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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
bounded_lattice_hom.as_boolring (f : bounded_lattice_hom α β) :
as_boolring α →+* as_boolring β | { to_fun := to_boolring ∘ f ∘ of_boolring,
map_zero' := f.map_bot',
map_one' := f.map_top',
map_add' := map_symm_diff' f,
map_mul' := f.map_inf' } | def | bounded_lattice_hom.as_boolring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"bounded_lattice_hom",
"map_symm_diff'",
"of_boolring",
"to_boolring"
] | Turn a bounded lattice homomorphism from Boolean algebras `α` to `β` into a ring homomorphism
from `α` to `β` considered as Boolean rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_lattice_hom.as_boolring_id :
(bounded_lattice_hom.id α).as_boolring = ring_hom.id _ | rfl | lemma | bounded_lattice_hom.as_boolring_id | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"bounded_lattice_hom.id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_lattice_hom.as_boolring_comp (g : bounded_lattice_hom β γ)
(f : bounded_lattice_hom α β) :
(g.comp f).as_boolring = g.as_boolring.comp f.as_boolring | rfl | lemma | bounded_lattice_hom.as_boolring_comp | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.as_boolalg_as_boolring (α : Type*) [boolean_algebra α] :
as_boolalg (as_boolring α) ≃o α | ⟨of_boolalg.trans of_boolring, λ a b,
of_boolring_le_of_boolring_iff.trans of_boolalg_mul_of_boolalg_eq_left_iff⟩ | def | order_iso.as_boolalg_as_boolring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"as_boolring",
"boolean_algebra",
"of_boolring"
] | Order isomorphism between `α` considered as a Boolean ring considered as a Boolean algebra and
`α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_equiv.as_boolring_as_boolalg (α : Type*) [boolean_ring α] :
as_boolring (as_boolalg α) ≃+* α | { map_mul' := λ a b, rfl,
map_add' := of_boolalg_symm_diff,
..of_boolring.trans of_boolalg } | def | ring_equiv.as_boolring_as_boolalg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"as_boolring",
"boolean_ring",
"of_boolalg",
"of_boolalg_symm_diff"
] | Ring isomorphism between `α` considered as a Boolean algebra considered as a Boolean ring and
`α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_right [distrib R] {a b c : R} :
commute a b → commute a c → commute a (b + c) | semiconj_by.add_right | theorem | commute.add_right | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"distrib",
"semiconj_by.add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left [distrib R] {a b c : R} :
commute a c → commute b c → commute (a + b) c | semiconj_by.add_left | theorem | commute.add_left | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"distrib",
"semiconj_by.add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_right [distrib R] {x y : R} (h : commute x y) : commute x (bit0 y) | h.add_right h | lemma | commute.bit0_right | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_left [distrib R] {x y : R} (h : commute x y) : commute (bit0 x) y | h.add_left h | lemma | commute.bit0_left | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_right [non_assoc_semiring R] {x y : R} (h : commute x y) : commute x (bit1 y) | h.bit0_right.add_right (commute.one_right x) | lemma | commute.bit1_right | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"commute.one_right",
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_left [non_assoc_semiring R] {x y : R} (h : commute x y) : commute (bit1 x) y | h.bit0_left.add_left (commute.one_left y) | lemma | commute.bit1_left | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"commute.one_left",
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_sub_mul_self_eq [non_unital_non_assoc_ring R] {a b : R} (h : commute a b) :
