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/Data/Finsupp/Interval.lean
|
Finsupp.card_Iic
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝¹ : CanonicallyOrderedAddMonoid α\ninst✝ : LocallyFiniteOrder α\nf : ι →₀ α\n⊢ card (Iic f) = ∏ i in f.support, card (Iic (↑f i))",
"tactic": "classical simp_rw [Iic_eq_Icc, card_Icc, Finsupp.bot_eq_zero, support_zero, empty_union,\n zero_apply, bot_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝¹ : CanonicallyOrderedAddMonoid α\ninst✝ : LocallyFiniteOrder α\nf : ι →₀ α\n⊢ card (Iic f) = ∏ i in f.support, card (Iic (↑f i))",
"tactic": "simp_rw [Iic_eq_Icc, card_Icc, Finsupp.bot_eq_zero, support_zero, empty_union,\nzero_apply, bot_eq_zero]"
}
] |
[
139,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Order/Disjoint.lean
|
disjoint_sup_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\na b c : α\n⊢ Disjoint a (b ⊔ c) ↔ Disjoint a b ∧ Disjoint a c",
"tactic": "simp only [disjoint_iff, inf_sup_left, sup_eq_bot_iff]"
}
] |
[
193,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
ite_pow
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.4429\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Pow M ℕ\nP : Prop\ninst✝ : Decidable P\na b : M\nc : ℕ\n⊢ (if P then a else b) ^ c = if P then a ^ c else b ^ c",
"tactic": "split_ifs <;> rfl"
}
] |
[
61,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Algebra/Lie/Free.lean
|
FreeLieAlgebra.liftAux_spec
|
[
{
"state_after": "case lie_self\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝ : lib R X\n⊢ ↑(liftAux R f) (a✝ * a✝) = ↑(liftAux R f) 0\n\ncase leibniz_lie\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝ b✝ c✝ : lib R X\n⊢ ↑(liftAux R f) (a✝ * (b✝ * c✝)) = ↑(liftAux R f) (a✝ * b✝ * c✝ + b✝ * (a✝ * c✝))\n\ncase smul\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nt✝ : R\na✝¹ b✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (t✝ • a✝¹) = ↑(liftAux R f) (t✝ • b✝)\n\ncase add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nh : Rel R X a b\n⊢ ↑(liftAux R f) a = ↑(liftAux R f) b",
"tactic": "induction h"
},
{
"state_after": "case leibniz_lie\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝ b✝ c✝ : lib R X\n⊢ ↑(liftAux R f) (a✝ * (b✝ * c✝)) = ↑(liftAux R f) (a✝ * b✝ * c✝ + b✝ * (a✝ * c✝))\n\ncase smul\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nt✝ : R\na✝¹ b✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (t✝ • a✝¹) = ↑(liftAux R f) (t✝ • b✝)\n\ncase add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"state_before": "case lie_self\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝ : lib R X\n⊢ ↑(liftAux R f) (a✝ * a✝) = ↑(liftAux R f) 0\n\ncase leibniz_lie\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝ b✝ c✝ : lib R X\n⊢ ↑(liftAux R f) (a✝ * (b✝ * c✝)) = ↑(liftAux R f) (a✝ * b✝ * c✝ + b✝ * (a✝ * c✝))\n\ncase smul\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nt✝ : R\na✝¹ b✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (t✝ • a✝¹) = ↑(liftAux R f) (t✝ • b✝)\n\ncase add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"tactic": "case lie_self a' => simp only [liftAux_map_mul, NonUnitalAlgHom.map_zero, lie_self]"
},
{
"state_after": "case smul\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nt✝ : R\na✝¹ b✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (t✝ • a✝¹) = ↑(liftAux R f) (t✝ • b✝)\n\ncase add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"state_before": "case leibniz_lie\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝ b✝ c✝ : lib R X\n⊢ ↑(liftAux R f) (a✝ * (b✝ * c✝)) = ↑(liftAux R f) (a✝ * b✝ * c✝ + b✝ * (a✝ * c✝))\n\ncase smul\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nt✝ : R\na✝¹ b✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (t✝ • a✝¹) = ↑(liftAux R f) (t✝ • b✝)\n\ncase add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"tactic": "case leibniz_lie a' b' c' =>\n simp only [liftAux_map_mul, liftAux_map_add, sub_add_cancel, lie_lie]"
},
{
"state_after": "case add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"state_before": "case smul\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nt✝ : R\na✝¹ b✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (t✝ • a✝¹) = ↑(liftAux R f) (t✝ • b✝)\n\ncase add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"tactic": "case smul t a' b' _ h₂ => simp only [liftAux_map_smul, h₂]"
},
{
"state_after": "case mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"state_before": "case add_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ + c✝) = ↑(liftAux R f) (b✝ + c✝)\n\ncase mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"tactic": "case add_right a' b' c' _ h₂ => simp only [liftAux_map_add, h₂]"
},
{
"state_after": "case mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"state_before": "case mul_left\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X b✝ c✝\na_ih✝ : ↑(liftAux R f) b✝ = ↑(liftAux R f) c✝\n⊢ ↑(liftAux R f) (a✝¹ * b✝) = ↑(liftAux R f) (a✝¹ * c✝)\n\ncase mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"tactic": "case mul_left a' b' c' _ h₂ => simp only [liftAux_map_mul, h₂]"
},
{
"state_after": "no goals",
"state_before": "case mul_right\nR : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a✝¹ b✝ c✝ : lib R X\na✝ : Rel R X a✝¹ b✝\na_ih✝ : ↑(liftAux R f) a✝¹ = ↑(liftAux R f) b✝\n⊢ ↑(liftAux R f) (a✝¹ * c✝) = ↑(liftAux R f) (b✝ * c✝)",
"tactic": "case mul_right a' b' c' _ h₂ => simp only [liftAux_map_mul, h₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a' : lib R X\n⊢ ↑(liftAux R f) (a' * a') = ↑(liftAux R f) 0",
"tactic": "simp only [liftAux_map_mul, NonUnitalAlgHom.map_zero, lie_self]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a' b' c' : lib R X\n⊢ ↑(liftAux R f) (a' * (b' * c')) = ↑(liftAux R f) (a' * b' * c' + b' * (a' * c'))",
"tactic": "simp only [liftAux_map_mul, liftAux_map_add, sub_add_cancel, lie_lie]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b : lib R X\nt : R\na' b' : lib R X\na✝ : Rel R X a' b'\nh₂ : ↑(liftAux R f) a' = ↑(liftAux R f) b'\n⊢ ↑(liftAux R f) (t • a') = ↑(liftAux R f) (t • b')",
"tactic": "simp only [liftAux_map_smul, h₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a' b' c' : lib R X\na✝ : Rel R X a' b'\nh₂ : ↑(liftAux R f) a' = ↑(liftAux R f) b'\n⊢ ↑(liftAux R f) (a' + c') = ↑(liftAux R f) (b' + c')",
"tactic": "simp only [liftAux_map_add, h₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a' b' c' : lib R X\na✝ : Rel R X b' c'\nh₂ : ↑(liftAux R f) b' = ↑(liftAux R f) c'\n⊢ ↑(liftAux R f) (a' * b') = ↑(liftAux R f) (a' * c')",
"tactic": "simp only [liftAux_map_mul, h₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nf : X → L\na b a' b' c' : lib R X\na✝ : Rel R X a' b'\nh₂ : ↑(liftAux R f) a' = ↑(liftAux R f) b'\n⊢ ↑(liftAux R f) (a' * c') = ↑(liftAux R f) (b' * c')",
"tactic": "simp only [liftAux_map_mul, h₂]"
}
] |
[
214,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.sublistsLen_sublist_sublists'
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nl : List α\n⊢ sublistsLen 0 l <+ sublists' l",
"tactic": "simp"
},
{
"state_after": "α✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nn : ℕ\na : α\nl : List α\n⊢ sublistsLen (n + 1) l ++ map (cons a) (sublistsLen n l) <+ sublists' l ++ map (cons a) (sublists' l)",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nn : ℕ\na : α\nl : List α\n⊢ sublistsLen (n + 1) (a :: l) <+ sublists' (a :: l)",
"tactic": "rw [sublistsLen_succ_cons, sublists'_cons]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nα : Type u_1\nn : ℕ\na : α\nl : List α\n⊢ sublistsLen (n + 1) l ++ map (cons a) (sublistsLen n l) <+ sublists' l ++ map (cons a) (sublists' l)",
"tactic": "exact (sublistsLen_sublist_sublists' _ _).append ((sublistsLen_sublist_sublists' _ _).map _)"
}
] |
[
295,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
zero_eq_nndist
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.552184\nι : Type ?u.552187\ninst✝¹ : PseudoMetricSpace α\nγ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ 0 = nndist x y ↔ x = y",
"tactic": "simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist]"
}
] |
[
2890,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2889,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Dense.exists_lt
|
[] |
[
757,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
11
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_indicator
|
[] |
[
297,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.coe_mono
|
[] |
[
334,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
Equiv.pointReflection_midpoint_right
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nV : Type u_2\nV' : Type ?u.28779\nP : Type u_1\nP' : Type ?u.28785\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx✝ y✝ z x y : P\n⊢ ↑(pointReflection (midpoint R x y)) y = x",
"tactic": "rw [midpoint_comm, Equiv.pointReflection_midpoint_left]"
}
] |
[
89,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.id_comp
|
[] |
[
712,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
712,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.equiv_congr_left
|
[] |
[
879,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
878,
1
] |
Mathlib/Topology/UnitInterval.lean
|
unitInterval.le_one
|
[] |
[
