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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|---|---|---|---|---|---|---|---|---|---|---|
is_smul_regular_of_group [mul_action G R] (g : G) : is_smul_regular R g | begin
intros x y h,
convert congr_arg ((•) g⁻¹) h using 1;
simp [←smul_assoc]
end | lemma | is_smul_regular_of_group | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"mul_action"
] | An element of a group acting on a Type is regular. This relies on the availability
of the inverse given by groups, since there is no `left_cancel_smul` typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.is_smul_regular (a : Rˣ) : is_smul_regular M (a : R) | is_smul_regular.of_mul_eq_one a.inv_val | lemma | units.is_smul_regular | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"is_smul_regular.of_mul_eq_one"
] | Any element in `Rˣ` is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit.is_smul_regular (ua : is_unit a) : is_smul_regular M a | begin
rcases ua with ⟨a, rfl⟩,
exact a.is_smul_regular M
end | lemma | is_unit.is_smul_regular | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"is_unit"
] | A unit is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left : Rˣ →* add_aut R | distrib_mul_action.to_add_aut _ _ | def | add_aut.mul_left | algebra.ring | src/algebra/ring/add_aut.lean | [
"group_theory.group_action.group",
"algebra.module.basic"
] | [
"distrib_mul_action.to_add_aut"
] | Left multiplication by a unit of a semiring as an additive automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right (u : Rˣ) : add_aut R | distrib_mul_action.to_add_aut Rᵐᵒᵖˣ R (units.op_equiv.symm $ mul_opposite.op u) | def | add_aut.mul_right | algebra.ring | src/algebra/ring/add_aut.lean | [
"group_theory.group_action.group",
"algebra.module.basic"
] | [
"distrib_mul_action.to_add_aut",
"mul_opposite.op"
] | Right multiplication by a unit of a semiring as an additive automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_apply (u : Rˣ) (x : R) : mul_right u x = x * u | rfl | lemma | add_aut.mul_right_apply | algebra.ring | src/algebra/ring/add_aut.lean | [
"group_theory.group_action.group",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_symm_apply (u : Rˣ) (x : R) : (mul_right u).symm x = x * ↑u⁻¹ | rfl | lemma | add_aut.mul_right_symm_apply | algebra.ring | src/algebra/ring/add_aut.lean | [
"group_theory.group_action.group",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_aut (R : Type*) [has_mul R] [has_add R] | ring_equiv R R | def | ring_aut | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"ring_equiv"
] | The group of ring automorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_aut : ring_aut R →* add_aut R | by refine_struct { to_fun := ring_equiv.to_add_equiv }; intros; refl | def | ring_aut.to_add_aut | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"ring_aut"
] | Monoid homomorphism from ring automorphisms to additive automorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_mul_aut : ring_aut R →* mul_aut R | by refine_struct { to_fun := ring_equiv.to_mul_equiv }; intros; refl | def | ring_aut.to_mul_aut | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"mul_aut",
"ring_aut"
] | Monoid homomorphism from ring automorphisms to multiplicative automorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_perm : ring_aut R →* equiv.perm R | by refine_struct { to_fun := ring_equiv.to_equiv }; intros; refl | def | ring_aut.to_perm | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"equiv.perm",
"ring_aut"
] | Monoid homomorphism from ring automorphisms to permutations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_mul_semiring_action : mul_semiring_action (ring_aut R) R | { smul := ($),
smul_zero := ring_equiv.map_zero,
smul_add := ring_equiv.map_add,
smul_one := ring_equiv.map_one,
smul_mul := ring_equiv.map_mul,
one_smul := λ _, rfl,
mul_smul := λ _ _ _, rfl } | instance | ring_aut.apply_mul_semiring_action | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"mul_semiring_action",
"one_smul",
"ring_aut",
"ring_equiv.map_add",
"ring_equiv.map_mul",
"ring_equiv.map_one",
"ring_equiv.map_zero",
"smul_add",
"smul_zero"
] | The tautological action by the group of automorphism of a ring `R` on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (f : ring_aut R) (r : R) : f • r = f r | rfl | lemma | ring_aut.smul_def | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"ring_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_has_faithful_smul : has_faithful_smul (ring_aut R) R | ⟨λ _ _, ring_equiv.ext⟩ | instance | ring_aut.apply_has_faithful_smul | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"has_faithful_smul",
"ring_aut"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mul_semiring_action.to_ring_aut [mul_semiring_action G R] : G →* ring_aut R | { to_fun := mul_semiring_action.to_ring_equiv G R,
map_mul' := λ g h, ring_equiv.ext $ mul_smul g h,
map_one' := ring_equiv.ext $ one_smul _, } | def | mul_semiring_action.to_ring_aut | algebra.ring | src/algebra/ring/aut.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.aut",
"algebra.ring.equiv"
] | [
"mul_semiring_action",
"mul_semiring_action.to_ring_equiv",
"one_smul",
"ring_aut",
"ring_equiv.ext"
] | Each element of the group defines a ring automorphism.
