statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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map (f : Set → Set) [H : definable 1 f] : Set → Set | image (λ y, pair y (f y)) | def | Set.map | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | Graph of a function: `map f x` is the ZFC function which maps `a ∈ x` to `f a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map {f : Set → Set} [H : definable 1 f] {x y : Set} :
y ∈ map f x ↔ ∃ z ∈ x, pair z (f z) = y | mem_image | theorem | Set.mem_map | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unique {f : Set.{u} → Set.{u}} [H : definable 1 f] {x z : Set.{u}} (zx : z ∈ x) :
∃! w, pair z w ∈ map f x | ⟨f z, image.mk _ _ zx, λ y yx, let ⟨w, wx, we⟩ := mem_image.1 yx, ⟨wz, fy⟩ := pair_injective we in
by rw[←fy, wz]⟩ | theorem | Set.map_unique | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_is_func {f : Set → Set} [H : definable 1 f] {x y : Set} :
is_func x y (map f x) ↔ ∀ z ∈ x, f z ∈ y | ⟨λ ⟨ss, h⟩ z zx, let ⟨t, t1, t2⟩ := h z zx in
(t2 (f z) (image.mk _ _ zx)).symm ▸ (pair_mem_prod.1 (ss t1)).right,
λ h, ⟨λ y yx, let ⟨z, zx, ze⟩ := mem_image.1 yx in ze ▸ pair_mem_prod.2 ⟨zx, h z zx⟩,
λ z, map_unique⟩⟩ | theorem | Set.map_is_func | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hereditarily (p : Set → Prop) : Set → Prop | | x := p x ∧ ∀ y ∈ x, hereditarily y
using_well_founded { dec_tac := `[assumption] } | def | Set.hereditarily | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | Given a predicate `p` on ZFC sets. `hereditarily p x` means that `x` has property `p` and the
members of `x` are all `hereditarily p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hereditarily_iff :
hereditarily p x ↔ p x ∧ ∀ y ∈ x, hereditarily p y | by rw [← hereditarily] | lemma | Set.hereditarily_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hereditarily.self (h : x.hereditarily p) : p x | h.def.1 | lemma | Set.hereditarily.self | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hereditarily.mem (h : x.hereditarily p) (hy : y ∈ x) : y.hereditarily p | h.def.2 _ hy | lemma | Set.hereditarily.mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hereditarily.empty : hereditarily p x → p ∅ | begin
apply x.induction_on,
intros y IH h,
rcases Set.eq_empty_or_nonempty y with (rfl|⟨a, ha⟩),
{ exact h.self },
{ exact IH a ha (h.mem ha) }
end | lemma | Set.hereditarily.empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.eq_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Class | set Set | def | Class | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set"
] | The collection of all classes.
We define `Class` as `set Set`, as this allows us to get many instances automatically. However, in
practice, we treat it as (the definitionally equal) `Set → Prop`. This means, the preferred way to
state that `x : Set` belongs to `A : Class` is to write `A x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {x y : Class.{u}} : (∀ z : Set.{u}, x z ↔ y z) → x = y | set.ext | theorem | Class.ext | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {x y : Class.{u}} : x = y ↔ ∀ z, x z ↔ y z | set.ext_iff | theorem | Class.ext_iff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"set.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_Set (x : Set.{u}) : Class.{u} | {y | y ∈ x} | def | Class.of_Set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Coerce a ZFC set into a class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
univ : Class | set.univ | def | Class.univ | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class"
] | The universal class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Set (B : Class.{u}) (A : Class.{u}) : Prop | ∃ x, ↑x = A ∧ B x | def | Class.to_Set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Assert that `A` is a ZFC set satisfying `B` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem (A B : Class.{u}) : Prop | to_Set.{u} B A | def | Class.mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | `A ∈ B` if `A` is a ZFC set which satisfies `B` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_def (A B : Class.{u}) : A ∈ B ↔ ∃ x, ↑x = A ∧ B x | iff.rfl | theorem | Class.mem_def | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_empty (x : Class.{u}) : x ∉ (∅ : Class.