a * a - b * b = (a + b) * (a - b) | by rw [add_mul, mul_sub, mul_sub, h.eq, sub_add_sub_cancel] | lemma | commute.mul_self_sub_mul_self_eq | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"non_unital_non_assoc_ring"
] | Representation of a difference of two squares of commuting elements as a product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_self_sub_mul_self_eq' [non_unital_non_assoc_ring R] {a b : R} (h : commute a b) :
a * a - b * b = (a - b) * (a + b) | by rw [mul_add, sub_mul, sub_mul, h.eq, sub_add_sub_cancel] | lemma | commute.mul_self_sub_mul_self_eq' | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"non_unital_non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_eq_mul_self_iff [non_unital_non_assoc_ring R] [no_zero_divisors R] {a b : R}
(h : commute a b) : a * a = b * b ↔ a = b ∨ a = -b | by rw [← sub_eq_zero, h.mul_self_sub_mul_self_eq, mul_eq_zero, or_comm, sub_eq_zero,
add_eq_zero_iff_eq_neg] | lemma | commute.mul_self_eq_mul_self_iff | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"mul_eq_zero",
"mul_self_eq_mul_self_iff",
"no_zero_divisors",
"non_unital_non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_right : commute a b → commute a (- b) | semiconj_by.neg_right | theorem | commute.neg_right | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.neg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_right_iff : commute a (-b) ↔ commute a b | semiconj_by.neg_right_iff | theorem | commute.neg_right_iff | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.neg_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_left : commute a b → commute (- a) b | semiconj_by.neg_left | theorem | commute.neg_left | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.neg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_left_iff : commute (-a) b ↔ commute a b | semiconj_by.neg_left_iff | theorem | commute.neg_left_iff | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.neg_left_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_right (a : R) : commute a (-1) | semiconj_by.neg_one_right a | theorem | commute.neg_one_right | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.neg_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_left (a : R): commute (-1) a | semiconj_by.neg_one_left a | theorem | commute.neg_one_left | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.neg_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_right : commute a b → commute a c → commute a (b - c) | semiconj_by.sub_right | theorem | commute.sub_right | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.sub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_left : commute a c → commute b c → commute (a - b) c | semiconj_by.sub_left | theorem | commute.sub_left | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute",
"semiconj_by.sub_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_sub_mul_self [comm_ring R] (a b : R) : a * a - b * b = (a + b) * (a - b) | (commute.all a b).mul_self_sub_mul_self_eq | theorem | mul_self_sub_mul_self | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"comm_ring",
"commute.all"
] | Representation of a difference of two squares in a commutative ring as a product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_self_sub_one [non_assoc_ring R] (a : R) : a * a - 1 = (a + 1) * (a - 1) | by rw [←(commute.one_right a).mul_self_sub_mul_self_eq, mul_one] | lemma | mul_self_sub_one | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute.one_right",
"mul_one",
"non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_eq_mul_self_iff [comm_ring R] [no_zero_divisors R] {a b : R} :
a * a = b * b ↔ a = b ∨ a = -b | (commute.all a b).mul_self_eq_mul_self_iff | lemma | mul_self_eq_mul_self_iff | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"comm_ring",