This is a stronger version of `distrib_mul_action.to_add_aut` and
`mul_distrib_mul_action.to_mul_aut`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left {R : Type*} [distrib R] (r : R) : add_hom R R | ⟨(*) r, mul_add r⟩ | def | add_hom.mul_left | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"add_hom",
"distrib"
] | Left multiplication by an element of a type with distributive multiplication is an `add_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right {R : Type*} [distrib R] (r : R) : add_hom R R | ⟨λ a, a * r, λ _ _, add_mul _ _ r⟩ | def | add_hom.mul_right | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"add_hom",
"distrib"
] | Left multiplication by an element of a type with distributive multiplication is an `add_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_bit0 (f : F) (a : α) : (f (bit0 a) : β) = bit0 (f a) | map_add _ _ _ | lemma | map_bit0 | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [] | Additive homomorphisms preserve `bit0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left {R : Type*} [non_unital_non_assoc_semiring R] (r : R) : R →+ R | { to_fun := (*) r,
map_zero' := mul_zero r,
map_add' := mul_add r } | def | add_monoid_hom.mul_left | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"mul_zero",
"non_unital_non_assoc_semiring"
] | Left multiplication by an element of a (semi)ring is an `add_monoid_hom` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_left {R : Type*} [non_unital_non_assoc_semiring R] (r : R) :
⇑(mul_left r) = (*) r | rfl | lemma | add_monoid_hom.coe_mul_left | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right {R : Type*} [non_unital_non_assoc_semiring R] (r : R) : R →+ R | { to_fun := λ a, a * r,
map_zero' := zero_mul r,
map_add' := λ _ _, add_mul _ _ r } | def | add_monoid_hom.mul_right | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"non_unital_non_assoc_semiring",
"zero_mul"
] | Right multiplication by an element of a (semi)ring is an `add_monoid_hom` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_right {R : Type*} [non_unital_non_assoc_semiring R] (r : R) :
⇑(mul_right r) = (* r) | rfl | lemma | add_monoid_hom.coe_mul_right | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_apply {R : Type*} [non_unital_non_assoc_semiring R] (a r : R) :
mul_right r a = a * r | rfl | lemma | add_monoid_hom.mul_right_apply | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_neg' (a : α) : (- a)⁻¹ = - a⁻¹ | by rw [eq_comm, eq_inv_iff_mul_eq_one, neg_mul, mul_neg,neg_neg, mul_left_inv] | lemma | inv_neg' | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"eq_inv_iff_mul_eq_one",
"mul_left_inv",
"mul_neg",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Vieta_formula_quadratic {b c x : α} (h : x * x - b * x + c = 0) :
∃ y : α, y * y - b * y + c = 0 ∧ x + y = b ∧ x * y = c | begin
have : c = x * (b - x) := (eq_neg_of_add_eq_zero_right h).trans (by simp [mul_sub, mul_comm]),
refine ⟨b - x, _, by simp, by rw this⟩,
rw [this, sub_add, ← sub_mul, sub_self]
end | lemma | Vieta_formula_quadratic | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"mul_comm"
] | Vieta's formula for a quadratic equation, relating the coefficients of the polynomial with
its roots. This particular version states that if we have a root `x` of a monic quadratic
polynomial, then there is another root `y` such that `x + y` is negative the `a_1` coefficient
and `x * y` is the `a_0` coefficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
succ_ne_self [non_assoc_ring α] [nontrivial α] (a : α) : a + 1 ≠ a | λ h, one_ne_zero ((add_right_inj a).mp (by simp [h])) | lemma | succ_ne_self | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"non_assoc_ring",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_ne_self [non_assoc_ring α] [nontrivial α] (a : α) : a - 1 ≠ a | λ h, one_ne_zero (neg_injective ((add_right_inj a).mp (by simpa [sub_eq_add_neg] using h))) | lemma | pred_ne_self | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"non_assoc_ring",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_cancel_mul_zero.to_no_zero_divisors [ring α] [is_left_cancel_mul_zero α] :
