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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|---|---|---|---|---|---|---|---|---|---|---|
mem_span_support (f : monoid_algebra k G) :
f ∈ submodule.span k (of k G '' (f.support : set G)) | by rw [of, monoid_hom.coe_mk, ← finsupp.supported_eq_span_single, finsupp.mem_supported] | lemma | monoid_algebra.mem_span_support | algebra.monoid_algebra | src/algebra/monoid_algebra/support.lean | [
"algebra.monoid_algebra.basic"
] | [
"finsupp.mem_supported",
"finsupp.supported_eq_span_single",
"monoid_algebra",
"monoid_hom.coe_mk",
"submodule.span"
] | An element of `monoid_algebra k G` is in the subalgebra generated by its support. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
support_mul [decidable_eq G] [has_add G] (a b : add_monoid_algebra k G) :
(a * b).support ⊆ a.support.bUnion (λa₁, b.support.bUnion $ λa₂, {a₁ + a₂}) | @monoid_algebra.support_mul k (multiplicative G) _ _ _ _ _ | lemma | add_monoid_algebra.support_mul | algebra.monoid_algebra | src/algebra/monoid_algebra/support.lean | [
"algebra.monoid_algebra.basic"
] | [
"add_monoid_algebra",
"monoid_algebra.support_mul",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_mul_single [add_right_cancel_semigroup G]
(f : add_monoid_algebra k G) (r : k) (hr : ∀ y, y * r = 0 ↔ y = 0) (x : G) :
(f * single x r : add_monoid_algebra k G).support = f.support.map (add_right_embedding x) | @monoid_algebra.support_mul_single k (multiplicative G) _ _ _ _ hr _ | lemma | add_monoid_algebra.support_mul_single | algebra.monoid_algebra | src/algebra/monoid_algebra/support.lean | [
"algebra.monoid_algebra.basic"
] | [
"add_monoid_algebra",
"add_right_cancel_semigroup",
"monoid_algebra.support_mul_single",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_single_mul [add_left_cancel_semigroup G]
(f : add_monoid_algebra k G) (r : k) (hr : ∀ y, r * y = 0 ↔ y = 0) (x : G) :
(single x r * f : add_monoid_algebra k G).support = f.support.map (add_left_embedding x) | @monoid_algebra.support_single_mul k (multiplicative G) _ _ _ _ hr _ | lemma | add_monoid_algebra.support_single_mul | algebra.monoid_algebra | src/algebra/monoid_algebra/support.lean | [
"algebra.monoid_algebra.basic"
] | [
"add_left_cancel_semigroup",
"add_monoid_algebra",
"monoid_algebra.support_single_mul",
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_support [add_zero_class G] (f : add_monoid_algebra k G) :
f ∈ submodule.span k (of k G '' (f.support : set G)) | by rw [of, monoid_hom.coe_mk, ← finsupp.supported_eq_span_single, finsupp.mem_supported] | lemma | add_monoid_algebra.mem_span_support | algebra.monoid_algebra | src/algebra/monoid_algebra/support.lean | [
"algebra.monoid_algebra.basic"
] | [
"add_monoid_algebra",
"add_zero_class",
"finsupp.mem_supported",
"finsupp.supported_eq_span_single",
"monoid_hom.coe_mk",
"submodule.span"
] | An element of `add_monoid_algebra k G` is in the submodule generated by its support. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_span_support' (f : add_monoid_algebra k G) :
f ∈ submodule.span k (of' k G '' (f.support : set G)) | by rw [of', ← finsupp.supported_eq_span_single, finsupp.mem_supported] | lemma | add_monoid_algebra.mem_span_support' | algebra.monoid_algebra | src/algebra/monoid_algebra/support.lean | [
"algebra.monoid_algebra.basic"
] | [
"add_monoid_algebra",
"finsupp.mem_supported",
"finsupp.supported_eq_span_single",
"submodule.span"
] | An element of `add_monoid_algebra k G` is in the subalgebra generated by its support, using
unbundled inclusion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_algebra.to_direct_sum [semiring M] (f : add_monoid_algebra M ι) : ⨁ i : ι, M | finsupp.to_dfinsupp f | def | add_monoid_algebra.to_direct_sum | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"finsupp.to_dfinsupp",
"semiring"
] | Interpret a `add_monoid_algebra` as a homogenous `direct_sum`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_algebra.to_direct_sum_single (i : ι) (m : M) :