"commute.all",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_eq_one_iff [non_assoc_ring R] [no_zero_divisors R] {a : R} :
a * a = 1 ↔ a = 1 ∨ a = -1 | by rw [←(commute.one_right a).mul_self_eq_mul_self_iff, mul_one] | lemma | mul_self_eq_one_iff | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"commute.one_right",
"mul_one",
"mul_self_eq_mul_self_iff",
"no_zero_divisors",
"non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_self_iff [ring R] [no_zero_divisors R] (u : Rˣ) : u⁻¹ = u ↔ u = 1 ∨ u = -1 | begin
rw inv_eq_iff_mul_eq_one,
simp only [ext_iff],
push_cast,
exact mul_self_eq_one_iff
end | lemma | units.inv_eq_self_iff | algebra.ring | src/algebra/ring/commute.lean | [
"algebra.ring.semiconj",
"algebra.ring.units",
"algebra.group.commute"
] | [
"inv_eq_iff_mul_eq_one",
"mul_self_eq_one_iff",
"no_zero_divisors",
"ring"
] | In the unit group of an integral domain, a unit is its own inverse iff the unit is one or
one's additive inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_comp_triple (σ₁₂ : R₁ →+* R₂) (σ₂₃ : R₂ →+* R₃)
(σ₁₃ : out_param (R₁ →+* R₃)) : Prop | (comp_eq : σ₂₃.comp σ₁₂ = σ₁₃) | class | ring_hom_comp_triple | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"comp_eq"
] | Class that expresses the fact that three ring homomorphisms form a composition triple. This is
used to handle composition of semilinear maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] {x : R₁} :
σ₂₃ (σ₁₂ x) = σ₁₃ x | ring_hom.congr_fun comp_eq x | lemma | ring_hom_comp_triple.comp_apply | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"comp_eq",
"ring_hom.congr_fun",
"ring_hom_comp_triple"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_inv_pair (σ : R₁ →+* R₂) (σ' : out_param (R₂ →+* R₁)) : Prop | (comp_eq : σ'.comp σ = ring_hom.id R₁)
(comp_eq₂ : σ.comp σ' = ring_hom.id R₂) | class | ring_hom_inv_pair | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"comp_eq",
"ring_hom.id"
] | Class that expresses the fact that two ring homomorphisms are inverses of each other. This is
used to handle `symm` for semilinear equivalences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply_eq {x : R₁} : σ' (σ x) = x | by { rw [← ring_hom.comp_apply, comp_eq], simp } | lemma | ring_hom_inv_pair.comp_apply_eq | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"comp_eq",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply_eq₂ {x : R₂} : σ (σ' x) = x | by { rw [← ring_hom.comp_apply, comp_eq₂], simp } | lemma | ring_hom_inv_pair.comp_apply_eq₂ | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ids : ring_hom_inv_pair (ring_hom.id R₁) (ring_hom.id R₁) | ⟨rfl, rfl⟩ | instance | ring_hom_inv_pair.ids | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom.id",
"ring_hom_inv_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triples {σ₂₁ : R₂ →+* R₁} [ring_hom_inv_pair σ₁₂ σ₂₁] :
ring_hom_comp_triple σ₁₂ σ₂₁ (ring_hom.id R₁) | ⟨by simp only [comp_eq]⟩ | instance | ring_hom_inv_pair.triples | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"comp_eq",
"ring_hom.id",
"ring_hom_comp_triple",
"ring_hom_inv_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triples₂ {σ₂₁ : R₂ →+* R₁} [ring_hom_inv_pair σ₁₂ σ₂₁] :
ring_hom_comp_triple σ₂₁ σ₁₂ (ring_hom.id R₂) | ⟨by simp only [comp_eq₂]⟩ | instance | ring_hom_inv_pair.triples₂ | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom.id",
"ring_hom_comp_triple",
"ring_hom_inv_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ring_equiv (e : R₁ ≃+* R₂) :
ring_hom_inv_pair (↑e : R₁ →+* R₂) ↑e.symm | ⟨e.symm_to_ring_hom_comp_to_ring_hom, e.symm.symm_to_ring_hom_comp_to_ring_hom⟩ | lemma | ring_hom_inv_pair.of_ring_equiv | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom_inv_pair"
] | Construct a `ring_hom_inv_pair` from both directions of a ring equiv.