no_zero_divisors α | begin
refine ⟨λ x y h, _⟩,
by_cases hx : x = 0,
{ left, exact hx },
{ right,
rw [← sub_zero (x * y), ← mul_zero x, ← mul_sub] at h,
convert (is_left_cancel_mul_zero.mul_left_cancel_of_ne_zero) hx h,
rw [sub_zero] }
end | lemma | is_left_cancel_mul_zero.to_no_zero_divisors | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"is_left_cancel_mul_zero",
"mul_zero",
"no_zero_divisors",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_right_cancel_mul_zero.to_no_zero_divisors [ring α] [is_right_cancel_mul_zero α] :
no_zero_divisors α | begin
refine ⟨λ x y h, _⟩,
by_cases hy : y = 0,
{ right, exact hy },
{ left,
rw [← sub_zero (x * y), ← zero_mul y, ← sub_mul] at h,
convert (is_right_cancel_mul_zero.mul_right_cancel_of_ne_zero) hy h,
rw [sub_zero] }
end | lemma | is_right_cancel_mul_zero.to_no_zero_divisors | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"is_right_cancel_mul_zero",
"no_zero_divisors",
"ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_divisors.to_is_cancel_mul_zero [ring α] [no_zero_divisors α] :
is_cancel_mul_zero α | { mul_left_cancel_of_ne_zero := λ a b c ha h,
begin
rw [← sub_eq_zero, ← mul_sub] at h,
exact sub_eq_zero.1 ((eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left ha)
end,
mul_right_cancel_of_ne_zero := λ a b c hb h,
begin
rw [← sub_eq_zero, ← sub_mul] at h,
exact sub_eq_zero.1 ((eq_zero_or_eq_zero... | instance | no_zero_divisors.to_is_cancel_mul_zero | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"is_cancel_mul_zero",
"no_zero_divisors",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_divisors.to_is_domain [ring α] [h : nontrivial α] [no_zero_divisors α] :
is_domain α | { .. no_zero_divisors.to_is_cancel_mul_zero α, .. h } | lemma | no_zero_divisors.to_is_domain | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"is_domain",
"no_zero_divisors",
"no_zero_divisors.to_is_cancel_mul_zero",
"nontrivial",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain.to_no_zero_divisors [ring α] [is_domain α] : no_zero_divisors α | is_right_cancel_mul_zero.to_no_zero_divisors α | instance | is_domain.to_no_zero_divisors | algebra.ring | src/algebra/ring/basic.lean | [
"algebra.ring.defs",
"algebra.hom.group",
"algebra.opposites"
] | [
"is_domain",
"is_right_cancel_mul_zero.to_no_zero_divisors",
"no_zero_divisors",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boolean_ring α extends ring α | (mul_self : ∀ a : α, a * a = a) | class | boolean_ring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"mul_self",
"ring"
] | A Boolean ring is a ring where multiplication is idempotent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_self : a * a = a | boolean_ring.mul_self _ | lemma | mul_self | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_self : a + a = 0 | have a + a = a + a + (a + a) :=
calc a + a = (a+a) * (a+a) : by rw mul_self
... = a*a + a*a + (a*a + a*a) : by rw [add_mul, mul_add]
... = a + a + (a + a) : by rw mul_self,
by rwa self_eq_add_left at this | lemma | add_self | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq : -a = a | calc -a = -a + 0 : by rw add_zero
... = -a + -a + a : by rw [←neg_add_self, add_assoc]
... = a : by rw [add_self, zero_add] | lemma | neg_eq | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_zero' : a + b = 0 ↔ a = b | calc a + b = 0 ↔ a = -b : add_eq_zero_iff_eq_neg
... ↔ a = b : by rw neg_eq | lemma | add_eq_zero' | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_mul : a*b + b*a = 0 | have a + b = a + b + (a*b + b*a) :=
calc a + b = (a + b) * (a + b) : by rw mul_self
... = a*a + a*b + (b*a + b*b) : by rw [add_mul, mul_add, mul_add]
... = a + a*b + (b*a + b) : by simp only [mul_self]
... = a + b + (a*b + b*a) : by abel,
