⊢ ↑(some s ⊓ none) = ↑(some s) ∩ ↑none",
"tactic": "rw [WithBot.none_eq_bot, inf_bot_eq]"
},
{
"state_after": "no goals",
"state_before": "case some.none\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns : NonemptyInterval α\n⊢ ↑⊥ = ↑(some s) ∩ ↑⊥",
"tactic": "exact (inter_empty _).symm"
},
{
"state_after": "case some.some\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : NonemptyInterval α\n⊢ ↑(if h : s.fst ≤ t.snd ∧ t.fst ≤ s.snd then\n some\n { toProd := (s.fst ⊔ t.fst, s.snd ⊓ t.snd),\n fst_le_snd := (_ : s.fst ⊔ t.fst ≤ (s.fst ⊔ t.fst, s.snd ⊓ t.snd).snd) }\n else ⊥) =\n Icc (s.fst ⊔ t.fst) (s.snd ⊓ t.snd)",
"state_before": "case some.some\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : NonemptyInterval α\n⊢ ↑(some s ⊓ some t) = ↑(some s) ∩ ↑(some t)",
"tactic": "refine' (_ : setLike.coe (dite\n (s.toProd.fst ≤ t.toProd.snd ∧ t.toProd.fst ≤ s.toProd.snd)\n _ _) = _).trans Icc_inter_Icc.symm"
},
{
"state_after": "case some.some.inl\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : NonemptyInterval α\nh : s.fst ≤ t.snd ∧ t.fst ≤ s.snd\n⊢ ↑(some\n { toProd := (s.fst ⊔ t.fst, s.snd ⊓ t.snd),\n fst_le_snd := (_ : s.fst ⊔ t.fst ≤ (s.fst ⊔ t.fst, s.snd ⊓ t.snd).snd) }) =\n Icc (s.fst ⊔ t.fst) (s.snd ⊓ t.snd)\n\ncase some.some.inr\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : NonemptyInterval α\nh : ¬(s.fst ≤ t.snd ∧ t.fst ≤ s.snd)\n⊢ ↑⊥ = Icc (s.fst ⊔ t.fst) (s.snd ⊓ t.snd)",
"state_before": "case some.some\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : NonemptyInterval α\n⊢ ↑(if h : s.fst ≤ t.snd ∧ t.fst ≤ s.snd then\n some\n { toProd := (s.fst ⊔ t.fst, s.snd ⊓ t.snd),\n fst_le_snd := (_ : s.fst ⊔ t.fst ≤ (s.fst ⊔ t.fst, s.snd ⊓ t.snd).snd) }\n else ⊥) =\n Icc (s.fst ⊔ t.fst) (s.snd ⊓ t.snd)",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case some.some.inl\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : NonemptyInterval α\nh : s.fst ≤ t.snd ∧ t.fst ≤ s.snd\n⊢ ↑(some\n { toProd := (s.fst ⊔ t.fst, s.snd ⊓ t.snd),\n fst_le_snd := (_ : s.fst ⊔ t.fst ≤ (s.fst ⊔ t.fst, s.snd ⊓ t.snd).snd) }) =\n Icc (s.fst ⊔ t.fst) (s.snd ⊓ t.snd)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case some.some.inr\nα : Type u_1\nβ : Type ?u.38930\nγ : Type ?u.38933\nδ : Type ?u.38936\nι : Sort ?u.38939\nκ : ι → Sort ?u.38944\ninst✝¹ : Lattice α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\ns t : NonemptyInterval α\nh : ¬(s.fst ≤ t.snd ∧ t.fst ≤ s.snd)\n⊢ ↑⊥ = Icc (s.fst ⊔ t.fst) (s.snd ⊓ t.snd)",
"tactic": "exact (Icc_eq_empty fun H =>\n h ⟨le_sup_left.trans <| H.trans inf_le_right,\n le_sup_right.trans <| H.trans inf_le_left⟩).symm"
}
] |
[
616,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
597,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.lower_le_lower
|
[] |
[
99,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.tendsto_lintegral_nn_filter_of_le_const
|
[
{
"state_after": "case refine_1\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\ninst✝¹ : IsCountablyGenerated L\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c\nf : Ω → ℝ≥0\nfs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (f ω))\n⊢ ∀ᶠ (n : ι) in L, ∀ᵐ (a : Ω) ∂μ, ↑(↑(fs n) a) ≤ (fun x => ↑c) a\n\ncase refine_2\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\ninst✝¹ : IsCountablyGenerated L\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c\nf : Ω → ℝ≥0\nfs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (f ω))\n⊢ ∀ᵐ (a : Ω) ∂μ, Tendsto (fun n => ↑(↑(fs n) a)) L (𝓝 ↑(f a))",
"state_before": "Ω : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\ninst✝¹ : IsCountablyGenerated L\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c\nf : Ω → ℝ≥0\nfs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (f ω))\n⊢ Tendsto (fun i => ∫⁻ (ω : Ω), ↑(↑(fs i) ω) ∂μ) L (𝓝 (∫⁻ (ω : Ω), ↑(f ω) ∂μ))",
"tactic": "refine tendsto_lintegral_filter_of_dominated_convergence (fun _ => c)\n (eventually_of_forall fun i => (ENNReal.continuous_coe.comp (fs i).continuous).measurable) ?_\n (@lintegral_const_lt_top _ _ μ _ _ (@ENNReal.coe_ne_top c)).ne ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\ninst✝¹ : IsCountablyGenerated L\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c\nf : Ω → ℝ≥0\nfs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (f ω))\n⊢ ∀ᶠ (n : ι) in L, ∀ᵐ (a : Ω) ∂μ, ↑(↑(fs n) a) ≤ (fun x => ↑c) a",
"tactic": "simpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nΩ : Type u_2\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_1\nL : Filter ι\ninst✝¹ : IsCountablyGenerated L\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, ↑(fs i) ω ≤ c\nf : Ω → ℝ≥0\nfs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => ↑(fs i) ω) L (𝓝 (f ω))\n⊢ ∀ᵐ (a : Ω) ∂μ, Tendsto (fun n => ↑(↑(fs n) a)) L (𝓝 ↑(f a))",
"tactic": "simpa only [Function.comp_apply, ENNReal.tendsto_coe] using fs_lim"
}
] |
[
587,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.mul_mem_left
|
[] |
[
68,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.funLeft_surjective_of_injective
|
[
{
"state_after": "R : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\n⊢ ∃ a, ↑(funLeft R M f) a = g",
"state_before": "R : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\n⊢ Surjective ↑(funLeft R M f)",
"tactic": "intro g"
},
{
"state_after": "R : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\n⊢ (↑(funLeft R M f) fun x => if h : ∃ y, f y = x then g (Exists.choose h) else 0) = g",
"state_before": "R : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\n⊢ ∃ a, ↑(funLeft R M f) a = g",
"tactic": "refine' ⟨fun x => if h : ∃ y, f y = x then g h.choose else 0, _⟩"
},
{
"state_after": "case h\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\n⊢ ↑(funLeft R M f) (fun x => if h : ∃ y, f y = x then g (Exists.choose h) else 0) x✝ = g x✝",
"state_before": "R : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\n⊢ (↑(funLeft R M f) fun x => if h : ∃ y, f y = x then g (Exists.choose h) else 0) = g",
"tactic": "ext"
},
{
"state_after": "case h\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\n⊢ (if h : ∃ y, f y = f x✝ then g (Exists.choose h) else 0) = g x✝",
"state_before": "case h\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\n⊢ ↑(funLeft R M f) (fun x => if h : ∃ y, f y = x then g (Exists.choose h) else 0) x✝ = g x✝",
"tactic": "dsimp only [funLeft_apply]"
},
{
"state_after": "case h.inl\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\nw : ∃ y, f y = f x✝\n⊢ g (Exists.choose w) = g x✝\n\ncase h.inr\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\nw : ¬∃ y, f y = f x✝\n⊢ 0 = g x✝",
"state_before": "case h\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\n⊢ (if h : ∃ y, f y = f x✝ then g (Exists.choose h) else 0) = g x✝",
"tactic": "split_ifs with w"
},
{
"state_after": "case h.inl.e_a\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\nw : ∃ y, f y = f x✝\n⊢ Exists.choose w = x✝",
"state_before": "case h.inl\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\nw : ∃ y, f y = f x✝\n⊢ g (Exists.choose w) = g x✝",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.inl.e_a\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\nw : ∃ y, f y = f x✝\n⊢ Exists.choose w = x✝",
"tactic": "exact hf w.choose_spec"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nR : Type u_4\nR₁ : Type ?u.2320959\nR₂ : Type ?u.2320962\nR₃ : Type ?u.2320965\nR₄ : Type ?u.2320968\nS : Type ?u.2320971\nK : Type ?u.2320974\nK₂ : Type ?u.2320977\nM : Type u_3\nM' : Type ?u.2320983\nM₁ : Type ?u.2320986\nM₂ : Type ?u.2320989\nM₃ : Type ?u.2320992\nM₄ : Type ?u.2320995\nN : Type ?u.2320998\nN₂ : Type ?u.2321001\nι : Type ?u.2321004\nV : Type ?u.2321007\nV₂ : Type ?u.2321010\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_1\nn : Type u_2\np : Type ?u.2321051\nf : m → n\nhf : Injective f\ng : m → M\nx✝ : m\nw : ¬∃ y, f y = f x✝\n⊢ 0 = g x✝",
"tactic": "simp only [not_true, exists_apply_eq_apply] at w"
}
] |
[
2629,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2619,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.nat_double_succ
|
[] |
[
860,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
859,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.closure_subsemigroup_closure
|
[] |
[
649,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.of_diff
|
[] |
[
772,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
771,
1
] |
Mathlib/Logic/Equiv/Fin.lean
|
finSuccEquiv'_last_apply
|
[
{
"state_after": "case intro\nm n : ℕ\ni : Fin n\nh : ↑Fin.castSucc i ≠ Fin.last n\n⊢ ↑(finSuccEquiv' (Fin.last n)) (↑Fin.castSucc i) = some (Fin.castLT (↑Fin.castSucc i) (_ : ↑(↑Fin.castSucc i) < n))",
"state_before": "m n : ℕ\ni : Fin (n + 1)\nh : i ≠ Fin.last n\n⊢ ↑(finSuccEquiv' (Fin.last n)) i = some (Fin.castLT i (_ : ↑i < n))",
"tactic": "rcases Fin.exists_castSucc_eq.2 h with ⟨i, rfl⟩"
},
{
"state_after": "case intro\nm n : ℕ\ni : Fin n\nh : ↑Fin.castSucc i ≠ Fin.last n\n⊢ some i = some (Fin.castLT (↑Fin.castSucc i) (_ : ↑(↑Fin.castSucc i) < n))",
"state_before": "case intro\nm n : ℕ\ni : Fin n\nh : ↑Fin.castSucc i ≠ Fin.last n\n⊢ ↑(finSuccEquiv' (Fin.last n)) (↑Fin.castSucc i) = some (Fin.castLT (↑Fin.castSucc i) (_ : ↑(↑Fin.castSucc i) < n))",
"tactic": "rw [finSuccEquiv'_last_apply_castSucc]"
},
{
"state_after": "no goals",
"state_before": "case intro\nm n : ℕ\ni : Fin n\nh : ↑Fin.castSucc i ≠ Fin.last n\n⊢ some i = some (Fin.castLT (↑Fin.castSucc i) (_ : ↑(↑Fin.castSucc i) < n))",
"tactic": "rfl"
}
] |
[
233,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.chain'_toList
|
[