add_monoid_algebra.to_direct_sum (finsupp.single i m) = direct_sum.of _ i m | finsupp.to_dfinsupp_single i m | lemma | add_monoid_algebra.to_direct_sum_single | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra.to_direct_sum",
"direct_sum.of",
"finsupp.single",
"finsupp.to_dfinsupp_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
direct_sum.to_add_monoid_algebra (f : ⨁ i : ι, M) :
add_monoid_algebra M ι | dfinsupp.to_finsupp f | def | direct_sum.to_add_monoid_algebra | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"dfinsupp.to_finsupp"
] | Interpret a homogenous `direct_sum` as a `add_monoid_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum.to_add_monoid_algebra_of (i : ι) (m : M) :
(direct_sum.of _ i m : ⨁ i : ι, M).to_add_monoid_algebra = finsupp.single i m | dfinsupp.to_finsupp_single i m | lemma | direct_sum.to_add_monoid_algebra_of | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"dfinsupp.to_finsupp_single",
"direct_sum.of",
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_algebra.to_direct_sum_to_add_monoid_algebra (f : add_monoid_algebra M ι) :
f.to_direct_sum.to_add_monoid_algebra = f | finsupp.to_dfinsupp_to_finsupp f | lemma | add_monoid_algebra.to_direct_sum_to_add_monoid_algebra | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"finsupp.to_dfinsupp_to_finsupp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
direct_sum.to_add_monoid_algebra_to_direct_sum (f : ⨁ i : ι, M) :
f.to_add_monoid_algebra.to_direct_sum = f | dfinsupp.to_finsupp_to_dfinsupp f | lemma | direct_sum.to_add_monoid_algebra_to_direct_sum | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"dfinsupp.to_finsupp_to_dfinsupp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_direct_sum_zero [semiring M] :
(0 : add_monoid_algebra M ι).to_direct_sum = 0 | finsupp.to_dfinsupp_zero | lemma | add_monoid_algebra.to_direct_sum_zero | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"finsupp.to_dfinsupp_zero",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_direct_sum_add [semiring M] (f g : add_monoid_algebra M ι) :
(f + g).to_direct_sum = f.to_direct_sum + g.to_direct_sum | finsupp.to_dfinsupp_add _ _ | lemma | add_monoid_algebra.to_direct_sum_add | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"finsupp.to_dfinsupp_add",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_direct_sum_mul [decidable_eq ι] [add_monoid ι] [semiring M]
(f g : add_monoid_algebra M ι) :
(f * g).to_direct_sum = f.to_direct_sum * g.to_direct_sum | begin
let to_hom : add_monoid_algebra M ι →+ (⨁ i : ι, M) :=
⟨to_direct_sum, to_direct_sum_zero, to_direct_sum_add⟩,
show to_hom (f * g) = to_hom f * to_hom g,
revert f g,
rw add_monoid_hom.map_mul_iff,
ext xi xv yi yv : 4,
dsimp only [add_monoid_hom.comp_apply, add_monoid_hom.compl₂_apply,
add_mono... | lemma | add_monoid_algebra.to_direct_sum_mul | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid",
"add_monoid_algebra",
"add_monoid_algebra.single_mul_single",
"add_monoid_algebra.to_direct_sum_single",
"add_monoid_hom.map_mul_iff",
"add_monoid_hom.mul_apply",
"direct_sum.of_mul_of",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_algebra_zero [semiring M] [Π m : M, decidable (m ≠ 0)] :
to_add_monoid_algebra 0 = (0 : add_monoid_algebra M ι) | dfinsupp.to_finsupp_zero | lemma | direct_sum.to_add_monoid_algebra_zero | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"dfinsupp.to_finsupp_zero",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_algebra_add [semiring M] [Π m : M, decidable (m ≠ 0)]
(f g : ⨁ i : ι, M) :
(f + g).to_add_monoid_algebra = to_add_monoid_algebra f + to_add_monoid_algebra g | dfinsupp.to_finsupp_add _ _ | lemma | direct_sum.to_add_monoid_algebra_add | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"dfinsupp.to_finsupp_add",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_algebra_mul [add_monoid ι] [semiring M] [Π m : M, decidable (m ≠ 0)]