149,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
AEMeasurable.smul_const
|
[] |
[
642,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
640,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.lookupAll_nodup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nh : NodupKeys l\n⊢ Nodup (lookupAll a l)",
"tactic": "(rw [lookupAll_eq_dlookup a h]; apply Option.toList_nodup)"
},
{
"state_after": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nh : NodupKeys l\n⊢ Nodup (Option.toList (dlookup a l))",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nh : NodupKeys l\n⊢ Nodup (lookupAll a l)",
"tactic": "rw [lookupAll_eq_dlookup a h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nh : NodupKeys l\n⊢ Nodup (Option.toList (dlookup a l))",
"tactic": "apply Option.toList_nodup"
}
] |
[
329,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.isFixedPt_gfp
|
[] |
[
123,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Data/List/Zip.lean
|
List.zip_cons_cons
|
[] |
[
43,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Quotient.rep_spec
|
[] |
[
686,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
685,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.neighborSet_singletonSubgraph
|
[] |
[
872,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.ToPartrec.Code.zero_eval
|
[
{
"state_after": "no goals",
"state_before": "v : List ℕ\n⊢ eval zero v = pure [0]",
"tactic": "simp [zero]"
}
] |
[
204,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.range_top_iff_surjective
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_2\nG' : Type ?u.475486\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.475495\ninst✝³ : AddGroup A\nN✝ : Type ?u.475501\nP : Type ?u.475504\ninst✝² : Group N✝\ninst✝¹ : Group P\nK : Subgroup G\nN : Type u_1\ninst✝ : Group N\nf : G →* N\n⊢ ↑(range f) = ↑⊤ ↔ Set.range ↑f = Set.univ",
"tactic": "rw [coe_range, coe_top]"
}
] |
[
2658,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2656,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_congr
|
[] |
[
814,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.Nontrivial.einfsep_exists_of_finite
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\n⊢ ∃ x x_1 y x_2 _hxy, einfsep s = edist x y",
"tactic": "classical\n cases nonempty_fintype s\n simp_rw [einfsep_of_fintype]\n rcases@Finset.exists_mem_eq_inf _ _ _ _ s.offDiag.toFinset (by simpa) (uncurry edist) with\n ⟨w, hxy, hed⟩\n simp_rw [mem_toFinset] at hxy\n refine' ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x x_1 y x_2 _hxy, einfsep s = edist x y",
"state_before": "α : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\n⊢ ∃ x x_1 y x_2 _hxy, einfsep s = edist x y",
"tactic": "cases nonempty_fintype s"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x h y h h, Finset.inf (toFinset (offDiag s)) (uncurry edist) = edist x y",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x x_1 y x_2 _hxy, einfsep s = edist x y",
"tactic": "simp_rw [einfsep_of_fintype]"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhxy : w ∈ toFinset (offDiag s)\nhed : Finset.inf (toFinset (offDiag s)) (uncurry edist) = uncurry edist w\n⊢ ∃ x h y h h, Finset.inf (toFinset (offDiag s)) (uncurry edist) = edist x y",
"state_before": "case intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ ∃ x h y h h, Finset.inf (toFinset (offDiag s)) (uncurry edist) = edist x y",
"tactic": "rcases@Finset.exists_mem_eq_inf _ _ _ _ s.offDiag.toFinset (by simpa) (uncurry edist) with\n ⟨w, hxy, hed⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhed : Finset.inf (toFinset (offDiag s)) (uncurry edist) = uncurry edist w\nhxy : w ∈ offDiag s\n⊢ ∃ x h y h h, Finset.inf (toFinset (offDiag s)) (uncurry edist) = edist x y",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhxy : w ∈ toFinset (offDiag s)\nhed : Finset.inf (toFinset (offDiag s)) (uncurry edist) = uncurry edist w\n⊢ ∃ x h y h h, Finset.inf (toFinset (offDiag s)) (uncurry edist) = edist x y",
"tactic": "simp_rw [mem_toFinset] at hxy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\nw : α × α\nhed : Finset.inf (toFinset (offDiag s)) (uncurry edist) = uncurry edist w\nhxy : w ∈ offDiag s\n⊢ ∃ x h y h h, Finset.inf (toFinset (offDiag s)) (uncurry edist) = edist x y",
"tactic": "refine' ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.36701\ninst✝¹ : EDist α\nx y : α\ns t : Set α\ninst✝ : Finite ↑s\nhs : Set.Nontrivial s\nval✝ : Fintype ↑s\n⊢ Finset.Nonempty (toFinset (offDiag s))",
"tactic": "simpa"
}
] |
[
202,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean
|
RingSubgroupsBasis.of_comm
|
[
{
"state_after": "A✝ : Type ?u.910\nι✝ : Type ?u.913\ninst✝¹ : Ring A✝\nA : Type u_1\nι : Type u_2\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\nx : A\ni : ι\n⊢ ∃ j, ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)",
"state_before": "A✝ : Type ?u.910\nι✝ : Type ?u.913\ninst✝¹ : Ring A✝\nA : Type u_1\nι : Type u_2\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\n⊢ ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)",
"tactic": "intro x i"
},
{
"state_after": "case intro\nA✝ : Type ?u.910\nι✝ : Type ?u.913\ninst✝¹ : Ring A✝\nA : Type u_1\nι : Type u_2\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\nx : A\ni j : ι\nhj : ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\n⊢ ∃ j, ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)",
"state_before": "A✝ : Type ?u.910\nι✝ : Type ?u.913\ninst✝¹ : Ring A✝\nA : Type u_1\nι : Type u_2\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\nx : A\ni : ι\n⊢ ∃ j, ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)",
"tactic": "cases' leftMul x i with j hj"
},
{
"state_after": "case intro\nA✝ : Type ?u.910\nι✝ : Type ?u.913\ninst✝¹ : Ring A✝\nA : Type u_1\nι : Type u_2\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\nx : A\ni j : ι\nhj : ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\n⊢ ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)",
"state_before": "case intro\nA✝ : Type ?u.910\nι✝ : Type ?u.913\ninst✝¹ : Ring A✝\nA : Type u_1\nι : Type u_2\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\nx : A\ni j : ι\nhj : ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\n⊢ ∃ j, ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)",
"tactic": "use j"
},
{
"state_after": "no goals",
"state_before": "case intro\nA✝ : Type ?u.910\nι✝ : Type ?u.913\ninst✝¹ : Ring A✝\nA : Type u_1\nι : Type u_2\ninst✝ : CommRing A\nB : ι → AddSubgroup A\ninter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j\nmul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)\nleftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\nx : A\ni j : ι\nhj : ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i)\n⊢ ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)",
"tactic": "simpa [mul_comm] using hj"
}
] |
[
69,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Iso.cancel_iso_inv_right
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf f' : X ⟶ Y\ng : Z ≅ Y\n⊢ f ≫ g.inv = f' ≫ g.inv ↔ f = f'",
"tactic": "simp only [cancel_mono]"
}
] |
[
556,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infiniteNeg_mul_of_infinitePos_not_infinitesimal_neg
|
[
{
"state_after": "x y : ℝ*\n⊢ InfinitePos x → ¬Infinitesimal (-y) → 0 < -y → InfinitePos (x * -y)",
"state_before": "x y : ℝ*\n⊢ InfinitePos x → ¬Infinitesimal y → y < 0 → InfiniteNeg (x * y)",
"tactic": "rw [← infinitePos_neg, ← neg_pos, neg_mul_eq_mul_neg, ← infinitesimal_neg]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ*\n⊢ InfinitePos x → ¬Infinitesimal (-y) → 0 < -y → InfinitePos (x * -y)",
"tactic": "exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos"
}
] |
[
843,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
840,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
LinearMap.mapMatrix_id
|
[] |
[
1473,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1472,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.iSupLift_mk
|
[
{
"state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → { x // x ∈ K i } →ₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = AlgHom.comp (f j) (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : ι\nx : { x // x ∈ K i }\nhx : ↑x ∈ T\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)\n (_ :\n ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj })\n ↑T (_ : ↑T ⊆ ⋃ (i : ι), ↑(K i)) { val := ↑x, property := hx } =\n ↑(f i) x",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → { x // x ∈ K i } →ₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = AlgHom.comp (f j) (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : ι\nx : { x // x ∈ K i }\nhx : ↑x ∈ T\n⊢ ↑(iSupLift K dir f hf T hT) { val := ↑x, property := hx } = ↑(f i) x",
"tactic": "dsimp [iSupLift, inclusion]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → { x // x ∈ K i } →ₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = AlgHom.comp (f j) (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : ι\nx : { x // x ∈ K i }\nhx : ↑x ∈ T\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)\n (_ :\n ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj })\n ↑T (_ : ↑T ⊆ ⋃ (i : ι), ↑(K i)) { val := ↑x, property := hx } =\n ↑(f i) x",
"tactic": "rw [Set.iUnionLift_mk]"
}
] |
[