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi : i < List.length (toList s) - 1\n⊢ IsMaximal (List.get (toList s) { val := i, isLt := (_ : i < List.length (toList s)) })\n (List.get (toList s) { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) })",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\n⊢ ∀ (i : ℕ) (h : i < List.length (toList s) - 1),\n IsMaximal (List.get (toList s) { val := i, isLt := (_ : i < List.length (toList s)) })\n (List.get (toList s) { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) })",
"tactic": "intro i hi"
},
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi : i < List.length (toList s) - 1\n⊢ IsMaximal\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i, isLt := (_ : i < List.length (toList s)) }))\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) }))",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi : i < List.length (toList s) - 1\n⊢ IsMaximal (List.get (toList s) { val := i, isLt := (_ : i < List.length (toList s)) })\n (List.get (toList s) { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) })",
"tactic": "simp only [toList, List.get_ofFn]"
},
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi✝ : i < List.length (toList s) - 1\nhi : i < s.length + 1 - 1\n⊢ IsMaximal\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i, isLt := (_ : i < List.length (toList s)) }))\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) }))",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi : i < List.length (toList s) - 1\n⊢ IsMaximal\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i, isLt := (_ : i < List.length (toList s)) }))\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) }))",
"tactic": "rw [length_toList] at hi"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi✝ : i < List.length (toList s) - 1\nhi : i < s.length + 1 - 1\n⊢ IsMaximal\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i, isLt := (_ : i < List.length (toList s)) }))\n (series s\n (↑(Fin.cast (_ : List.length (List.ofFn s.series) = s.length + 1))\n { val := i + 1, isLt := (_ : Nat.succ i < List.length (toList s)) }))",
"tactic": "exact s.step ⟨i, hi⟩"
}
] |
[
254,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_eq_empty
|
[] |
[
528,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.leadingCoeff_of_a_ne_zero'
|
[] |
[
209,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Algebra/Ring/Divisibility.lean
|
dvd_sub_self_right
|
[] |
[
158,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Algebra/Lie/IdealOperations.lean
|
LieSubmodule.lie_inf
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅I, N ⊓ N'⁆ ≤ ⁅I, N⁆ ∧ ⁅I, N ⊓ N'⁆ ≤ ⁅I, N'⁆",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅I, N ⊓ N'⁆ ≤ ⁅I, N⁆ ⊓ ⁅I, N'⁆",
"tactic": "rw [le_inf_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\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 M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅I, N ⊓ N'⁆ ≤ ⁅I, N⁆ ∧ ⁅I, N ⊓ N'⁆ ≤ ⁅I, N'⁆",
"tactic": "constructor <;>\napply mono_lie_right <;> [exact inf_le_left; exact inf_le_right]"
}
] |
[
199,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffAt.sin
|
[] |
[
1040,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1039,
1
] |
Mathlib/LinearAlgebra/TensorPower.lean
|
TensorPower.tprod_mul_tprod
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ ↑(reindex R M finSumFinEquiv) (↑(tmulEquiv R M) (↑(tprod R) a ⊗ₜ[R] ↑(tprod R) b)) = ↑(tprod R) (Fin.append a b)",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ GradedMonoid.GMul.mul (↑(tprod R) a) (↑(tprod R) b) = ↑(tprod R) (Fin.append a b)",
"tactic": "dsimp [gMul_def, mulEquiv]"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ ↑(reindex R M finSumFinEquiv) (⨂ₜ[R] (i : Fin na ⊕ Fin nb), Sum.elim a b i) = ↑(tprod R) (Fin.append a b)",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ ↑(reindex R M finSumFinEquiv) (↑(tmulEquiv R M) (↑(tprod R) a ⊗ₜ[R] ↑(tprod R) b)) = ↑(tprod R) (Fin.append a b)",
"tactic": "rw [tmulEquiv_apply R M a b]"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ (⨂ₜ[R] (i : Fin (na + nb)), Sum.elim a b (↑finSumFinEquiv.symm i)) = ↑(tprod R) (Fin.append a b)",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ ↑(reindex R M finSumFinEquiv) (⨂ₜ[R] (i : Fin na ⊕ Fin nb), Sum.elim a b i) = ↑(tprod R) (Fin.append a b)",
"tactic": "refine' (reindex_tprod _ _).trans _"
},
{
"state_after": "case h.e_6.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ (fun i => Sum.elim a b (↑finSumFinEquiv.symm i)) = Fin.append a b",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ (⨂ₜ[R] (i : Fin (na + nb)), Sum.elim a b (↑finSumFinEquiv.symm i)) = ↑(tprod R) (Fin.append a b)",
"tactic": "congr 1"
},
{
"state_after": "case h.e_6.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ (fun i => Sum.elim a b (Fin.addCases Sum.inl Sum.inr i)) = Fin.addCases a b",
"state_before": "case h.e_6.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ (fun i => Sum.elim a b (↑finSumFinEquiv.symm i)) = Fin.append a b",
"tactic": "dsimp only [Fin.append, finSumFinEquiv, Equiv.coe_fn_symm_mk]"
},
{
"state_after": "case h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ ∀ (x : Fin (na + nb)), Sum.elim a b (Fin.addCases Sum.inl Sum.inr x) = Fin.addCases a b x",
"state_before": "case h.e_6.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ (fun i => Sum.elim a b (Fin.addCases Sum.inl Sum.inr i)) = Fin.addCases a b",
"tactic": "apply funext"
},
{
"state_after": "no goals",
"state_before": "case h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nna nb : ℕ\na : Fin na → M\nb : Fin nb → M\n⊢ ∀ (x : Fin (na + nb)), Sum.elim a b (Fin.addCases Sum.inl Sum.inr x) = Fin.addCases a b x",
"tactic": "apply Fin.addCases <;> simp"
}
] |
[
167,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Logic/Nontrivial.lean
|
Pi.nontrivial_at
|
[
{
"state_after": "α : Type ?u.4851\nβ : Type ?u.4854\nI : Type u_2\nf : I → Type u_1\ni' : I\ninst : ∀ (i : I), Nonempty (f i)\ninst✝ : Nontrivial (f i')\nthis : DecidableEq ((i : I) → f i) := Classical.decEq ((i : I) → f i)\n⊢ Nontrivial ((i : I) → f i)",
"state_before": "α : Type ?u.4851\nβ : Type ?u.4854\nI : Type u_2\nf : I → Type u_1\ni' : I\ninst : ∀ (i : I), Nonempty (f i)\ninst✝ : Nontrivial (f i')\n⊢ Nontrivial ((i : I) → f i)",
"tactic": "letI := Classical.decEq (∀ i : I, f i)"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.4851\nβ : Type ?u.4854\nI : Type u_2\nf : I → Type u_1\ni' : I\ninst : ∀ (i : I), Nonempty (f i)\ninst✝ : Nontrivial (f i')\nthis : DecidableEq ((i : I) → f i) := Classical.decEq ((i : I) → f i)\n⊢ Nontrivial ((i : I) → f i)",
"tactic": "exact (Function.update_injective (fun i ↦ Classical.choice (inst i)) i').nontrivial"
}
] |
[
188,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Init/CcLemmas.lean
|
and_eq_of_eq
|
[] |
[
34,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
normUnit_mul_normUnit
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\n⊢ normUnit (a * ↑(normUnit a)) = 1",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n⊢ normUnit (a * ↑(normUnit a)) = 1",
"tactic": "nontriviality α using Subsingleton.elim a 0"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\n✝ : Nontrivial α\n⊢ normUnit (0 * ↑(normUnit 0)) = 1\n\ncase inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nh : a ≠ 0\n⊢ normUnit (a * ↑(normUnit a)) = 1",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\n⊢ normUnit (a * ↑(normUnit a)) = 1",
"tactic": "obtain rfl | h := eq_or_ne a 0"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\n✝ : Nontrivial α\n⊢ normUnit (0 * ↑(normUnit 0)) = 1",
"tactic": "rw [normUnit_zero, zero_mul, normUnit_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\n✝ : Nontrivial α\nh : a ≠ 0\n⊢ normUnit (a * ↑(normUnit a)) = 1",
"tactic": "rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]"
}
] |
[
169,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.subtype_eq_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.127828\nγ : Type ?u.127831\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\n⊢ Finset.subtype p s = ∅ ↔ ∀ (x : α), p x → ¬x ∈ s",
"tactic": "simp [ext_iff, Subtype.forall, Subtype.coe_mk]"
}
] |
[
670,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/Topology/Sequences.lean
|
tendsto_subseq_of_frequently_bounded
|
[] |
[
426,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.tendsto_comap'_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.311794\nι : Sort x\nm : α → β\nf : Filter α\ng : Filter β\ni : γ → α\nh : range i ∈ f\n⊢ (map m ∘ map i) (comap i f) ≤ g ↔ Tendsto m f g",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.311794\nι : Sort x\nm : α → β\nf : Filter α\ng : Filter β\ni : γ → α\nh : range i ∈ f\n⊢ Tendsto (m ∘ i) (comap i f) g ↔ Tendsto m f g",
"tactic": "rw [Tendsto, ← map_compose]"
}
] |
[
2961,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2958,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
Ideal.lcm_eq_inf
|
[] |
[
879,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.invRev_invRev
|
[] |
[
567,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_right_inj
|
[] |
[
395,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.coe_ringHom_trans
|
[] |
[
549,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11546\nι : Type ?u.11549\nM : Type u_2\nN : Type ?u.11555\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf g : α → M\na : α\ns : Set α\n⊢ Disjoint (mulSupport fun x => 1) s",
"tactic": "simp only [mulSupport_one, empty_disjoint]"
}
] |
[