(f g : ⨁ i : ι, M) :
(f * g).to_add_monoid_algebra = to_add_monoid_algebra f * to_add_monoid_algebra g | begin
apply_fun add_monoid_algebra.to_direct_sum,
{ simp },
{ apply function.left_inverse.injective,
apply add_monoid_algebra.to_direct_sum_to_add_monoid_algebra }
end | lemma | direct_sum.to_add_monoid_algebra_mul | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid",
"add_monoid_algebra.to_direct_sum",
"add_monoid_algebra.to_direct_sum_to_add_monoid_algebra",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_algebra_equiv_direct_sum [decidable_eq ι] [semiring M] [Π m : M, decidable (m ≠ 0)] :
add_monoid_algebra M ι ≃ (⨁ i : ι, M) | { to_fun := add_monoid_algebra.to_direct_sum, inv_fun := direct_sum.to_add_monoid_algebra,
..finsupp_equiv_dfinsupp } | def | add_monoid_algebra_equiv_direct_sum | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"add_monoid_algebra.to_direct_sum",
"direct_sum.to_add_monoid_algebra",
"finsupp_equiv_dfinsupp",
"inv_fun",
"semiring"
] | `add_monoid_algebra.to_direct_sum` and `direct_sum.to_add_monoid_algebra` together form an
equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_algebra_add_equiv_direct_sum
[decidable_eq ι] [semiring M] [Π m : M, decidable (m ≠ 0)] :
add_monoid_algebra M ι ≃+ (⨁ i : ι, M) | { to_fun := add_monoid_algebra.to_direct_sum, inv_fun := direct_sum.to_add_monoid_algebra,
map_add' := add_monoid_algebra.to_direct_sum_add,
.. add_monoid_algebra_equiv_direct_sum} | def | add_monoid_algebra_add_equiv_direct_sum | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid_algebra",
"add_monoid_algebra.to_direct_sum",
"add_monoid_algebra.to_direct_sum_add",
"add_monoid_algebra_equiv_direct_sum",
"direct_sum.to_add_monoid_algebra",
"inv_fun",
"semiring"
] | The additive version of `add_monoid_algebra.to_add_monoid_algebra`. Note that this is
`noncomputable` because `add_monoid_algebra.has_add` is noncomputable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_algebra_ring_equiv_direct_sum
[decidable_eq ι] [add_monoid ι] [semiring M]
[Π m : M, decidable (m ≠ 0)] :
add_monoid_algebra M ι ≃+* ⨁ i : ι, M | { to_fun := add_monoid_algebra.to_direct_sum, inv_fun := direct_sum.to_add_monoid_algebra,
map_mul' := add_monoid_algebra.to_direct_sum_mul,
..(add_monoid_algebra_add_equiv_direct_sum : add_monoid_algebra M ι ≃+ ⨁ i : ι, M) } | def | add_monoid_algebra_ring_equiv_direct_sum | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid",
"add_monoid_algebra",
"add_monoid_algebra.to_direct_sum",
"add_monoid_algebra.to_direct_sum_mul",
"add_monoid_algebra_add_equiv_direct_sum",
"direct_sum.to_add_monoid_algebra",
"inv_fun",
"semiring"
] | The ring version of `add_monoid_algebra.to_add_monoid_algebra`. Note that this is
`noncomputable` because `add_monoid_algebra.has_add` is noncomputable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_algebra_alg_equiv_direct_sum
[decidable_eq ι] [add_monoid ι] [comm_semiring R] [semiring A] [algebra R A]
[Π m : A, decidable (m ≠ 0)] :
add_monoid_algebra A ι ≃ₐ[R] ⨁ i : ι, A | { to_fun := add_monoid_algebra.to_direct_sum, inv_fun := direct_sum.to_add_monoid_algebra,
commutes' := λ r, add_monoid_algebra.to_direct_sum_single _ _,
..(add_monoid_algebra_ring_equiv_direct_sum : add_monoid_algebra A ι ≃+* ⨁ i : ι, A) } | def | add_monoid_algebra_alg_equiv_direct_sum | algebra.monoid_algebra | src/algebra/monoid_algebra/to_direct_sum.lean | [
"algebra.direct_sum.algebra",
"algebra.monoid_algebra.basic",
"data.finsupp.to_dfinsupp"
] | [
"add_monoid",
"add_monoid_algebra",
"add_monoid_algebra.to_direct_sum",
"add_monoid_algebra.to_direct_sum_single",
"add_monoid_algebra_ring_equiv_direct_sum",
"algebra",
"comm_semiring",
"direct_sum.to_add_monoid_algebra",
"inv_fun",
"semiring"
] | The algebra version of `add_monoid_algebra.to_add_monoid_algebra`. Note that this is
`noncomputable` because `add_monoid_algebra.has_add` is noncomputable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absolute_value (R S : Type*) [semiring R] [ordered_semiring S]