This is not an instance, as for equivalences that are involutions, a better instance
would be `ring_hom_inv_pair e e`. Indeed, this declaration is not currently used in mathlib.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm (σ₁₂ : R₁ →+* R₂) (σ₂₁ : R₂ →+* R₁) [ring_hom_inv_pair σ₁₂ σ₂₁] :
ring_hom_inv_pair σ₂₁ σ₁₂ | ⟨ring_hom_inv_pair.comp_eq₂, ring_hom_inv_pair.comp_eq⟩ | lemma | ring_hom_inv_pair.symm | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom_inv_pair"
] | Swap the direction of a `ring_hom_inv_pair`. This is not an instance as it would loop, and better
instances are often available and may often be preferrable to using this one. Indeed, this
declaration is not currently used in mathlib.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ids : ring_hom_comp_triple (ring_hom.id R₁) σ₁₂ σ₁₂ | ⟨by { ext, simp }⟩ | instance | ring_hom_comp_triple.ids | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom.id",
"ring_hom_comp_triple"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_ids : ring_hom_comp_triple σ₁₂ (ring_hom.id R₂) σ₁₂ | ⟨by { ext, simp }⟩ | instance | ring_hom_comp_triple.right_ids | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom.id",
"ring_hom_comp_triple"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_surjective (σ : R₁ →+* R₂) : Prop | (is_surjective : function.surjective σ) | class | ring_hom_surjective | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [] | Class expressing the fact that a `ring_hom` is surjective. This is needed in the context
of semilinear maps, where some lemmas require this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_surjective (σ : R₁ →+* R₂) [t : ring_hom_surjective σ] : function.surjective σ | t.is_surjective | lemma | ring_hom.is_surjective | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pair {σ₁ : R₁ →+* R₂} {σ₂ : R₂ →+* R₁}
[ring_hom_inv_pair σ₁ σ₂] : ring_hom_surjective σ₁ | ⟨λ x, ⟨σ₂ x, ring_hom_inv_pair.comp_apply_eq₂⟩⟩ | instance | ring_hom_surjective.inv_pair | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom_inv_pair",
"ring_hom_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ids : ring_hom_surjective (ring_hom.id R₁) | ⟨is_surjective⟩ | instance | ring_hom_surjective.ids | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom.id",
"ring_hom_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [ring_hom_surjective σ₁₂] [ring_hom_surjective σ₂₃] :
ring_hom_surjective σ₁₃ | { is_surjective := begin
have := σ₂₃.is_surjective.comp σ₁₂.is_surjective,
rwa [← ring_hom.coe_comp, ring_hom_comp_triple.comp_eq] at this,
end } | lemma | ring_hom_surjective.comp | algebra.ring | src/algebra/ring/comp_typeclasses.lean | [
"algebra.ring.equiv"
] | [
"ring_hom.coe_comp",
"ring_hom_comp_triple",
"ring_hom_surjective"
] | This cannot be an instance as there is no way to infer `σ₁₂` and `σ₂₃`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distrib (R : Type*) extends has_mul R, has_add R | (left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c)
(right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c) | class | distrib | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"left_distrib",
"right_distrib"
] | A typeclass stating that multiplication is left and right distributive
over addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_distrib_class (R : Type*) [has_mul R] [has_add R] | (left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c) | class | left_distrib_class | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"left_distrib"
] | A typeclass stating that multiplication is left distributive over addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_distrib_class (R : Type*) [has_mul R] [has_add R] | (right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c) | class | right_distrib_class | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"right_distrib"
] | A typeclass stating that multiplication is right distributive over addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distrib.left_distrib_class (R : Type*) [distrib R] : left_distrib_class R | ⟨distrib.left_distrib⟩ | instance | distrib.left_distrib_class | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"distrib",
"left_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib.right_distrib_class (R : Type*) [distrib R] : right_distrib_class R | ⟨distrib.right_distrib⟩ | instance | distrib.right_distrib_class | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"distrib",
"right_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_distrib [has_mul R] [has_add R] [left_distrib_class R] (a b c : R) :
a * (b + c) = a * b + a * c | left_distrib_class.left_distrib a b c | lemma | left_distrib | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"left_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_distrib [has_mul R] [has_add R] [right_distrib_class R] (a b c : R) :
(a + b) * c = a * c + b * c | right_distrib_class.right_distrib a b c | lemma | right_distrib | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"right_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_three_right [has_mul R] [has_add R] [right_distrib_class R] (a b c d : R) :
(a + b + c) * d = a * d + b * d + c * d | by simp [right_distrib] | lemma | distrib_three_right | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"right_distrib",
