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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coe_id : ⇑(centroid_hom.id α) = id | rfl | lemma | centroid_hom.coe_id | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_monoid_hom_id :
(centroid_hom.id α : α →+ α) = add_monoid_hom.id α | rfl | lemma | centroid_hom.coe_to_add_monoid_hom_id | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : centroid_hom.id α a = a | rfl | lemma | centroid_hom.id_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g f : centroid_hom α) : centroid_hom α | { map_mul_left' := λ a b, (congr_arg g $ f.map_mul_left' _ _).trans $ g.map_mul_left' _ _,
map_mul_right' := λ a b, (congr_arg g $ f.map_mul_right' _ _).trans $ g.map_mul_right' _ _,
.. g.to_add_monoid_hom.comp f.to_add_monoid_hom } | def | centroid_hom.comp | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | Composition of `centroid_hom`s as a `centroid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (g f : centroid_hom α) : ⇑(g.comp f) = g ∘ f | rfl | lemma | centroid_hom.coe_comp | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (g f : centroid_hom α) (a : α) : g.comp f a = g (f a) | rfl | lemma | centroid_hom.comp_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_add_monoid_hom (g f : centroid_hom α) :
(g.comp f : α →+ α) = (g : α →+ α).comp f | rfl | lemma | centroid_hom.coe_comp_add_monoid_hom | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (h g f : centroid_hom α) : (h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | centroid_hom.comp_assoc | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : centroid_hom α) : f.comp (centroid_hom.id α) = f | ext $ λ a, rfl | lemma | centroid_hom.comp_id | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"centroid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : centroid_hom α) : (centroid_hom.id α).comp f = f | ext $ λ a, rfl | lemma | centroid_hom.id_comp | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"centroid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ f : centroid_hom α} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | centroid_hom.cancel_right | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g f₁ f₂ : centroid_hom α} (hg : injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | centroid_hom.cancel_left | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nsmul : has_smul ℕ (centroid_hom α) | ⟨λ n f,
{ map_mul_left' := λ a b,
by { change n • f (a * b) = a * n • f b, rw [map_mul_left f, ←mul_smul_comm] },
map_mul_right' := λ a b,
by { change n • f (a * b) = n • f a * b, rw [map_mul_right f, ←smul_mul_assoc] },
.. (n • f : α →+ α) }⟩ | instance | centroid_hom.has_nsmul | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_npow_nat : has_pow (centroid_hom α) ℕ | ⟨λ f n,
{ map_mul_left' := λ a b, begin
induction n with n ih,
{ simp },
{ rw pow_succ,
exact (congr_arg f.to_End ih).trans (f.map_mul_left' _ _) }
end,
map_mul_right' := λ a b, begin
induction n with n ih,
{ simp },
{ rw pow_succ,
exact (congr_arg f.to_End ih).trans (f.map_mul_... | instance | centroid_hom.has_npow_nat | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"add_monoid.End",
"centroid_hom",
"ih",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ⇑(0 : centroid_hom α) = 0 | rfl | lemma | centroid_hom.coe_zero | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : centroid_hom α) = id | rfl | lemma | centroid_hom.coe_one | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (f g : centroid_hom α) : ⇑(f + g) = f + g | rfl | lemma | centroid_hom.coe_add | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : centroid_hom α) : ⇑(f * g) = f ∘ g | rfl | lemma | centroid_hom.coe_mul | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul (f : centroid_hom α) (n : ℕ) :