1226,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1223,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_left_inj
|
[] |
[
537,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/CategoryTheory/Category/Basic.lean
|
CategoryTheory.epi_of_epi_fac
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : Epi (f ≫ g)\n⊢ Epi g",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh : X ⟶ Z\ninst✝ : Epi h\nw : f ≫ g = h\n⊢ Epi g",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nX✝ Y✝ Z✝ X Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : Epi (f ≫ g)\n⊢ Epi g",
"tactic": "exact epi_of_epi f g"
}
] |
[
347,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.inv_image_trans_target
|
[] |
[
739,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
738,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_comp'
|
[] |
[
788,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
Algebra.commutes
|
[] |
[
368,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
le_one_of_mul_le_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.17091\ninst✝² : MulOneClass α\ninst✝¹ : LE α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b : α\nh : a * b ≤ b\n⊢ a * ?m.17465 h ≤ 1 * ?m.17465 h",
"tactic": "simpa only [one_mul]"
}
] |
[
394,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
|
MvQPF.recF_eq
|
[
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n⊢ g\n (abs\n { fst := a,\n snd :=\n splitFun f' fun i =>\n MvPFunctor.wRec (P F) (fun a f' _f rec => g (abs { fst := a, snd := splitFun f' rec })) (f i) }) =\n g\n (abs\n { fst := a,\n snd :=\n splitFun f' ((MvPFunctor.wRec (P F) fun a f' _f rec => g (abs { fst := a, snd := splitFun f' rec })) ∘ f) })",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n⊢ recF g (MvPFunctor.wMk (P F) a f' f) = g (abs { fst := a, snd := splitFun f' (recF g ∘ f) })",
"tactic": "rw [recF, MvPFunctor.wRec_eq]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n⊢ g\n (abs\n { fst := a,\n snd :=\n splitFun f' fun i =>\n MvPFunctor.wRec (P F) (fun a f' _f rec => g (abs { fst := a, snd := splitFun f' rec })) (f i) }) =\n g\n (abs\n { fst := a,\n snd :=\n splitFun f' ((MvPFunctor.wRec (P F) fun a f' _f rec => g (abs { fst := a, snd := splitFun f' rec })) ∘ f) })",
"tactic": "rfl"
}
] |
[
70,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin_finset_isCompactElement
|
[
{
"state_after": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS✝ : Set E\nα : E\nS : Finset E\nkey : adjoin F ↑S = ⨆ (x : E) (_ : x ∈ S), F⟮x⟯\n⊢ IsCompactElement (adjoin F ↑S)",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS✝ : Set E\nα : E\nS : Finset E\n⊢ IsCompactElement (adjoin F ↑S)",
"tactic": "have key : adjoin F ↑S = ⨆ x ∈ S, F⟮x⟯ := by\nrefine' le_antisymm (adjoin_le_iff.mpr fun x hx => SetLike.mem_coe.mpr\n (adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_iff.2 (fun E1 hE1 => hE1 hx)))))\n (iSup_le fun x => iSup_le fun hx => adjoin_simple_le_iff.mpr (subset_adjoin F S hx))"
},
{
"state_after": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS✝ : Set E\nα : E\nS : Finset E\nkey : adjoin F ↑S = ⨆ (x : E) (_ : x ∈ S), F⟮x⟯\n⊢ IsCompactElement (Finset.sup S fun x => F⟮x⟯)",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS✝ : Set E\nα : E\nS : Finset E\nkey : adjoin F ↑S = ⨆ (x : E) (_ : x ∈ S), F⟮x⟯\n⊢ IsCompactElement (adjoin F ↑S)",
"tactic": "rw [key, ← Finset.sup_eq_iSup]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS✝ : Set E\nα : E\nS : Finset E\nkey : adjoin F ↑S = ⨆ (x : E) (_ : x ∈ S), F⟮x⟯\n⊢ IsCompactElement (Finset.sup S fun x => F⟮x⟯)",
"tactic": "exact finset_sup_compact_of_compact S fun x _ => adjoin_simple_isCompactElement x"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS✝ : Set E\nα : E\nS : Finset E\n⊢ adjoin F ↑S = ⨆ (x : E) (_ : x ∈ S), F⟮x⟯",
"tactic": "refine' le_antisymm (adjoin_le_iff.mpr fun x hx => SetLike.mem_coe.mpr\n (adjoin_simple_le_iff.mp (le_iSup_of_le x (le_iSup_iff.2 (fun E1 hE1 => hE1 hx)))))\n (iSup_le fun x => iSup_le fun hx => adjoin_simple_le_iff.mpr (subset_adjoin F S hx))"
}
] |
[
621,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.SignedMeasure.toJordanDecomposition_zero
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.74138\ninst✝ : MeasurableSpace α\n⊢ toSignedMeasure (toJordanDecomposition 0) = toSignedMeasure 0",
"state_before": "α : Type u_1\nβ : Type ?u.74138\ninst✝ : MeasurableSpace α\n⊢ toJordanDecomposition 0 = 0",
"tactic": "apply toSignedMeasure_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.74138\ninst✝ : MeasurableSpace α\n⊢ toSignedMeasure (toJordanDecomposition 0) = toSignedMeasure 0",
"tactic": "simp [toSignedMeasure_zero]"
}
] |
[
457,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Comp.lean
|
DifferentiableOn.iterate
|
[] |
[
212,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
11
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
NormedAddCommGroup.cauchy_series_of_le_geometric''
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\n⊢ CauchySeq fun n => ∑ k in Finset.range (n + 1), u k",
"state_before": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\n⊢ CauchySeq fun n => ∑ k in Finset.range (n + 1), u k",
"tactic": "set v : ℕ → α := fun n ↦ if n < N then 0 else u n"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\n⊢ CauchySeq fun n => ∑ k in Finset.range (n + 1), u k",
"state_before": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\n⊢ CauchySeq fun n => ∑ k in Finset.range (n + 1), u k",
"tactic": "have hC : 0 ≤ C :=\n (zero_le_mul_right <| pow_pos hr₀ N).mp ((norm_nonneg _).trans <| h N <| le_refl N)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\n⊢ CauchySeq fun n => ∑ k in Finset.range (n + 1), u k",
"state_before": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\n⊢ CauchySeq fun n => ∑ k in Finset.range (n + 1), u k",
"tactic": "have : ∀ n ≥ N, u n = v n := by\n intro n hn\n simp [hn, if_neg (not_lt.mpr hn)]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\n⊢ ℝ\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\n⊢ ∀ (n : ℕ), ‖v n‖ ≤ ?refine'_1 * r ^ n",
"state_before": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\n⊢ CauchySeq fun n => ∑ k in Finset.range (n + 1), u k",
"tactic": "refine'\n cauchySeq_sum_of_eventually_eq this (NormedAddCommGroup.cauchy_series_of_le_geometric' hr₁ _)"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\n⊢ ‖v n‖ ≤ C * r ^ n",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\n⊢ ∀ (n : ℕ), ‖v n‖ ≤ C * r ^ n",
"tactic": "intro n"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\n⊢ ‖if n < N then 0 else u n‖ ≤ C * r ^ n",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\n⊢ ‖v n‖ ≤ C * r ^ n",
"tactic": "simp only"
},
{
"state_after": "case refine'_2.inl\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : n < N\n⊢ ‖0‖ ≤ C * r ^ n\n\ncase refine'_2.inr\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : ¬n < N\n⊢ ‖u n‖ ≤ C * r ^ n",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\n⊢ ‖if n < N then 0 else u n‖ ≤ C * r ^ n",
"tactic": "split_ifs with H"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nn : ℕ\nhn : n ≥ N\n⊢ u n = v n",
"state_before": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\n⊢ ∀ (n : ℕ), n ≥ N → u n = v n",
"tactic": "intro n hn"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nn : ℕ\nhn : n ≥ N\n⊢ u n = v n",
"tactic": "simp [hn, if_neg (not_lt.mpr hn)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\n⊢ ℝ",
"tactic": "exact C"
},
{
"state_after": "case refine'_2.inl\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : n < N\n⊢ 0 ≤ C * r ^ n",
"state_before": "case refine'_2.inl\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : n < N\n⊢ ‖0‖ ≤ C * r ^ n",
"tactic": "rw [norm_zero]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inl\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : n < N\n⊢ 0 ≤ C * r ^ n",
"tactic": "exact mul_nonneg hC (pow_nonneg hr₀.le _)"
},
{
"state_after": "case refine'_2.inr\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : N ≤ n\n⊢ ‖u n‖ ≤ C * r ^ n",
"state_before": "case refine'_2.inr\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : ¬n < N\n⊢ ‖u n‖ ≤ C * r ^ n",
"tactic": "push_neg at H"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inr\nα : Type u_1\nβ : Type ?u.1175572\nι : Type ?u.1175575\ninst✝ : SeminormedAddCommGroup α\nr✝ C✝ : ℝ\nf : ℕ → α\nC : ℝ\nu : ℕ → α\nN : ℕ\nr : ℝ\nhr₀ : 0 < r\nhr₁ : r < 1\nh : ∀ (n : ℕ), n ≥ N → ‖u n‖ ≤ C * r ^ n\nv : ℕ → α := fun n => if n < N then 0 else u n\nhC : 0 ≤ C\nthis : ∀ (n : ℕ), n ≥ N → u n = v n\nn : ℕ\nH : N ≤ n\n⊢ ‖u n‖ ≤ C * r ^ n",
"tactic": "exact h _ H"
}
] |
[
448,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.map_eq_of_subset
|
[] |
[
279,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
Real.hasStrictDerivAt_log
|
[
{
"state_after": "case inl\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\n⊢ HasStrictDerivAt log x⁻¹ x\n\ncase inr\nx : ℝ\nhx✝ : x ≠ 0\nhx : 0 < x\n⊢ HasStrictDerivAt log x⁻¹ x",
"state_before": "x : ℝ\nhx : x ≠ 0\n⊢ HasStrictDerivAt log x⁻¹ x",
"tactic": "cases' hx.lt_or_lt with hx hx"
},
{
"state_after": "case h.e'_6\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\n⊢ log = log ∘ Neg.neg\n\ncase h.e'_7\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\n⊢ x⁻¹ = (-x)⁻¹ * -1",