by rwa self_eq_add_right at this | lemma | mul_add_mul | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_eq_add : a - b = a + b | by rw [sub_eq_add_neg, add_right_inj, neg_eq] | lemma | sub_eq_add | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_add_self : a * (1 + a) = 0 | by rw [mul_add, mul_one, mul_self, add_self] | lemma | mul_one_add_self | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self",
"mul_one",
"mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
boolean_ring.to_comm_ring : comm_ring α | { mul_comm := λ a b, by rw [←add_eq_zero', mul_add_mul],
.. (infer_instance : boolean_ring α) } | instance | boolean_ring.to_comm_ring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"boolean_ring",
"comm_ring",
"mul_add_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_boolalg (α : Type*) | α | def | as_boolalg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [] | Type synonym to view a Boolean ring as a Boolean algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_boolalg : α ≃ as_boolalg α | equiv.refl _ | def | to_boolalg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"equiv.refl"
] | The "identity" equivalence between `as_boolalg α` and `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_boolalg : as_boolalg α ≃ α | equiv.refl _ | def | of_boolalg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"equiv.refl"
] | The "identity" equivalence between `α` and `as_boolalg α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_boolalg_symm_eq : (@to_boolalg α).symm = of_boolalg | rfl | lemma | to_boolalg_symm_eq | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"of_boolalg",
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_symm_eq : (@of_boolalg α).symm = to_boolalg | rfl | lemma | of_boolalg_symm_eq | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"of_boolalg",
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolalg_of_boolalg (a : as_boolalg α) : to_boolalg (of_boolalg a) = a | rfl | lemma | to_boolalg_of_boolalg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg",
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_to_boolalg (a : α) : of_boolalg (to_boolalg a) = a | rfl | lemma | of_boolalg_to_boolalg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"of_boolalg",
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolalg_inj {a b : α} : to_boolalg a = to_boolalg b ↔ a = b | iff.rfl | lemma | to_boolalg_inj | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_inj {a b : as_boolalg α} : of_boolalg a = of_boolalg b ↔ a = b | iff.rfl | lemma | of_boolalg_inj | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sup : has_sup α | ⟨λ x y, x + y + x * y⟩ | def | boolean_ring.has_sup | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"has_sup"
] | The join operation in a Boolean ring is `x + y + x * y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_inf : has_inf α | ⟨(*)⟩ | def | boolean_ring.has_inf | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"has_inf"
] | The meet operation in a Boolean ring is `x * y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_comm (a b : α) : a ⊔ b = b ⊔ a | by { dsimp only [(⊔)], ring } | lemma | boolean_ring.sup_comm | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"ring",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_comm (a b : α) : a ⊓ b = b ⊓ a | by { dsimp only [(⊓)], ring } | lemma | boolean_ring.inf_comm | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"inf_comm",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_assoc (a b c : α) : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) | by { dsimp only [(⊔)], ring } | lemma | boolean_ring.sup_assoc | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"ring",
"sup_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_assoc (a b c : α) : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) | by { dsimp only [(⊓)], ring } | lemma | boolean_ring.inf_assoc | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"inf_assoc",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_inf_self (a b : α) : a ⊔ a ⊓ b = a | by { dsimp only [(⊔), (⊓)], assoc_rw [mul_self, add_self, add_zero] } | lemma | boolean_ring.sup_inf_self | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self",
