Proof Assistant Projects
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Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
has_abs (α : Type*) := (abs : α → α) | has_abs (α : Type*) | (abs : α → α) | class | has_abs | algebra | src/algebra/abs.lean | [] | [] | Absolute value is a unary operator with properties similar to the absolute value of a real number. | 39 | 39 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pos_part (α : Type*) := (pos : α → α) | has_pos_part (α : Type*) | (pos : α → α) | class | has_pos_part | algebra | src/algebra/abs.lean | [] | [] | The positive part of an element admiting a decomposition into positive and negative parts. | 45 | 45 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_neg_part (α : Type*) := (neg : α → α) | has_neg_part (α : Type*) | (neg : α → α) | class | has_neg_part | algebra | src/algebra/abs.lean | [] | [] | The negative part of an element admiting a decomposition into positive and negative parts. | 50 | 50 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_torsor (G : out_param Type*) (P : Type*) [out_param $ add_group G]
extends add_action G P, has_vsub G P :=
[nonempty : nonempty P]
(vsub_vadd' : ∀ (p1 p2 : P), (p1 -ᵥ p2 : G) +ᵥ p2 = p1)
(vadd_vsub' : ∀ (g : G) (p : P), g +ᵥ p -ᵥ p = g) | add_torsor (G : out_param Type*) (P : Type*) [out_param $ add_group G]
extends add_action G P, has_vsub G P | [nonempty : nonempty P]
(vsub_vadd' : ∀ (p1 p2 : P), (p1 -ᵥ p2 : G) +ᵥ p2 = p1)
(vadd_vsub' : ∀ (g : G) (p : P), g +ᵥ p -ᵥ p = g) | class | add_torsor | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"add_action",
"add_group",
"has_vsub"
] | An `add_torsor G P` gives a structure to the nonempty type `P`,
acted on by an `add_group G` with a transitive and free action given
by the `+ᵥ` operation and a corresponding subtraction given by the
`-ᵥ` operation. In the case of a vector space, it is an affine
space. | 48 | 52 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group_is_add_torsor (G : Type*) [add_group G] :
add_torsor G G :=
{ vsub := has_sub.sub,
vsub_vadd' := sub_add_cancel,
vadd_vsub' := add_sub_cancel } | add_group_is_add_torsor (G : Type*) [add_group G] :
add_torsor G G | { vsub := has_sub.sub,
vsub_vadd' := sub_add_cancel,
vadd_vsub' := add_sub_cancel } | instance | add_group_is_add_torsor | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"add_group",
"add_torsor"
] | An `add_group G` is a torsor for itself. | 58 | 63 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_eq_sub {G : Type*} [add_group G] (g1 g2 : G) : g1 -ᵥ g2 = g1 - g2 :=
rfl | vsub_eq_sub {G : Type*} [add_group G] (g1 g2 : G) : g1 -ᵥ g2 = g1 - g2 | rfl | lemma | vsub_eq_sub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"add_group"
] | Simplify subtraction for a torsor for an `add_group G` over
itself. | 67 | 68 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_vadd (p1 p2 : P) : p1 -ᵥ p2 +ᵥ p2 = p1 :=
add_torsor.vsub_vadd' p1 p2 | vsub_vadd (p1 p2 : P) : p1 -ᵥ p2 +ᵥ p2 = p1 | add_torsor.vsub_vadd' p1 p2 | lemma | vsub_vadd | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | Adding the result of subtracting from another point produces that
point. | 77 | 78 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_vsub (g : G) (p : P) : g +ᵥ p -ᵥ p = g :=
add_torsor.vadd_vsub' g p | vadd_vsub (g : G) (p : P) : g +ᵥ p -ᵥ p = g | add_torsor.vadd_vsub' g p | lemma | vadd_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | Adding a group element then subtracting the original point
produces that group element. | 82 | 83 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_right_cancel {g1 g2 : G} (p : P) (h : g1 +ᵥ p = g2 +ᵥ p) : g1 = g2 :=
by rw [←vadd_vsub g1, h, vadd_vsub] | vadd_right_cancel {g1 g2 : G} (p : P) (h : g1 +ᵥ p = g2 +ᵥ p) : g1 = g2 | by rw [←vadd_vsub g1, h, vadd_vsub] | lemma | vadd_right_cancel | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_vsub"
] | If the same point added to two group elements produces equal
results, those group elements are equal. | 87 | 88 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_right_cancel_iff {g1 g2 : G} (p : P) : g1 +ᵥ p = g2 +ᵥ p ↔ g1 = g2 :=
⟨vadd_right_cancel p, λ h, h ▸ rfl⟩ | vadd_right_cancel_iff {g1 g2 : G} (p : P) : g1 +ᵥ p = g2 +ᵥ p ↔ g1 = g2 | ⟨vadd_right_cancel p, λ h, h ▸ rfl⟩ | lemma | vadd_right_cancel_iff | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 90 | 91 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_right_injective (p : P) : function.injective ((+ᵥ p) : G → P) :=
λ g1 g2, vadd_right_cancel p | vadd_right_injective (p : P) : function.injective ((+ᵥ p) : G → P) | λ g1 g2, vadd_right_cancel p | lemma | vadd_right_injective | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_right_cancel"
] | Adding a group element to the point `p` is an injective
function. | 95 | 96 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_vsub_assoc (g : G) (p1 p2 : P) : g +ᵥ p1 -ᵥ p2 = g + (p1 -ᵥ p2) :=
begin
apply vadd_right_cancel p2,
rw [vsub_vadd, add_vadd, vsub_vadd]
end | vadd_vsub_assoc (g : G) (p1 p2 : P) : g +ᵥ p1 -ᵥ p2 = g + (p1 -ᵥ p2) | begin
apply vadd_right_cancel p2,
rw [vsub_vadd, add_vadd, vsub_vadd]
end | lemma | vadd_vsub_assoc | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_right_cancel",
"vsub_vadd"
] | Adding a group element to a point, then subtracting another point,
produces the same result as subtracting the points then adding the
group element. | 101 | 105 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_self (p : P) : p -ᵥ p = (0 : G) :=
by rw [←zero_add (p -ᵥ p), ←vadd_vsub_assoc, vadd_vsub] | vsub_self (p : P) : p -ᵥ p = (0 : G) | by rw [←zero_add (p -ᵥ p), ←vadd_vsub_assoc, vadd_vsub] | lemma | vsub_self | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_vsub"
] | Subtracting a point from itself produces 0. | 108 | 109 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_vsub_eq_zero {p1 p2 : P} (h : p1 -ᵥ p2 = (0 : G)) : p1 = p2 :=
by rw [←vsub_vadd p1 p2, h, zero_vadd] | eq_of_vsub_eq_zero {p1 p2 : P} (h : p1 -ᵥ p2 = (0 : G)) : p1 = p2 | by rw [←vsub_vadd p1 p2, h, zero_vadd] | lemma | eq_of_vsub_eq_zero | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | If subtracting two points produces 0, they are equal. | 112 | 113 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_eq_zero_iff_eq {p1 p2 : P} : p1 -ᵥ p2 = (0 : G) ↔ p1 = p2 :=
iff.intro eq_of_vsub_eq_zero (λ h, h ▸ vsub_self _) | vsub_eq_zero_iff_eq {p1 p2 : P} : p1 -ᵥ p2 = (0 : G) ↔ p1 = p2 | iff.intro eq_of_vsub_eq_zero (λ h, h ▸ vsub_self _) | lemma | vsub_eq_zero_iff_eq | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"eq_of_vsub_eq_zero",