222,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.supported_iUnion
|
[
{
"state_after": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\n⊢ supported M R (⋃ (i : δ), s i) ≤ ⨆ (i : δ), supported M R (s i)",
"state_before": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\n⊢ supported M R (⋃ (i : δ), s i) = ⨆ (i : δ), supported M R (s i)",
"tactic": "refine' le_antisymm _ (iSup_le fun i => supported_mono <| Set.subset_iUnion _ _)"
},
{
"state_after": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\n⊢ supported M R (⋃ (i : δ), s i) ≤ ⨆ (i : δ), supported M R (s i)",
"state_before": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\n⊢ supported M R (⋃ (i : δ), s i) ≤ ⨆ (i : δ), supported M R (s i)",
"tactic": "haveI := Classical.decPred fun x => x ∈ ⋃ i, s i"
},
{
"state_after": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\n⊢ LinearMap.range (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i))) ≤\n ⨆ (i : δ), supported M R (s i)",
"state_before": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\n⊢ supported M R (⋃ (i : δ), s i) ≤ ⨆ (i : δ), supported M R (s i)",
"tactic": "suffices\n LinearMap.range ((Submodule.subtype _).comp (restrictDom M R (⋃ i, s i))) ≤\n ⨆ i, supported M R (s i) by\n rwa [LinearMap.range_comp, range_restrictDom, Submodule.map_top, range_subtype] at this"
},
{
"state_after": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\n⊢ ⊤ ≤\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"state_before": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\n⊢ LinearMap.range (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i))) ≤\n ⨆ (i : δ), supported M R (s i)",
"tactic": "rw [range_le_iff_comap, eq_top_iff]"
},
{
"state_after": "case intro\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl : α →₀ M\n⊢ l ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"state_before": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\n⊢ ⊤ ≤\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"tactic": "rintro l ⟨⟩"
},
{
"state_after": "case intro.refine_1\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl : α →₀ M\n⊢ 0 ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))\n\ncase intro.refine_2\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl : α →₀ M\n⊢ ∀ (a : α) (b : M) (f : α →₀ M),\n ¬a ∈ f.support →\n b ≠ 0 →\n f ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i)) →\n single a b + f ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"state_before": "case intro\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl : α →₀ M\n⊢ l ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"tactic": "refine Finsupp.induction l ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis✝ : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nthis :\n LinearMap.range (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i))) ≤\n ⨆ (i : δ), supported M R (s i)\n⊢ supported M R (⋃ (i : δ), s i) ≤ ⨆ (i : δ), supported M R (s i)",
"tactic": "rwa [LinearMap.range_comp, range_restrictDom, Submodule.map_top, range_subtype] at this"
},
{
"state_after": "no goals",
"state_before": "case intro.refine_1\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl : α →₀ M\n⊢ 0 ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"tactic": "exact zero_mem _"
},
{
"state_after": "case intro.refine_2\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl✝ : α →₀ M\nx : α\na : M\nl : α →₀ M\nx✝¹ : ¬x ∈ l.support\nx✝ : a ≠ 0\n⊢ single x a ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"state_before": "case intro.refine_2\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl : α →₀ M\n⊢ ∀ (a : α) (b : M) (f : α →₀ M),\n ¬a ∈ f.support →\n b ≠ 0 →\n f ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i)) →\n single a b + f ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"tactic": "refine' fun x a l _ _ => add_mem _"
},
{
"state_after": "case pos\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl✝ : α →₀ M\nx : α\na : M\nl : α →₀ M\nx✝¹ : ¬x ∈ l.support\nx✝ : a ≠ 0\nh : ∃ i, x ∈ s i\n⊢ single x a ∈ ⨆ (i : δ), supported M R (s i)",
"state_before": "case intro.refine_2\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl✝ : α →₀ M\nx : α\na : M\nl : α →₀ M\nx✝¹ : ¬x ∈ l.support\nx✝ : a ≠ 0\n⊢ single x a ∈\n comap (comp (Submodule.subtype (supported M R (⋃ (i : δ), s i))) (restrictDom M R (⋃ (i : δ), s i)))\n (⨆ (i : δ), supported M R (s i))",
"tactic": "by_cases h : ∃ i, x ∈ s i <;> simp [h]"
},
{
"state_after": "case pos.intro\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl✝ : α →₀ M\nx : α\na : M\nl : α →₀ M\nx✝¹ : ¬x ∈ l.support\nx✝ : a ≠ 0\ni : δ\nhi : x ∈ s i\n⊢ single x a ∈ ⨆ (i : δ), supported M R (s i)",
"state_before": "case pos\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl✝ : α →₀ M\nx : α\na : M\nl : α →₀ M\nx✝¹ : ¬x ∈ l.support\nx✝ : a ≠ 0\nh : ∃ i, x ∈ s i\n⊢ single x a ∈ ⨆ (i : δ), supported M R (s i)",
"tactic": "cases' h with i hi"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nα : Type u_2\nM : Type u_3\nN : Type ?u.93459\nP : Type ?u.93462\nR : Type u_4\nS : Type ?u.93468\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nδ : Type u_1\ns : δ → Set α\nthis : DecidablePred fun x => x ∈ ⋃ (i : δ), s i\nl✝ : α →₀ M\nx : α\na : M\nl : α →₀ M\nx✝¹ : ¬x ∈ l.support\nx✝ : a ≠ 0\ni : δ\nhi : x ∈ s i\n⊢ single x a ∈ ⨆ (i : δ), supported M R (s i)",
"tactic": "exact le_iSup (fun i => supported M R (s i)) i (single_mem_supported R _ hi)"
}
] |
[
297,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap
|
[
{
"state_after": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\n⊢ ∀ (x : PrimeSpectrum R), (∃ x_1, I ≤ x_1.asIdeal ∧ ↑(comap f) x_1 = x) ↔ Ideal.comap f I ≤ x.asIdeal",
"state_before": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\n⊢ ↑(comap f) '' zeroLocus ↑I = zeroLocus ↑(Ideal.comap f I)",
"tactic": "simp only [Set.ext_iff, Set.mem_image, mem_zeroLocus, SetLike.coe_subset_coe]"
},
{
"state_after": "case refine'_1\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\n⊢ (∃ x, I ≤ x.asIdeal ∧ ↑(comap f) x = p) → Ideal.comap f I ≤ p.asIdeal\n\ncase refine'_2\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\n⊢ ∃ x, I ≤ x.asIdeal ∧ ↑(comap f) x = p",
"state_before": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\n⊢ ∀ (x : PrimeSpectrum R), (∃ x_1, I ≤ x_1.asIdeal ∧ ↑(comap f) x_1 = x) ↔ Ideal.comap f I ≤ x.asIdeal",
"tactic": "refine' fun p => ⟨_, fun h_I_p => _⟩"
},
{
"state_after": "case refine'_1.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum S\nhp : I ≤ p.asIdeal\na : R\nha : a ∈ Ideal.comap f I\n⊢ a ∈ (↑(comap f) p).asIdeal",
"state_before": "case refine'_1\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\n⊢ (∃ x, I ≤ x.asIdeal ∧ ↑(comap f) x = p) → Ideal.comap f I ≤ p.asIdeal",
"tactic": "rintro ⟨p, hp, rfl⟩ a ha"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum S\nhp : I ≤ p.asIdeal\na : R\nha : a ∈ Ideal.comap f I\n⊢ a ∈ (↑(comap f) p).asIdeal",
"tactic": "exact hp ha"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\n⊢ ∃ x, I ≤ x.asIdeal ∧ ↑(comap f) x = p",
"state_before": "case refine'_2\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\n⊢ ∃ x, I ≤ x.asIdeal ∧ ↑(comap f) x = p",
"tactic": "have hp : ker f ≤ p.asIdeal := (Ideal.comap_mono bot_le).trans h_I_p"
},
{
"state_after": "case refine'_2.refine'_1\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : S\nhx : x ∈ I\n⊢ x ∈ { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) }.asIdeal\n\ncase refine'_2.refine'_2\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\n⊢ ↑(comap f) { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) } = p",
"state_before": "case refine'_2\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\n⊢ ∃ x, I ≤ x.asIdeal ∧ ↑(comap f) x = p",
"tactic": "refine' ⟨⟨p.asIdeal.map f, Ideal.map_isPrime_of_surjective hf hp⟩, fun x hx => _, _⟩"
},
{
"state_after": "case refine'_2.refine'_1.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx' : R\nhx : ↑f x' ∈ I\n⊢ ↑f x' ∈ { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) }.asIdeal",
"state_before": "case refine'_2.refine'_1\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : S\nhx : x ∈ I\n⊢ x ∈ { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) }.asIdeal",
"tactic": "obtain ⟨x', rfl⟩ := hf x"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_1.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx' : R\nhx : ↑f x' ∈ I\n⊢ ↑f x' ∈ { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) }.asIdeal",
"tactic": "exact Ideal.mem_map_of_mem f (h_I_p hx)"
},
{
"state_after": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ x ∈\n (↑(comap f)\n { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) }).asIdeal ↔\n x ∈ p.asIdeal",
"state_before": "case refine'_2.refine'_2\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\n⊢ ↑(comap f) { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) } = p",
"tactic": "ext x"
},
{
"state_after": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ ↑f x ∈ Ideal.map f p.asIdeal ↔ x ∈ p.asIdeal",
"state_before": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ x ∈\n (↑(comap f)\n { asIdeal := Ideal.map f p.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map f p.asIdeal)) }).asIdeal ↔\n x ∈ p.asIdeal",
"tactic": "change f x ∈ p.asIdeal.map f ↔ _"
},
{
"state_after": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ (∃ x_1, x_1 ∈ p.asIdeal ∧ ↑f x_1 = ↑f x) ↔ x ∈ p.asIdeal",
"state_before": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ ↑f x ∈ Ideal.map f p.asIdeal ↔ x ∈ p.asIdeal",