extends R →ₙ* S | (nonneg' : ∀ x, 0 ≤ to_fun x)
(eq_zero' : ∀ x, to_fun x = 0 ↔ x = 0)
(add_le' : ∀ x y, to_fun (x + y) ≤ to_fun x + to_fun y) | structure | absolute_value | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"ordered_semiring",
"semiring"
] | `absolute_value R S` is the type of absolute values on `R` mapping to `S`:
the maps that preserve `*`, are nonnegative, positive definite and satisfy the triangle equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_hom_class : zero_hom_class (absolute_value R S) R S | { coe := λ f, f.to_fun,
coe_injective' := λ f g h, by { obtain ⟨⟨_, _⟩, _⟩ := f, obtain ⟨⟨_, _⟩, _⟩ := g, congr' },
map_zero := λ f, (f.eq_zero' _).2 rfl } | instance | absolute_value.zero_hom_class | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value",
"zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom_class : mul_hom_class (absolute_value R S) R S | { map_mul := λ f, f.map_mul'
..absolute_value.zero_hom_class } | instance | absolute_value.mul_hom_class | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value",
"absolute_value.zero_hom_class",
"map_mul",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_hom_class : nonneg_hom_class (absolute_value R S) R S | { map_nonneg := λ f, f.nonneg',
..absolute_value.zero_hom_class } | instance | absolute_value.nonneg_hom_class | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value",
"absolute_value.zero_hom_class",
"map_nonneg",
"nonneg_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subadditive_hom_class : subadditive_hom_class (absolute_value R S) R S | { map_add_le_add := λ f, f.add_le',
..absolute_value.zero_hom_class } | instance | absolute_value.subadditive_hom_class | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value",
"absolute_value.zero_hom_class",
"subadditive_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : R →ₙ* S) {h₁ h₂ h₃} : ((absolute_value.mk f h₁ h₂ h₃) : R → S) = f | rfl | lemma | absolute_value.coe_mk | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext ⦃f g : absolute_value R S⦄ : (∀ x, f x = g x) → f = g | fun_like.ext _ _ | lemma | absolute_value.ext | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (f : absolute_value R S) : R → S | f
initialize_simps_projections absolute_value (to_mul_hom_to_fun → apply) | def | absolute_value.simps.apply | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value"
] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_mul_hom : ⇑abv.to_mul_hom = abv | rfl | lemma | absolute_value.coe_to_mul_hom | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg (x : R) : 0 ≤ abv x | abv.nonneg' x | theorem | absolute_value.nonneg | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero {x : R} : abv x = 0 ↔ x = 0 | abv.eq_zero' x | theorem | absolute_value.eq_zero | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le (x y : R) : abv (x + y) ≤ abv x + abv y | abv.add_le' x y | theorem | absolute_value.add_le | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"add_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul (x y : R) : abv (x * y) = abv x * abv y | abv.map_mul' x y | theorem | absolute_value.map_mul | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_iff {x : R} : abv x ≠ 0 ↔ x ≠ 0 | abv.eq_zero.not | theorem | absolute_value.ne_zero_iff | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"ne_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos {x : R} (hx : x ≠ 0) : 0 < abv x | lt_of_le_of_ne (abv.nonneg x) (ne.symm $ mt abv.eq_zero.mp hx) | theorem | absolute_value.pos | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_iff {x : R} : 0 < abv x ↔ x ≠ 0 | ⟨λ h₁, mt abv.eq_zero.mpr h₁.ne', abv.pos⟩ | theorem | absolute_value.pos_iff | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero {x : R} (hx : x ≠ 0) : abv x ≠ 0 | (abv.pos hx).ne' | theorem | absolute_value.ne_zero | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one_of_is_regular (h : is_left_regular (abv 1)) : abv 1 = 1 | h $ by simp [←abv.map_mul] | theorem | absolute_value.map_one_of_is_regular | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"is_left_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : abv 0 = 0 | abv.eq_zero.2 rfl | theorem | absolute_value.map_zero | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_le (a b c : R) : abv (a - c) ≤ abv (a - b) + abv (b - c) | by simpa [sub_eq_add_neg, add_assoc] using abv.add_le (a - b) (b - c) | lemma | absolute_value.sub_le | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub_eq_zero_iff (a b : R) : abv (a - b) = 0 ↔ a = b | abv.eq_zero.trans sub_eq_zero | lemma | absolute_value.map_sub_eq_zero_iff | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one : abv 1 = 1 | abv.map_one_of_is_regular ((is_regular_of_ne_zero $ abv.ne_zero one_ne_zero).left) | theorem | absolute_value.map_one | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"is_regular_of_ne_zero",
"map_one",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_with_zero_hom : R →*₀ S | abv | def | absolute_value.to_monoid_with_zero_hom | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | Absolute values from a nontrivial `R` to a linear ordered ring preserve `*`, `0` and `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_monoid_with_zero_hom : ⇑abv.to_monoid_with_zero_hom = abv | rfl | lemma | absolute_value.coe_to_monoid_with_zero_hom | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_hom : R →* S | abv | def | absolute_value.to_monoid_hom | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | Absolute values from a nontrivial `R` to a linear ordered ring preserve `*` and `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_monoid_hom : ⇑abv.to_monoid_hom = abv | rfl | lemma | absolute_value.coe_to_monoid_hom | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pow (a : R) (n : ℕ) : abv (a ^ n) = abv a ^ n | abv.to_monoid_hom.map_pow a n | lemma | absolute_value.map_pow | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sub (a b : R) : abv a - abv b ≤ abv (a - b) | sub_le_iff_le_add.2 $ by simpa using abv.add_le (a - b) b | lemma | absolute_value.le_sub | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (a : R) : abv (-a) = abv a | begin
by_cases ha : a = 0, { simp [ha] },
refine (mul_self_eq_mul_self_iff.mp
(by rw [← abv.map_mul, neg_mul_neg, abv.map_mul])).resolve_right _,
exact ((neg_lt_zero.mpr (abv.pos ha)).trans (abv.pos (neg_ne_zero.mpr ha))).ne'
end | theorem | absolute_value.map_neg | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"neg_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (a b : R) : abv (a - b) = abv (b - a) | by rw [← neg_sub, abv.map_neg] | theorem | absolute_value.map_sub | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs : absolute_value S S | { to_fun := abs,
nonneg' := abs_nonneg,
eq_zero' := λ _, abs_eq_zero,
add_le' := abs_add,
map_mul' := abs_mul } | def | absolute_value.abs | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"abs_add",
"abs_eq_zero",
"abs_mul",
"abs_nonneg",
"absolute_value"
] | `absolute_value.abs` is `abs` as a bundled `absolute_value`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_abv_sub_le_abv_sub (a b : R) :
abs (abv a - abv b) ≤ abv (a - b) | abs_sub_le_iff.2 ⟨abv.le_sub _ _, by rw abv.map_sub; apply abv.le_sub⟩ | lemma | absolute_value.abs_abv_sub_le_abv_sub | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_absolute_value {S} [ordered_semiring S]
{R} [semiring R] (f : R → S) : Prop | (abv_nonneg [] : ∀ x, 0 ≤ f x)
(abv_eq_zero [] : ∀ {x}, f x = 0 ↔ x = 0)
(abv_add [] : ∀ x y, f (x + y) ≤ f x + f y)
(abv_mul [] : ∀ x y, f (x * y) = f x * f y) | class | is_absolute_value | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"ordered_semiring",
"semiring"
] | A function `f` is an absolute value if it is nonnegative, zero only at 0, additive, and
multiplicative.