"vsub_self"
] | Subtracting two points produces 0 if and only if they are
equal. | 117 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_ne_zero {p q : P} : p -ᵥ q ≠ (0 : G) ↔ p ≠ q :=
not_congr vsub_eq_zero_iff_eq | vsub_ne_zero {p q : P} : p -ᵥ q ≠ (0 : G) ↔ p ≠ q | not_congr vsub_eq_zero_iff_eq | lemma | vsub_ne_zero | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_eq_zero_iff_eq"
] | null | 120 | 121 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_add_vsub_cancel (p1 p2 p3 : P) : p1 -ᵥ p2 + (p2 -ᵥ p3) = (p1 -ᵥ p3) :=
begin
apply vadd_right_cancel p3,
rw [add_vadd, vsub_vadd, vsub_vadd, vsub_vadd]
end | vsub_add_vsub_cancel (p1 p2 p3 : P) : p1 -ᵥ p2 + (p2 -ᵥ p3) = (p1 -ᵥ p3) | begin
apply vadd_right_cancel p3,
rw [add_vadd, vsub_vadd, vsub_vadd, vsub_vadd]
end | lemma | vsub_add_vsub_cancel | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_right_cancel",
"vsub_vadd"
] | Cancellation adding the results of two subtractions. | 124 | 128 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_vsub_eq_vsub_rev (p1 p2 : P) : -(p1 -ᵥ p2) = (p2 -ᵥ p1) :=
begin
refine neg_eq_of_add_eq_zero_right (vadd_right_cancel p1 _),
rw [vsub_add_vsub_cancel, vsub_self],
end | neg_vsub_eq_vsub_rev (p1 p2 : P) : -(p1 -ᵥ p2) = (p2 -ᵥ p1) | begin
refine neg_eq_of_add_eq_zero_right (vadd_right_cancel p1 _),
rw [vsub_add_vsub_cancel, vsub_self],
end | lemma | neg_vsub_eq_vsub_rev | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_right_cancel",
"vsub_add_vsub_cancel",
"vsub_self"
] | Subtracting two points in the reverse order produces the negation
of subtracting them. | 132 | 136 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_vsub_eq_sub_vsub (g : G) (p q : P) : g +ᵥ p -ᵥ q = g - (q -ᵥ p) :=
by rw [vadd_vsub_assoc, sub_eq_add_neg, neg_vsub_eq_vsub_rev] | vadd_vsub_eq_sub_vsub (g : G) (p q : P) : g +ᵥ p -ᵥ q = g - (q -ᵥ p) | by rw [vadd_vsub_assoc, sub_eq_add_neg, neg_vsub_eq_vsub_rev] | lemma | vadd_vsub_eq_sub_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"neg_vsub_eq_vsub_rev",
"vadd_vsub_assoc"
] | null | 138 | 139 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_vadd_eq_vsub_sub (p1 p2 : P) (g : G) : p1 -ᵥ (g +ᵥ p2) = (p1 -ᵥ p2) - g :=
by rw [←add_right_inj (p2 -ᵥ p1 : G), vsub_add_vsub_cancel, ←neg_vsub_eq_vsub_rev, vadd_vsub,
←add_sub_assoc, ←neg_vsub_eq_vsub_rev, neg_add_self, zero_sub] | vsub_vadd_eq_vsub_sub (p1 p2 : P) (g : G) : p1 -ᵥ (g +ᵥ p2) = (p1 -ᵥ p2) - g | by rw [←add_right_inj (p2 -ᵥ p1 : G), vsub_add_vsub_cancel, ←neg_vsub_eq_vsub_rev, vadd_vsub,
←add_sub_assoc, ←neg_vsub_eq_vsub_rev, neg_add_self, zero_sub] | lemma | vsub_vadd_eq_vsub_sub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_vsub",
"vsub_add_vsub_cancel"
] | Subtracting the result of adding a group element produces the same result
as subtracting the points and subtracting that group element. | 143 | 145 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_sub_vsub_cancel_right (p1 p2 p3 : P) :
(p1 -ᵥ p3) - (p2 -ᵥ p3) = (p1 -ᵥ p2) :=
by rw [←vsub_vadd_eq_vsub_sub, vsub_vadd] | vsub_sub_vsub_cancel_right (p1 p2 p3 : P) :
(p1 -ᵥ p3) - (p2 -ᵥ p3) = (p1 -ᵥ p2) | by rw [←vsub_vadd_eq_vsub_sub, vsub_vadd] | lemma | vsub_sub_vsub_cancel_right | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_vadd"
] | Cancellation subtracting the results of two subtractions. | 148 | 150 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_vadd_iff_vsub_eq (p1 : P) (g : G) (p2 : P) : p1 = g +ᵥ p2 ↔ p1 -ᵥ p2 = g :=
⟨λ h, h.symm ▸ vadd_vsub _ _, λ h, h ▸ (vsub_vadd _ _).symm⟩ | eq_vadd_iff_vsub_eq (p1 : P) (g : G) (p2 : P) : p1 = g +ᵥ p2 ↔ p1 -ᵥ p2 = g | ⟨λ h, h.symm ▸ vadd_vsub _ _, λ h, h ▸ (vsub_vadd _ _).symm⟩ | lemma | eq_vadd_iff_vsub_eq | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_vsub",
"vsub_vadd"
] | Convert between an equality with adding a group element to a point
and an equality of a subtraction of two points with a group
element. | 155 | 156 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_eq_vadd_iff_neg_add_eq_vsub {v₁ v₂ : G} {p₁ p₂ : P} :
v₁ +ᵥ p₁ = v₂ +ᵥ p₂ ↔ - v₁ + v₂ = p₁ -ᵥ p₂ :=
by rw [eq_vadd_iff_vsub_eq, vadd_vsub_assoc, ← add_right_inj (-v₁), neg_add_cancel_left, eq_comm] | vadd_eq_vadd_iff_neg_add_eq_vsub {v₁ v₂ : G} {p₁ p₂ : P} :
v₁ +ᵥ p₁ = v₂ +ᵥ p₂ ↔ - v₁ + v₂ = p₁ -ᵥ p₂ | by rw [eq_vadd_iff_vsub_eq, vadd_vsub_assoc, ← add_right_inj (-v₁), neg_add_cancel_left, eq_comm] | lemma | vadd_eq_vadd_iff_neg_add_eq_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"eq_vadd_iff_vsub_eq",
"vadd_vsub_assoc"
] | null | 158 | 160 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
singleton_vsub_self (p : P) : ({p} : set P) -ᵥ {p} = {(0:G)} :=
by rw [set.singleton_vsub_singleton, vsub_self] | singleton_vsub_self (p : P) : ({p} : set P) -ᵥ {p} = {(0:G)} | by rw [set.singleton_vsub_singleton, vsub_self] | lemma | set.singleton_vsub_self | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"set.singleton_vsub_singleton",
"vsub_self"
] | null | 165 | 166 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_vsub_vadd_cancel_right (v₁ v₂ : G) (p : P) :
(v₁ +ᵥ p) -ᵥ (v₂ +ᵥ p) = v₁ - v₂ :=
by rw [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, vsub_self, add_zero] | vadd_vsub_vadd_cancel_right (v₁ v₂ : G) (p : P) :
(v₁ +ᵥ p) -ᵥ (v₂ +ᵥ p) = v₁ - v₂ | by rw [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, vsub_self, add_zero] | lemma | vadd_vsub_vadd_cancel_right | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_vsub_assoc",
"vsub_self",
"vsub_vadd_eq_vsub_sub"
] | null | 170 | 172 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_left_cancel {p1 p2 p : P} (h : p1 -ᵥ p = p2 -ᵥ p) : p1 = p2 :=
by rwa [←sub_eq_zero, vsub_sub_vsub_cancel_right, vsub_eq_zero_iff_eq] at h | vsub_left_cancel {p1 p2 p : P} (h : p1 -ᵥ p = p2 -ᵥ p) : p1 = p2 | by rwa [←sub_eq_zero, vsub_sub_vsub_cancel_right, vsub_eq_zero_iff_eq] at h | lemma | vsub_left_cancel | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_eq_zero_iff_eq",
"vsub_sub_vsub_cancel_right"
] | If the same point subtracted from two points produces equal
results, those points are equal. | 176 | 177 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_left_cancel_iff {p1 p2 p : P} : (p1 -ᵥ p) = p2 -ᵥ p ↔ p1 = p2 :=
⟨vsub_left_cancel, λ h, h ▸ rfl⟩ | vsub_left_cancel_iff {p1 p2 p : P} : (p1 -ᵥ p) = p2 -ᵥ p ↔ p1 = p2 | ⟨vsub_left_cancel, λ h, h ▸ rfl⟩ | lemma | vsub_left_cancel_iff | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | The same point subtracted from two points produces equal results