⇑(n • f) = n • f | rfl | lemma | centroid_hom.coe_nsmul | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (a : α) : (0 : centroid_hom α) a = 0 | rfl | lemma | centroid_hom.zero_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (a : α) : (1 : centroid_hom α) a = a | rfl | lemma | centroid_hom.one_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (f g : centroid_hom α) (a : α) : (f + g) a = f a + g a | rfl | lemma | centroid_hom.add_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (f g : centroid_hom α) (a : α) : (f * g) a = f (g a) | rfl | lemma | centroid_hom.mul_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_apply (f : centroid_hom α) (n : ℕ) (a : α) : (n • f) a = n • f a | rfl | lemma | centroid_hom.nsmul_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_zero : (0 : centroid_hom α).to_End = 0 | rfl | lemma | centroid_hom.to_End_zero | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_add (x y : centroid_hom α) : (x + y).to_End = x.to_End + y.to_End | rfl | lemma | centroid_hom.to_End_add | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_nsmul (x : centroid_hom α) (n : ℕ) : (n • x).to_End = n • x.to_End | rfl | lemma | centroid_hom.to_End_nsmul | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_cast (n : ℕ) : ⇑(n : centroid_hom α) = n • id | rfl | lemma | centroid_hom.coe_nat_cast | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_apply (n : ℕ) (m : α):
(n : centroid_hom α) m = n • m | rfl | lemma | centroid_hom.nat_cast_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_one : (1 : centroid_hom α).to_End = 1 | rfl | lemma | centroid_hom.to_End_one | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_mul (x y : centroid_hom α) : (x * y).to_End = x.to_End * y.to_End | rfl | lemma | centroid_hom.to_End_mul | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_pow (x : centroid_hom α) (n : ℕ) : (x ^ n).to_End = x.to_End ^ n | by { ext, refl } | lemma | centroid_hom.to_End_pow | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_nat_cast (n : ℕ) : (n : centroid_hom α).to_End = ↑n | rfl | lemma | centroid_hom.to_End_nat_cast | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul_comm (T S : centroid_hom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) | by rw [comp_app, map_mul_right, map_mul_left, ←map_mul_right, ←map_mul_left] | lemma | centroid_hom.comp_mul_comm | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zsmul : has_smul ℤ (centroid_hom α) | ⟨λ n f,
{ map_mul_left' := λ a b,
by { change n • f (a * b) = a * n • f b, rw [map_mul_left f, ←mul_smul_comm] },
map_mul_right' := λ a b,
by { change n • f (a * b) = n • f a * b, rw [map_mul_right f, ←smul_mul_assoc] },
.. (n • f : α →+ α) }⟩ | instance | centroid_hom.has_zsmul | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_cast (z : ℤ) : ⇑(z : centroid_hom α) = z • id | rfl | lemma | centroid_hom.coe_int_cast | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_apply (z : ℤ) (m : α) :
(z : centroid_hom α) m = z • m | rfl | lemma | centroid_hom.int_cast_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_neg (x : centroid_hom α) : (-x).to_End = -x.to_End | rfl | lemma | centroid_hom.to_End_neg | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_sub (x y : centroid_hom α) : (x - y).to_End = x.to_End - y.to_End | rfl | lemma | centroid_hom.to_End_sub | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_zsmul (x : centroid_hom α) (n : ℤ) : (n • x).to_End = n • x.to_End | rfl | lemma | centroid_hom.to_End_zsmul | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (f : centroid_hom α) : ⇑(-f) = -f | rfl | lemma | centroid_hom.coe_neg | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (f g : centroid_hom α) : ⇑(f - g) = f - g | rfl | lemma | centroid_hom.coe_sub | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply (f : centroid_hom α) (a : α) : (-f) a = - f a | rfl | lemma | centroid_hom.neg_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply (f g : centroid_hom α) (a : α) : (f - g) a = f a - g a | rfl | lemma | centroid_hom.sub_apply | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_End_int_cast (z : ℤ) : (z : centroid_hom α).to_End = ↑z | rfl | lemma | centroid_hom.to_End_int_cast | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) :
comm_ring (centroid_hom α) | { mul_comm := λ f g, begin
ext,
refine sub_eq_zero.1 ((or_self _).1 $ h _ _ $ λ r, _),