"state_before": "case inl\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\n⊢ HasStrictDerivAt log x⁻¹ x",
"tactic": "convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1"
},
{
"state_after": "case h.e'_6.h\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\ny : ℝ\n⊢ log y = (log ∘ Neg.neg) y",
"state_before": "case h.e'_6\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\n⊢ log = log ∘ Neg.neg",
"tactic": "ext y"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6.h\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\ny : ℝ\n⊢ log y = (log ∘ Neg.neg) y",
"tactic": "exact (log_neg_eq_log y).symm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\nx : ℝ\nhx✝ : x ≠ 0\nhx : x < 0\n⊢ x⁻¹ = (-x)⁻¹ * -1",
"tactic": "field_simp [hx.ne]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nhx✝ : x ≠ 0\nhx : 0 < x\n⊢ HasStrictDerivAt log x⁻¹ x",
"tactic": "exact hasStrictDerivAt_log_of_pos hx"
}
] |
[
50,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.uniformContinuous_equiv
|
[] |
[
452,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Algebra/Invertible.lean
|
mul_invOf_self'
|
[] |
[
112,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.toWithTop_zero
|
[] |
[
565,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
562,
1
] |
src/lean/Init/SimpLemmas.lean
|
Bool.and_false
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ (b && false) = false",
"tactic": "cases b <;> rfl"
}
] |
[
110,
87
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
110,
9
] |
Mathlib/Algebra/DirectSum/Ring.lean
|
DirectSum.one_mul
|
[
{
"state_after": "ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\n⊢ ↑(mulHom A) One.one = AddMonoidHom.id (⨁ (i : ι), A i)",
"state_before": "ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\n⊢ 1 * x = x",
"tactic": "suffices mulHom A One.one = AddMonoidHom.id (⨁ i, A i) from FunLike.congr_fun this x"
},
{
"state_after": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\n⊢ ∀ (i : ι) (y : A i),\n ↑(↑(mulHom A) One.one) (↑(of (fun i => A i) i) y) = ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) y)",
"state_before": "ι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\n⊢ ↑(mulHom A) One.one = AddMonoidHom.id (⨁ (i : ι), A i)",
"tactic": "apply addHom_ext"
},
{
"state_after": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ↑(↑(mulHom A) One.one) (↑(of (fun i => A i) i) xi) = ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) xi)",
"state_before": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\n⊢ ∀ (i : ι) (y : A i),\n ↑(↑(mulHom A) One.one) (↑(of (fun i => A i) i) y) = ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) y)",
"tactic": "intro i xi"
},
{
"state_after": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ↑(↑(mulHom A) (↑(of (fun i => A i) 0) GradedMonoid.GOne.one)) (↑(of (fun i => A i) i) xi) =\n ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) xi)",
"state_before": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ↑(↑(mulHom A) One.one) (↑(of (fun i => A i) i) xi) = ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) xi)",
"tactic": "simp only [One.one]"
},
{
"state_after": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ↑(of A (0 + i)) (GradedMonoid.GMul.mul GradedMonoid.GOne.one xi) =\n ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) xi)",
"state_before": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ↑(↑(mulHom A) (↑(of (fun i => A i) 0) GradedMonoid.GOne.one)) (↑(of (fun i => A i) i) xi) =\n ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) xi)",
"tactic": "rw [mulHom_of_of]"
},
{
"state_after": "no goals",
"state_before": "case H\nι : Type u_1\ninst✝³ : DecidableEq ι\nA : ι → Type u_2\ninst✝² : (i : ι) → AddCommMonoid (A i)\ninst✝¹ : AddMonoid ι\ninst✝ : GSemiring A\nx : ⨁ (i : ι), A i\ni : ι\nxi : A i\n⊢ ↑(of A (0 + i)) (GradedMonoid.GMul.mul GradedMonoid.GOne.one xi) =\n ↑(AddMonoidHom.id (⨁ (i : ι), A i)) (↑(of (fun i => A i) i) xi)",
"tactic": "exact of_eq_of_gradedMonoid_eq (one_mul <| GradedMonoid.mk i xi)"
}
] |
[
234,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
16
] |
Mathlib/Combinatorics/Quiver/Cast.lean
|
Quiver.Path.cast_nil
|
[
{
"state_after": "U : Type u_1\ninst✝ : Quiver U\nu' : U\n⊢ cast (_ : u' = u') (_ : u' = u') nil = nil",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu u' : U\nhu : u = u'\n⊢ cast hu hu nil = nil",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu' : U\n⊢ cast (_ : u' = u') (_ : u' = u') nil = nil",
"tactic": "rfl"
}
] |
[
112,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/W/Basic.lean
|
WType.ofSigma_toSigma
|
[] |
[
65,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.min_erase_ne_self
|
[
{
"state_after": "case h.e'_2.h\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝ : WithTop α = WithBot αᵒᵈ\n⊢ Finset.min (erase s x) = Finset.max (erase (map (Equiv.toEmbedding toDual) s) (↑toDual x))",
"state_before": "F : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\n⊢ Finset.min (erase s x) ≠ ↑x",
"tactic": "convert @max_erase_ne_self αᵒᵈ _ (toDual x) (s.map toDual.toEmbedding) using 1"
},
{
"state_after": "case h.e'_2.h.h\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝ : WithTop α = WithBot αᵒᵈ\n⊢ erase s x = erase (map (Equiv.toEmbedding toDual) s) (↑toDual x)",
"state_before": "case h.e'_2.h\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝ : WithTop α = WithBot αᵒᵈ\n⊢ Finset.min (erase s x) = Finset.max (erase (map (Equiv.toEmbedding toDual) s) (↑toDual x))",
"tactic": "apply congr_arg"
},
{
"state_after": "case h.e'_2.h.h.h.e'_3.h\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝¹ : WithTop α = WithBot αᵒᵈ\ne_1✝ : α = αᵒᵈ\n⊢ s = map (Equiv.toEmbedding toDual) s",
"state_before": "case h.e'_2.h.h\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝ : WithTop α = WithBot αᵒᵈ\n⊢ erase s x = erase (map (Equiv.toEmbedding toDual) s) (↑toDual x)",
"tactic": "congr!"
},
{
"state_after": "case h.e'_2.h.h.h.e'_3.h.a\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝¹ : WithTop α = WithBot αᵒᵈ\ne_1✝ : α = αᵒᵈ\na✝ : α\n⊢ a✝ ∈ s ↔ a✝ ∈ map (Equiv.toEmbedding toDual) s",
"state_before": "case h.e'_2.h.h.h.e'_3.h\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝¹ : WithTop α = WithBot αᵒᵈ\ne_1✝ : α = αᵒᵈ\n⊢ s = map (Equiv.toEmbedding toDual) s",
"tactic": "ext"
},
{
"state_after": "case h.e'_2.h.h.h.e'_3.h.a\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝¹ : WithTop α = WithBot αᵒᵈ\ne_1✝ : α = αᵒᵈ\na✝ : α\n⊢ a✝ ∈ s ↔ ↑toDual.symm a✝ ∈ s",
"state_before": "case h.e'_2.h.h.h.e'_3.h.a\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝¹ : WithTop α = WithBot αᵒᵈ\ne_1✝ : α = αᵒᵈ\na✝ : α\n⊢ a✝ ∈ s ↔ a✝ ∈ map (Equiv.toEmbedding toDual) s",
"tactic": "simp only [mem_map_equiv]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.h.h.e'_3.h.a\nF : Type ?u.373369\nα : Type u_1\nβ : Type ?u.373375\nγ : Type ?u.373378\nι : Type ?u.373381\nκ : Type ?u.373384\ninst✝ : LinearOrder α\ns✝ : Finset α\nH : Finset.Nonempty s✝\nx : α\ns : Finset α\ne_1✝¹ : WithTop α = WithBot αᵒᵈ\ne_1✝ : α = αᵒᵈ\na✝ : α\n⊢ a✝ ∈ s ↔ ↑toDual.symm a✝ ∈ s",
"tactic": "exact Iff.rfl"
}
] |
[
1564,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1559,
1
] |
Mathlib/RingTheory/Ideal/Over.lean
|
Ideal.Quotient.mk_smul_mk_quotient_map_quotient
|
[] |
[
176,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Analysis/SpecialFunctions/Polynomials.lean
|
Polynomial.div_tendsto_atTop_of_degree_gt
|
[] |
[
195,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.roots_X_sub_C
|
[
{
"state_after": "case a\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nr s : R\n⊢ count s (roots (X - ↑C r)) = count s {r}",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nr : R\n⊢ roots (X - ↑C r) = {r}",
"tactic": "ext s"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nr s : R\n⊢ count s (roots (X - ↑C r)) = count s {r}",
"tactic": "rw [count_roots, rootMultiplicity_X_sub_C, count_singleton]"
}
] |
[
650,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
648,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.ringHom_surjective
|
[] |
[
1176,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1175,
1
] |
Mathlib/Data/List/Basic.lean
|
List.length_modifyNthTail
|
[] |
[
1549,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1545,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.ToPartrec.Code.id_eval
|
[
{
"state_after": "no goals",
"state_before": "v : List ℕ\n⊢ eval id v = pure v",
"tactic": "simp [id]"
}
] |
[
186,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.neg_snd
|
[] |
[
277,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Data/Bool/Count.lean
|
List.Chain'.count_true_le_count_false_add_one
|
[] |
[
100,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Algebra/Associated.lean
|
prime_pow_succ_dvd_mul
|
[
{
"state_after": "α✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni : ℕ\nhxy : p ^ (i + 1) ∣ x * y\n⊢ ¬p ∣ y → p ^ (i + 1) ∣ x",
"state_before": "α✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni : ℕ\nhxy : p ^ (i + 1) ∣ x * y\n⊢ p ^ (i + 1) ∣ x ∨ p ∣ y",
"tactic": "rw [or_iff_not_imp_right]"
},
{
"state_after": "α✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni : ℕ\nhxy : p ^ (i + 1) ∣ x * y\nhy : ¬p ∣ y\n⊢ p ^ (i + 1) ∣ x",