if and only if those points are equal. | 181 | 182 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_left_injective (p : P) : function.injective ((-ᵥ p) : P → G) :=
λ p2 p3, vsub_left_cancel | vsub_left_injective (p : P) : function.injective ((-ᵥ p) : P → G) | λ p2 p3, vsub_left_cancel | lemma | vsub_left_injective | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_left_cancel"
] | Subtracting the point `p` is an injective function. | 185 | 186 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_right_cancel {p1 p2 p : P} (h : p -ᵥ p1 = p -ᵥ p2) : p1 = p2 :=
begin
refine vadd_left_cancel (p -ᵥ p2) _,
rw [vsub_vadd, ← h, vsub_vadd]
end | vsub_right_cancel {p1 p2 p : P} (h : p -ᵥ p1 = p -ᵥ p2) : p1 = p2 | begin
refine vadd_left_cancel (p -ᵥ p2) _,
rw [vsub_vadd, ← h, vsub_vadd]
end | lemma | vsub_right_cancel | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_vadd"
] | If subtracting two points from the same point produces equal
results, those points are equal. | 190 | 194 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_right_cancel_iff {p1 p2 p : P} : p -ᵥ p1 = p -ᵥ p2 ↔ p1 = p2 :=
⟨vsub_right_cancel, λ h, h ▸ rfl⟩ | vsub_right_cancel_iff {p1 p2 p : P} : p -ᵥ p1 = p -ᵥ p2 ↔ p1 = p2 | ⟨vsub_right_cancel, λ h, h ▸ rfl⟩ | lemma | vsub_right_cancel_iff | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | Subtracting two points from the same point produces equal results
if and only if those points are equal. | 198 | 199 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_right_injective (p : P) : function.injective ((-ᵥ) p : P → G) :=
λ p2 p3, vsub_right_cancel | vsub_right_injective (p : P) : function.injective ((-ᵥ) p : P → G) | λ p2 p3, vsub_right_cancel | lemma | vsub_right_injective | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_right_cancel"
] | Subtracting a point from the point `p` is an injective
function. | 203 | 204 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_sub_vsub_cancel_left (p1 p2 p3 : P) :
(p3 -ᵥ p2) - (p3 -ᵥ p1) = (p1 -ᵥ p2) :=
by rw [sub_eq_add_neg, neg_vsub_eq_vsub_rev, add_comm, vsub_add_vsub_cancel] | vsub_sub_vsub_cancel_left (p1 p2 p3 : P) :
(p3 -ᵥ p2) - (p3 -ᵥ p1) = (p1 -ᵥ p2) | by rw [sub_eq_add_neg, neg_vsub_eq_vsub_rev, add_comm, vsub_add_vsub_cancel] | lemma | vsub_sub_vsub_cancel_left | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"neg_vsub_eq_vsub_rev",
"vsub_add_vsub_cancel"
] | Cancellation subtracting the results of two subtractions. | 215 | 217 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_vsub_vadd_cancel_left (v : G) (p1 p2 : P) :
(v +ᵥ p1) -ᵥ (v +ᵥ p2) = p1 -ᵥ p2 :=
by rw [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_sub_cancel'] | vadd_vsub_vadd_cancel_left (v : G) (p1 p2 : P) :
(v +ᵥ p1) -ᵥ (v +ᵥ p2) = p1 -ᵥ p2 | by rw [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_sub_cancel'] | lemma | vadd_vsub_vadd_cancel_left | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_vsub_assoc",
"vsub_vadd_eq_vsub_sub"
] | null | 219 | 221 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_vadd_comm (p1 p2 p3 : P) : (p1 -ᵥ p2 : G) +ᵥ p3 = p3 -ᵥ p2 +ᵥ p1 :=
begin
rw [←@vsub_eq_zero_iff_eq G, vadd_vsub_assoc, vsub_vadd_eq_vsub_sub],
simp
end | vsub_vadd_comm (p1 p2 p3 : P) : (p1 -ᵥ p2 : G) +ᵥ p3 = p3 -ᵥ p2 +ᵥ p1 | begin
rw [←@vsub_eq_zero_iff_eq G, vadd_vsub_assoc, vsub_vadd_eq_vsub_sub],
simp
end | lemma | vsub_vadd_comm | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_vsub_assoc",
"vsub_eq_zero_iff_eq",
"vsub_vadd_eq_vsub_sub"
] | null | 223 | 227 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vadd_eq_vadd_iff_sub_eq_vsub {v₁ v₂ : G} {p₁ p₂ : P} :
v₁ +ᵥ p₁ = v₂ +ᵥ p₂ ↔ v₂ - v₁ = p₁ -ᵥ p₂ :=
by rw [vadd_eq_vadd_iff_neg_add_eq_vsub, neg_add_eq_sub] | vadd_eq_vadd_iff_sub_eq_vsub {v₁ v₂ : G} {p₁ p₂ : P} :
v₁ +ᵥ p₁ = v₂ +ᵥ p₂ ↔ v₂ - v₁ = p₁ -ᵥ p₂ | by rw [vadd_eq_vadd_iff_neg_add_eq_vsub, neg_add_eq_sub] | lemma | vadd_eq_vadd_iff_sub_eq_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vadd_eq_vadd_iff_neg_add_eq_vsub"
] | null | 229 | 231 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vsub_sub_vsub_comm (p₁ p₂ p₃ p₄ : P) :
(p₁ -ᵥ p₂) - (p₃ -ᵥ p₄) = (p₁ -ᵥ p₃) - (p₂ -ᵥ p₄) :=
by rw [← vsub_vadd_eq_vsub_sub, vsub_vadd_comm, vsub_vadd_eq_vsub_sub] | vsub_sub_vsub_comm (p₁ p₂ p₃ p₄ : P) :
(p₁ -ᵥ p₂) - (p₃ -ᵥ p₄) = (p₁ -ᵥ p₃) - (p₂ -ᵥ p₄) | by rw [← vsub_vadd_eq_vsub_sub, vsub_vadd_comm, vsub_vadd_eq_vsub_sub] | lemma | vsub_sub_vsub_comm | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_vadd_comm",
"vsub_vadd_eq_vsub_sub"
] | null | 233 | 235 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: add_torsor (G × G') (P × P') :=
{ vadd := λ v p, (v.1 +ᵥ p.1, v.2 +ᵥ p.2),
zero_vadd := λ p, by simp,
add_vadd := by simp [add_vadd],
vsub := λ p₁ p₂, (p₁.1 -ᵥ p₂.1, p₁.2 -ᵥ p₂.2),
nonempty := prod.nonempty,
vsub_vadd' := λ p₁ p₂, show (p₁.1 -ᵥ p₂.1 +ᵥ p₂.1, _) = p₁, by simp,
vadd_vsub' := λ v p, show (v.... | : add_torsor (G × G') (P × P') | { vadd := λ v p, (v.1 +ᵥ p.1, v.2 +ᵥ p.2),
zero_vadd := λ p, by simp,
add_vadd := by simp [add_vadd],
vsub := λ p₁ p₂, (p₁.1 -ᵥ p₂.1, p₁.2 -ᵥ p₂.2),
nonempty := prod.nonempty,
vsub_vadd' := λ p₁ p₂, show (p₁.1 -ᵥ p₂.1 +ᵥ p₂.1, _) = p₁, by simp,
vadd_vsub' := λ v p, show (v.1 +ᵥ p.1 -ᵥ p.1, v.2 +ᵥ p.2 -ᵥ p.2... | instance | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"add_torsor"
] | null | 244 | 251 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_vadd (v : G × G') (p : P × P') : (v +ᵥ p).1 = v.1 +ᵥ p.1 := rfl | fst_vadd (v : G × G') (p : P × P') : (v +ᵥ p).1 = v.1 +ᵥ p.1 | rfl | lemma | prod.fst_vadd | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 253 | 253 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_vadd (v : G × G') (p : P × P') : (v +ᵥ p).2 = v.2 +ᵥ p.2 := rfl | snd_vadd (v : G × G') (p : P × P') : (v +ᵥ p).2 = v.2 +ᵥ p.2 | rfl | lemma | prod.snd_vadd | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 254 | 254 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_vadd_mk (v : G) (v' : G') (p : P) (p' : P') :
(v, v') +ᵥ (p, p') = (v +ᵥ p, v' +ᵥ p') := rfl | mk_vadd_mk (v : G) (v' : G') (p : P) (p' : P') :