rw [mul_assoc, sub_mul, sub_eq_zero, ← map_mul_right, ← map_mul_right, coe_mul, coe_mul,
comp_mul_comm],
end,
..centroid_hom.ring } | def | centroid_hom.comm_ring | algebra.hom | src/algebra/hom/centroid.lean | [
"algebra.group_power.lemmas",
"algebra.hom.group_instances"
] | [
"centroid_hom",
"comm_ring",
"mul_assoc",
"mul_comm"
] | A prime associative ring has commutative centroid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semiconj_by.map [mul_hom_class F M N] (h : semiconj_by a x y) (f : F) :
semiconj_by (f a) (f x) (f y) | by simpa only [semiconj_by, map_mul] using congr_arg f h | lemma | semiconj_by.map | algebra.hom | src/algebra/hom/commute.lean | [
"algebra.hom.group",
"algebra.group.commute"
] | [
"map_mul",
"mul_hom_class",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.map [mul_hom_class F M N] (h : commute x y) (f : F) :
commute (f x) (f y) | h.map f | lemma | commute.map | algebra.hom | src/algebra/hom/commute.lean | [
"algebra.hom.group",
"algebra.group.commute"
] | [
"commute",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_embedding {G : Type*} [left_cancel_semigroup G] (g : G) : G ↪ G | { to_fun := λ h, g * h, inj' := mul_right_injective g } | def | mul_left_embedding | algebra.hom | src/algebra/hom/embedding.lean | [
"algebra.group.defs",
"logic.embedding.basic"
] | [
"left_cancel_semigroup",
"mul_right_injective"
] | The embedding of a left cancellative semigroup into itself
by left multiplication by a fixed element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_embedding {G : Type*} [right_cancel_semigroup G] (g : G) : G ↪ G | { to_fun := λ h, h * g, inj' := mul_left_injective g } | def | mul_right_embedding | algebra.hom | src/algebra/hom/embedding.lean | [
"algebra.group.defs",
"logic.embedding.basic"
] | [
"mul_left_injective",
"right_cancel_semigroup"
] | The embedding of a right cancellative semigroup into itself
by right multiplication by a fixed element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_embedding_eq_mul_right_embedding {G : Type*} [cancel_comm_monoid G] (g : G) :
mul_left_embedding g = mul_right_embedding g | by { ext, exact mul_comm _ _ } | lemma | mul_left_embedding_eq_mul_right_embedding | algebra.hom | src/algebra/hom/embedding.lean | [
"algebra.group.defs",
"logic.embedding.basic"
] | [
"cancel_comm_monoid",
"mul_comm",
"mul_left_embedding",
"mul_right_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_freiman_hom (A : set α) (β : Type*) [add_comm_monoid α] [add_comm_monoid β] (n : ℕ) | (to_fun : α → β)
(map_sum_eq_map_sum' {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : s.card = n) (ht : t.card = n) (h : s.sum = t.sum) :
(s.map to_fun).sum = (t.map to_fun).sum) | structure | add_freiman_hom | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"add_comm_monoid",
"multiset"
] | An additive `n`-Freiman homomorphism is a map which preserves sums of `n` elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
freiman_hom (A : set α) (β : Type*) [comm_monoid α] [comm_monoid β] (n : ℕ) | (to_fun : α → β)
(map_prod_eq_map_prod' {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : s.card = n) (ht : t.card = n) (h : s.prod = t.prod) :
(s.map to_fun).prod = (t.map to_fun).prod) | structure | freiman_hom | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"comm_monoid",
"multiset"
] | A `n`-Freiman homomorphism on a set `A` is a map which preserves products of `n` elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_freiman_hom_class (F : Type*) (A : out_param $ set α) (β : out_param $ Type*)
[add_comm_monoid α] [add_comm_monoid β] (n : ℕ) [fun_like F α (λ _, β)] : Prop | (map_sum_eq_map_sum' (f : F) {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A)
(htA : ∀ ⦃x⦄, x ∈ t → x ∈ A) (hs : s.card = n) (ht : t.card = n) (h : s.sum = t.sum) :
(s.map f).sum = (t.map f).sum) | class | add_freiman_hom_class | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"add_comm_monoid",
"fun_like",
"multiset"
] | `add_freiman_hom_class F s β n` states that `F` is a type of `n`-ary sums-preserving morphisms.