"tactic": "rw [Ideal.mem_map_iff_of_surjective f hf]"
},
{
"state_after": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ (∃ x_1, x_1 ∈ p.asIdeal ∧ ↑f x_1 = ↑f x) → x ∈ p.asIdeal",
"state_before": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ (∃ x_1, x_1 ∈ p.asIdeal ∧ ↑f x_1 = ↑f x) ↔ x ∈ p.asIdeal",
"tactic": "refine' ⟨_, fun hx => ⟨x, hx, rfl⟩⟩"
},
{
"state_after": "case refine'_2.refine'_2.asIdeal.h.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx x' : R\nhx' : x' ∈ p.asIdeal\nheq : ↑f x' = ↑f x\n⊢ x ∈ p.asIdeal",
"state_before": "case refine'_2.refine'_2.asIdeal.h\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx : R\n⊢ (∃ x_1, x_1 ∈ p.asIdeal ∧ ↑f x_1 = ↑f x) → x ∈ p.asIdeal",
"tactic": "rintro ⟨x', hx', heq⟩"
},
{
"state_after": "case refine'_2.refine'_2.asIdeal.h.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx x' : R\nhx' : x' ∈ p.asIdeal\nheq : ↑f x' = ↑f x\n⊢ x' - (x' - x) ∈ p.asIdeal",
"state_before": "case refine'_2.refine'_2.asIdeal.h.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx x' : R\nhx' : x' ∈ p.asIdeal\nheq : ↑f x' = ↑f x\n⊢ x ∈ p.asIdeal",
"tactic": "rw [← sub_sub_cancel x' x]"
},
{
"state_after": "case refine'_2.refine'_2.asIdeal.h.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx x' : R\nhx' : x' ∈ p.asIdeal\nheq : ↑f x' = ↑f x\n⊢ x' - x ∈ ker f",
"state_before": "case refine'_2.refine'_2.asIdeal.h.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx x' : R\nhx' : x' ∈ p.asIdeal\nheq : ↑f x' = ↑f x\n⊢ x' - (x' - x) ∈ p.asIdeal",
"tactic": "refine' p.asIdeal.sub_mem hx' (hp _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_2.asIdeal.h.intro.intro\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nS' : Type ?u.338029\ninst✝ : CommRing S'\nf : R →+* S\nhf : Surjective ↑f\nI : Ideal S\np : PrimeSpectrum R\nh_I_p : Ideal.comap f I ≤ p.asIdeal\nhp : ker f ≤ p.asIdeal\nx x' : R\nhx' : x' ∈ p.asIdeal\nheq : ↑f x' = ↑f x\n⊢ x' - x ∈ ker f",
"tactic": "rwa [mem_ker, map_sub, sub_eq_zero]"
}
] |
[
736,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
719,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
Set.Nonempty.bddAbove_lowerBounds
|
[] |
[
266,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
Subgroup.smul_normal
|
[] |
[
395,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
Bundle.zeroSection_snd
|
[] |
[
390,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.Parallel.refl
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\n⊢ s = map (↑(constVAdd k P 0)) s",
"tactic": "simp"
}
] |
[
1735,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1734,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.complete'
|
[
{
"state_after": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\n⊢ ∃ N, ∀ (i : ℕ), i ≥ N → ↑padicNormE (Padic.lim f - ↑f i) < ε",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\n⊢ ∃ N, ∀ (i : ℕ), i ≥ N → ↑padicNormE (Padic.lim f - ↑f i) < ε",
"tactic": "obtain ⟨N, hN⟩ := exi_rat_seq_conv f (half_pos hε)"
},
{
"state_after": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\n⊢ ∃ N, ∀ (i : ℕ), i ≥ N → ↑padicNormE (Padic.lim f - ↑f i) < ε",
"state_before": "case intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\n⊢ ∃ N, ∀ (i : ℕ), i ≥ N → ↑padicNormE (Padic.lim f - ↑f i) < ε",
"tactic": "obtain ⟨N2, hN2⟩ := padicNormE.defn (lim' f) (half_pos hε)"
},
{
"state_after": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑f i) < ε",
"state_before": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\n⊢ ∃ N, ∀ (i : ℕ), i ≥ N → ↑padicNormE (Padic.lim f - ↑f i) < ε",
"tactic": "refine' ⟨max N N2, fun i hi ↦ _⟩"
},
{
"state_after": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i) + (↑(↑(Padic.lim' f) i) - ↑f i)) < ε",
"state_before": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑f i) < ε",
"tactic": "rw [← sub_add_sub_cancel _ (lim' f i : ℚ_[p]) _]"
},
{
"state_after": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) + ↑padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε",
"state_before": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i) + (↑(↑(Padic.lim' f) i) - ↑f i)) < ε",
"tactic": "refine' (padicNormE.add_le _ _).trans_lt _"
},
{
"state_after": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) + ↑padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε / 2 + ε / 2",
"state_before": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) + ↑padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε",
"tactic": "rw [← add_halves ε]"
},
{
"state_after": "case intro.intro.h₁\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) < ε / 2\n\ncase intro.intro.h₂\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε / 2",
"state_before": "case intro.intro\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) + ↑padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε / 2 + ε / 2",
"tactic": "apply _root_.add_lt_add"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h₁\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (Padic.lim f - ↑(↑(Padic.lim' f) i)) < ε / 2",
"tactic": "apply hN2 _ (le_of_max_le_right hi)"
},
{
"state_after": "case intro.intro.h₂\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (↑f i - ↑(↑(Padic.lim' f) i)) < ε / 2",
"state_before": "case intro.intro.h₂\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (↑(↑(Padic.lim' f) i) - ↑f i) < ε / 2",
"tactic": "rw [padicNormE.map_sub]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h₂\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nε : ℚ\nhε : ε > 0\nN : ℕ\nhN : ∀ (i : ℕ), i ≥ N → ↑padicNormE (↑f i - ↑(limSeq f i)) < ε / 2\nN2 : ℕ\nhN2 : ∀ (i : ℕ), i ≥ N2 → ↑padicNormE (mk (Padic.lim' f) - ↑(↑(Padic.lim' f) i)) < ε / 2\ni : ℕ\nhi : i ≥ max N N2\n⊢ ↑padicNormE (↑f i - ↑(↑(Padic.lim' f) i)) < ε / 2",
"tactic": "exact hN _ (le_of_max_le_left hi)"
}
] |
[
748,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
737,
1
] |
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
isCyclic_of_card_pow_eq_one_le
|
[
{
"state_after": "α : Type u\na : α\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → card (filter (fun a => a ^ n = 1) univ) ≤ n\n⊢ 0 < φ (Fintype.card α)",
"state_before": "α : Type u\na : α\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → card (filter (fun a => a ^ n = 1) univ) ≤ n\n⊢ 0 < card (filter (fun a => orderOf a = Fintype.card α) univ)",
"tactic": "rw [card_orderOf_eq_totient_aux₂ hn dvd_rfl]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\na : α\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → card (filter (fun a => a ^ n = 1) univ) ≤ n\n⊢ 0 < φ (Fintype.card α)",
"tactic": "exact totient_pos (Fintype.card_pos_iff.2 ⟨1⟩)"
}
] |
[
404,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
398,
1
] |
Mathlib/Data/List/Basic.lean
|
List.takeWhile_takeWhile
|
[
{
"state_after": "case nil\nι : Type ?u.417345\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np✝ : α → Bool\nl : List α\np q : α → Bool\n⊢ takeWhile p (takeWhile q []) = takeWhile (fun a => decide (p a = true ∧ q a = true)) []\n\ncase cons\nι : Type ?u.417345\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np✝ : α → Bool\nl : List α\np q : α → Bool\nhd : α\ntl : List α\nIH : takeWhile p (takeWhile q tl) = takeWhile (fun a => decide (p a = true ∧ q a = true)) tl\n⊢ takeWhile p (takeWhile q (hd :: tl)) = takeWhile (fun a => decide (p a = true ∧ q a = true)) (hd :: tl)",
"state_before": "ι : Type ?u.417345\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np✝ : α → Bool\nl✝ : List α\np q : α → Bool\nl : List α\n⊢ takeWhile p (takeWhile q l) = takeWhile (fun a => decide (p a = true ∧ q a = true)) l",
"tactic": "induction' l with hd tl IH"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.417345\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np✝ : α → Bool\nl : List α\np q : α → Bool\n⊢ takeWhile p (takeWhile q []) = takeWhile (fun a => decide (p a = true ∧ q a = true)) []",
"tactic": "simp [takeWhile]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.417345\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\np✝ : α → Bool\nl : List α\np q : α → Bool\nhd : α\ntl : List α\nIH : takeWhile p (takeWhile q tl) = takeWhile (fun a => decide (p a = true ∧ q a = true)) tl\n⊢ takeWhile p (takeWhile q (hd :: tl)) = takeWhile (fun a => decide (p a = true ∧ q a = true)) (hd :: tl)",
"tactic": "by_cases hp : p hd <;> by_cases hq : q hd <;> simp [takeWhile, hp, hq, IH]"
}
] |
[
3666,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3662,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.subset_iUnion
|
[] |
[
326,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.map_sub_swap
|
[] |
[
296,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Topology/Connected.lean
|
preimage_connectedComponent_connected
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\n⊢ IsConnected (f ⁻¹' connectedComponent t)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\n⊢ IsConnected (f ⁻¹' connectedComponent t)",
"tactic": "have hf : Surjective f := Surjective.of_comp fun t : β => (connected_fibers t).1"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\n⊢ IsPreconnected (f ⁻¹' connectedComponent t)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\n⊢ IsConnected (f ⁻¹' connectedComponent t)",
"tactic": "refine ⟨Nonempty.preimage connectedComponent_nonempty hf, ?_⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\n⊢ IsPreconnected (f ⁻¹' connectedComponent t)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\n⊢ IsPreconnected (f ⁻¹' connectedComponent t)",