"mul_self",
"sup_inf_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_sup_self (a b : α) : a ⊓ (a ⊔ b) = a | begin
dsimp only [(⊔), (⊓)],
rw [mul_add, mul_add, mul_self, ←mul_assoc, mul_self, add_assoc, add_self, add_zero]
end | lemma | boolean_ring.inf_sup_self | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self",
"inf_sup_self",
"mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sup_inf_aux (a b c : α) : (a + b + a * b) * (a + c + a * c) = a + b * c + a * (b * c) | calc (a + b + a * b) * (a + c + a * c) =
a * a + b * c + a * (b * c) +
(a * b + (a * a) * b) +
(a * c + (a * a) * c) +
(a * b * c + (a * a) * b * c) : by ring
... = a + b * c + a * (b * c) : by simp only [mul_self, add_self, add_zero] | lemma | boolean_ring.le_sup_inf_aux | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self",
"mul_self",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sup_inf (a b c : α) : (a ⊔ b) ⊓ (a ⊔ c) ⊔ (a ⊔ b ⊓ c) = a ⊔ b ⊓ c | by { dsimp only [(⊔), (⊓)], rw [le_sup_inf_aux, add_self, mul_self, zero_add] } | lemma | boolean_ring.le_sup_inf | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self",
"le_sup_inf",
"mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolean_algebra : boolean_algebra α | { le_sup_inf := le_sup_inf,
top := 1,
le_top := λ a, show a + 1 + a * 1 = 1, by assoc_rw [mul_one, add_comm, add_self, add_zero],
bot := 0,
bot_le := λ a, show 0 + a + 0 * a = a, by rw [zero_mul, zero_add, add_zero],
compl := λ a, 1 + a,
inf_compl_le_bot := λ a,
show a*(1+a) + 0 + a*(1+a)*0 = 0,
by ... | def | boolean_ring.to_boolean_algebra | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self",
"boolean_algebra",
"bot_le",
"inf_assoc",
"inf_comm",
"inf_sup_self",
"lattice.mk'",
"le_sup_inf",
"le_top",
"mul_one",
"mul_self",
"sup_assoc",
"sup_comm",
"sup_inf_self",
"zero_mul"
] | The Boolean algebra structure on a Boolean ring.
The data is defined so that:
* `a ⊔ b` unfolds to `a + b + a * b`
* `a ⊓ b` unfolds to `a * b`
* `a ≤ b` unfolds to `a + b + a * b = b`
* `⊥` unfolds to `0`
* `⊤` unfolds to `1`
* `aᶜ` unfolds to `1 + a`
* `a \ b` unfolds to `a * (1 + b)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_boolalg_top : of_boolalg (⊤ : as_boolalg α) = 1 | rfl | lemma | of_boolalg_top | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_bot : of_boolalg (⊥ : as_boolalg α) = 0 | rfl | lemma | of_boolalg_bot | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_sup (a b : as_boolalg α) :
of_boolalg (a ⊔ b) = of_boolalg a + of_boolalg b + of_boolalg a * of_boolalg b | rfl | lemma | of_boolalg_sup | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_inf (a b : as_boolalg α) :
of_boolalg (a ⊓ b) = of_boolalg a * of_boolalg b | rfl | lemma | of_boolalg_inf | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_compl (a : as_boolalg α) : of_boolalg aᶜ = 1 + of_boolalg a | rfl | lemma | of_boolalg_compl | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_sdiff (a b : as_boolalg α) :
of_boolalg (a \ b) = of_boolalg a * (1 + of_boolalg b) | rfl | lemma | of_boolalg_sdiff | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_symm_diff_aux (a b : α) : (a + b + a * b) * (1 + a * b) = a + b | calc (a + b + a * b) * (1 + a * b)
= a + b + (a * b + (a * b) * (a * b)) + (a * (b * b) + (a * a) * b) : by ring
... = a + b : by simp only [mul_self, add_self, add_zero] | lemma | of_boolalg_symm_diff_aux | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"add_self",
"mul_self",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_symm_diff (a b : as_boolalg α) :
of_boolalg (a ∆ b) = of_boolalg a + of_boolalg b | by { rw symm_diff_eq_sup_sdiff_inf, exact of_boolalg_symm_diff_aux _ _ } | lemma | of_boolalg_symm_diff | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"of_boolalg",
"of_boolalg_symm_diff_aux",