{u}) | λ ⟨_, _, h⟩, h | theorem | Class.not_mem_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_empty_hom (x : Set.{u}) : ¬ (∅ : Class.{u}) x | id | theorem | Class.not_empty_hom | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_univ {A : Class.{u}} : A ∈ univ.{u} ↔ ∃ x : Set.{u}, ↑x = A | exists_congr $ λ x, and_true _ | theorem | Class.mem_univ | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_univ_hom (x : Set.{u}) : univ.{u} x | trivial | theorem | Class.mem_univ_hom | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_univ_iff_forall {A : Class.{u}} : A = univ ↔ ∀ x : Set, A x | set.eq_univ_iff_forall | theorem | Class.eq_univ_iff_forall | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"set.eq_univ_iff_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_univ_of_forall {A : Class.{u}} : (∀ x : Set, A x) → A = univ | set.eq_univ_of_forall | theorem | Class.eq_univ_of_forall | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"set.eq_univ_of_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_wf : @well_founded Class.{u} (∈) | ⟨begin
have H : ∀ x : Set.{u}, @acc Class.{u} (∈) ↑x,
{ refine λ a, Set.induction_on a (λ x IH, ⟨x, _⟩),
rintros A ⟨z, rfl, hz⟩,
exact IH z hz },
{ refine λ A, ⟨A, _⟩,
rintros B ⟨x, rfl, hx⟩,
exact H x }
end⟩ | theorem | Class.mem_wf | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_asymm {x y : Class} : x ∈ y → y ∉ x | asymm | theorem | Class.mem_asymm | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_irrefl (x : Class) : x ∉ x | irrefl x | theorem | Class.mem_irrefl | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
univ_not_mem_univ : univ ∉ univ | mem_irrefl _ | theorem | Class.univ_not_mem_univ | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | **There is no universal set.**
This is stated as `univ ∉ univ`, meaning that `univ` (the class of all sets) is proper (does not
belong to the class of all sets). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Cong_to_Class (x : set Class.{u}) : Class.{u} | {y | ↑y ∈ x} | def | Class.Cong_to_Class | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Convert a conglomerate (a collection of classes) into a class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Cong_to_Class_empty : Cong_to_Class ∅ = ∅ | by { ext, simp [Cong_to_Class] } | theorem | Class.Cong_to_Class_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Class_to_Cong (x : Class.{u}) : set Class.{u} | {y | y ∈ x} | def | Class.Class_to_Cong | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Convert a class into a conglomerate (a collection of classes) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Class_to_Cong_empty : Class_to_Cong ∅ = ∅ | by { ext, simp [Class_to_Cong] } | theorem | Class.Class_to_Cong_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerset (x : Class) : Class | Cong_to_Class (set.powerset x) | def | Class.powerset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class",
"set.powerset"
] | The power class of a class is the class of all subclasses that are ZFC sets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sUnion (x : Class) : Class | ⋃₀ (Class_to_Cong x) | def | Class.sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class"
] | The union of a class is the class of all members of ZFC sets in the class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sInter (x : Class) : Class | ⋂₀ Class_to_Cong x | def | Class.sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class"
] | The intersection of a class is the class of all members of ZFC sets in the class | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_Set.inj {x y : Set.{u}} (h : (x : Class.{u}) = y) : x = y | Set.ext $ λ z, by { change (x : Class.{u}) z ↔ (y : Class.{u}) z, rw h } | theorem | Class.of_Set.inj | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Set_of_Set (A : Class.{u}) (x : Set.{u}) : to_Set A x ↔ A x | ⟨λ ⟨y, yx, py⟩, by rwa of_Set.inj yx at py, λ px, ⟨x, rfl, px⟩⟩ | theorem | Class.to_Set_of_Set | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mem {x : Set.{u}} {A : Class.{u}} : (x : Class.{u}) ∈ A ↔ A x | to_Set_of_Set _ _ | theorem | Class.coe_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_apply {x y : Set.{u}} : (y : Class.{u}) x ↔ x ∈ y | iff.rfl | theorem | Class.coe_apply | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subset (x y : Set.{u}) : (x : Class.{u}) ⊆ y ↔ x ⊆ y | iff.rfl | theorem | Class.coe_subset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sep (p : Class.{u}) (x : Set.{u}) :