"right_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_semiring (α : Type u) extends
add_comm_monoid α, distrib α, mul_zero_class α | class | non_unital_non_assoc_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"add_comm_monoid",
"distrib",
"mul_zero_class"
] | A not-necessarily-unital, not-necessarily-associative semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_semiring (α : Type u) extends
non_unital_non_assoc_semiring α, semigroup_with_zero α | class | non_unital_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"non_unital_non_assoc_semiring",
"semigroup_with_zero"
] | An associative but not-necessarily unital semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_assoc_semiring (α : Type u) extends
non_unital_non_assoc_semiring α, mul_zero_one_class α, add_comm_monoid_with_one α | class | non_assoc_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"add_comm_monoid_with_one",
"mul_zero_one_class",
"non_unital_non_assoc_semiring"
] | A unital but not-necessarily-associative semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring (α : Type u) extends non_unital_semiring α, non_assoc_semiring α, monoid_with_zero α | class | semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"monoid_with_zero",
"non_assoc_semiring",
"non_unital_semiring"
] | A semiring is a type with the following structures: additive commutative monoid
(`add_comm_monoid`), multiplicative monoid (`monoid`), distributive laws (`distrib`), and
multiplication by zero law (`mul_zero_class`). The actual definition extends `monoid_with_zero`
instead of `monoid` and `mul_zero_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_one_eq_two : 1 + 1 = (2 : α) | rfl | lemma | one_add_one_eq_two | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_mul [right_distrib_class α] (a b : α) : (a + 1) * b = a * b + b | by rw [add_mul, one_mul] | lemma | add_one_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"one_mul",
"right_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_one [left_distrib_class α] (a b : α) : a * (b + 1) = a * b + a | by rw [mul_add, mul_one] | lemma | mul_add_one | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"left_distrib_class",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_mul [right_distrib_class α] (a b : α) : (1 + a) * b = b + a * b | by rw [add_mul, one_mul] | lemma | one_add_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"one_mul",
"right_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_add [left_distrib_class α] (a b : α) : a * (1 + b) = a + a * b | by rw [mul_add, mul_one] | lemma | mul_one_add | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"left_distrib_class",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_mul [right_distrib_class α] (n : α) : 2 * n = n + n | eq.trans (right_distrib 1 1 n) (by simp) | theorem | two_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"right_distrib",
"right_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_eq_two_mul [right_distrib_class α] (n : α) : bit0 n = 2 * n | (two_mul _).symm | theorem | bit0_eq_two_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"right_distrib_class",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_two [left_distrib_class α] (n : α) : n * 2 = n + n | (left_distrib n 1 1).trans (by simp) | theorem | mul_two | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"left_distrib",
"left_distrib_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ite {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
a * (if P then b else c) = if P then a * b else a * c | by split_ifs; refl | lemma | mul_ite | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_mul {α} [has_mul α] (P : Prop) [decidable P] (a b c : α) :
(if P then a else b) * c = if P then a * c else b * c | by split_ifs; refl | lemma | ite_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_boole {α} [mul_zero_one_class α] (P : Prop) [decidable P] (a : α) :
a * (if P then 1 else 0) = if P then a else 0 | by simp | lemma | mul_boole | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boole_mul {α} [mul_zero_one_class α] (P : Prop) [decidable P] (a : α) :
(if P then 1 else 0) * a = if P then a else 0 | by simp | lemma | boole_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_mul_zero_left {α : Type*} [mul_zero_class α] (P : Prop) [decidable P] (a b : α) :
ite P (a * b) 0 = ite P a 0 * b | by { by_cases h : P; simp [h], } | lemma | ite_mul_zero_left | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_mul_zero_right {α : Type*} [mul_zero_class α] (P : Prop) [decidable P] (a b : α) :
ite P (a * b) 0 = a * ite P b 0 | by { by_cases h : P; simp [h], } | lemma | ite_mul_zero_right | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_and_mul_zero {α : Type*} [mul_zero_class α]
(P Q : Prop) [decidable P] [decidable Q] (a b : α) :
ite (P ∧ Q) (a * b) 0 = ite P a 0 * ite Q b 0 | by simp only [←ite_and, ite_mul, mul_ite, mul_zero, zero_mul, and_comm] | lemma | ite_and_mul_zero | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"ite_mul",
"mul_ite",
"mul_zero",
"mul_zero_class",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_comm_semiring (α : Type u) extends non_unital_semiring α, comm_semigroup α | class | non_unital_comm_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_semigroup",
"non_unital_semiring"
] | A non-unital commutative semiring is a `non_unital_semiring` with commutative multiplication.