"tactic": "have hT : IsClosed (f ⁻¹' connectedComponent t) :=\n (hcl (connectedComponent t)).1 isClosed_connectedComponent"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\n⊢ ∀ (u v : Set α),\n IsClosed u →\n IsClosed v →\n f ⁻¹' connectedComponent t ⊆ u ∪ v →\n Disjoint u v → f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\n⊢ IsPreconnected (f ⁻¹' connectedComponent t)",
"tactic": "rw [isPreconnected_iff_subset_of_fully_disjoint_closed hT]"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\n⊢ ∀ (u v : Set α),\n IsClosed u →\n IsClosed v →\n f ⁻¹' connectedComponent t ⊆ u ∪ v →\n Disjoint u v → f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "intro u v hu hv huv uv_disj"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "let T₁ := { t' ∈ connectedComponent t | f ⁻¹' {t'} ⊆ u }"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "let T₂ := { t' ∈ connectedComponent t | f ⁻¹' {t'} ⊆ v }"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v\nhT₁ : IsClosed T₁\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "have hT₁ : IsClosed T₁ := (hcl T₁).2 (T₁_u.symm ▸ IsClosed.inter hT hu)"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v\nhT₁ : IsClosed T₁\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "have hT₂ : IsClosed T₂ := (hcl T₂).2 (T₂_v.symm ▸ IsClosed.inter hT hv)"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\nT_decomp : connectedComponent t ⊆ T₁ ∪ T₂\nT_disjoint : Disjoint T₁ T₂\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\nT_decomp : connectedComponent t ⊆ T₁ ∪ T₂\n⊢ f ⁻¹' connectedComponent t ⊆ u ∨ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "have T_disjoint : Disjoint T₁ T₂ := by\n refine' Disjoint.of_preimage hf _\n rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq,\n inter_empty]"
},
{
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{
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"tactic": "cases' fiber_decomp t' ht' with htu htv"
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{
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"tactic": "right"
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"tactic": "exact ⟨ht', htv⟩"
},
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"tactic": "rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq,\n inter_empty]"
},
{
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"tactic": "rw [Subset.antisymm_iff] at T₁_u"
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{
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"tactic": "suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₁\n from (this.trans T₁_u.1).trans (inter_subset_right _ _)"
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"tactic": "right"
},
{
"state_after": "case inr.h\nα : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\nT_decomp : connectedComponent t ⊆ T₁ ∪ T₂\nT_disjoint : Disjoint T₁ T₂\nh : connectedComponent t ⊆ T₂\n⊢ f ⁻¹' connectedComponent t ⊆ v",
"state_before": "case inr.h\nα : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\nT_decomp : connectedComponent t ⊆ T₁ ∪ T₂\nT_disjoint : Disjoint T₁ T₂\nh : connectedComponent t ⊆ T₂\n⊢ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "rw [Subset.antisymm_iff] at T₂_v"
},
{
"state_after": "case inr.h\nα : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\nT_decomp : connectedComponent t ⊆ T₁ ∪ T₂\nT_disjoint : Disjoint T₁ T₂\nh : connectedComponent t ⊆ T₂\n⊢ f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂",
"state_before": "case inr.h\nα : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\nT_decomp : connectedComponent t ⊆ T₁ ∪ T₂\nT_disjoint : Disjoint T₁ T₂\nh : connectedComponent t ⊆ T₂\n⊢ f ⁻¹' connectedComponent t ⊆ v",
"tactic": "suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂\n from (this.trans T₂_v.1).trans (inter_subset_right _ _)"
},
{
"state_after": "no goals",
"state_before": "case inr.h\nα : Type u\nβ : Type v\nι : Type ?u.107968\nπ : ι → Type ?u.107973\ninst✝¹ : TopologicalSpace α\ns t✝ u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)\nt : β\nhf : Surjective f\nhT : IsClosed (f ⁻¹' connectedComponent t)\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhuv : f ⁻¹' connectedComponent t ⊆ u ∪ v\nuv_disj : Disjoint u v\nT₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}\nT₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}\nfiber_decomp : ∀ (t' : β), t' ∈ connectedComponent t → f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v\nT₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u\nT₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂\nhT₁ : IsClosed T₁\nhT₂ : IsClosed T₂\nT_decomp : connectedComponent t ⊆ T₁ ∪ T₂\nT_disjoint : Disjoint T₁ T₂\nh : connectedComponent t ⊆ T₂\n⊢ f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂",
"tactic": "exact preimage_mono h"
}
] |
[
1084,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1010,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Int.cast_list_prod
|
[] |
[
2232,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2231,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.lxor_bit
|
[
{
"state_after": "no goals",
"state_before": "a : Bool\nm : ℤ\nb : Bool\nn : ℤ\n⊢ lxor' (bit a m) (bit b n) = bit (xor a b) (lxor' m n)",
"tactic": "rw [← bitwise_xor, bitwise_bit]"
}
] |
[
325,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.id
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.22649\nσ : Type ?u.22652\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\n⊢ ∀ (n : ℕ), encode (decode n) = encode (Option.map id (decode n))",
"tactic": "simp"
}
] |
[
259,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
11
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.dist_le_distortion_mul
|
[
{
"state_after": "ι : Type u_1\nI✝ J : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI : Box ι\ni : ι\nA : lower I i - upper I i < 0\n⊢ dist I.lower I.upper ≤ ↑(distortion I) * (upper I i - lower I i)",
"state_before": "ι : Type u_1\nI✝ J : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI : Box ι\ni : ι\n⊢ dist I.lower I.upper ≤ ↑(distortion I) * (upper I i - lower I i)",
"tactic": "have A : I.lower i - I.upper i < 0 := sub_neg.2 (I.lower_lt_upper i)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI✝ J : Box ι\nx y : ι → ℝ\ninst✝ : Fintype ι\nI : Box ι\ni : ι\nA : lower I i - upper I i < 0\n⊢ dist I.lower I.upper ≤ ↑(distortion I) * (upper I i - lower I i)",
"tactic": "simpa only [← NNReal.coe_le_coe, ← dist_nndist, NNReal.coe_mul, Real.dist_eq, abs_of_neg A,\n neg_sub] using I.nndist_le_distortion_mul i"
}
] |
[
528,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
MvPolynomial.prime_C_iff
|
[
{
"state_after": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : ↑C r = 0\n⊢ 0 = ↑C 0",
"state_before": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : ↑C r = 0\n⊢ r = 0",
"tactic": "rw [← C_inj, h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : ↑C r = 0\n⊢ 0 = ↑C 0",
"tactic": "simp"
},
{
"state_after": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : IsUnit (↑C r)\n⊢ IsUnit (↑constantCoeff (↑C r))\n\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : IsUnit (↑C r)\n⊢ Type ?u.531262",
"state_before": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : IsUnit (↑C r)\n⊢ IsUnit r",
"tactic": "rw [← constantCoeff_C _ r]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : IsUnit (↑C r)\n⊢ IsUnit (↑constantCoeff (↑C r))\n\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\nh : IsUnit (↑C r)\n⊢ Type ?u.531262",
"tactic": "exact h.map _"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑C r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"state_before": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\na b : (fun x => MvPolynomial σ R) r\nhd : ↑C r ∣ a * b\n⊢ ↑C r ∣ a ∨ ↑C r ∣ b",
"tactic": "obtain ⟨s, a', b', rfl, rfl⟩ := exists_finset_rename₂ a b"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑C r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"tactic": "rw [← algebraMap_eq] at hd"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"tactic": "have := (@killCompl s σ R _ ((↑) : s → σ) Subtype.coe_injective).toRingHom.map_dvd hd"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis✝ :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\nthis : ↑(algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"tactic": "have : algebraMap R _ r ∣ a' * b' := by convert this <;> simp"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis✝ :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\nthis : ↑(algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'\n⊢ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) a' ∨ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) b'",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis✝ :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\nthis : ↑(algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'\n⊢ ↑C r ∣ ↑(rename Subtype.val) a' ∨ ↑C r ∣ ↑(rename Subtype.val) b'",
"tactic": "rw [← rename_C ((↑) : s → σ)]"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis✝ :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\nthis : ↑(algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'\nf : MvPolynomial { x // x ∈ s } R →+* MvPolynomial σ R := ↑(rename Subtype.val)\n⊢ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) a' ∨ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) b'",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis✝ :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\nthis : ↑(algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'\n⊢ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) a' ∨ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) b'",