(↑{y ∈ x | p y} : Class.{u}) = {y ∈ x | p y} | ext $ λ y, Set.mem_sep | theorem | Class.coe_sep | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.mem_sep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_empty : ↑(∅ : Set.{u}) = (∅ : Class.{u}) | ext $ λ y, (iff_false _).2 $ Set.not_mem_empty y | theorem | Class.coe_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.not_mem_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_insert (x y : Set.{u}) :
↑(insert x y) = @insert Set.{u} Class.{u} _ x y | ext $ λ z, Set.mem_insert_iff | theorem | Class.coe_insert | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.mem_insert_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_union (x y : Set.{u}) : ↑(x ∪ y) = (x : Class.{u}) ∪ y | ext $ λ z, Set.mem_union | theorem | Class.coe_union | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.mem_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inter (x y : Set.{u}) : ↑(x ∩ y) = (x : Class.{u}) ∩ y | ext $ λ z, Set.mem_inter | theorem | Class.coe_inter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.mem_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_diff (x y : Set.{u}) : ↑(x \ y) = (x : Class.{u}) \ y | ext $ λ z, Set.mem_diff | theorem | Class.coe_diff | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.mem_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_powerset (x : Set.{u}) : ↑x.powerset = powerset.{u} x | ext $ λ z, Set.mem_powerset | theorem | Class.coe_powerset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.mem_powerset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerset_apply {A : Class.{u}} {x : Set.{u}} : powerset A x ↔ ↑x ⊆ A | iff.rfl | theorem | Class.powerset_apply | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion_apply {x : Class} {y : Set} : (⋃₀ x) y ↔ ∃ z : Set, x z ∧ y ∈ z | begin
split,
{ rintro ⟨-, ⟨z, rfl, hxz⟩, hyz⟩,
exact ⟨z, hxz, hyz⟩ },
{ exact λ ⟨z, hxz, hyz⟩, ⟨_, coe_mem.2 hxz, hyz⟩ }
end | theorem | Class.sUnion_apply | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class",
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sUnion (x : Set.{u}) : ↑(⋃₀ x) = ⋃₀ (x : Class.{u}) | ext $ λ y, Set.mem_sUnion.trans (sUnion_apply.trans $ by simp_rw [coe_apply, exists_prop]).symm | theorem | Class.coe_sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sUnion {x y : Class.{u}} : y ∈ ⋃₀ x ↔ ∃ z, z ∈ x ∧ y ∈ z | begin
split,
{ rintro ⟨w, rfl, z, hzx, hwz⟩,
exact ⟨z, hzx, coe_mem.2 hwz⟩ },
{ rintro ⟨w, hwx, z, rfl, hwz⟩,
exact ⟨z, rfl, w, hwx, hwz⟩ }
end | theorem | Class.mem_sUnion | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sInter_apply {x : Class.{u}} {y : Set.{u}} :
(⋂₀ x) y ↔ ∀ z : Set.{u}, x z → y ∈ z | begin
refine ⟨λ hxy z hxz, hxy _ ⟨z, rfl, hxz⟩, _⟩,
rintro H - ⟨z, rfl, hxz⟩,
exact H _ hxz
end | theorem | Class.sInter_apply | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sInter {x : Set.{u}} (h : x.nonempty) :
↑(⋂₀ x) = ⋂₀ (x : Class.{u}) | set.ext $ λ y, (Set.mem_sInter h).trans sInter_apply.symm | theorem | Class.coe_sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.mem_sInter",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_mem_sInter {x y z : Class} (hy : y ∈ ⋂₀ x) (hz : z ∈ x) : y ∈ z | by { obtain ⟨w, rfl, hw⟩ := hy, exact coe_mem.2 (hw z hz) } | theorem | Class.mem_of_mem_sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sInter {x y : Class.{u}} (h : x.nonempty) : y ∈ ⋂₀ x ↔ ∀ z, z ∈ x → y ∈ z | begin
refine ⟨λ hy z, mem_of_mem_sInter hy, λ H, _⟩,
simp_rw [mem_def, sInter_apply],
obtain ⟨z, hz⟩ := h,
obtain ⟨y, rfl, hzy⟩ := H z (coe_mem.2 hz),
refine ⟨y, rfl, λ w hxw, _⟩,
simpa only [coe_mem, coe_apply] using H w (coe_mem.2 hxw),
end | theorem | Class.mem_sInter | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sUnion_empty : ⋃₀ (∅ : Class.{u}) = ∅ | by { ext, simp } | theorem | Class.sUnion_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sInter_empty : ⋂₀ (∅ : Class.{u}) = univ | by { ext, simp [sInter, ←univ] } | theorem | Class.sInter_empty | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_univ_of_powerset_subset {A : Class} (hA : powerset A ⊆ A) : A = univ | eq_univ_of_forall begin