You should extend this class when you extend `add_freiman_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
freiman_hom_class (F : Type*) (A : out_param $ set α) (β : out_param $ Type*) [comm_monoid α]
[comm_monoid β] (n : ℕ) [fun_like F α (λ _, β)] : Prop | (map_prod_eq_map_prod' (f : F) {s t : multiset α} (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A)
(htA : ∀ ⦃x⦄, x ∈ t → x ∈ A) (hs : s.card = n) (ht : t.card = n) (h : s.prod = t.prod) :
(s.map f).prod = (t.map f).prod) | class | freiman_hom_class | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"comm_monoid",
"fun_like",
"multiset"
] | `freiman_hom_class F A β n` states that `F` is a type of `n`-ary products-preserving morphisms.
You should extend this class when you extend `freiman_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_prod_eq_map_prod [freiman_hom_class F A β n] (f : F) {s t : multiset α}
(hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A) (hs : s.card = n) (ht : t.card = n)
(h : s.prod = t.prod) :
(s.map f).prod = (t.map f).prod | freiman_hom_class.map_prod_eq_map_prod' f hsA htA hs ht h | lemma | map_prod_eq_map_prod | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom_class",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul_map_eq_map_mul_map [freiman_hom_class F A β 2] (f : F) (ha : a ∈ A) (hb : b ∈ A)
(hc : c ∈ A) (hd : d ∈ A) (h : a * b = c * d) :
f a * f b = f c * f d | begin
simp_rw ←prod_pair at ⊢ h,
refine map_prod_eq_map_prod f _ _ (card_pair _ _) (card_pair _ _) h; simp [ha, hb, hc, hd],
end | lemma | map_mul_map_eq_map_mul_map | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom_class",
"map_prod_eq_map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fun_like : fun_like (A →*[n] β) α (λ _, β) | { coe := to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr' } | instance | freiman_hom.fun_like | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"fun_like"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
freiman_hom_class : freiman_hom_class (A →*[n] β) A β n | { map_prod_eq_map_prod' := map_prod_eq_map_prod' } | instance | freiman_hom.freiman_hom_class | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : A →*[n] β) : f.to_fun = f | rfl | lemma | freiman_hom.to_fun_eq_coe | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext ⦃f g : A →*[n] β⦄ (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | lemma | freiman_hom.ext | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : α → β) (h : ∀ s t : multiset α, (∀ ⦃x⦄, x ∈ s → x ∈ A) → (∀ ⦃x⦄, x ∈ t → x ∈ A) →
s.card = n → t.card = n → s.prod = t.prod → (s.map f).prod = (t.map f).prod) :
⇑(mk f h) = f | rfl | lemma | freiman_hom.coe_mk | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : A →*[n] β) (h) : mk f h = f | ext $ λ _, rfl | lemma | freiman_hom.mk_coe | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (A : set α) (n : ℕ) : A →*[n] α | { to_fun := λ x, x, map_prod_eq_map_prod' := λ s t _ _ _ _ h, by rw [map_id', map_id', h] } | def | freiman_hom.id | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | The identity map from a commutative monoid to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (f : B →*[n] γ) (g : A →*[n] β) (hAB : A.maps_to g B) : A →*[n] γ | { to_fun := f ∘ g,
map_prod_eq_map_prod' := λ s t hsA htA hs ht h, begin
rw [←map_map,
map_prod_eq_map_prod f _ _ ((s.card_map _).trans hs) ((t.card_map _).trans ht)
(map_prod_eq_map_prod g hsA htA hs ht h), map_map],
{ simpa using (λ a h, hAB (hsA h)) },
{ simpa using (λ a h, hAB (htA h)) }
e... | def | freiman_hom.comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"map_prod_eq_map_prod"