"state_before": "α✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni : ℕ\nhxy : p ^ (i + 1) ∣ x * y\n⊢ ¬p ∣ y → p ^ (i + 1) ∣ x",
"tactic": "intro hy"
},
{
"state_after": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ^ (Nat.zero + 1) ∣ x * y\n⊢ p ^ (Nat.zero + 1) ∣ x\n\ncase succ\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x✝ * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx : α\nhxy : p ^ (Nat.succ i + 1) ∣ x * y\n⊢ p ^ (Nat.succ i + 1) ∣ x",
"state_before": "α✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni : ℕ\nhxy : p ^ (i + 1) ∣ x * y\nhy : ¬p ∣ y\n⊢ p ^ (i + 1) ∣ x",
"tactic": "induction' i with i ih generalizing x"
},
{
"state_after": "case succ\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x✝ * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx : α\nhxy : p * p ^ (i + 1) ∣ x * y\n⊢ p * p ^ (i + 1) ∣ x",
"state_before": "case succ\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x✝ * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx : α\nhxy : p ^ (Nat.succ i + 1) ∣ x * y\n⊢ p ^ (Nat.succ i + 1) ∣ x",
"tactic": "rw [pow_succ] at hxy⊢"
},
{
"state_after": "case succ.intro\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx' : α\nhxy : p * p ^ (i + 1) ∣ p * x' * y\n⊢ p * p ^ (i + 1) ∣ p * x'",
"state_before": "case succ\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x✝ * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx : α\nhxy : p * p ^ (i + 1) ∣ x * y\n⊢ p * p ^ (i + 1) ∣ x",
"tactic": "obtain ⟨x', rfl⟩ := (h.dvd_or_dvd (dvd_of_mul_right_dvd hxy)).resolve_right hy"
},
{
"state_after": "case succ.intro\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx' : α\nhxy : p * p ^ (i + 1) ∣ p * (x' * y)\n⊢ p * p ^ (i + 1) ∣ p * x'",
"state_before": "case succ.intro\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx' : α\nhxy : p * p ^ (i + 1) ∣ p * x' * y\n⊢ p * p ^ (i + 1) ∣ p * x'",
"tactic": "rw [mul_assoc] at hxy"
},
{
"state_after": "no goals",
"state_before": "case succ.intro\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x y : α\nh : Prime p\ni✝ : ℕ\nhxy✝ : p ^ (i✝ + 1) ∣ x * y\nhy : ¬p ∣ y\ni : ℕ\nih : ∀ {x : α}, p ^ (i + 1) ∣ x * y → p ^ (i + 1) ∣ x\nx' : α\nhxy : p * p ^ (i + 1) ∣ p * (x' * y)\n⊢ p * p ^ (i + 1) ∣ p * x'",
"tactic": "exact mul_dvd_mul_left p (ih ((mul_dvd_mul_iff_left h.ne_zero).mp hxy))"
},
{
"state_after": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ^ (Nat.zero + 1) ∣ x * y\n⊢ p ^ (Nat.zero + 1) ∣ x",
"state_before": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ^ (Nat.zero + 1) ∣ x * y\n⊢ p ^ (Nat.zero + 1) ∣ x",
"tactic": "simp only [zero_add, pow_one] at *"
},
{
"state_after": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ^ (Nat.zero + 1) ∣ x * y\n⊢ p ∣ x",
"state_before": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ^ (Nat.zero + 1) ∣ x * y\n⊢ p ^ (Nat.zero + 1) ∣ x",
"tactic": "rw [pow_one]"
},
{
"state_after": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ∣ x * y\n⊢ p ∣ x",
"state_before": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ^ (Nat.zero + 1) ∣ x * y\n⊢ p ∣ x",
"tactic": "rw [pow_one] at hxy"
},
{
"state_after": "no goals",
"state_before": "case zero\nα✝ : Type ?u.61415\nβ : Type ?u.61418\nγ : Type ?u.61421\nδ : Type ?u.61424\nα : Type u_1\ninst✝ : CancelCommMonoidWithZero α\np x✝ y : α\nh : Prime p\ni : ℕ\nhxy✝ : p ^ (i + 1) ∣ x✝ * y\nhy : ¬p ∣ y\nx : α\nhxy : p ∣ x * y\n⊢ p ∣ x",
"tactic": "exact (h.dvd_or_dvd hxy).resolve_right hy"
}
] |
[
158,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Algebra/CharP/MixedCharZero.lean
|
isEmpty_algebraRat_iff_mixedCharZero
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ ¬IsEmpty (Algebra ℚ R) ↔ ¬∃ p, p > 0 ∧ MixedCharZero R p",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ IsEmpty (Algebra ℚ R) ↔ ∃ p, p > 0 ∧ MixedCharZero R p",
"tactic": "rw [← not_iff_not]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ ¬IsEmpty (Algebra ℚ R) ↔ ∀ (p : ℕ), p > 0 → ¬MixedCharZero R p",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ ¬IsEmpty (Algebra ℚ R) ↔ ¬∃ p, p > 0 ∧ MixedCharZero R p",
"tactic": "push_neg"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ Nonempty (Algebra ℚ R) ↔ ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ ¬IsEmpty (Algebra ℚ R) ↔ ∀ (p : ℕ), p > 0 → ¬MixedCharZero R p",
"tactic": "rw [not_isEmpty_iff, ← EqualCharZero.iff_not_mixedCharZero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\n⊢ Nonempty (Algebra ℚ R) ↔ ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)",
"tactic": "apply EqualCharZero.nonempty_algebraRat_iff"
}
] |
[
310,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.iSup_biUnion
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.455051\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\nι : Type ?u.455063\nκ : Type ?u.455066\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\ns : Finset γ\nt : γ → Finset α\nf : α → β\n⊢ (⨆ (y : α) (_ : y ∈ Finset.biUnion s t), f y) = ⨆ (x : γ) (_ : x ∈ s) (y : α) (_ : y ∈ t x), f y",
"tactic": "simp [@iSup_comm _ α, iSup_and]"
}
] |
[
1988,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1987,
1
] |
Mathlib/CategoryTheory/Simple.lean
|
CategoryTheory.simple_of_isSimpleOrder_subobject
|
[
{
"state_after": "case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\n⊢ ∀ {Y : C} (f : Y ⟶ X) [inst : Mono f], IsIso f ↔ f ≠ 0",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\n⊢ Simple X",
"tactic": "constructor"
},
{
"state_after": "case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ IsIso f ↔ f ≠ 0",
"state_before": "case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\n⊢ ∀ {Y : C} (f : Y ⟶ X) [inst : Mono f], IsIso f ↔ f ≠ 0",
"tactic": "intros Y f hf"
},
{
"state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ IsIso f → f ≠ 0\n\ncase mono_isIso_iff_nonzero.mpr\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ f ≠ 0 → IsIso f",
"state_before": "case mono_isIso_iff_nonzero\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ IsIso f ↔ f ≠ 0",
"tactic": "constructor"
},
{
"state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : IsIso f\n⊢ f ≠ 0",
"state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ IsIso f → f ≠ 0",
"tactic": "intro i"
},
{
"state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : mk f = ⊤\n⊢ f ≠ 0",
"state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : IsIso f\n⊢ f ≠ 0",
"tactic": "rw [Subobject.isIso_iff_mk_eq_top] at i"
},
{
"state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : mk f = ⊤\nw : f = 0\n⊢ False",
"state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : mk f = ⊤\n⊢ f ≠ 0",
"tactic": "intro w"
},
{
"state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : mk f = ⊤\nw✝ : f = 0\nw : mk f = ⊥\n⊢ False",
"state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : mk f = ⊤\nw : f = 0\n⊢ False",
"tactic": "rw [← Subobject.mk_eq_bot_iff_zero] at w"
},
{
"state_after": "no goals",
"state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : mk f = ⊤\nw✝ : f = 0\nw : mk f = ⊥\n⊢ False",
"tactic": "exact IsSimpleOrder.bot_ne_top (w.symm.trans i)"
},
{
"state_after": "case mono_isIso_iff_nonzero.mpr\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\n⊢ IsIso f",
"state_before": "case mono_isIso_iff_nonzero.mpr\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\n⊢ f ≠ 0 → IsIso f",
"tactic": "intro i"
},
{
"state_after": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\nh : mk f = ⊥\n⊢ IsIso f\n\ncase mono_isIso_iff_nonzero.mpr.inr\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\nh : mk f = ⊤\n⊢ IsIso f",
"state_before": "case mono_isIso_iff_nonzero.mpr\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\n⊢ IsIso f",
"tactic": "rcases IsSimpleOrder.eq_bot_or_eq_top (Subobject.mk f) with (h | h)"
},
{
"state_after": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\nh✝ : mk f = ⊥\nh : f = 0\n⊢ IsIso f",
"state_before": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\nh : mk f = ⊥\n⊢ IsIso f",
"tactic": "rw [Subobject.mk_eq_bot_iff_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\nh✝ : mk f = ⊥\nh : f = 0\n⊢ IsIso f",
"tactic": "exact False.elim (i h)"
},
{
"state_after": "no goals",
"state_before": "case mono_isIso_iff_nonzero.mpr.inr\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : IsSimpleOrder (Subobject X)\nY : C\nf : Y ⟶ X\nhf : Mono f\ni : f ≠ 0\nh : mk f = ⊤\n⊢ IsIso f",
"tactic": "exact (Subobject.isIso_iff_mk_eq_top _).mpr h"
}
] |
[
252,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.closure_eq_image_prod
|
[
{
"state_after": "M : Type u_1\nA : Type ?u.78612\nB : Type ?u.78615\ninst✝ : Monoid M\ns : Set M\n⊢ Set.range ↑(↑FreeMonoid.lift Subtype.val) = Set.range (List.prod ∘ List.map Subtype.val)",
"state_before": "M : Type u_1\nA : Type ?u.78612\nB : Type ?u.78615\ninst✝ : Monoid M\ns : Set M\n⊢ ↑(closure s) = List.prod '' {l | ∀ (x : M), x ∈ l → x ∈ s}",
"tactic": "rw [closure_eq_mrange, coe_mrange, ← Set.range_list_map_coe, ← Set.range_comp]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nA : Type ?u.78612\nB : Type ?u.78615\ninst✝ : Monoid M\ns : Set M\n⊢ Set.range ↑(↑FreeMonoid.lift Subtype.val) = Set.range (List.prod ∘ List.map Subtype.val)",
"tactic": "exact congrArg _ (funext <| FreeMonoid.lift_apply _)"
}
] |
[