"tactic": "let f := @AlgHom.toRingHom R (MvPolynomial s R)\n (MvPolynomial σ R) _ _ _ _ _ (@rename _ _ R _ ((↑) : s → σ))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis✝ :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\nthis : ↑(algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'\nf : MvPolynomial { x // x ∈ s } R →+* MvPolynomial σ R := ↑(rename Subtype.val)\n⊢ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) a' ∨ ↑(rename Subtype.val) (↑C r) ∣ ↑(rename Subtype.val) b'",
"tactic": "exact (((prime_C_iff_of_fintype s).2 hr).2.2 a' b' this).imp f.map_dvd f.map_dvd"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.467790\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\nhr : Prime r\ns : Finset σ\na' b' : MvPolynomial { x // x ∈ s } R\nhd : ↑(algebraMap R (MvPolynomial σ R)) r ∣ ↑(rename Subtype.val) a' * ↑(rename Subtype.val) b'\nthis :\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(algebraMap R (MvPolynomial σ R)) r) ∣\n ↑↑(killCompl (_ : Function.Injective fun a => ↑a)) (↑(rename Subtype.val) a' * ↑(rename Subtype.val) b')\n⊢ ↑(algebraMap R (MvPolynomial { x // x ∈ s } R)) r ∣ a' * b'",
"tactic": "convert this <;> simp"
}
] |
[
811,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/Topology/Order/Basic.lean
|
interior_Ici
|
[] |
[
2277,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2276,
1
] |
Mathlib/Topology/Algebra/FilterBasis.lean
|
ModuleFilterBasis.smul_right
|
[] |
[
366,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
BumpCovering.eventuallyEq_one
|
[] |
[
360,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Data/Matrix/Rank.lean
|
Matrix.rank_one
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.4278\nm : Type ?u.4281\nn : Type u_2\no : Type ?u.4287\nR : Type u_1\nm_fin : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : Fintype o\ninst✝² : CommRing R\ninst✝¹ : StrongRankCondition R\ninst✝ : DecidableEq n\n⊢ rank 1 = Fintype.card n",
"tactic": "rw [rank, mulVecLin_one, LinearMap.range_id, finrank_top, finrank_pi]"
}
] |
[
56,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Algebra/Ring/Divisibility.lean
|
two_dvd_bit1
|
[] |
[
134,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.NumDenSameDeg.num_zero
|
[] |
[
152,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
LipschitzOnWith.extend_finite_dimension
|
[
{
"state_after": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"state_before": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"tactic": "let ι : Type _ := Basis.ofVectorSpaceIndex ℝ E'"
},
{
"state_after": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"state_before": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"tactic": "let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv"
},
{
"state_after": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"state_before": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"tactic": "have LA : LipschitzWith ‖A.toContinuousLinearMap‖₊ A := by apply A.lipschitz"
},
{
"state_after": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"state_before": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"tactic": "have L : LipschitzOnWith (‖A.toContinuousLinearMap‖₊ * K) (A ∘ f) s :=\n LA.comp_lipschitzOnWith hf"
},
{
"state_after": "case intro.intro\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"state_before": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"tactic": "obtain ⟨g, hg, gs⟩ :\n ∃ g : α → ι → ℝ, LipschitzWith (‖A.toContinuousLinearMap‖₊ * K) g ∧ EqOn (A ∘ f) g s :=\n L.extend_pi"
},
{
"state_after": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\n⊢ LipschitzWith (lipschitzExtensionConstant E' * K) (↑(ContinuousLinearEquiv.symm A) ∘ g)\n\ncase intro.intro.refine'_2\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\n⊢ EqOn f (↑(ContinuousLinearEquiv.symm A) ∘ g) s",
"state_before": "case intro.intro\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\n⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s",
"tactic": "refine' ⟨A.symm ∘ g, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\n⊢ LipschitzWith ‖↑A‖₊ ↑A",
"tactic": "apply A.lipschitz"
},
{
"state_after": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nLAsymm : LipschitzWith ‖↑(ContinuousLinearEquiv.symm A)‖₊ ↑(ContinuousLinearEquiv.symm A)\n⊢ LipschitzWith (lipschitzExtensionConstant E' * K) (↑(ContinuousLinearEquiv.symm A) ∘ g)",
"state_before": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\n⊢ LipschitzWith (lipschitzExtensionConstant E' * K) (↑(ContinuousLinearEquiv.symm A) ∘ g)",
"tactic": "have LAsymm : LipschitzWith ‖A.symm.toContinuousLinearMap‖₊ A.symm := by\n apply A.symm.lipschitz"
},
{
"state_after": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nLAsymm : LipschitzWith ‖↑(ContinuousLinearEquiv.symm A)‖₊ ↑(ContinuousLinearEquiv.symm A)\n⊢ ‖↑(ContinuousLinearEquiv.symm A)‖₊ * (‖↑A‖₊ * K) ≤ lipschitzExtensionConstant E' * K",
"state_before": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nLAsymm : LipschitzWith ‖↑(ContinuousLinearEquiv.symm A)‖₊ ↑(ContinuousLinearEquiv.symm A)\n⊢ LipschitzWith (lipschitzExtensionConstant E' * K) (↑(ContinuousLinearEquiv.symm A) ∘ g)",
"tactic": "apply (LAsymm.comp hg).weaken"
},
{
"state_after": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nLAsymm : LipschitzWith ‖↑(ContinuousLinearEquiv.symm A)‖₊ ↑(ContinuousLinearEquiv.symm A)\n⊢ ‖↑(ContinuousLinearEquiv.symm A)‖₊ * ‖↑A‖₊ * K ≤\n (let A := LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'));\n max (‖↑(ContinuousLinearEquiv.symm A)‖₊ * ‖↑A‖₊) 1) *\n K",
"state_before": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nLAsymm : LipschitzWith ‖↑(ContinuousLinearEquiv.symm A)‖₊ ↑(ContinuousLinearEquiv.symm A)\n⊢ ‖↑(ContinuousLinearEquiv.symm A)‖₊ * (‖↑A‖₊ * K) ≤ lipschitzExtensionConstant E' * K",
"tactic": "rw [lipschitzExtensionConstant, ← mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nLAsymm : LipschitzWith ‖↑(ContinuousLinearEquiv.symm A)‖₊ ↑(ContinuousLinearEquiv.symm A)\n⊢ ‖↑(ContinuousLinearEquiv.symm A)‖₊ * ‖↑A‖₊ * K ≤\n (let A := LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'));\n max (‖↑(ContinuousLinearEquiv.symm A)‖₊ * ‖↑A‖₊) 1) *\n K",
"tactic": "refine' mul_le_mul' (le_max_left _ _) le_rfl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\n⊢ LipschitzWith ‖↑(ContinuousLinearEquiv.symm A)‖₊ ↑(ContinuousLinearEquiv.symm A)",
"tactic": "apply A.symm.lipschitz"
},
{
"state_after": "case intro.intro.refine'_2\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nx : α\nhx : x ∈ s\n⊢ f x = (↑(ContinuousLinearEquiv.symm A) ∘ g) x",
"state_before": "case intro.intro.refine'_2\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\n⊢ EqOn f (↑(ContinuousLinearEquiv.symm A) ∘ g) s",
"tactic": "intro x hx"
},
{
"state_after": "case intro.intro.refine'_2\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nx : α\nhx : x ∈ s\nthis : ↑A (f x) = g x\n⊢ f x = (↑(ContinuousLinearEquiv.symm A) ∘ g) x",
"state_before": "case intro.intro.refine'_2\n𝕜 : Type u\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nF : Type w\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁹ : AddCommGroup F'\ninst✝⁸ : Module 𝕜 F'\ninst✝⁷ : TopologicalSpace F'\ninst✝⁶ : TopologicalAddGroup F'\ninst✝⁵ : ContinuousSMul 𝕜 F'\ninst✝⁴ : CompleteSpace 𝕜\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nE' : Type u_2\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\ns : Set α\nf : α → E'\nK : ℝ≥0\nhf : LipschitzOnWith K f s\nι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')\nA : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ :=\n LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'))\nLA : LipschitzWith ‖↑A‖₊ ↑A\nL : LipschitzOnWith (‖↑A‖₊ * K) (↑A ∘ f) s\ng : α → ι → ℝ\nhg : LipschitzWith (‖↑A‖₊ * K) g\ngs : EqOn (↑A ∘ f) g s\nx : α\nhx : x ∈ s\n⊢ f x = (↑(ContinuousLinearEquiv.symm A) ∘ g) x",
"tactic": "have : A (f x) = g x := gs hx"
}
] |
[
223,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.cocone_inl
|
[] |
[
342,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift_iInf_le
|
[] |
[
200,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isCompact_of_finite_subfamily_closed
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.27539\nπ : ι → Type ?u.27544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nh :\n ∀ {ι : Type u} (Z : ι → Set α),\n (∀ (i : ι), IsClosed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → ∃ t, (s ∩ ⋂ (i : ι) (_ : i ∈ t), Z i) = ∅\nι✝ : Type u\nU : ι✝ → Set α\nhUo : ∀ (i : ι✝), IsOpen (U i)\nhsU : (s ⊓ ⋂ (i : ι✝), U iᶜ) = ⊥\n⊢ ∃ t, s ⊆ ⋃ (i : ι✝) (_ : i ∈ t), U i",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.27539\nπ : ι → Type ?u.27544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nh :\n ∀ {ι : Type u} (Z : ι → Set α),\n (∀ (i : ι), IsClosed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → ∃ t, (s ∩ ⋂ (i : ι) (_ : i ∈ t), Z i) = ∅\nι✝ : Type u\nU : ι✝ → Set α\nhUo : ∀ (i : ι✝), IsOpen (U i)\nhsU : s ⊆ ⋃ (i : ι✝), U i\n⊢ ∃ t, s ⊆ ⋃ (i : ι✝) (_ : i ∈ t), U i",