In other words, it is a type with the following structures: additive commutative monoid
(`add_comm_monoid`), commutative semigroup (`comm_semigroup`), distributive laws (`distrib`), and
multiplication by zero law (`mul_zero_cl... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring (α : Type u) extends semiring α, comm_monoid α | class | comm_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_monoid",
"semiring"
] | A commutative semiring is a `semiring` with commutative multiplication. In other words, it is a
type with the following structures: additive commutative monoid (`add_comm_monoid`), multiplicative
commutative monoid (`comm_monoid`), distributive laws (`distrib`), and multiplication by zero law
(`mul_zero_class`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring.to_non_unital_comm_semiring [comm_semiring α] : non_unital_comm_semiring α | { .. comm_semiring.to_comm_monoid α, .. comm_semiring.to_semiring α } | instance | comm_semiring.to_non_unital_comm_semiring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_semiring",
"non_unital_comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring.to_comm_monoid_with_zero [comm_semiring α] : comm_monoid_with_zero α | { .. comm_semiring.to_comm_monoid α, .. comm_semiring.to_semiring α } | instance | comm_semiring.to_comm_monoid_with_zero | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"comm_monoid_with_zero",
"comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_self_eq (a b : α) : (a + b) * (a + b) = a*a + 2*a*b + b*b | by simp only [two_mul, add_mul, mul_add, add_assoc, mul_comm b] | lemma | add_mul_self_eq | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_comm",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_distrib_neg (α : Type*) [has_mul α] extends has_involutive_neg α | (neg_mul : ∀ x y : α, -x * y = -(x * y))
(mul_neg : ∀ x y : α, x * -y = -(x * y)) | class | has_distrib_neg | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"has_involutive_neg",
"mul_neg",
"neg_mul"
] | Typeclass for a negation operator that distributes across multiplication.
This is useful for dealing with submonoids of a ring that contain `-1` without having to duplicate
lemmas. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_mul (a b : α) : - a * b = - (a * b) | has_distrib_neg.neg_mul _ _ | lemma | neg_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_neg (a b : α) : a * - b = - (a * b) | has_distrib_neg.mul_neg _ _ | lemma | mul_neg | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mul_neg (a b : α) : -a * -b = a * b | by simp | lemma | neg_mul_neg | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mul_eq_neg_mul (a b : α) : -(a * b) = -a * b | (neg_mul _ _).symm | lemma | neg_mul_eq_neg_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mul_eq_mul_neg (a b : α) : -(a * b) = a * -b | (mul_neg _ _).symm | lemma | neg_mul_eq_mul_neg | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mul_comm (a b : α) : -a * b = a * -b | by simp | lemma | neg_mul_comm | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_neg_one_mul (a : α) : -a = -1 * a | by simp | theorem | neg_eq_neg_one_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_neg_one (a : α) : a * -1 = -a | by simp | lemma | mul_neg_one | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | An element of a ring multiplied by the additive inverse of one is the element's additive
inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_one_mul (a : α) : -1 * a = -a | by simp | lemma | neg_one_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | The additive inverse of one multiplied by an element of a ring is the element's additive
inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_zero_class.neg_zero_class : neg_zero_class α | { neg_zero := by rw [←zero_mul (0 : α), ←neg_mul, mul_zero, mul_zero],
..mul_zero_class.to_has_zero α,
..has_distrib_neg.to_has_involutive_neg α } | instance | mul_zero_class.neg_zero_class | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_zero",
"neg_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_ring (α : Type u) extends
add_comm_group α, non_unital_non_assoc_semiring α | class | non_unital_non_assoc_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"add_comm_group",