] | Composition of Freiman homomorphisms as a Freiman homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : B →*[n] γ) (g : A →*[n] β) {hfg} : ⇑(f.comp g hfg) = f ∘ g | rfl | lemma | freiman_hom.coe_comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : B →*[n] γ) (g : A →*[n] β) {hfg} (x : α) : f.comp g hfg x = f (g x) | rfl | lemma | freiman_hom.comp_apply | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : A →*[n] β) (g : B →*[n] γ) (h : C →*[n] δ) {hf hhg hgf}
{hh : A.maps_to (g.comp f hgf) C} :
(h.comp g hhg).comp f hf = h.comp (g.comp f hgf) hh | rfl | lemma | freiman_hom.comp_assoc | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : B →*[n] γ} {f : A →*[n] β} (hf : function.surjective f) {hg₁ hg₂} :
g₁.comp f hg₁ = g₂.comp f hg₂ ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, λ h, h ▸ rfl⟩ | lemma | freiman_hom.cancel_right | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right_on {g₁ g₂ : B →*[n] γ} {f : A →*[n] β} (hf : A.surj_on f B) {hf'} :
A.eq_on (g₁.comp f hf') (g₂.comp f hf') ↔ B.eq_on g₁ g₂ | hf.cancel_right hf' | lemma | freiman_hom.cancel_right_on | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left_on {g : B →*[n] γ} {f₁ f₂ : A →*[n] β} (hg : B.inj_on g) {hf₁ hf₂} :
A.eq_on (g.comp f₁ hf₁) (g.comp f₂ hf₂) ↔ A.eq_on f₁ f₂ | hg.cancel_left hf₁ hf₂ | lemma | freiman_hom.cancel_left_on | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : A →*[n] β) {hf} : f.comp (freiman_hom.id A n) hf = f | ext $ λ x, rfl | lemma | freiman_hom.comp_id | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : A →*[n] β) {hf} : (freiman_hom.id B n).comp f hf = f | ext $ λ x, rfl | lemma | freiman_hom.id_comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (A : set α) (n : ℕ) (b : β) : A →*[n] β | { to_fun := λ _, b,
map_prod_eq_map_prod' := λ s t _ _ hs ht _,
by rw [multiset.map_const, multiset.map_const, prod_replicate, prod_replicate, hs, ht] } | def | freiman_hom.const | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"multiset.map_const"
] | `freiman_hom.const A n b` is the Freiman homomorphism sending everything to `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_apply (n : ℕ) (b : β) (x : α) : const A n b x = b | rfl | lemma | freiman_hom.const_apply | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_comp (n : ℕ) (c : γ) (f : A →*[n] β) {hf} : (const B n c).comp f hf = const A n c | rfl | lemma | freiman_hom.const_comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (x : α) : (1 : A →*[n] β) x = 1 | rfl | lemma | freiman_hom.one_apply | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_comp (f : A →*[n] β) {hf} : (1 : B →*[n] γ).comp f hf = 1 | rfl | lemma | freiman_hom.one_comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (f g : A →*[n] β) (x : α) : (f * g) x = f x * g x | rfl | lemma | freiman_hom.mul_apply | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comp (g₁ g₂ : B →*[n] γ) (f : A →*[n] β) {hg hg₁ hg₂} :
(g₁ * g₂).comp f hg = g₁.comp f hg₁ * g₂.comp f hg₂ | rfl | lemma | freiman_hom.mul_comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply (f : A →*[n] G) (x : α) : f⁻¹ x = (f x)⁻¹ | rfl | lemma | freiman_hom.inv_apply | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp (f : B →*[n] G) (g : A →*[n] β) {hf hf'} :
f⁻¹.comp g hf = (f.comp g hf')⁻¹ | ext $ λ x, rfl | lemma | freiman_hom.inv_comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_apply (f g : A →*[n] G) (x : α) : (f / g) x = f x / g x | rfl | lemma | freiman_hom.div_apply | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_comp (f₁ f₂ : B →*[n] G) (g : A →*[n] β) {hf hf₁ hf₂} :
(f₁ / f₂).comp g hf = f₁.comp g hf₁ / f₂.comp g hf₂ | ext $ λ x, rfl | lemma | freiman_hom.div_comp | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.freiman_hom_class : freiman_hom_class (α →* β) set.univ β n | { map_prod_eq_map_prod' := λ f s t _ _ _ _ h, by rw [←f.map_multiset_prod, h, f.map_multiset_prod] } | instance | monoid_hom.freiman_hom_class | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom_class"
] | A monoid homomorphism is naturally a `freiman_hom` on its entire domain.