"symm_diff_eq_sup_sdiff_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolalg_mul_of_boolalg_eq_left_iff {a b : as_boolalg α} :
of_boolalg a * of_boolalg b = of_boolalg a ↔ a ≤ b | @inf_eq_left (as_boolalg α) _ _ _ | lemma | of_boolalg_mul_of_boolalg_eq_left_iff | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"inf_eq_left",
"of_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolalg_zero : to_boolalg (0 : α) = ⊥ | rfl | lemma | to_boolalg_zero | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolalg_one : to_boolalg (1 : α) = ⊤ | rfl | lemma | to_boolalg_one | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolalg_mul (a b : α) :
to_boolalg (a * b) = to_boolalg a ⊓ to_boolalg b | rfl | lemma | to_boolalg_mul | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolalg_add_add_mul (a b : α) :
to_boolalg (a + b + a * b) = to_boolalg a ⊔ to_boolalg b | rfl | lemma | to_boolalg_add_add_mul | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolalg_add (a b : α) : to_boolalg (a + b) = to_boolalg a ∆ to_boolalg b | (of_boolalg_symm_diff _ _).symm | lemma | to_boolalg_add | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"of_boolalg_symm_diff",
"to_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.as_boolalg (f : α →+* β) :
bounded_lattice_hom (as_boolalg α) (as_boolalg β) | { to_fun := to_boolalg ∘ f ∘ of_boolalg,
map_sup' := λ a b, begin
dsimp,
simp_rw [map_add f, map_mul f],
refl,
end,
map_inf' := f.map_mul',
map_top' := f.map_one',
map_bot' := f.map_zero' } | def | ring_hom.as_boolalg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"bounded_lattice_hom",
"map_mul",
"of_boolalg",
"to_boolalg"
] | Turn a ring homomorphism from Boolean rings `α` to `β` into a bounded lattice homomorphism
from `α` to `β` considered as Boolean algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.as_boolalg_id : (ring_hom.id α).as_boolalg = bounded_lattice_hom.id _ | rfl | lemma | ring_hom.as_boolalg_id | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg",
"bounded_lattice_hom.id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.as_boolalg_comp (g : β →+* γ) (f : α →+* β) :
(g.comp f).as_boolalg = g.as_boolalg.comp f.as_boolalg | rfl | lemma | ring_hom.as_boolalg_comp | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolalg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_boolring (α : Type*) | α | def | as_boolring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [] | Type synonym to view a Boolean ring as a Boolean algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_boolring : α ≃ as_boolring α | equiv.refl _ | def | to_boolring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"equiv.refl"
] | The "identity" equivalence between `as_boolring α` and `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_boolring : as_boolring α ≃ α | equiv.refl _ | def | of_boolring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"equiv.refl"
] | The "identity" equivalence between `α` and `as_boolring α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_boolring_symm_eq : (@to_boolring α).symm = of_boolring | rfl | lemma | to_boolring_symm_eq | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"of_boolring",
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_symm_eq : (@of_boolring α).symm = to_boolring | rfl | lemma | of_boolring_symm_eq | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"of_boolring",
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolring_of_boolring (a : as_boolring α) : to_boolring (of_boolring a) = a | rfl | lemma | to_boolring_of_boolring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring",
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_to_boolring (a : α) : of_boolring (to_boolring a) = a | rfl | lemma | of_boolring_to_boolring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"of_boolring",