366,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
hasSum_subtype_iff_of_support_subset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.26169\nδ : Type ?u.26172\ninst✝¹ : AddCommMonoid α\ninst✝ : TopologicalSpace α\nf g : β → α\na b : α\ns✝ : Finset β\ns : Set β\nhf : support f ⊆ s\n⊢ ∀ (x : β), (¬x ∈ Set.range fun a => ↑a) → f x = 0",
"tactic": "simpa using support_subset_iff'.1 hf"
}
] |
[
154,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
|
CategoryTheory.Functor.mapBiproduct_hom
|
[] |
[
307,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.sub_self
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ a - a = 0",
"tactic": "rw [sub_def, ← Category.comp_id a, ← prod.comp_lift, Category.assoc, diag_σ, comp_zero]"
}
] |
[
363,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean
|
PFunctor.M.casesOn_mk'
|
[] |
[
461,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Analysis/NormedSpace/MStructure.lean
|
IsLprojection.coe_bot
|
[] |
[
253,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.map_one
|
[] |
[
724,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
11
] |
Std/Data/List/Init/Lemmas.lean
|
List.foldl_reverse
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nl : List α\nf : β → α → β\nb : β\n⊢ foldl f b (reverse l) = foldr (fun x y => f y x) b l",
"tactic": "simp [foldl_eq_foldlM, foldr_eq_foldrM]"
}
] |
[
186,
101
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
185,
9
] |
Std/Data/Nat/Lemmas.lean
|
Nat.mul_mod_mul_left
|
[
{
"state_after": "no goals",
"state_before": "z x y : Nat\ny0 : y = 0\n⊢ z * x % (z * y) = z * (x % y)",
"tactic": "rw [y0, Nat.mul_zero, mod_zero, mod_zero]"
},
{
"state_after": "no goals",
"state_before": "z x y : Nat\ny0 : ¬y = 0\nz0 : z = 0\n⊢ z * x % (z * y) = z * (x % y)",
"tactic": "rw [z0, Nat.zero_mul, Nat.zero_mul, Nat.zero_mul, mod_zero]"
},
{
"state_after": "no goals",
"state_before": "z x y : Nat\ny0 : ¬y = 0\nz0 : ¬z = 0\n⊢ z * x % (z * y) = z * (x % y)",
"tactic": "induction x using Nat.strongInductionOn with\n| _ n IH =>\n have y0 : y > 0 := Nat.pos_of_ne_zero y0\n have z0 : z > 0 := Nat.pos_of_ne_zero z0\n cases Nat.lt_or_ge n y with\n | inl yn => rw [mod_eq_of_lt yn, mod_eq_of_lt (Nat.mul_lt_mul_of_pos_left yn z0)]\n | inr yn =>\n rw [mod_eq_sub_mod yn, mod_eq_sub_mod (Nat.mul_le_mul_left z yn),\n ← Nat.mul_sub_left_distrib]\n exact IH _ (sub_lt (Nat.lt_of_lt_of_le y0 yn) y0)"
},
{
"state_after": "case ind\nz y : Nat\ny0✝ : ¬y = 0\nz0 : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\n⊢ z * n % (z * y) = z * (n % y)",
"state_before": "case ind\nz y : Nat\ny0 : ¬y = 0\nz0 : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\n⊢ z * n % (z * y) = z * (n % y)",
"tactic": "have y0 : y > 0 := Nat.pos_of_ne_zero y0"
},
{
"state_after": "case ind\nz y : Nat\ny0✝ : ¬y = 0\nz0✝ : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\nz0 : z > 0\n⊢ z * n % (z * y) = z * (n % y)",
"state_before": "case ind\nz y : Nat\ny0✝ : ¬y = 0\nz0 : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\n⊢ z * n % (z * y) = z * (n % y)",
"tactic": "have z0 : z > 0 := Nat.pos_of_ne_zero z0"
},
{
"state_after": "no goals",
"state_before": "case ind\nz y : Nat\ny0✝ : ¬y = 0\nz0✝ : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\nz0 : z > 0\n⊢ z * n % (z * y) = z * (n % y)",
"tactic": "cases Nat.lt_or_ge n y with\n| inl yn => rw [mod_eq_of_lt yn, mod_eq_of_lt (Nat.mul_lt_mul_of_pos_left yn z0)]\n| inr yn =>\n rw [mod_eq_sub_mod yn, mod_eq_sub_mod (Nat.mul_le_mul_left z yn),\n ← Nat.mul_sub_left_distrib]\n exact IH _ (sub_lt (Nat.lt_of_lt_of_le y0 yn) y0)"
},
{
"state_after": "no goals",
"state_before": "case ind.inl\nz y : Nat\ny0✝ : ¬y = 0\nz0✝ : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\nz0 : z > 0\nyn : n < y\n⊢ z * n % (z * y) = z * (n % y)",
"tactic": "rw [mod_eq_of_lt yn, mod_eq_of_lt (Nat.mul_lt_mul_of_pos_left yn z0)]"
},
{
"state_after": "case ind.inr\nz y : Nat\ny0✝ : ¬y = 0\nz0✝ : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\nz0 : z > 0\nyn : n ≥ y\n⊢ z * (n - y) % (z * y) = z * ((n - y) % y)",
"state_before": "case ind.inr\nz y : Nat\ny0✝ : ¬y = 0\nz0✝ : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\nz0 : z > 0\nyn : n ≥ y\n⊢ z * n % (z * y) = z * (n % y)",
"tactic": "rw [mod_eq_sub_mod yn, mod_eq_sub_mod (Nat.mul_le_mul_left z yn),\n ← Nat.mul_sub_left_distrib]"
},
{
"state_after": "no goals",
"state_before": "case ind.inr\nz y : Nat\ny0✝ : ¬y = 0\nz0✝ : ¬z = 0\nn : Nat\nIH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)\ny0 : y > 0\nz0 : z > 0\nyn : n ≥ y\n⊢ z * (n - y) % (z * y) = z * ((n - y) % y)",
"tactic": "exact IH _ (sub_lt (Nat.lt_of_lt_of_le y0 yn) y0)"
}
] |
[
525,
58
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
510,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fintype.univ_ofSubsingleton
|
[] |
[
584,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Topology/OmegaCompletePartialOrder.lean
|
Scott.isOpen_univ
|
[] |
[
57,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Module.map_eval_injective
|
[
{
"state_after": "case hf\nK : Type u₁\nV : Type u₂\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ Function.Injective ↑(eval K V)",
"state_before": "K : Type u₁\nV : Type u₂\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ Function.Injective (Submodule.map (eval K V))",
"tactic": "apply Submodule.map_injective_of_injective"
},
{
"state_after": "case hf\nK : Type u₁\nV : Type u₂\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ ker (eval K V) = ⊥",
"state_before": "case hf\nK : Type u₁\nV : Type u₂\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ Function.Injective ↑(eval K V)",
"tactic": "rw [← LinearMap.ker_eq_bot]"
},
{
"state_after": "no goals",
"state_before": "case hf\nK : Type u₁\nV : Type u₂\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ ker (eval K V) = ⊥",
"tactic": "apply eval_ker K V"
}
] |
[
551,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Algebra/Star/Unitary.lean
|
unitary.star_eq_inv
|
[] |
[
125,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Sbtw.not_swap_left
|
[] |
[
404,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_empty_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.24579\nβ : Type u_3\nβ' : Type ?u.24585\nγ : Type u_1\nγ' : Type ?u.24591\nδ : Type ?u.24594\nδ' : Type ?u.24597\nε : Type ?u.24600\nε' : Type ?u.24603\nζ : Type ?u.24606\nζ' : Type ?u.24609\nν : Type ?u.24612\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\n⊢ ↑(image₂ f s ∅) = ↑∅",
"tactic": "simp"
}
] |
[
145,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Analysis/Calculus/Deriv/Comp.lean
|
fderivWithin.comp_derivWithin
|
[] |
[
271,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Computability/Reduce.lean
|
equivalence_of_manyOneEquiv
|
[] |
[
180,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.ValidFor.next_stop
|
[
{
"state_after": "no goals",
"state_before": "l m r : List Char\n⊢ Substring.next\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } }\n { byteIdx := utf8Len m } =\n { byteIdx := utf8Len m }",
"tactic": "simp [Substring.next, Pos.add_eq]"
}
] |
[
841,
50
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
840,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.isUnit_iff_eq_bot
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.294633\nγ : Type ?u.294636\nδ : Type ?u.294639\ninst✝ : CommMonoid α\na : Associates α\n⊢ IsUnit a ↔ a = ⊥",
"tactic": "rw [Associates.isUnit_iff_eq_one, bot_eq_one]"
}
] |
[
890,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
889,
1
] |
Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean
|
FiniteDimensional.Submodule.finrank_map_subtype_eq
|
[] |
[
48,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingDist_nonneg
|
[] |
[
56,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.subset_adjoin
|
[] |
[
321,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
OpenSubgroup.coe_comap
|
[] |
[
279,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.signBijAux_mem
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\n⊢ (if x : ↑f { fst := a₁, snd := a₂ }.snd < ↑f { fst := a₁, snd := a₂ }.fst then\n { fst := ↑f { fst := a₁, snd := a₂ }.fst, snd := ↑f { fst := a₁, snd := a₂ }.snd }\n else { fst := ↑f { fst := a₁, snd := a₂ }.snd, snd := ↑f { fst := a₁, snd := a₂ }.fst }) ∈\n finPairsLT n",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\n⊢ signBijAux f { fst := a₁, snd := a₂ } ∈ finPairsLT n",
"tactic": "unfold signBijAux"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nh : ↑f { fst := a₁, snd := a₂ }.snd < ↑f { fst := a₁, snd := a₂ }.fst\n⊢ { fst := ↑f { fst := a₁, snd := a₂ }.fst, snd := ↑f { fst := a₁, snd := a₂ }.snd } ∈ finPairsLT n\n\ncase inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nh : ¬↑f { fst := a₁, snd := a₂ }.snd < ↑f { fst := a₁, snd := a₂ }.fst\n⊢ { fst := ↑f { fst := a₁, snd := a₂ }.snd, snd := ↑f { fst := a₁, snd := a₂ }.fst } ∈ finPairsLT n",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\n⊢ (if x : ↑f { fst := a₁, snd := a₂ }.snd < ↑f { fst := a₁, snd := a₂ }.fst then\n { fst := ↑f { fst := a₁, snd := a₂ }.fst, snd := ↑f { fst := a₁, snd := a₂ }.snd }\n else { fst := ↑f { fst := a₁, snd := a₂ }.snd, snd := ↑f { fst := a₁, snd := a₂ }.fst }) ∈\n finPairsLT n",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nh : ↑f { fst := a₁, snd := a₂ }.snd < ↑f { fst := a₁, snd := a₂ }.fst\n⊢ { fst := ↑f { fst := a₁, snd := a₂ }.fst, snd := ↑f { fst := a₁, snd := a₂ }.snd } ∈ finPairsLT n",