We can't leave the domain `A : set α` of the `freiman_hom` a free variable, since it wouldn't be
inferrable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.to_freiman_hom (A : set α) (n : ℕ) (f : α →* β) : A →*[n] β | { to_fun := f,
map_prod_eq_map_prod' := λ s t hsA htA, map_prod_eq_map_prod f
(λ _ _, set.mem_univ _) (λ _ _, set.mem_univ _) } | def | monoid_hom.to_freiman_hom | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"map_prod_eq_map_prod",
"set.mem_univ"
] | A `monoid_hom` is naturally a `freiman_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.to_freiman_hom_coe (f : α →* β) : (f.to_freiman_hom A n : α → β) = f | rfl | lemma | monoid_hom.to_freiman_hom_coe | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.to_freiman_hom_injective :
function.injective (monoid_hom.to_freiman_hom A n : (α →* β) → A →*[n] β) | λ f g h, monoid_hom.ext $ show _, from fun_like.ext_iff.mp h | lemma | monoid_hom.to_freiman_hom_injective | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"monoid_hom.ext",
"monoid_hom.to_freiman_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod_eq_map_prod_of_le [freiman_hom_class F A β n] (f : F) {s t : multiset α}
(hsA : ∀ x ∈ s, x ∈ A) (htA : ∀ x ∈ t, x ∈ A) (hs : s.card = m)
(ht : t.card = m) (hst : s.prod = t.prod) (h : m ≤ n) :
(s.map f).prod = (t.map f).prod | begin
obtain rfl | hm := m.eq_zero_or_pos,
{ rw card_eq_zero at hs ht,
rw [hs, ht] },
rw [←hs, card_pos_iff_exists_mem] at hm,
obtain ⟨a, ha⟩ := hm,
suffices : ((s + replicate (n - m) a).map f).prod = ((t + replicate (n - m) a).map f).prod,
{ simp_rw [multiset.map_add, prod_add] at this,
exact mul_r... | lemma | map_prod_eq_map_prod_of_le | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"add_tsub_cancel_of_le",
"freiman_hom_class",
"map_prod_eq_map_prod",
"mul_right_cancel",
"multiset",
"multiset.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
freiman_hom.to_freiman_hom (h : m ≤ n) (f : A →*[n] β) : A →*[m] β | { to_fun := f,
map_prod_eq_map_prod' := λ s t hsA htA hs ht hst,
map_prod_eq_map_prod_of_le f hsA htA hs ht hst h } | def | freiman_hom.to_freiman_hom | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"map_prod_eq_map_prod_of_le"
] | `α →*[n] β` is naturally included in `A →*[m] β` for any `m ≤ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
freiman_hom.freiman_hom_class_of_le [freiman_hom_class F A β n] (h : m ≤ n) :
freiman_hom_class F A β m | { map_prod_eq_map_prod' := λ f s t hsA htA hs ht hst,
map_prod_eq_map_prod_of_le f hsA htA hs ht hst h } | lemma | freiman_hom.freiman_hom_class_of_le | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom_class",
"map_prod_eq_map_prod_of_le"
] | A `n`-Freiman homomorphism is also a `m`-Freiman homomorphism for any `m ≤ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
freiman_hom.to_freiman_hom_coe (h : m ≤ n) (f : A →*[n] β) :
(f.to_freiman_hom h : α → β) = f | rfl | lemma | freiman_hom.to_freiman_hom_coe | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
freiman_hom.to_freiman_hom_injective (h : m ≤ n) :
function.injective (freiman_hom.to_freiman_hom h : (A →*[n] β) → A →*[m] β) | λ f g hfg, freiman_hom.ext $ by convert fun_like.ext_iff.1 hfg | lemma | freiman_hom.to_freiman_hom_injective | algebra.hom | src/algebra/hom/freiman.lean | [
"algebra.big_operators.multiset.basic",