"tactic": "rw [← disjoint_compl_right_iff_subset, compl_iUnion, disjoint_iff] at hsU"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Type ?u.27539\nπ : ι → Type ?u.27544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nh :\n ∀ {ι : Type u} (Z : ι → Set α),\n (∀ (i : ι), IsClosed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → ∃ t, (s ∩ ⋂ (i : ι) (_ : i ∈ t), Z i) = ∅\nι✝ : Type u\nU : ι✝ → Set α\nhUo : ∀ (i : ι✝), IsOpen (U i)\nhsU : (s ⊓ ⋂ (i : ι✝), U iᶜ) = ⊥\nt : Finset ι✝\nht : (s ∩ ⋂ (i : ι✝) (_ : i ∈ t), U iᶜ) = ∅\n⊢ ∃ t, s ⊆ ⋃ (i : ι✝) (_ : i ∈ t), U i",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.27539\nπ : ι → Type ?u.27544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nh :\n ∀ {ι : Type u} (Z : ι → Set α),\n (∀ (i : ι), IsClosed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → ∃ t, (s ∩ ⋂ (i : ι) (_ : i ∈ t), Z i) = ∅\nι✝ : Type u\nU : ι✝ → Set α\nhUo : ∀ (i : ι✝), IsOpen (U i)\nhsU : (s ⊓ ⋂ (i : ι✝), U iᶜ) = ⊥\n⊢ ∃ t, s ⊆ ⋃ (i : ι✝) (_ : i ∈ t), U i",
"tactic": "rcases h (fun i => (U i)ᶜ) (fun i => (hUo _).isClosed_compl) hsU with ⟨t, ht⟩"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Type ?u.27539\nπ : ι → Type ?u.27544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nh :\n ∀ {ι : Type u} (Z : ι → Set α),\n (∀ (i : ι), IsClosed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → ∃ t, (s ∩ ⋂ (i : ι) (_ : i ∈ t), Z i) = ∅\nι✝ : Type u\nU : ι✝ → Set α\nhUo : ∀ (i : ι✝), IsOpen (U i)\nhsU : (s ⊓ ⋂ (i : ι✝), U iᶜ) = ⊥\nt : Finset ι✝\nht : (s ∩ ⋂ (i : ι✝) (_ : i ∈ t), U iᶜ) = ∅\n⊢ s ⊆ ⋃ (i : ι✝) (_ : i ∈ t), U i",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Type ?u.27539\nπ : ι → Type ?u.27544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nh :\n ∀ {ι : Type u} (Z : ι → Set α),\n (∀ (i : ι), IsClosed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → ∃ t, (s ∩ ⋂ (i : ι) (_ : i ∈ t), Z i) = ∅\nι✝ : Type u\nU : ι✝ → Set α\nhUo : ∀ (i : ι✝), IsOpen (U i)\nhsU : (s ⊓ ⋂ (i : ι✝), U iᶜ) = ⊥\nt : Finset ι✝\nht : (s ∩ ⋂ (i : ι✝) (_ : i ∈ t), U iᶜ) = ∅\n⊢ ∃ t, s ⊆ ⋃ (i : ι✝) (_ : i ∈ t), U i",
"tactic": "refine ⟨t, ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Type ?u.27539\nπ : ι → Type ?u.27544\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nh :\n ∀ {ι : Type u} (Z : ι → Set α),\n (∀ (i : ι), IsClosed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → ∃ t, (s ∩ ⋂ (i : ι) (_ : i ∈ t), Z i) = ∅\nι✝ : Type u\nU : ι✝ → Set α\nhUo : ∀ (i : ι✝), IsOpen (U i)\nhsU : (s ⊓ ⋂ (i : ι✝), U iᶜ) = ⊥\nt : Finset ι✝\nht : (s ∩ ⋂ (i : ι✝) (_ : i ∈ t), U iᶜ) = ∅\n⊢ s ⊆ ⋃ (i : ι✝) (_ : i ∈ t), U i",
"tactic": "rwa [← disjoint_compl_right_iff_subset, compl_iUnion₂, disjoint_iff]"
}
] |
[
348,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Algebra/Group/Pi.lean
|
Pi.mulSingle_strictMono
|
[] |
[
715,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
714,
1
] |
Mathlib/Data/Finset/Pi.lean
|
Finset.pi_disjoint_of_disjoint
|
[] |
[
138,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Algebra/Module/Projective.lean
|
Module.Projective.of_lifting_property'
|
[] |
[
188,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.rightInverse_iff_comp
|
[] |
[
340,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Data/List/Lattice.lean
|
List.inter_reverse
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl l₁ l₂ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nxs ys : List α\n⊢ List.inter xs (reverse ys) = List.inter xs ys",
"tactic": "simp only [List.inter, mem_reverse]"
}
] |
[
198,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Topology/Covering.lean
|
IsCoveringMapOn.continuousAt
|
[] |
[
100,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
11
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.measurable_uncurry_of_continuous_of_measurable
|
[
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\n⊢ Measurable (uncurry u)",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\n⊢ Measurable (uncurry u)",
"tactic": "obtain ⟨t_sf, ht_sf⟩ :\n ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by\n have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id\n refine' ⟨h_str_meas.approx, fun j x => _⟩\n exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)"
},
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ Measurable (uncurry u)",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\n⊢ Measurable (uncurry u)",
"tactic": "let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd"
},
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\n⊢ Measurable (uncurry u)",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ Measurable (uncurry u)",
"tactic": "have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by\n rw [tendsto_pi_nhds]\n exact fun p => ht_sf p.fst p.snd"
},
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ Measurable (U n)",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\n⊢ Measurable (uncurry u)",
"tactic": "refine' measurable_of_tendsto_metrizable (fun n => _) h_tendsto"
},
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\n⊢ Measurable (U n)",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ Measurable (U n)",
"tactic": "have h_meas : Measurable fun p : (t_sf n).range × α => u (↑p.fst) p.snd := by\n have :\n (fun p : ↥(t_sf n).range × α => u (↑p.fst) p.snd) =\n (fun p : α × (t_sf n).range => u (↑p.snd) p.fst) ∘ Prod.swap :=\n rfl\n rw [this, @measurable_swap_iff α (↥(t_sf n).range) β m]\n exact measurable_from_prod_countable fun j => h j"
},
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable (U n)",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\n⊢ Measurable (U n)",
"tactic": "have :\n (fun p : ι × α => u (t_sf n p.fst) p.snd) =\n (fun p : ↥(t_sf n).range × α => u p.fst p.snd) ∘ fun p : ι × α =>\n (⟨t_sf n p.fst, SimpleFunc.mem_range_self _ _⟩, p.snd) :=\n rfl"
},
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable\n ((fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd))",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable (U n)",
"tactic": "simp_rw [this]"
},
{
"state_after": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable fun p => { val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable\n ((fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd))",
"tactic": "refine' h_meas.comp (Measurable.prod_mk _ measurable_snd)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_meas : Measurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable fun p => { val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }",
"tactic": "exact ((t_sf n).measurable.comp measurable_fst).subtype_mk"
},
{
"state_after": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nh_str_meas : StronglyMeasurable id\n⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t n) j) x) atTop (𝓝 (u j x))",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\n⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t n) j) x) atTop (𝓝 (u j x))",
"tactic": "have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id"
},
{
"state_after": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nh_str_meas : StronglyMeasurable id\nj : ι\nx : α\n⊢ Tendsto (fun n => u (↑(StronglyMeasurable.approx h_str_meas n) j) x) atTop (𝓝 (u j x))",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nh_str_meas : StronglyMeasurable id\n⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t n) j) x) atTop (𝓝 (u j x))",
"tactic": "refine' ⟨h_str_meas.approx, fun j x => _⟩"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nh_str_meas : StronglyMeasurable id\nj : ι\nx : α\n⊢ Tendsto (fun n => u (↑(StronglyMeasurable.approx h_str_meas n) j) x) atTop (𝓝 (u j x))",
"tactic": "exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)"
},
{
"state_after": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ ∀ (x : ι × α), Tendsto (fun i => U i x) atTop (𝓝 (u x.fst x.snd))",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ Tendsto U atTop (𝓝 fun p => u p.fst p.snd)",
"tactic": "rw [tendsto_pi_nhds]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ ∀ (x : ι × α), Tendsto (fun i => U i x) atTop (𝓝 (u x.fst x.snd))",
"tactic": "exact fun p => ht_sf p.fst p.snd"
},
{
"state_after": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.fst) p.snd",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ Measurable fun p => u (↑p.fst) p.snd",
"tactic": "have :\n (fun p : ↥(t_sf n).range × α => u (↑p.fst) p.snd) =\n (fun p : α × (t_sf n).range => u (↑p.snd) p.fst) ∘ Prod.swap :=\n rfl"
},
{
"state_after": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.snd) p.fst",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.fst) p.snd",
"tactic": "rw [this, @measurable_swap_iff α (↥(t_sf n).range) β m]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.556281\nβ✝ : Type ?u.556284\nγ : Type ?u.556287\nι✝ : Type ?u.556290\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), Measurable (u i)\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.snd) p.fst",
"tactic": "exact measurable_from_prod_countable fun j => h j"
}
] |
[
2008,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1979,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.exists_mem_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.438021\nγ : Type ?u.438024\np : α → Prop\n⊢ (∃ x, x ∈ ∅ ∧ p x) ↔ False",
"tactic": "simp only [not_mem_empty, false_and_iff, exists_false]"
}
] |
[
3068,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3067,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.Subset.trans
|
[] |
[
396,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Combinatorics/Additive/Energy.lean
|
Finset.multiplicativeEnergy_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns s₁ s₂ t t₁ t₂ : Finset α\n⊢ multiplicativeEnergy s t = 0 ↔ s = ∅ ∨ t = ∅",
"tactic": "simp [← (Nat.zero_le _).not_gt_iff_eq, not_and_or, imp_iff_or_not, or_comm]"
}
] |
[
122,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_zero
|
[] |
[
98,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Order/Basic.lean