(v, v') +ᵥ (p, p') = (v +ᵥ p, v' +ᵥ p') | rfl | lemma | prod.mk_vadd_mk | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 255 | 256 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_vsub (p₁ p₂ : P × P') : (p₁ -ᵥ p₂ : G × G').1 = p₁.1 -ᵥ p₂.1 := rfl | fst_vsub (p₁ p₂ : P × P') : (p₁ -ᵥ p₂ : G × G').1 = p₁.1 -ᵥ p₂.1 | rfl | lemma | prod.fst_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 258 | 258 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_vsub (p₁ p₂ : P × P') : (p₁ -ᵥ p₂ : G × G').2 = p₁.2 -ᵥ p₂.2 := rfl | snd_vsub (p₁ p₂ : P × P') : (p₁ -ᵥ p₂ : G × G').2 = p₁.2 -ᵥ p₂.2 | rfl | lemma | prod.snd_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 259 | 259 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_vsub_mk (p₁ p₂ : P) (p₁' p₂' : P') :
((p₁, p₁') -ᵥ (p₂, p₂') : G × G') = (p₁ -ᵥ p₂, p₁' -ᵥ p₂') := rfl | mk_vsub_mk (p₁ p₂ : P) (p₁' p₂' : P') :
((p₁, p₁') -ᵥ (p₂, p₂') : G × G') = (p₁ -ᵥ p₂, p₁' -ᵥ p₂') | rfl | lemma | prod.mk_vsub_mk | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 260 | 261 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[T : ∀ i, add_torsor (fg i) (fp i)] : add_torsor (Π i, fg i) (Π i, fp i) :=
{ vadd := λ g p, λ i, g i +ᵥ p i,
zero_vadd := λ p, funext $ λ i, zero_vadd (fg i) (p i),
add_vadd := λ g₁ g₂ p, funext $ λ i, add_vadd (g₁ i) (g₂ i) (p i),
vsub := λ p₁ p₂, λ i, p₁ i -ᵥ p₂ i,
nonempty := ⟨λ i, classical.choice (T i).no... | [T : ∀ i, add_torsor (fg i) (fp i)] : add_torsor (Π i, fg i) (Π i, fp i) | { vadd := λ g p, λ i, g i +ᵥ p i,
zero_vadd := λ p, funext $ λ i, zero_vadd (fg i) (p i),
add_vadd := λ g₁ g₂ p, funext $ λ i, add_vadd (g₁ i) (g₂ i) (p i),
vsub := λ p₁ p₂, λ i, p₁ i -ᵥ p₂ i,
nonempty := ⟨λ i, classical.choice (T i).nonempty⟩,
vsub_vadd' := λ p₁ p₂, funext $ λ i, vsub_vadd (p₁ i) (p₂ i),
v... | instance | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"add_torsor",
"vadd_vsub",
"vsub_vadd"
] | A product of `add_torsor`s is an `add_torsor`. | 273 | 280 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_const (p : P) : G ≃ P :=
{ to_fun := λ v, v +ᵥ p,
inv_fun := λ p', p' -ᵥ p,
left_inv := λ v, vadd_vsub _ _,
right_inv := λ p', vsub_vadd _ _ } | vadd_const (p : P) : G ≃ P | { to_fun := λ v, v +ᵥ p,
inv_fun := λ p', p' -ᵥ p,
left_inv := λ v, vadd_vsub _ _,
right_inv := λ p', vsub_vadd _ _ } | def | equiv.vadd_const | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"inv_fun",
"vadd_vsub",
"vsub_vadd"
] | `v ↦ v +ᵥ p` as an equivalence. | 291 | 295 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_vadd_const (p : P) : ⇑(vadd_const p) = λ v, v+ᵥ p := rfl | coe_vadd_const (p : P) : ⇑(vadd_const p) = λ v, v+ᵥ p | rfl | lemma | equiv.coe_vadd_const | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 297 | 297 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_vadd_const_symm (p : P) : ⇑(vadd_const p).symm = λ p', p' -ᵥ p := rfl | coe_vadd_const_symm (p : P) : ⇑(vadd_const p).symm = λ p', p' -ᵥ p | rfl | lemma | equiv.coe_vadd_const_symm | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 299 | 299 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_vsub (p : P) : P ≃ G :=
{ to_fun := (-ᵥ) p,
inv_fun := λ v, -v +ᵥ p,
left_inv := λ p', by simp,
right_inv := λ v, by simp [vsub_vadd_eq_vsub_sub] } | const_vsub (p : P) : P ≃ G | { to_fun := (-ᵥ) p,
inv_fun := λ v, -v +ᵥ p,
left_inv := λ p', by simp,
right_inv := λ v, by simp [vsub_vadd_eq_vsub_sub] } | def | equiv.const_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"inv_fun",
"vsub_vadd_eq_vsub_sub"
] | `p' ↦ p -ᵥ p'` as an equivalence. | 302 | 306 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const_vsub (p : P) : ⇑(const_vsub p) = (-ᵥ) p := rfl | coe_const_vsub (p : P) : ⇑(const_vsub p) = (-ᵥ) p | rfl | lemma | equiv.coe_const_vsub | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 308 | 308 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const_vsub_symm (p : P) : ⇑(const_vsub p).symm = λ v, -v +ᵥ p := rfl | coe_const_vsub_symm (p : P) : ⇑(const_vsub p).symm = λ v, -v +ᵥ p | rfl | lemma | equiv.coe_const_vsub_symm | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 310 | 310 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_vadd (v : G) : equiv.perm P :=
{ to_fun := (+ᵥ) v,
inv_fun := (+ᵥ) (-v),
left_inv := λ p, by simp [vadd_vadd],
right_inv := λ p, by simp [vadd_vadd] } | const_vadd (v : G) : equiv.perm P | { to_fun := (+ᵥ) v,
inv_fun := (+ᵥ) (-v),
left_inv := λ p, by simp [vadd_vadd],
right_inv := λ p, by simp [vadd_vadd] } | def | equiv.const_vadd | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"equiv.perm",
"inv_fun"
] | The permutation given by `p ↦ v +ᵥ p`. | 315 | 319 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const_vadd (v : G) : ⇑(const_vadd P v) = (+ᵥ) v := rfl | coe_const_vadd (v : G) : ⇑(const_vadd P v) = (+ᵥ) v | rfl | lemma | equiv.coe_const_vadd | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 321 | 321 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_vadd_zero : const_vadd P (0:G) = 1 := ext $ zero_vadd G | const_vadd_zero : const_vadd P (0:G) = 1 | ext $ zero_vadd G | lemma | equiv.const_vadd_zero | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 325 | 325 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_vadd_add (v₁ v₂ : G) :
const_vadd P (v₁ + v₂) = const_vadd P v₁ * const_vadd P v₂ :=
ext $ add_vadd v₁ v₂ | const_vadd_add (v₁ v₂ : G) :
const_vadd P (v₁ + v₂) = const_vadd P v₁ * const_vadd P v₂ | ext $ add_vadd v₁ v₂ | lemma | equiv.const_vadd_add | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 329 | 331 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_vadd_hom : multiplicative G →* equiv.perm P :=
{ to_fun := λ v, const_vadd P v.to_add,
map_one' := const_vadd_zero G P,
map_mul' := const_vadd_add P } | const_vadd_hom : multiplicative G →* equiv.perm P | { to_fun := λ v, const_vadd P v.to_add,
map_one' := const_vadd_zero G P,
map_mul' := const_vadd_add P } | def | equiv.const_vadd_hom | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"equiv.perm",
"multiplicative"
] | `equiv.const_vadd` as a homomorphism from `multiplicative G` to `equiv.perm P` | 334 | 337 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection (x : P) : perm P := (const_vsub x).trans (vadd_const x) | point_reflection (x : P) : perm P | (const_vsub x).trans (vadd_const x) | def | equiv.point_reflection | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | Point reflection in `x` as a permutation. | 344 | 344 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection_apply (x y : P) : point_reflection x y = x -ᵥ y +ᵥ x := rfl | point_reflection_apply (x y : P) : point_reflection x y = x -ᵥ y +ᵥ x | rfl | lemma | equiv.point_reflection_apply | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 346 | 346 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection_symm (x : P) : (point_reflection x).symm = point_reflection x :=