See also the type `absolute_value` which represents a bundled version of absolute values. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.absolute_value.is_absolute_value
(abv : absolute_value R S) : is_absolute_value abv | { abv_nonneg := abv.nonneg,
abv_eq_zero := λ _, abv.eq_zero,
abv_add := abv.add_le,
abv_mul := abv.map_mul } | instance | absolute_value.is_absolute_value | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value",
"is_absolute_value"
] | A bundled absolute value is an absolute value. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_absolute_value : absolute_value R S | { to_fun := abv,
add_le' := abv_add abv,
eq_zero' := λ _, abv_eq_zero abv,
nonneg' := abv_nonneg abv,
map_mul' := abv_mul abv } | def | is_absolute_value.to_absolute_value | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"absolute_value"
] | Convert an unbundled `is_absolute_value` to a bundled `absolute_value`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abv_zero : abv 0 = 0 | (to_absolute_value abv).map_zero | theorem | is_absolute_value.abv_zero | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_pos {a : R} : 0 < abv a ↔ a ≠ 0 | (to_absolute_value abv).pos_iff | theorem | is_absolute_value.abv_pos | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_is_absolute_value : is_absolute_value (abs : S → S) | absolute_value.abs.is_absolute_value | instance | is_absolute_value.abs_is_absolute_value | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"is_absolute_value"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_one [nontrivial R] : abv 1 = 1 | (to_absolute_value abv).map_one | theorem | is_absolute_value.abv_one | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"map_one",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_hom [nontrivial R] : R →*₀ S | (to_absolute_value abv).to_monoid_with_zero_hom | def | is_absolute_value.abv_hom | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"nontrivial"
] | `abv` as a `monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abv_pow [nontrivial R] (abv : R → S) [is_absolute_value abv]
(a : R) (n : ℕ) : abv (a ^ n) = abv a ^ n | (to_absolute_value abv).map_pow a n | lemma | is_absolute_value.abv_pow | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"is_absolute_value",
"map_pow",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_sub_le (a b c : R) : abv (a - c) ≤ abv (a - b) + abv (b - c) | by simpa [sub_eq_add_neg, add_assoc] using abv_add abv (a - b) (b - c) | lemma | is_absolute_value.abv_sub_le | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_abv_le_abv_sub (a b : R) : abv a - abv b ≤ abv (a - b) | (to_absolute_value abv).le_sub a b | lemma | is_absolute_value.sub_abv_le_abv_sub | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_neg (a : R) : abv (-a) = abv a | (to_absolute_value abv).map_neg a | theorem | is_absolute_value.abv_neg | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_sub (a b : R) : abv (a - b) = abv (b - a) | (to_absolute_value abv).map_sub a b | theorem | is_absolute_value.abv_sub | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_abv_sub_le_abv_sub (a b : R) :
abs (abv a - abv b) ≤ abv (a - b) | (to_absolute_value abv).abs_abv_sub_le_abv_sub a b | lemma | is_absolute_value.abs_abv_sub_le_abv_sub | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_one' : abv 1 = 1 | (to_absolute_value abv).map_one_of_is_regular
$ (is_regular_of_ne_zero $ (to_absolute_value abv).ne_zero one_ne_zero).left | lemma | is_absolute_value.abv_one' | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"is_regular_of_ne_zero",