"data.fun_like.basic"
] | [
"freiman_hom.ext",
"freiman_hom.to_freiman_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_hom (M : Type*) (N : Type*) [has_zero M] [has_zero N] | (to_fun : M → N)
(map_zero' : to_fun 0 = 0) | structure | zero_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | `zero_hom M N` is the type of functions `M → N` that preserve zero.
When possible, instead of parametrizing results over `(f : zero_hom M N)`,
you should parametrize over `(F : Type*) [zero_hom_class F M N] (f : F)`.
When you extend this structure, make sure to also extend `zero_hom_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_hom_class (F : Type*) (M N : out_param $ Type*)
[has_zero M] [has_zero N] extends fun_like F M (λ _, N) | (map_zero : ∀ (f : F), f 0 = 0) | class | zero_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like"
] | `zero_hom_class F M N` states that `F` is a type of zero-preserving homomorphisms.
You should extend this typeclass when you extend `zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_map {R M} [has_zero R] [has_zero M] [zero_hom_class F R M] (f : F) {r : R}
[ne_zero (f r)] : ne_zero r | ⟨λ h, ne (f r) $ by convert zero_hom_class.map_zero f⟩ | lemma | ne_zero.of_map | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"ne_zero",
"zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_injective {R M} [has_zero R] {r : R} [ne_zero r] [has_zero M] [zero_hom_class F R M]
{f : F} (hf : function.injective f) : ne_zero (f r) | ⟨by { rw ← zero_hom_class.map_zero f, exact hf.ne (ne r) }⟩ | lemma | ne_zero.of_injective | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"ne_zero",
"zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_hom (M : Type*) (N : Type*) [has_add M] [has_add N] | (to_fun : M → N)
(map_add' : ∀ x y, to_fun (x + y) = to_fun x + to_fun y) | structure | add_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | `add_hom M N` is the type of functions `M → N` that preserve addition.
When possible, instead of parametrizing results over `(f : add_hom M N)`,
you should parametrize over `(F : Type*) [add_hom_class F M N] (f : F)`.
When you extend this structure, make sure to extend `add_hom_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_hom_class (F : Type*) (M N : out_param $ Type*)
[has_add M] [has_add N] extends fun_like F M (λ _, N) | (map_add : ∀ (f : F) (x y : M), f (x + y) = f x + f y) | class | add_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like"
] | `add_hom_class F M N` states that `F` is a type of addition-preserving homomorphisms.
You should declare an instance of this typeclass when you extend `add_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom (M : Type*) (N : Type*) [add_zero_class M] [add_zero_class N]
extends zero_hom M N, add_hom M N | structure | add_monoid_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"add_hom",
"add_zero_class",
"zero_hom"
] | `M →+ N` is the type of functions `M → N` that preserve the `add_zero_class` structure.
`add_monoid_hom` is also used for group homomorphisms.
When possible, instead of parametrizing results over `(f : M →+ N)`,
you should parametrize over `(F : Type*) [add_monoid_hom_class F M N] (f : F)`.
When you extend this stru... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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