ext $ by simp [point_reflection] | point_reflection_symm (x : P) : (point_reflection x).symm = point_reflection x | ext $ by simp [point_reflection] | lemma | equiv.point_reflection_symm | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [] | null | 348 | 349 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection_self (x : P) : point_reflection x x = x := vsub_vadd _ _ | point_reflection_self (x : P) : point_reflection x x = x | vsub_vadd _ _ | lemma | equiv.point_reflection_self | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"vsub_vadd"
] | null | 351 | 351 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection_involutive (x : P) : involutive (point_reflection x : P → P) :=
λ y, (equiv.apply_eq_iff_eq_symm_apply _).2 $ by rw point_reflection_symm | point_reflection_involutive (x : P) : involutive (point_reflection x : P → P) | λ y, (equiv.apply_eq_iff_eq_symm_apply _).2 $ by rw point_reflection_symm | lemma | equiv.point_reflection_involutive | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"equiv.apply_eq_iff_eq_symm_apply",
"involutive"
] | null | 353 | 354 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
point_reflection_fixed_iff_of_injective_bit0 {x y : P} (h : injective (bit0 : G → G)) :
point_reflection x y = y ↔ y = x :=
by rw [point_reflection_apply, eq_comm, eq_vadd_iff_vsub_eq, ← neg_vsub_eq_vsub_rev,
neg_eq_iff_add_eq_zero, ← bit0, ← bit0_zero, h.eq_iff, vsub_eq_zero_iff_eq, eq_comm] | point_reflection_fixed_iff_of_injective_bit0 {x y : P} (h : injective (bit0 : G → G)) :
point_reflection x y = y ↔ y = x | by rw [point_reflection_apply, eq_comm, eq_vadd_iff_vsub_eq, ← neg_vsub_eq_vsub_rev,
neg_eq_iff_add_eq_zero, ← bit0, ← bit0_zero, h.eq_iff, vsub_eq_zero_iff_eq, eq_comm] | lemma | equiv.point_reflection_fixed_iff_of_injective_bit0 | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"bit0_zero",
"eq_vadd_iff_vsub_eq",
"neg_vsub_eq_vsub_rev",
"vsub_eq_zero_iff_eq"
] | `x` is the only fixed point of `point_reflection x`. This lemma requires
`x + x = y + y ↔ x = y`. There is no typeclass to use here, so we add it as an explicit argument. | 358 | 361 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_point_reflection_left_of_injective_bit0 {G P : Type*} [add_comm_group G]
[add_torsor G P] (h : injective (bit0 : G → G)) (y : P) :
injective (λ x : P, point_reflection x y) :=
λ x₁ x₂ (hy : point_reflection x₁ y = point_reflection x₂ y),
by rwa [point_reflection_apply, point_reflection_apply, vadd_eq_va... | injective_point_reflection_left_of_injective_bit0 {G P : Type*} [add_comm_group G]
[add_torsor G P] (h : injective (bit0 : G → G)) (y : P) :
injective (λ x : P, point_reflection x y) | λ x₁ x₂ (hy : point_reflection x₁ y = point_reflection x₂ y),
by rwa [point_reflection_apply, point_reflection_apply, vadd_eq_vadd_iff_sub_eq_vsub,
vsub_sub_vsub_cancel_right, ← neg_vsub_eq_vsub_rev, neg_eq_iff_add_eq_zero, ← bit0, ← bit0_zero,
h.eq_iff, vsub_eq_zero_iff_eq] at hy | lemma | equiv.injective_point_reflection_left_of_injective_bit0 | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"add_comm_group",
"add_torsor",
"bit0_zero",
"neg_vsub_eq_vsub_rev",
"vadd_eq_vadd_iff_sub_eq_vsub",
"vsub_eq_zero_iff_eq",
"vsub_sub_vsub_cancel_right"
] | null | 365 | 371 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_torsor.subsingleton_iff (G P : Type*) [add_group G] [add_torsor G P] :
subsingleton G ↔ subsingleton P :=
begin
inhabit P,
exact (equiv.vadd_const default).subsingleton_congr,
end | add_torsor.subsingleton_iff (G P : Type*) [add_group G] [add_torsor G P] :
subsingleton G ↔ subsingleton P | begin
inhabit P,
exact (equiv.vadd_const default).subsingleton_congr,
end | lemma | add_torsor.subsingleton_iff | algebra | src/algebra/add_torsor.lean | [
"data.set.pointwise.smul"
] | [
"add_group",
"add_torsor",
"equiv.vadd_const"
] | null | 375 | 380 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infinite_of_char_zero (R A : Type*) [comm_ring R] [is_domain R]
[ring A] [algebra R A] [char_zero A] : {x : A | is_algebraic R x}.infinite :=
infinite_of_injective_forall_mem nat.cast_injective is_algebraic_nat | infinite_of_char_zero (R A : Type*) [comm_ring R] [is_domain R]
[ring A] [algebra R A] [char_zero A] : {x : A | is_algebraic R x}.infinite | infinite_of_injective_forall_mem nat.cast_injective is_algebraic_nat | lemma | algebraic.infinite_of_char_zero | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"algebra",
"char_zero",
"comm_ring",
"infinite",
"is_algebraic",
"is_algebraic_nat",
"is_domain",
"nat.cast_injective",
"ring"
] | null | 30 | 32 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph_0_le_cardinal_mk_of_char_zero (R A : Type*) [comm_ring R] [is_domain R]
[ring A] [algebra R A] [char_zero A] : ℵ₀ ≤ #{x : A // is_algebraic R x} :=
infinite_iff.1 (set.infinite_coe_iff.2 $ infinite_of_char_zero R A) | aleph_0_le_cardinal_mk_of_char_zero (R A : Type*) [comm_ring R] [is_domain R]
[ring A] [algebra R A] [char_zero A] : ℵ₀ ≤ #{x : A // is_algebraic R x} | infinite_iff.1 (set.infinite_coe_iff.2 $ infinite_of_char_zero R A) | theorem | algebraic.aleph_0_le_cardinal_mk_of_char_zero | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"algebra",
"char_zero",
"comm_ring",
"is_algebraic",
"is_domain",
"ring"
] | null | 34 | 36 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal_mk_lift_le_mul :
cardinal.lift.{u} (#{x : A // is_algebraic R x}) ≤ cardinal.lift.{v} #(R[X]) * ℵ₀ :=
begin
rw [←mk_ulift, ←mk_ulift],
choose g hg₁ hg₂ using λ x : {x : A | is_algebraic R x}, x.coe_prop,
refine lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g (λ f, _),
rw [lift_le_aleph_0, le_aleph_0_... | cardinal_mk_lift_le_mul :
cardinal.lift.{u} (#{x : A // is_algebraic R x}) ≤ cardinal.lift.{v} #(R[X]) * ℵ₀ | begin
rw [←mk_ulift, ←mk_ulift],
choose g hg₁ hg₂ using λ x : {x : A | is_algebraic R x}, x.coe_prop,
refine lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g (λ f, _),
rw [lift_le_aleph_0, le_aleph_0_iff_set_countable],
suffices : maps_to coe (g ⁻¹' {f}) (f.root_set A),
from this.countable_of_inj_on (subty... | theorem | algebraic.cardinal_mk_lift_le_mul | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"countable",
"is_algebraic"
] | null | 43 | 54 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal_mk_lift_le_max :
cardinal.lift.{u} (#{x : A // is_algebraic R x}) ≤ max (cardinal.lift.{v} (#R)) ℵ₀ :=
(cardinal_mk_lift_le_mul R A).trans $
(mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans $ by simp | cardinal_mk_lift_le_max :
cardinal.lift.{u} (#{x : A // is_algebraic R x}) ≤ max (cardinal.lift.{v} (#R)) ℵ₀ | (cardinal_mk_lift_le_mul R A).trans $
(mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans $ by simp | theorem | algebraic.cardinal_mk_lift_le_max | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"is_algebraic",
"mul_le_mul_right'"
] | null | 56 | 59 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal_mk_lift_of_infinite [infinite R] :
cardinal.lift.{u} (#{x : A // is_algebraic R x}) = cardinal.lift.{v} (#R) :=
((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left $ aleph_0_le_mk _)).antisymm $
lift_mk_le'.2 ⟨⟨λ x, ⟨algebra_map R A x, is_algebraic_algebra_map _⟩,
λ x y h, no_zero_smul_divisors.algebr... | cardinal_mk_lift_of_infinite [infinite R] :
cardinal.lift.{u} (#{x : A // is_algebraic R x}) = cardinal.lift.{v} (#R) | ((cardinal_mk_lift_le_max R A).trans_eq (max_eq_left $ aleph_0_le_mk _)).antisymm $
lift_mk_le'.2 ⟨⟨λ x, ⟨algebra_map R A x, is_algebraic_algebra_map _⟩,
λ x y h, no_zero_smul_divisors.algebra_map_injective R A (subtype.ext_iff.1 h)⟩⟩ | lemma | algebraic.cardinal_mk_lift_of_infinite | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"infinite",
"is_algebraic",
"is_algebraic_algebra_map",
"no_zero_smul_divisors.algebra_map_injective"
] | null | 61 | 65 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable : set.countable {x : A | is_algebraic R x} :=
begin
rw [←le_aleph_0_iff_set_countable, ←lift_le],
apply (cardinal_mk_lift_le_max R A).trans,
simp
end | countable : set.countable {x : A | is_algebraic R x} | begin
rw [←le_aleph_0_iff_set_countable, ←lift_le],
apply (cardinal_mk_lift_le_max R A).trans,
simp
end | theorem | algebraic.countable | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"countable",
"is_algebraic",
"set.countable"
] | null | 69 | 74 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal_mk_of_countble_of_char_zero [char_zero A] [is_domain R] :
#{x : A // is_algebraic R x} = ℵ₀ :=
(algebraic.countable R A).le_aleph_0.antisymm (aleph_0_le_cardinal_mk_of_char_zero R A) | cardinal_mk_of_countble_of_char_zero [char_zero A] [is_domain R] :
#{x : A // is_algebraic R x} = ℵ₀ | (algebraic.countable R A).le_aleph_0.antisymm (aleph_0_le_cardinal_mk_of_char_zero R A) | theorem | algebraic.cardinal_mk_of_countble_of_char_zero | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"algebraic.countable",
"char_zero",
"is_algebraic",
"is_domain"
] | null | 76 | 78 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal_mk_le_mul : #{x : A // is_algebraic R x} ≤ #R[X] * ℵ₀ :=
by { rw [←lift_id (#_), ←lift_id #R[X]], exact cardinal_mk_lift_le_mul R A } | cardinal_mk_le_mul : #{x : A // is_algebraic R x} ≤ #R[X] * ℵ₀ | by { rw [←lift_id (#_), ←lift_id #R[X]], exact cardinal_mk_lift_le_mul R A } | theorem | algebraic.cardinal_mk_le_mul | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"is_algebraic"
] | null | 87 | 88 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal_mk_le_max : #{x : A // is_algebraic R x} ≤ max (#R) ℵ₀ :=
by { rw [←lift_id (#_), ←lift_id (#R)], exact cardinal_mk_lift_le_max R A } | cardinal_mk_le_max : #{x : A // is_algebraic R x} ≤ max (#R) ℵ₀ | by { rw [←lift_id (#_), ←lift_id (#R)], exact cardinal_mk_lift_le_max R A } | theorem | algebraic.cardinal_mk_le_max | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"is_algebraic"
] | null | 90 | 91 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cardinal_mk_of_infinite [infinite R] : #{x : A // is_algebraic R x} = #R :=
lift_inj.1 $ cardinal_mk_lift_of_infinite R A | cardinal_mk_of_infinite [infinite R] : #{x : A // is_algebraic R x} = #R | lift_inj.1 $ cardinal_mk_lift_of_infinite R A | theorem | algebraic.cardinal_mk_of_infinite | algebra | src/algebra/algebraic_card.lean | [
"data.polynomial.cardinal",
"ring_theory.algebraic"
] | [
"infinite",
"is_algebraic"
] | null | 93 | 94 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime (p : α) : Prop :=
p ≠ 0 ∧ ¬ is_unit p ∧ (∀a b, p ∣ a * b → p ∣ a ∨ p ∣ b) | prime (p : α) : Prop | p ≠ 0 ∧ ¬ is_unit p ∧ (∀a b, p ∣ a * b → p ∣ a ∨ p ∣ b) | def | prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit"
] | prime element of a `comm_monoid_with_zero` | 23 | 24 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero : p ≠ 0 :=
hp.1 | ne_zero : p ≠ 0 | hp.1 | lemma | prime.ne_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"ne_zero"
] | null | 30 | 31 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_unit : ¬ is_unit p :=
hp.2.1 | not_unit : ¬ is_unit p | hp.2.1 | lemma | prime.not_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit"
] | null | 33 | 34 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_dvd_one : ¬ p ∣ 1 :=
mt (is_unit_of_dvd_one _) hp.not_unit | not_dvd_one : ¬ p ∣ 1 | mt (is_unit_of_dvd_one _) hp.not_unit | lemma | prime.not_dvd_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit_of_dvd_one"
] | null | 36 | 37 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_one : p ≠ 1 :=
λ h, hp.2.1 (h.symm ▸ is_unit_one) | ne_one : p ≠ 1 | λ h, hp.2.1 (h.symm ▸ is_unit_one) | lemma | prime.ne_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit_one"
] | null | 39 | 40 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_or_dvd (hp : prime p) {a b : α} (h : p ∣ a * b) :
p ∣ a ∨ p ∣ b :=
hp.2.2 a b h | dvd_or_dvd (hp : prime p) {a b : α} (h : p ∣ a * b) :
p ∣ a ∨ p ∣ b | hp.2.2 a b h | lemma | prime.dvd_or_dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"prime"
] | null | 42 | 44 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_of_dvd_pow (hp : prime p) {a : α} {n : ℕ} (h : p ∣ a^n) :
p ∣ a :=
begin
induction n with n ih,
{ rw pow_zero at h,
have := is_unit_of_dvd_one _ h,
have := not_unit hp,
contradiction },
rw pow_succ at h,
cases dvd_or_dvd hp h with dvd_a dvd_pow,
{ assumption },
exact ih dvd_pow
end | dvd_of_dvd_pow (hp : prime p) {a : α} {n : ℕ} (h : p ∣ a^n) :
p ∣ a | begin
induction n with n ih,
{ rw pow_zero at h,
have := is_unit_of_dvd_one _ h,
have := not_unit hp,
contradiction },
rw pow_succ at h,
cases dvd_or_dvd hp h with dvd_a dvd_pow,
{ assumption },
exact ih dvd_pow
end | lemma | prime.dvd_of_dvd_pow | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"dvd_pow",
"ih",
"is_unit_of_dvd_one",
"pow_succ",
"pow_zero",
"prime"
] | null | 46 | 58 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_prime_zero : ¬ prime (0 : α) :=
λ h, h.ne_zero rfl | not_prime_zero : ¬ prime (0 : α) | λ h, h.ne_zero rfl | lemma | not_prime_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"prime"
] | null | 62 | 63 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_prime_one : ¬ prime (1 : α) :=