"tactic": "exact mem_finPairsLT.2 h"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nh : ¬↑f { fst := a₁, snd := a₂ }.snd < ↑f { fst := a₁, snd := a₂ }.fst\n⊢ { fst := ↑f { fst := a₁, snd := a₂ }.snd, snd := ↑f { fst := a₁, snd := a₂ }.fst } ∈ finPairsLT n",
"tactic": "exact mem_finPairsLT.2\n ((le_of_not_gt h).lt_of_ne fun h => (mem_finPairsLT.1 ha).ne (f.injective h.symm))"
}
] |
[
369,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.ofLe_injective
|
[] |
[
633,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.eq_zero_or_eq_zero_of_mul_eq_zero
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\n⊢ ¬φ = 0 → ψ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\n⊢ φ = 0 ∨ ψ = 0",
"tactic": "rw [or_iff_not_imp_left]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\n⊢ ψ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\n⊢ ¬φ = 0 → ψ = 0",
"tactic": "intro H"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\n⊢ ψ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\n⊢ ψ = 0",
"tactic": "have ex : ∃ m, coeff R m φ ≠ 0 := by\n contrapose! H\n exact ext H"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\n⊢ ψ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\n⊢ ψ = 0",
"tactic": "let m := Nat.find ex"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\n⊢ ψ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\n⊢ ψ = 0",
"tactic": "have hm₁ : coeff R m φ ≠ 0 := Nat.find_spec ex"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\n⊢ ψ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\n⊢ ψ = 0",
"tactic": "have hm₂ : ∀ k < m, ¬coeff R k φ ≠ 0 := fun k => Nat.find_min ex"
},
{
"state_after": "case h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\n⊢ ↑(coeff R n) ψ = ↑(coeff R n) 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\n⊢ ψ = 0",
"tactic": "ext n"
},
{
"state_after": "case h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\n⊢ ↑(coeff R n) ψ = 0",
"state_before": "case h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\n⊢ ↑(coeff R n) ψ = ↑(coeff R n) 0",
"tactic": "rw [(coeff R n).map_zero]"
},
{
"state_after": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\n⊢ ↑(coeff R n) ψ = 0",
"state_before": "case h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\n⊢ ↑(coeff R n) ψ = 0",
"tactic": "induction' n using Nat.strong_induction_on with n ih"
},
{
"state_after": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ↑(coeff R (m + n)) (φ * ψ) = ↑(coeff R (m + n)) 0\n⊢ ↑(coeff R n) ψ = 0",
"state_before": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\n⊢ ↑(coeff R n) ψ = 0",
"tactic": "replace h := congr_arg (coeff R (m + n)) h"
},
{
"state_after": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ↑(coeff R (m, n).fst) φ * ↑(coeff R (m, n).snd) ψ = 0\n⊢ ↑(coeff R n) ψ = 0\n\ncase h.h.h₀\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\n⊢ ∀ (b : ℕ × ℕ), b ∈ Finset.Nat.antidiagonal (m + n) → b ≠ (m, n) → ↑(coeff R b.fst) φ * ↑(coeff R b.snd) ψ = 0\n\ncase h.h.h₁\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\n⊢ ¬(m, n) ∈ Finset.Nat.antidiagonal (m + n) → ↑(coeff R (m, n).fst) φ * ↑(coeff R (m, n).snd) ψ = 0",
"state_before": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ↑(coeff R (m + n)) (φ * ψ) = ↑(coeff R (m + n)) 0\n⊢ ↑(coeff R n) ψ = 0",
"tactic": "rw [LinearMap.map_zero, coeff_mul, Finset.sum_eq_single (m, n)] at h"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ∀ (m : ℕ), ↑(coeff R m) φ = 0\n⊢ φ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ¬φ = 0\n⊢ ∃ m, ↑(coeff R m) φ ≠ 0",
"tactic": "contrapose! H"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nh : φ * ψ = 0\nH : ∀ (m : ℕ), ↑(coeff R m) φ = 0\n⊢ φ = 0",
"tactic": "exact ext H"
},
{
"state_after": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ↑(coeff R (m, n).fst) φ = 0 ∨ ↑(coeff R (m, n).snd) ψ = 0\n⊢ ↑(coeff R n) ψ = 0",
"state_before": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ↑(coeff R (m, n).fst) φ * ↑(coeff R (m, n).snd) ψ = 0\n⊢ ↑(coeff R n) ψ = 0",
"tactic": "replace h := NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h"
},
{
"state_after": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ¬↑(coeff R (m, n).fst) φ = 0 → ↑(coeff R (m, n).snd) ψ = 0\n⊢ ↑(coeff R n) ψ = 0",
"state_before": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ↑(coeff R (m, n).fst) φ = 0 ∨ ↑(coeff R (m, n).snd) ψ = 0\n⊢ ↑(coeff R n) ψ = 0",
"tactic": "rw [or_iff_not_imp_left] at h"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ¬↑(coeff R (m, n).fst) φ = 0 → ↑(coeff R (m, n).snd) ψ = 0\n⊢ ↑(coeff R n) ψ = 0",
"tactic": "exact h hm₁"
},
{
"state_after": "case h.h.h₀.mk\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "case h.h.h₀\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\n⊢ ∀ (b : ℕ × ℕ), b ∈ Finset.Nat.antidiagonal (m + n) → b ≠ (m, n) → ↑(coeff R b.fst) φ * ↑(coeff R b.snd) ψ = 0",
"tactic": "rintro ⟨i, j⟩ hij hne"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : j < n\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0\n\ncase neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "case h.h.h₀.mk\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "by_cases hj : j < n"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : i < m\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0\n\ncase neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : ¬i < m\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "by_cases hi : i < m"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : ¬i < m\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : ¬i < m\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "rw [Finset.Nat.mem_antidiagonal] at hij"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : ¬i < m\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "push_neg at hi hj"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\n⊢ m < i",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "suffices m < i by\n have : m + n < i + j := add_lt_add_of_lt_of_le this hj\n exfalso\n exact ne_of_lt this hij.symm"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhi : Nat.find ex ≤ i\nhj : n ≤ j\nhne : i ≤ Nat.find ex\n⊢ (i, j) = (Nat.find ex, n)",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\n⊢ m < i",
"tactic": "contrapose! hne"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\nj : ℕ\nhj : n ≤ j\nhij : (Nat.find ex, j).fst + (Nat.find ex, j).snd = m + n\nhi hne : Nat.find ex ≤ Nat.find ex\n⊢ (Nat.find ex, j) = (Nat.find ex, n)",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhi : Nat.find ex ≤ i\nhj : n ≤ j\nhne : i ≤ Nat.find ex\n⊢ (i, j) = (Nat.find ex, n)",
"tactic": "obtain rfl := le_antisymm hi hne"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\nj : ℕ\nhj : n ≤ j\nhij : (Nat.find ex, j).fst + (Nat.find ex, j).snd = m + n\nhi hne : Nat.find ex ≤ Nat.find ex\n⊢ (Nat.find ex, j) = (Nat.find ex, n)",
"tactic": "simpa [Ne.def, Prod.mk.inj_iff] using (add_right_inj m).mp hij"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : j < n\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "rw [ih j hj, MulZeroClass.mul_zero]"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : i < m\nhm₂ : ¬↑(coeff R i) φ ≠ 0\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : i < m\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "specialize hm₂ _ hi"
},
{
"state_after": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : i < m\nhm₂ : ↑(coeff R i) φ = 0\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : i < m\nhm₂ : ¬↑(coeff R i) φ ≠ 0\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "push_neg at hm₂"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j) ∈ Finset.Nat.antidiagonal (m + n)\nhne : (i, j) ≠ (m, n)\nhj : ¬j < n\nhi : i < m\nhm₂ : ↑(coeff R i) φ = 0\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "rw [hm₂, MulZeroClass.zero_mul]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\nthis✝ : m < i\nthis : m + n < i + j\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\nthis : m < i\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "have : m + n < i + j := add_lt_add_of_lt_of_le this hj"
},
{
"state_after": "case h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\nthis✝ : m < i\nthis : m + n < i + j\n⊢ False",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\nthis✝ : m < i\nthis : m + n < i + j\n⊢ ↑(coeff R (i, j).fst) φ * ↑(coeff R (i, j).snd) ψ = 0",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\ni j : ℕ\nhij : (i, j).fst + (i, j).snd = m + n\nhne : (i, j) ≠ (m, n)\nhi : Nat.find ex ≤ i\nhj : n ≤ j\nthis✝ : m < i\nthis : m + n < i + j\n⊢ False",
"tactic": "exact ne_of_lt this hij.symm"
},
{
"state_after": "case h.h.h₁\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\n⊢ ↑(coeff R (Nat.find ex)) φ * ↑(coeff R n) ψ ≠ 0 → (Nat.find ex, n) ∈ Finset.Nat.antidiagonal (Nat.find ex + n)",
"state_before": "case h.h.h₁\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\n⊢ ¬(m, n) ∈ Finset.Nat.antidiagonal (m + n) → ↑(coeff R (m, n).fst) φ * ↑(coeff R (m, n).snd) ψ = 0",
"tactic": "contrapose!"