by_contra' hnA,
exact well_founded.min_mem Set.mem_wf _ hnA (hA $ λ x hx, not_not.1 $
λ hB, well_founded.not_lt_min Set.mem_wf _ hnA hB $ coe_apply.1 hx)
end | theorem | Class.eq_univ_of_powerset_subset | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class",
"Set.mem_wf",
"well_founded.min_mem",
"well_founded.not_lt_min"
] | An induction principle for sets. If every subset of a class is a member, then the class is
universal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iota (A : Class) : Class | ⋃₀ {x | ∀ y, A y ↔ y = x} | def | Class.iota | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class"
] | The definite description operator, which is `{x}` if `{y | A y} = {x}` and `∅` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iota_val (A : Class) (x : Set) (H : ∀ y, A y ↔ y = x) : iota A = ↑x | ext $ λ y, ⟨λ ⟨._, ⟨x', rfl, h⟩, yx'⟩, by rwa ←((H x').1 $ (h x').2 rfl),
λ yx, ⟨_, ⟨x, rfl, H⟩, yx⟩⟩ | theorem | Class.iota_val | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class",
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iota_ex (A) : iota.{u} A ∈ univ.{u} | mem_univ.2 $ or.elim (classical.em $ ∃ x, ∀ y, A y ↔ y = x)
(λ ⟨x, h⟩, ⟨x, eq.symm $ iota_val A x h⟩)
(λ hn, ⟨∅, ext (λ z, coe_empty.symm ▸ ⟨false.rec _, λ ⟨._, ⟨x, rfl, H⟩, zA⟩, hn ⟨x, H⟩⟩)⟩) | theorem | Class.iota_ex | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | Unlike the other set constructors, the `iota` definite descriptor
is a set for any set input, but not constructively so, so there is no
associated `Class → Set` function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fval (F A : Class.{u}) : Class.{u} | iota (λ y, to_Set (λ x, F (Set.pair x y)) A) | def | Class.fval | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set.pair"
] | Function value | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fval_ex (F A : Class.{u}) : F ′ A ∈ univ.{u} | iota_ex _ | theorem | Class.fval_ex | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_fval {f : Set.{u} → Set.{u}} [H : pSet.definable 1 f]
{x y : Set.{u}} (h : y ∈ x) :
(Set.map f x ′ y : Class.{u}) = f y | Class.iota_val _ _ (λ z, by { rw [Class.to_Set_of_Set, Class.coe_apply, mem_map], exact
⟨λ ⟨w, wz, pr⟩, let ⟨wy, fw⟩ := Set.pair_injective pr in by rw[←fw, wy],
λ e, by { subst e, exact ⟨_, h, rfl⟩ }⟩ }) | theorem | Set.map_fval | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class.coe_apply",
"Class.iota_val",
"Class.to_Set_of_Set",
"Set.map",
"Set.pair_injective",
"mem_map",
"pSet.definable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
choice : Set | @map (λ y, classical.epsilon (λ z, z ∈ y)) (classical.all_definable _) x | def | Set.choice | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Set",
"classical.all_definable"
] | A choice function on the class of nonempty ZFC sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
choice_mem_aux (y : Set.{u}) (yx : y ∈ x) : classical.epsilon (λ z : Set.{u}, z ∈ y) ∈ y | @classical.epsilon_spec _ (λ z : Set.{u}, z ∈ y) $ classical.by_contradiction $ λ n, h $
by rwa ←((eq_empty y).2 $ λ z zx, n ⟨z, zx⟩) | theorem | Set.choice_mem_aux | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
choice_is_func : is_func x (⋃₀ x) (choice x) | (@map_is_func _ (classical.all_definable _) _ _).2 $
λ y yx, mem_sUnion.2 ⟨y, yx, choice_mem_aux x h y yx⟩ | theorem | Set.choice_is_func | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"classical.all_definable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
choice_mem (y : Set.{u}) (yx : y ∈ x) : (choice x ′ y : Class.{u}) ∈ (y : Class.{u}) | begin
delta choice,
rw [map_fval yx, Class.coe_mem, Class.coe_apply],
exact choice_mem_aux x h y yx
end | theorem | Set.choice_mem | set_theory.zfc | src/set_theory/zfc/basic.lean | [
"data.set.lattice",
"logic.small.basic",
"order.well_founded"
] | [
"Class.coe_apply",
"Class.coe_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive (x : Set) : Prop | ∀ y ∈ x, y ⊆ x | def | Set.is_transitive | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [
"Set"