λ h, h.not_unit is_unit_one | not_prime_one : ¬ prime (1 : α) | λ h, h.not_unit is_unit_one | lemma | not_prime_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit_one",
"prime"
] | null | 65 | 66 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_prime (hinv : ∀ a, g (f a : β) = a) (hp : prime (f p)) : prime p :=
⟨ λ h, hp.1 $ by simp [h], λ h, hp.2.1 $ h.map f, λ a b h, by
{ refine (hp.2.2 (f a) (f b) $ by { convert map_dvd f h, simp }).imp _ _;
{ intro h, convert ← map_dvd g h; apply hinv } } ⟩ | comap_prime (hinv : ∀ a, g (f a : β) = a) (hp : prime (f p)) : prime p | ⟨ λ h, hp.1 $ by simp [h], λ h, hp.2.1 $ h.map f, λ a b h, by
{ refine (hp.2.2 (f a) (f b) $ by { convert map_dvd f h, simp }).imp _ _;
{ intro h, convert ← map_dvd g h; apply hinv } } ⟩ | lemma | comap_prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"map_dvd",
"prime"
] | null | 72 | 75 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.prime_iff (e : α ≃* β) : prime p ↔ prime (e p) :=
⟨ λ h, comap_prime e.symm e (λ a, by simp) $ (e.symm_apply_apply p).substr h,
comap_prime e e.symm (λ a, by simp) ⟩ | mul_equiv.prime_iff (e : α ≃* β) : prime p ↔ prime (e p) | ⟨ λ h, comap_prime e.symm e (λ a, by simp) $ (e.symm_apply_apply p).substr h,
comap_prime e e.symm (λ a, by simp) ⟩ | lemma | mul_equiv.prime_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comap_prime",
"prime"
] | null | 77 | 79 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.left_dvd_or_dvd_right_of_dvd_mul [cancel_comm_monoid_with_zero α] {p : α}
(hp : prime p) {a b : α} : a ∣ p * b → p ∣ a ∨ a ∣ b :=
begin
rintro ⟨c, hc⟩,
rcases hp.2.2 a c (hc ▸ dvd_mul_right _ _) with h | ⟨x, rfl⟩,
{ exact or.inl h },
{ rw [mul_left_comm, mul_right_inj' hp.ne_zero] at hc,
exact or.in... | prime.left_dvd_or_dvd_right_of_dvd_mul [cancel_comm_monoid_with_zero α] {p : α}
(hp : prime p) {a b : α} : a ∣ p * b → p ∣ a ∨ a ∣ b | begin
rintro ⟨c, hc⟩,
rcases hp.2.2 a c (hc ▸ dvd_mul_right _ _) with h | ⟨x, rfl⟩,
{ exact or.inl h },
{ rw [mul_left_comm, mul_right_inj' hp.ne_zero] at hc,
exact or.inr (hc.symm ▸ dvd_mul_right _ _) }
end | lemma | prime.left_dvd_or_dvd_right_of_dvd_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"dvd_mul_right",
"mul_left_comm",
"mul_right_inj'",
"prime"
] | null | 85 | 93 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.pow_dvd_of_dvd_mul_left
[cancel_comm_monoid_with_zero α]
{p a b : α} (hp : prime p) (n : ℕ) (h : ¬p ∣ a) (h' : p ^ n ∣ a * b) : p ^ n ∣ b :=
begin
induction n with n ih,
{ rw pow_zero, exact one_dvd b },
{ obtain ⟨c, rfl⟩ := ih (dvd_trans (pow_dvd_pow p n.le_succ) h'),
rw pow_succ',
apply mul_dv... | prime.pow_dvd_of_dvd_mul_left
[cancel_comm_monoid_with_zero α]
{p a b : α} (hp : prime p) (n : ℕ) (h : ¬p ∣ a) (h' : p ^ n ∣ a * b) : p ^ n ∣ b | begin
induction n with n ih,
{ rw pow_zero, exact one_dvd b },
{ obtain ⟨c, rfl⟩ := ih (dvd_trans (pow_dvd_pow p n.le_succ) h'),
rw pow_succ',
apply mul_dvd_mul_left _ ((hp.dvd_or_dvd _).resolve_left h),
rwa [←mul_dvd_mul_iff_left (pow_ne_zero n hp.ne_zero), ←pow_succ', mul_left_comm] }
end | lemma | prime.pow_dvd_of_dvd_mul_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"dvd_trans",
"ih",
"mul_dvd_mul_left",
"mul_left_comm",
"one_dvd",
"pow_dvd_pow",
"pow_ne_zero",
"pow_succ'",
"pow_zero",
"prime"
] | null | 95 | 105 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.pow_dvd_of_dvd_mul_right
[cancel_comm_monoid_with_zero α]
{p a b : α} (hp : prime p) (n : ℕ) (h : ¬p ∣ b) (h' : p ^ n ∣ a * b) : p ^ n ∣ a :=
by { rw [mul_comm] at h', exact hp.pow_dvd_of_dvd_mul_left n h h' } | prime.pow_dvd_of_dvd_mul_right
[cancel_comm_monoid_with_zero α]
{p a b : α} (hp : prime p) (n : ℕ) (h : ¬p ∣ b) (h' : p ^ n ∣ a * b) : p ^ n ∣ a | by { rw [mul_comm] at h', exact hp.pow_dvd_of_dvd_mul_left n h h' } | lemma | prime.pow_dvd_of_dvd_mul_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"mul_comm",
"prime"
] | null | 107 | 110 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd [cancel_comm_monoid_with_zero α]
{p a b : α} {n : ℕ} (hp : prime p) (hpow : p ^ n.succ ∣ a ^ n.succ * b ^ n)
(hb : ¬ p ^ 2 ∣ b) : p ∣ a :=
begin
-- Suppose `p ∣ b`, write `b = p * x` and `hy : a ^ n.succ * b ^ n = p ^ n.succ * y`.
cases (hp.dvd_or_dvd ((dvd_pow... | prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd [cancel_comm_monoid_with_zero α]
{p a b : α} {n : ℕ} (hp : prime p) (hpow : p ^ n.succ ∣ a ^ n.succ * b ^ n)
(hb : ¬ p ^ 2 ∣ b) : p ∣ a | begin
-- Suppose `p ∣ b`, write `b = p * x` and `hy : a ^ n.succ * b ^ n = p ^ n.succ * y`.
cases (hp.dvd_or_dvd ((dvd_pow_self p (nat.succ_ne_zero n)).trans hpow)) with H hbdiv,
{ exact hp.dvd_of_dvd_pow H },
obtain ⟨x, rfl⟩ := hp.dvd_of_dvd_pow hbdiv,
obtain ⟨y, hy⟩ := hpow,
-- Then we can divide out a co... | lemma | prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"dvd_mul_right",
"dvd_pow_self",
"mul_assoc",
"mul_comm",
"mul_left_cancel₀",
"mul_pow",
"pow_ne_zero",
"pow_succ'",
"pow_two",
"prime"
] | null | 112 | 131 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_pow_succ_dvd_mul {α : Type*} [cancel_comm_monoid_with_zero α]
{p x y : α} (h : prime p) {i : ℕ} (hxy : p ^ (i + 1) ∣ x * y) :
p ^ (i + 1) ∣ x ∨ p ∣ y :=
begin
rw or_iff_not_imp_right,
intro hy,
induction i with i ih generalizing x,
{ simp only [zero_add, pow_one] at *,
exact (h.dvd_or_dvd hxy).res... | prime_pow_succ_dvd_mul {α : Type*} [cancel_comm_monoid_with_zero α]
{p x y : α} (h : prime p) {i : ℕ} (hxy : p ^ (i + 1) ∣ x * y) :
p ^ (i + 1) ∣ x ∨ p ∣ y | begin
rw or_iff_not_imp_right,
intro hy,
induction i with i ih generalizing x,
{ simp only [zero_add, pow_one] at *,
exact (h.dvd_or_dvd hxy).resolve_right hy },
rw pow_succ at hxy ⊢,
obtain ⟨x', rfl⟩ := (h.dvd_or_dvd (dvd_of_mul_right_dvd hxy)).resolve_right hy,
rw mul_assoc at hxy,
exact mul_dvd_m... | lemma | prime_pow_succ_dvd_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"dvd_of_mul_right_dvd",
"ih",
"mul_assoc",
"mul_dvd_mul_iff_left",
"mul_dvd_mul_left",
"or_iff_not_imp_right",
"pow_one",
"pow_succ",
"prime"
] | null | 133 | 146 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible [monoid α] (p : α) : Prop :=
(not_unit : ¬ is_unit p)
(is_unit_or_is_unit' : ∀a b, p = a * b → is_unit a ∨ is_unit b) | irreducible [monoid α] (p : α) : Prop | (not_unit : ¬ is_unit p)
(is_unit_or_is_unit' : ∀a b, p = a * b → is_unit a ∨ is_unit b) | structure | irreducible | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"is_unit",
"monoid"
] | `irreducible p` states that `p` is non-unit and only factors into units.