},
{
"state_after": "case h.h.h₁\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\na✝ : ↑(coeff R (Nat.find ex)) φ * ↑(coeff R n) ψ ≠ 0\n⊢ (Nat.find ex, n) ∈ Finset.Nat.antidiagonal (Nat.find ex + n)",
"state_before": "case h.h.h₁\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\n⊢ ↑(coeff R (Nat.find ex)) φ * ↑(coeff R n) ψ ≠ 0 → (Nat.find ex, n) ∈ Finset.Nat.antidiagonal (Nat.find ex + n)",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case h.h.h₁\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : NoZeroDivisors R\nφ ψ : PowerSeries R\nH : ¬φ = 0\nex : ∃ m, ↑(coeff R m) φ ≠ 0\nm : ℕ := Nat.find ex\nhm₁ : ↑(coeff R m) φ ≠ 0\nhm₂ : ∀ (k : ℕ), k < m → ¬↑(coeff R k) φ ≠ 0\nn : ℕ\nih : ∀ (m : ℕ), m < n → ↑(coeff R m) ψ = 0\nh : ∑ p in Finset.Nat.antidiagonal (m + n), ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = 0\na✝ : ↑(coeff R (Nat.find ex)) φ * ↑(coeff R n) ψ ≠ 0\n⊢ (Nat.find ex, n) ∈ Finset.Nat.antidiagonal (Nat.find ex + n)",
"tactic": "rw [Finset.Nat.mem_antidiagonal]"
}
] |
[
2021,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1985,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
PosMulReflectLT.toPosMulMonoRev
|
[] |
[
1006,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1003,
1
] |
Mathlib/Data/Quot.lean
|
Quotient.out_eq'
|
[] |
[
792,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/Data/PNat/Basic.lean
|
PNat.natPred_inj
|
[] |
[
72,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.pointReflection_symm
|
[] |
[
567,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.sum_left
|
[] |
[
429,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/GroupTheory/CommutingProbability.lean
|
commProb_def'
|
[
{
"state_after": "M : Type ?u.17813\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\n⊢ ↑(Nat.card (ConjClasses G)) * ↑(Nat.card G) / (↑(Nat.card G) * ↑(Nat.card G)) =\n ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"state_before": "M : Type ?u.17813\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\n⊢ commProb G = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"tactic": "rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.17813\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\n⊢ ↑(Nat.card (ConjClasses G)) * ↑(Nat.card G) / (↑(Nat.card G) * ↑(Nat.card G)) =\n ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"tactic": "exact mul_div_mul_right _ _ (Nat.cast_ne_zero.mpr Finite.card_pos.ne')"
}
] |
[
92,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.bit1_im
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.1882781\ninst✝ : IsROrC K\nz : K\n⊢ ↑im (bit1 z) = bit0 (↑im z)",
"tactic": "simp only [bit1, map_add, bit0_im, one_im, add_zero]"
}
] |
[
196,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
iSup_eq_of_forall_le_of_tendsto
|
[] |
[
430,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean
|
CategoryTheory.hasProduct_fin
|
[
{
"state_after": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nf : Fin 0 → C\nthis : HasLimitsOfShape (Discrete (Fin 0)) C :=\n hasLimitsOfShape_of_equivalence (Discrete.equivalence finZeroEquiv'.symm)\n⊢ HasProduct f",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nf : Fin 0 → C\n⊢ HasProduct f",
"tactic": "letI : HasLimitsOfShape (Discrete (Fin 0)) C :=\n hasLimitsOfShape_of_equivalence (Discrete.equivalence.{0} finZeroEquiv'.symm)"
},
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nf : Fin 0 → C\nthis : HasLimitsOfShape (Discrete (Fin 0)) C :=\n hasLimitsOfShape_of_equivalence (Discrete.equivalence finZeroEquiv'.symm)\n⊢ HasProduct f",
"tactic": "infer_instance"
},
{
"state_after": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nf : Fin (n + 1) → C\nthis : ∀ (f : Fin n → C), HasProduct f\n⊢ HasProduct f",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nf : Fin (n + 1) → C\n⊢ HasProduct f",
"tactic": "haveI := hasProduct_fin n"
},
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝⁴ : SmallCategory J\nC : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasBinaryProducts C\ninst✝ : HasTerminal C\nn : ℕ\nf : Fin (n + 1) → C\nthis : ∀ (f : Fin n → C), HasProduct f\n⊢ HasProduct f",
"tactic": "apply HasLimit.mk ⟨_, extendFanIsLimit f (limit.isLimit _) (limit.isLimit _)⟩"
}
] |
[
108,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
9
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
hasDerivWithinAt_Iio_iff_Iic
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : PartialOrder 𝕜\n⊢ HasDerivWithinAt f f' (Iio x) x ↔ HasDerivWithinAt f f' (Iic x) x",
"tactic": "rw [← Iic_diff_right, hasDerivWithinAt_diff_singleton]"
}
] |
[
339,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/CategoryTheory/Monoidal/Mon_.lean
|
Mon_.one_rightUnitor
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nM : Mon_ C\n⊢ (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).hom ≫ M.one = M.one",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nM : Mon_ C\n⊢ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one",
"tactic": "slice_lhs 2 3 => rw [rightUnitor_naturality, ← unitors_equal]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nM : Mon_ C\n⊢ (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).hom ≫ M.one = M.one",
"tactic": "simp"
}
] |
[
405,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.equivOfRoot_symm
|
[] |
[
388,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Data/Fintype/Sum.lean
|
Fintype.card_subtype_or
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13939\np q : α → Prop\ninst✝² : Fintype { x // p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // p x ∨ q x }\n⊢ card { x // p x ∨ q x } ≤ card { x // p x } + card { x // q x }",
"tactic": "classical\n convert Fintype.card_le_of_embedding (subtypeOrLeftEmbedding p q)\n rw [Fintype.card_sum]"
},
{
"state_after": "case h.e'_4\nα : Type u_1\nβ : Type ?u.13939\np q : α → Prop\ninst✝² : Fintype { x // p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // p x ∨ q x }\n⊢ card { x // p x } + card { x // q x } = card ({ x // p x } ⊕ { x // q x })",
"state_before": "α : Type u_1\nβ : Type ?u.13939\np q : α → Prop\ninst✝² : Fintype { x // p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // p x ∨ q x }\n⊢ card { x // p x ∨ q x } ≤ card { x // p x } + card { x // q x }",
"tactic": "convert Fintype.card_le_of_embedding (subtypeOrLeftEmbedding p q)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\nβ : Type ?u.13939\np q : α → Prop\ninst✝² : Fintype { x // p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // p x ∨ q x }\n⊢ card { x // p x } + card { x // q x } = card ({ x // p x } ⊕ { x // q x })",
"tactic": "rw [Fintype.card_sum]"
}
] |
[
126,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.