] | A transitive set is one where every element is a subset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty_is_transitive : is_transitive ∅ | λ y hy, (not_mem_empty y hy).elim | theorem | Set.empty_is_transitive | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive.subset_of_mem (h : x.is_transitive) : y ∈ x → y ⊆ x | h y | theorem | Set.is_transitive.subset_of_mem | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive_iff_mem_trans : z.is_transitive ↔ ∀ {x y : Set}, x ∈ y → y ∈ z → x ∈ z | ⟨λ h x y hx hy, h.subset_of_mem hy hx, λ H x hx y hy, H hy hx⟩ | theorem | Set.is_transitive_iff_mem_trans | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [
"Set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive.inter (hx : x.is_transitive) (hy : y.is_transitive) :
(x ∩ y).is_transitive | λ z hz w hw, by { rw mem_inter at hz ⊢, exact ⟨hx.mem_trans hw hz.1, hy.mem_trans hw hz.2⟩ } | theorem | Set.is_transitive.inter | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive.sUnion (h : x.is_transitive) : (⋃₀ x).is_transitive | λ y hy z hz, begin
rcases mem_sUnion.1 hy with ⟨w, hw, hw'⟩,
exact mem_sUnion_of_mem hz (h.mem_trans hw' hw)
end | theorem | Set.is_transitive.sUnion | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive.sUnion' (H : ∀ y ∈ x, is_transitive y) : (⋃₀ x).is_transitive | λ y hy z hz, begin
rcases mem_sUnion.1 hy with ⟨w, hw, hw'⟩,
exact mem_sUnion_of_mem ((H w hw).mem_trans hz hw') hw
end | theorem | Set.is_transitive.sUnion' | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive.union (hx : x.is_transitive) (hy : y.is_transitive) :
(x ∪ y).is_transitive | begin
rw ←sUnion_pair,
apply is_transitive.sUnion' (λ z, _),
rw mem_pair,
rintro (rfl | rfl),
assumption'
end | theorem | Set.is_transitive.union | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive.powerset (h : x.is_transitive) : (powerset x).is_transitive | λ y hy z hz, by { rw mem_powerset at ⊢ hy, exact h.subset_of_mem (hy hz) } | theorem | Set.is_transitive.powerset | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive_iff_sUnion_subset : x.is_transitive ↔ ⋃₀ x ⊆ x | ⟨λ h y hy, by { rcases mem_sUnion.1 hy with ⟨z, hz, hz'⟩, exact h.mem_trans hz' hz },
λ H y hy z hz, H $ mem_sUnion_of_mem hz hy⟩ | theorem | Set.is_transitive_iff_sUnion_subset | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_transitive_iff_subset_powerset : x.is_transitive ↔ x ⊆ powerset x | ⟨λ h y hy, mem_powerset.2 $ h.subset_of_mem hy, λ H y hy z hz, mem_powerset.1 (H hy) hz⟩ | theorem | Set.is_transitive_iff_subset_powerset | set_theory.zfc | src/set_theory/zfc/ordinal.lean | [
"set_theory.zfc.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
context | (red : transparency)
(α : expr)
(univ : level)
(α0 : expr)
(is_group : bool)
(inst : expr) | structure | tactic.abel.context | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | The `context` for a call to `abel`.
Stores a few options for this call, and caches some common subexpressions
such as typeclass instances and `0 : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_context (red : transparency) (e : expr) : tactic context | do α ← infer_type e,
c ← mk_app ``add_comm_monoid [α] >>= mk_instance,
cg ← try_core (mk_app ``add_comm_group [α] >>= mk_instance),
u ← mk_meta_univ,
infer_type α >>= unify (expr.sort (level.succ u)),
u ← get_univ_assignment u,
α0 ← expr.of_nat α 0,
match cg with
| (some cg) := return ⟨red, α, u... | def | tactic.abel.mk_context | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"add_comm_group",
"add_comm_monoid",
"expr.of_nat"
] | Populate a `context` object for evaluating `e`, up to reducibility level `red`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
context.app (c : context) (n : name) (inst : expr) : list expr → expr | (@expr.const tt n [c.univ] c.α inst).mk_app | def | tactic.abel.context.app | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | Apply the function `n : ∀ {α} [inst : add_whatever α], _` to the
implicit parameters in the context, and the given list of arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
context.mk_app (c : context) (n inst : name) (l : list expr) : tactic expr | do m ← mk_instance ((expr.const inst [c.univ] : expr) c.α), return $ c.app n m l | def | tactic.abel.context.mk_app | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | Apply the function `n : ∀ {α} [inst α], _` to the implicit parameters in the
context, and the given list of arguments.