We explicitly avoid stating that `p` is non-zero, this would require a semiring. Assuming only a
monoid allows us to reuse irreducible for associated elements. | 153 | 155 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_dvd_one [comm_monoid α] {p : α} (hp : irreducible p) : ¬ p ∣ 1 :=
mt (is_unit_of_dvd_one _) hp.not_unit | not_dvd_one [comm_monoid α] {p : α} (hp : irreducible p) : ¬ p ∣ 1 | mt (is_unit_of_dvd_one _) hp.not_unit | lemma | irreducible.not_dvd_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid",
"irreducible",
"is_unit_of_dvd_one"
] | null | 159 | 160 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_or_is_unit [monoid α] {p : α} (hp : irreducible p) {a b : α} (h : p = a * b) :
is_unit a ∨ is_unit b :=
hp.is_unit_or_is_unit' a b h | is_unit_or_is_unit [monoid α] {p : α} (hp : irreducible p) {a b : α} (h : p = a * b) :
is_unit a ∨ is_unit b | hp.is_unit_or_is_unit' a b h | lemma | irreducible.is_unit_or_is_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"is_unit",
"monoid"
] | null | 162 | 164 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_iff [monoid α] {p : α} :
irreducible p ↔ ¬ is_unit p ∧ ∀a b, p = a * b → is_unit a ∨ is_unit b :=
⟨λ h, ⟨h.1, h.2⟩, λ h, ⟨h.1, h.2⟩⟩ | irreducible_iff [monoid α] {p : α} :
irreducible p ↔ ¬ is_unit p ∧ ∀a b, p = a * b → is_unit a ∨ is_unit b | ⟨λ h, ⟨h.1, h.2⟩, λ h, ⟨h.1, h.2⟩⟩ | lemma | irreducible_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"is_unit",
"monoid"
] | null | 168 | 170 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_irreducible_one [monoid α] : ¬ irreducible (1 : α) :=
by simp [irreducible_iff] | not_irreducible_one [monoid α] : ¬ irreducible (1 : α) | by simp [irreducible_iff] | theorem | not_irreducible_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"irreducible_iff",
"monoid"
] | null | 172 | 173 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible.ne_one [monoid α] : ∀ {p:α}, irreducible p → p ≠ 1
| _ hp rfl := not_irreducible_one hp | irreducible.ne_one [monoid α] : ∀ {p:α}, irreducible p → p ≠ 1
| _ hp rfl | not_irreducible_one hp | theorem | irreducible.ne_one | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"monoid",
"not_irreducible_one"
] | null | 175 | 176 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_irreducible_zero [monoid_with_zero α] : ¬ irreducible (0 : α)
| ⟨hn0, h⟩ := have is_unit (0:α) ∨ is_unit (0:α), from h 0 0 ((mul_zero 0).symm),
this.elim hn0 hn0 | not_irreducible_zero [monoid_with_zero α] : ¬ irreducible (0 : α)
| ⟨hn0, h⟩ | have is_unit (0:α) ∨ is_unit (0:α), from h 0 0 ((mul_zero 0).symm),
this.elim hn0 hn0 | theorem | not_irreducible_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"is_unit",
"monoid_with_zero",
"mul_zero"
] | null | 178 | 180 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible.ne_zero [monoid_with_zero α] : ∀ {p:α}, irreducible p → p ≠ 0
| _ hp rfl := not_irreducible_zero hp | irreducible.ne_zero [monoid_with_zero α] : ∀ {p:α}, irreducible p → p ≠ 0
| _ hp rfl | not_irreducible_zero hp | theorem | irreducible.ne_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"monoid_with_zero",
"not_irreducible_zero"
] | null | 182 | 183 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_irreducible_mul {α} [monoid α] {x y : α} :
irreducible (x * y) → is_unit x ∨ is_unit y
| ⟨_, h⟩ := h _ _ rfl | of_irreducible_mul {α} [monoid α] {x y : α} :
irreducible (x * y) → is_unit x ∨ is_unit y
| ⟨_, h⟩ | h _ _ rfl | theorem | of_irreducible_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"is_unit",
"monoid"
] | null | 185 | 187 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_irreducible_pow {α} [monoid α] {x : α} {n : ℕ} (hn : n ≠ 1) :
irreducible (x ^ n) → is_unit x :=
begin
obtain hn|hn := hn.lt_or_lt,
{ simp only [nat.lt_one_iff.mp hn, is_empty.forall_iff, not_irreducible_one, pow_zero] },
intro h,
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_lt hn,
rw [pow_succ, add_comm] at ... | of_irreducible_pow {α} [monoid α] {x : α} {n : ℕ} (hn : n ≠ 1) :
irreducible (x ^ n) → is_unit x | begin
obtain hn|hn := hn.lt_or_lt,
{ simp only [nat.lt_one_iff.mp hn, is_empty.forall_iff, not_irreducible_one, pow_zero] },
intro h,
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_lt hn,
rw [pow_succ, add_comm] at h,
exact (or_iff_left_of_imp is_unit_pow_succ_iff.mp).mp (of_irreducible_mul h)
end | theorem | of_irreducible_pow | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"irreducible",
"is_empty.forall_iff",
"is_unit",
"monoid",
"nat.exists_eq_add_of_lt",
"not_irreducible_one",
"of_irreducible_mul",
"pow_succ",
"pow_zero"
] | null | 189 | 198 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_or_factor {α} [monoid α] (x : α) (h : ¬ is_unit x) :
irreducible x ∨ ∃ a b, ¬ is_unit a ∧ ¬ is_unit b ∧ a * b = x :=
begin
haveI := classical.dec,
refine or_iff_not_imp_right.2 (λ H, _),
simp [h, irreducible_iff] at H ⊢,
refine λ a b h, classical.by_contradiction $ λ o, _,
simp [not_or_distrib] ... | irreducible_or_factor {α} [monoid α] (x : α) (h : ¬ is_unit x) :
irreducible x ∨ ∃ a b, ¬ is_unit a ∧ ¬ is_unit b ∧ a * b = x | begin
haveI := classical.dec,
refine or_iff_not_imp_right.2 (λ H, _),
simp [h, irreducible_iff] at H ⊢,
refine λ a b h, classical.by_contradiction $ λ o, _,
simp [not_or_distrib] at o,
exact H _ o.1 _ o.2 h.symm
end | theorem | irreducible_or_factor | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"classical.dec",
"irreducible",
"irreducible_iff",
"is_unit",
"monoid",
"not_or_distrib"
] | null | 200 | 209 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
End of preview. Expand in Data Studio
Lean3-Mathlib
Declarations from mathlib3, the Lean 3 mathematical library (archived).
Source
- Repository: https://github.com/leanprover-community/mathlib
- Commit:
65a1391a0106c9204fe45bc73a039f056558cb83 - Files: 3220
- License: apache-2.0
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Verbatim declaration with the leading keyword removed: signature and body/proof joined |
| statement | string | Signature with the leading keyword removed (verbatim slice) |
| proof | string | Verbatim proof/body, empty if none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, null if absent |
| line_start | int | First source line |
| line_end | int | Last source line |
| has_proof | bool | Whether a proof block was captured |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 133,201
- With proof: 132,770 (99.7%)
- With docstring: 35,100 (26.4%)
- Libraries: 417
By type
| Type | Count |
|---|---|
| lemma | 83,206 |
| def | 19,387 |
| theorem | 14,273 |
| instance | 13,632 |
| class | 1,010 |
| structure | 732 |
| abbreviation | 496 |
| inductive | 252 |
| example | 199 |
| definition | 10 |
| class inductive | 3 |
| macro | 1 |
Example
add_torsor (G : out_param Type*) (P : Type*) [out_param $ add_group G]
extends add_action G P, has_vsub G P :=
[nonempty : nonempty P]
(vsub_vadd' : ∀ (p1 p2 : P), (p1 -ᵥ p2 : G) +ᵥ p2 = p1)
(vadd_vsub' : ∀ (g : G) (p : P), g +ᵥ p -ᵥ p = g)
- type: class | symbolic_name:
add_torsor| src/algebra/add_torsor.lean:48
Use
Statement and proof are available both joined (fact) and split (statement, proof) for
proof-term modeling, autoformalization, retrieval, and dependency analysis via deps.
Citation
@misc{lean3_mathlib_dataset,
title = {Lean3-Mathlib},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/leanprover-community/mathlib, commit 65a1391a0106},
url = {https://huggingface.co/datasets/phanerozoic/Lean3-Mathlib}
}
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