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolring_inj {a b : α} : to_boolring a = to_boolring b ↔ a = b | iff.rfl | lemma | to_boolring_inj | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_inj {a b : as_boolring α} : of_boolring a = of_boolring b ↔ a = b | iff.rfl | lemma | of_boolring_inj | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_boolean_algebra.to_non_unital_comm_ring [generalized_boolean_algebra α] :
non_unital_comm_ring α | { add := (∆),
add_assoc := symm_diff_assoc,
zero := ⊥,
zero_add := bot_symm_diff,
add_zero := symm_diff_bot,
zero_mul := λ _, bot_inf_eq,
mul_zero := λ _, inf_bot_eq,
neg := id,
add_left_neg := symm_diff_self,
add_comm := symm_diff_comm,
mul := (⊓),
mul_assoc := λ _ _ _, inf_assoc,
mul_comm := λ... | def | generalized_boolean_algebra.to_non_unital_comm_ring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"bot_inf_eq",
"bot_symm_diff",
"generalized_boolean_algebra",
"inf_assoc",
"inf_bot_eq",
"inf_comm",
"inf_symm_diff_distrib_left",
"inf_symm_diff_distrib_right",
"left_distrib",
"mul_assoc",
"mul_comm",
"mul_zero",
"non_unital_comm_ring",
"right_distrib",
"symm_diff_assoc",
"symm_diff_... | Every generalized Boolean algebra has the structure of a non unital commutative ring with the
following data:
* `a + b` unfolds to `a ∆ b` (symmetric difference)
* `a * b` unfolds to `a ⊓ b`
* `-a` unfolds to `a`
* `0` unfolds to `⊥` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
boolean_algebra.to_boolean_ring : boolean_ring α | { one := ⊤,
one_mul := λ _, top_inf_eq,
mul_one := λ _, inf_top_eq,
mul_self := λ b, inf_idem,
..generalized_boolean_algebra.to_non_unital_comm_ring } | def | boolean_algebra.to_boolean_ring | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"boolean_ring",
"generalized_boolean_algebra.to_non_unital_comm_ring",
"inf_idem",
"inf_top_eq",
"mul_one",
"mul_self",
"one_mul",
"top_inf_eq"
] | Every Boolean algebra has the structure of a Boolean ring with the following data:
* `a + b` unfolds to `a ∆ b` (symmetric difference)
* `a * b` unfolds to `a ⊓ b`
* `-a` unfolds to `a`
* `0` unfolds to `⊥`
* `1` unfolds to `⊤` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_boolring_zero : of_boolring (0 : as_boolring α) = ⊥ | rfl | lemma | of_boolring_zero | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_one : of_boolring (1 : as_boolring α) = ⊤ | rfl | lemma | of_boolring_one | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_neg (a : as_boolring α) :
of_boolring (-a) = of_boolring a | rfl | lemma | of_boolring_neg | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_add (a b : as_boolring α) :
of_boolring (a + b) = of_boolring a ∆ of_boolring b | rfl | lemma | of_boolring_add | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_sub (a b : as_boolring α) :
of_boolring (a - b) = of_boolring a ∆ of_boolring b | rfl | lemma | of_boolring_sub | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_mul (a b : as_boolring α) :
of_boolring (a * b) = of_boolring a ⊓ of_boolring b | rfl | lemma | of_boolring_mul | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_boolring_le_of_boolring_iff {a b : as_boolring α} :
of_boolring a ≤ of_boolring b ↔ a * b = a | inf_eq_left.symm | lemma | of_boolring_le_of_boolring_iff | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"as_boolring",
"of_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolring_bot : to_boolring (⊥ : α) = 0 | rfl | lemma | to_boolring_bot | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolring_top : to_boolring (⊤ : α) = 1 | rfl | lemma | to_boolring_top | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolring_inf (a b : α) : to_boolring (a ⊓ b) = to_boolring a * to_boolring b | rfl | lemma | to_boolring_inf | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_boolring_symm_diff (a b : α) :
to_boolring (a ∆ b) = to_boolring a + to_boolring b | rfl | lemma | to_boolring_symm_diff | algebra.ring | src/algebra/ring/boolean_ring.lean | [
"algebra.punit_instances",
"tactic.abel",
"tactic.ring",
"order.hom.lattice"
] | [
"to_boolring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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