Compared to `context.app`, this takes the name of the typeclass, rather than an
inferred typeclass instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_g : name → name | | (name.mk_string s p) := name.mk_string (s ++ "g") p
| n := n | def | tactic.abel.add_g | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | Add the letter "g" to the end of the name, e.g. turning `term` into `termg`.
This is used to choose between declarations taking `add_comm_monoid` and those
taking `add_comm_group` instances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
context.iapp (c : context) (n : name) : list expr → expr | c.app (if c.is_group then add_g n else n) c.inst | def | tactic.abel.context.iapp | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | Apply the function `n : ∀ {α} [add_comm_{monoid,group} α]` to the given
list of arguments.
Will use the `add_comm_{monoid,group}` instance that has been cached in the context. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
term {α} [add_comm_monoid α] (n : ℕ) (x a : α) : α | n • x + a | def | tactic.abel.term | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
termg {α} [add_comm_group α] (n : ℤ) (x a : α) : α | n • x + a | def | tactic.abel.termg | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
context.mk_term (c : context) (n x a : expr) : expr | c.iapp ``term [n, x, a] | def | tactic.abel.context.mk_term | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | Evaluate a term with coefficient `n`, atom `x` and successor terms `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
context.int_to_expr (c : context) (n : ℤ) : tactic expr | expr.of_int (if c.is_group then `(ℤ) else `(ℕ)) n | def | tactic.abel.context.int_to_expr | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"expr.of_int"
] | Interpret an integer as a coefficient to a term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_expr : Type
| zero (e : expr) : normal_expr
| nterm (e : expr) (n : expr × ℤ) (x : expr) (a : normal_expr) : normal_expr | inductive | tactic.abel.normal_expr | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | ||
normal_expr.e : normal_expr → expr | | (normal_expr.zero e) := e
| (normal_expr.nterm e _ _ _) := e | def | tactic.abel.normal_expr.e | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_expr.term' (c : context) (n : expr × ℤ) (x : expr) (a : normal_expr) :
normal_expr | normal_expr.nterm (c.mk_term n.1 x a) n x a | def | tactic.abel.normal_expr.term' | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_expr.zero' (c : context) : normal_expr | normal_expr.zero c.α0 | def | tactic.abel.normal_expr.zero' | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_expr.to_list : normal_expr → list (ℤ × expr) | | (normal_expr.zero _) := []
| (normal_expr.nterm _ (_, n) x a) := (n, x) :: a.to_list | def | tactic.abel.normal_expr.to_list | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_expr.to_string (e : normal_expr) : string | " + ".intercalate $ (to_list e).map $
λ ⟨n, e⟩, to_string n ++ " • (" ++ to_string e ++ ")" | def | tactic.abel.normal_expr.to_string | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_expr.pp (e : normal_expr) : tactic format | do l ← (to_list e).mmap (λ ⟨n, e⟩, do
pe ← pp e, return (to_fmt n ++ " • (" ++ pe ++ ")")),
return $ format.join $ l.intersperse ↑" + " | def | tactic.abel.normal_expr.pp | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_expr.refl_conv (e : normal_expr) : tactic (normal_expr × expr) | do p ← mk_eq_refl e, return (e, p) | def | tactic.abel.normal_expr.refl_conv | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_add_term {α} [add_comm_monoid α] (k n x a a') (h : k + a = a') :
k + @term α _ n x a = term n x a' | by simp [h.symm, term]; ac_refl | theorem | tactic.abel.const_add_term | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_add_termg {α} [add_comm_group α] (k n x a a') (h : k + a = a') :
k + @termg α _ n x a = termg n x a' | by simp [h.symm, termg]; ac_refl | theorem | tactic.abel.const_add_termg | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
term_add_const {α} [add_comm_monoid α] (n x a k a') (h : a + k = a') :
@term α _ n x a + k = term n x a' | by simp [h.symm, term, add_assoc] | theorem | tactic.abel.term_add_const | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"add_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
term_add_constg {α} [add_comm_group α] (n x a k a') (h : a + k = a') :
@termg α _ n x a + k = termg n x a' | by simp [h.symm, termg, add_assoc] | theorem | tactic.abel.term_add_constg | tactic | src/tactic/abel.lean | [
"tactic.norm_num"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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