"non_unital_non_assoc_semiring"
] | A not-necessarily-unital, not-necessarily-associative ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_ring (α : Type*) extends
non_unital_non_assoc_ring α, non_unital_semiring α | class | non_unital_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"non_unital_non_assoc_ring",
"non_unital_semiring"
] | An associative but not-necessarily unital ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_assoc_ring (α : Type*) extends
non_unital_non_assoc_ring α, non_assoc_semiring α, add_comm_group_with_one α | class | non_assoc_ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"add_comm_group_with_one",
"non_assoc_semiring",
"non_unital_non_assoc_ring"
] | A unital but not-necessarily-associative ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring (α : Type u) extends add_comm_group_with_one α, monoid α, distrib α | class | ring | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"add_comm_group_with_one",
"distrib",
"monoid"
] | A ring is a type with the following structures: additive commutative group (`add_comm_group`),
multiplicative monoid (`monoid`), and distributive laws (`distrib`). Equivalently, a ring is a
`semiring` with a negation operation making it an additive group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_ring.to_has_distrib_neg : has_distrib_neg α | { neg := has_neg.neg,
neg_neg := neg_neg,
neg_mul := λ a b, eq_neg_of_add_eq_zero_left $ by rw [←right_distrib, add_left_neg, zero_mul],
mul_neg := λ a b, eq_neg_of_add_eq_zero_left $ by rw [←left_distrib, add_left_neg, mul_zero] } | instance | non_unital_non_assoc_ring.to_has_distrib_neg | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"has_distrib_neg",
"mul_neg",
"mul_zero",
"neg_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_left_distrib (a b c : α) : a * (b - c) = a * b - a * c | by simpa only [sub_eq_add_neg, neg_mul_eq_mul_neg] using mul_add a b (-c) | lemma | mul_sub_left_distrib | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"neg_mul_eq_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_right_distrib (a b c : α) : (a - b) * c = a * c - b * c | by simpa only [sub_eq_add_neg, neg_mul_eq_neg_mul] using add_mul a (-b) c | lemma | mul_sub_right_distrib | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"neg_mul_eq_neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_eq_mul_add_iff_sub_mul_add_eq : a * e + c = b * e + d ↔ (a - b) * e + c = d | calc
a * e + c = b * e + d ↔ a * e + c = d + b * e : by simp [add_comm]
... ↔ a * e + c - b * e = d : iff.intro (λ h, begin rw h, simp end) (λ h,
begin rw ← h, simp end)
... ↔ (a - b) * e + c = d : begin simp [sub_mul, sub_add_eq_add_sub] end | theorem | mul_add_eq_mul_add_iff_sub_mul_add_eq | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | An iff statement following from right distributivity in rings and the definition
of subtraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_mul_add_eq_of_mul_add_eq_mul_add : a * e + c = b * e + d → (a - b) * e + c = d | assume h,
calc
(a - b) * e + c = (a * e + c) - b * e : begin simp [sub_mul, sub_add_eq_add_sub] end
... = d : begin rw h, simp [@add_sub_cancel α] end | theorem | sub_mul_add_eq_of_mul_add_eq_mul_add | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [] | A simplification of one side of an equation exploiting right distributivity in rings
and the definition of subtraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_one_mul (a b : α) : (a - 1) * b = a * b - b | by rw [sub_mul, one_mul] | lemma | sub_one_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_one (a b : α) : a * (b - 1) = a * b - a | by rw [mul_sub, mul_one] | lemma | mul_sub_one | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_mul (a b : α) : (1 - a) * b = b - a * b | by rw [sub_mul, one_mul] | lemma | one_sub_mul | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_sub (a b : α) : a * (1 - b) = a - a * b | by rw [mul_sub, mul_one] | lemma | mul_one_sub | algebra.ring | src/algebra/ring/defs.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.int.cast.defs"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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