|
Ne.lt_or_lt
|
[] |
[
480,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/GroupTheory/Abelianization.lean
|
commutator_def
|
[] |
[
47,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.mem_coeIdeal
|
[] |
[
259,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
quotient_norm_add_le
|
[
{
"state_after": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n⊢ ‖↑x + ↑y‖ ≤ ‖↑x‖ + ‖↑y‖",
"state_before": "M : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M ⧸ S\n⊢ ‖x + y‖ ≤ ‖x‖ + ‖y‖",
"tactic": "rcases And.intro (mk_surjective x) (mk_surjective y) with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩"
},
{
"state_after": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n⊢ (⨅ (a : ↑↑S), ‖x + y + ↑a‖) ≤ (⨅ (a : ↑↑S), ‖x + ↑a‖) + ⨅ (a : ↑↑S), ‖y + ↑a‖",
"state_before": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n⊢ ‖↑x + ↑y‖ ≤ ‖↑x‖ + ‖↑y‖",
"tactic": "simp only [← mk'_apply, ← map_add, quotient_norm_mk_eq, sInf_image']"
},
{
"state_after": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\na b : ↑↑S\n⊢ (⨅ (a : ↑↑S), ‖x + y + ↑a‖) ≤ ‖x + ↑a‖ + ‖y + ↑b‖",
"state_before": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n⊢ (⨅ (a : ↑↑S), ‖x + y + ↑a‖) ≤ (⨅ (a : ↑↑S), ‖x + ↑a‖) + ⨅ (a : ↑↑S), ‖y + ↑a‖",
"tactic": "refine le_ciInf_add_ciInf fun a b ↦ ?_"
},
{
"state_after": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\na b : ↑↑S\n⊢ ‖x + y + ↑(a + b)‖ ≤ ‖x + ↑a‖ + ‖y + ↑b‖",
"state_before": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\na b : ↑↑S\n⊢ (⨅ (a : ↑↑S), ‖x + y + ↑a‖) ≤ ‖x + ↑a‖ + ‖y + ↑b‖",
"tactic": "refine ciInf_le_of_le ⟨0, forall_range_iff.2 fun _ ↦ norm_nonneg _⟩ (a + b) ?_"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nM : Type u_1\nN : Type ?u.346908\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\na b : ↑↑S\n⊢ ‖x + y + ↑(a + b)‖ ≤ ‖x + ↑a‖ + ‖y + ↑b‖",
"tactic": "exact (congr_arg norm (add_add_add_comm _ _ _ _)).trans_le (norm_add_le _ _)"
}
] |
[
217,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.disjoint_ker'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1371370\nR₂ : Type u_3\nR₃ : Type ?u.1371376\nR₄ : Type ?u.1371379\nS : Type ?u.1371382\nK : Type ?u.1371385\nK₂ : Type ?u.1371388\nM : Type u_2\nM' : Type ?u.1371394\nM₁ : Type ?u.1371397\nM₂ : Type u_4\nM₃ : Type ?u.1371403\nM₄ : Type ?u.1371406\nN : Type ?u.1371409\nN₂ : Type ?u.1371412\nι : Type ?u.1371415\nV : Type ?u.1371418\nV₂ : Type ?u.1371421\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R₂\ninst✝⁷ : Ring R₃\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : AddCommGroup M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_5\nsc : SemilinearMapClass F τ₁₂ M M₂\nf : F\np : Submodule R M\nH : ∀ (x : M), x ∈ p → ↑f x = 0 → x = 0\nx : M\nhx : x ∈ p\ny : M\nhy : y ∈ p\nh : ↑f x = ↑f y\n⊢ ↑f (x - y) = 0",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1371370\nR₂ : Type u_3\nR₃ : Type ?u.1371376\nR₄ : Type ?u.1371379\nS : Type ?u.1371382\nK : Type ?u.1371385\nK₂ : Type ?u.1371388\nM : Type u_2\nM' : Type ?u.1371394\nM₁ : Type ?u.1371397\nM₂ : Type u_4\nM₃ : Type ?u.1371403\nM₄ : Type ?u.1371406\nN : Type ?u.1371409\nN₂ : Type ?u.1371412\nι : Type ?u.1371415\nV : Type ?u.1371418\nV₂ : Type ?u.1371421\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R₂\ninst✝⁷ : Ring R₃\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M₂\ninst✝⁴ : AddCommGroup M₃\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nτ₁₂ : R →+* R₂\nτ₂₃ : R₂ →+* R₃\nτ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃\nF : Type u_5\nsc : SemilinearMapClass F τ₁₂ M M₂\nf : F\np : Submodule R M\nH : ∀ (x : M), x ∈ p → ∀ (y : M), y ∈ p → ↑f x = ↑f y → x = y\nx : M\nh₁ : x ∈ p\nh₂ : ↑f x = 0\n⊢ ↑f x = ↑f 0",
"tactic": "simpa using h₂"
}
] |
[
1495,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1491,
1
] |
Mathlib/RingTheory/Ideal/Over.lean
|
Ideal.exists_nonzero_mem_of_ne_bot
|
[
{
"state_after": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\n⊢ ∃ p, p ∈ P ∧ Polynomial.map (Quotient.mk (comap C P)) p ≠ 0",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\n⊢ ∃ p, p ∈ P ∧ Polynomial.map (Quotient.mk (comap C P)) p ≠ 0",
"tactic": "obtain ⟨m, hm⟩ := Submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb)"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ ↑m = 0",
"state_before": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\n⊢ ∃ p, p ∈ P ∧ Polynomial.map (Quotient.mk (comap C P)) p ≠ 0",
"tactic": "refine' ⟨m, Submodule.coe_mem m, fun pp0 => hm (Submodule.coe_eq_zero.mp _)⟩"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ Function.Injective ↑(mapRingHom (Quotient.mk (comap C P)))",
"state_before": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ ↑m = 0",
"tactic": "refine'\n (injective_iff_map_eq_zero (Polynomial.mapRingHom (Ideal.Quotient.mk\n (P.comap (C : R →+* R[X]))))).mp\n _ _ pp0"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ RingHom.ker (Quotient.mk (comap C P)) = ⊥",
"state_before": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ Function.Injective ↑(mapRingHom (Quotient.mk (comap C P)))",
"tactic": "refine' map_injective _ ((Ideal.Quotient.mk (P.comap C)).injective_iff_ker_eq_bot.mpr _)"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ comap C P = ⊥",
"state_before": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ RingHom.ker (Quotient.mk (comap C P)) = ⊥",
"tactic": "rw [mk_ker]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝¹ : CommRing R\nS : Type ?u.73967\ninst✝ : CommRing S\nf : R →+* S\nI J : Ideal S\nP : Ideal R[X]\nPb : P ≠ ⊥\nhP : ∀ (x : R), ↑C x ∈ P → x = 0\nm : { x // x ∈ P }\nhm : m ≠ 0\npp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0\n⊢ comap C P = ⊥",
"tactic": "exact (Submodule.eq_bot_iff _).mpr fun x hx => hP x (mem_comap.mp hx)"
}
] |
[
129,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.nat_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.96089\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : Semiring α\nh : Periodic f c\nn : ℕ\n⊢ Periodic f (↑n * c)",
"tactic": "simpa only [nsmul_eq_mul] using h.nsmul n"
}
] |
[
206,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sigmaToiUnion_injective
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.300743\nι : Sort ?u.300746\nι' : Sort ?u.300749\nι₂ : Sort ?u.300752\nκ : ι → Sort ?u.300757\nκ₁ : ι → Sort ?u.300762\nκ₂ : ι → Sort ?u.300767\nκ' : ι' → Sort ?u.300772\nt : α → Set β\nh : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)\na₁ : α\nb₁ : β\nh₁ : b₁ ∈ t a₁\na₂ : α\na_eq : a₁ = a₂\nh₂ : b₁ ∈ t a₂\neq :\n sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =\n sigmaToiUnion t { fst := a₂, snd := { val := b₁, property := h₂ } }\n⊢ ↑(Eq.recOn a_eq { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =\n ↑{ fst := a₂, snd := { val := b₁, property := h₂ } }.snd",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.300743\nι : Sort ?u.300746\nι' : Sort ?u.300749\nι₂ : Sort ?u.300752\nκ : ι → Sort ?u.300757\nκ₁ : ι → Sort ?u.300762\nκ₂ : ι → Sort ?u.300767\nκ' : ι' → Sort ?u.300772\nt : α → Set β\nh : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)\na₁ : α\nb₁ : β\nh₁ : b₁ ∈ t a₁\na₂ : α\nb₂ : β\nh₂ : b₂ ∈ t a₂\neq :\n sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =\n sigmaToiUnion t { fst := a₂, snd := { val := b₂, property := h₂ } }\nb_eq : b₁ = b₂\na_eq : a₁ = a₂\n⊢ ↑(Eq.recOn a_eq { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =\n ↑{ fst := a₂, snd := { val := b₂, property := h₂ } }.snd",
"tactic": "subst b_eq"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.300743\nι : Sort ?u.300746\nι' : Sort ?u.300749\nι₂ : Sort ?u.300752\nκ : ι → Sort ?u.300757\nκ₁ : ι → Sort ?u.300762\nκ₂ : ι → Sort ?u.300767\nκ' : ι' → Sort ?u.300772\nt : α → Set β\nh : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)\na₁ : α\nb₁ : β\nh₁ h₂ : b₁ ∈ t a₁\neq :\n sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =\n sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₂ } }\n⊢ ↑(Eq.recOn (_ : a₁ = a₁) { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =\n ↑{ fst := a₁, snd := { val := b₁, property := h₂ } }.snd",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.300743\nι : Sort ?u.300746\nι' : Sort ?u.300749\nι₂ : Sort ?u.300752\nκ : ι → Sort ?u.300757\nκ₁ : ι → Sort ?u.300762\nκ₂ : ι → Sort ?u.300767\nκ' : ι' → Sort ?u.300772\nt : α → Set β\nh : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)\na₁ : α\nb₁ : β\nh₁ : b₁ ∈ t a₁\na₂ : α\na_eq : a₁ = a₂\nh₂ : b₁ ∈ t a₂\neq :\n sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =\n sigmaToiUnion t { fst := a₂, snd := { val := b₁, property := h₂ } }\n⊢ ↑(Eq.recOn a_eq { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =\n ↑{ fst := a₂, snd := { val := b₁, property := h₂ } }.snd",
"tactic": "subst a_eq"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.300743\nι : Sort ?u.300746\nι' : Sort ?u.300749\nι₂ : Sort ?u.300752\nκ : ι → Sort ?u.300757\nκ₁ : ι → Sort ?u.300762\nκ₂ : ι → Sort ?u.300767\nκ' : ι' → Sort ?u.300772\nt : α → Set β\nh : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)\na₁ : α\nb₁ : β\nh₁ h₂ : b₁ ∈ t a₁\neq :\n sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =\n sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₂ } }\n⊢ ↑(Eq.recOn (_ : a₁ = a₁) { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =\n ↑{ fst := a₁, snd := { val := b₁, property := h₂ } }.snd",
"tactic": "rfl"
}
] |
[
2183,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2175,
1
] |
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