"ne_zero",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_hom' : R →*₀ S | ⟨abv, abv_zero abv, abv_one' abv, abv_mul abv⟩ | def | is_absolute_value.abv_hom' | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [] | An absolute value as a monoid with zero homomorphism, assuming the target is a semifield. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abv_inv (a : R) : abv a⁻¹ = (abv a)⁻¹ | map_inv₀ (abv_hom' abv) a | theorem | is_absolute_value.abv_inv | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"map_inv₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abv_div (a b : R) : abv (a / b) = abv a / abv b | map_div₀ (abv_hom' abv) a b | theorem | is_absolute_value.abv_div | algebra.order | src/algebra/order/absolute_value.lean | [
"algebra.group_with_zero.units.lemmas",
"algebra.order.field.defs",
"algebra.order.hom.basic",
"algebra.ring.regular"
] | [
"map_div₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_monotone : monotone (algebra_map R A) | λ a b h,
begin
rw [algebra.algebra_map_eq_smul_one, algebra.algebra_map_eq_smul_one, ←sub_nonneg, ←sub_smul],
transitivity (b - a) • (0 : A),
{ simp, },
{ exact smul_le_smul_of_nonneg zero_le_one (sub_nonneg.mpr h) }
end | lemma | algebra_map_monotone | algebra.order | src/algebra/order/algebra.lean | [
"algebra.algebra.basic",
"algebra.order.smul"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"monotone",
"smul_le_smul_of_nonneg",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean (α) [ordered_add_comm_monoid α] : Prop | (arch : ∀ (x : α) {y}, 0 < y → ∃ n : ℕ, x ≤ n • y) | class | archimedean | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"ordered_add_comm_monoid"
] | An ordered additive commutative monoid is called `archimedean` if for any two elements `x`, `y`
such that `0 < y` there exists a natural number `n` such that `x ≤ n • y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_dual.archimedean [ordered_add_comm_group α] [archimedean α] : archimedean αᵒᵈ | ⟨λ x y hy, let ⟨n, hn⟩ := archimedean.arch (-x : α) (neg_pos.2 hy) in
⟨n, by rwa [neg_nsmul, neg_le_neg_iff] at hn⟩⟩ | instance | order_dual.archimedean | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"ordered_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique_zsmul_near_of_pos {a : α} (ha : 0 < a) (g : α) :
∃! k : ℤ, k • a ≤ g ∧ g < (k + 1) • a | begin
let s : set ℤ := {n : ℤ | n • a ≤ g},
obtain ⟨k, hk : -g ≤ k • a⟩ := archimedean.arch (-g) ha,
have h_ne : s.nonempty := ⟨-k, by simpa using neg_le_neg hk⟩,
obtain ⟨k, hk⟩ := archimedean.arch g ha,
have h_bdd : ∀ n ∈ s, n ≤ (k : ℤ),
{ assume n hn,
apply (zsmul_le_zsmul_iff ha).mp,
rw ← coe_nat... | lemma | exists_unique_zsmul_near_of_pos | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"int.exists_greatest_of_bdd",
"lt_add_one"
] | An archimedean decidable linearly ordered `add_comm_group` has a version of the floor: for
`a > 0`, any `g` in the group lies between some two consecutive multiples of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_unique_zsmul_near_of_pos' {a : α} (ha : 0 < a) (g : α) :
∃! k : ℤ, 0 ≤ g - k • a ∧ g - k • a < a | by simpa only [sub_nonneg, add_zsmul, one_zsmul, sub_lt_iff_lt_add']
using exists_unique_zsmul_near_of_pos ha g | lemma | exists_unique_zsmul_near_of_pos' | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_unique_zsmul_near_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique_sub_zsmul_mem_Ico {a : α} (ha : 0 < a) (b c : α) :
∃! m : ℤ, b - m • a ∈ set.Ico c (c + a) | by simpa only [mem_Ico, le_sub_iff_add_le, zero_add, add_comm c, sub_lt_iff_lt_add', add_assoc]
using exists_unique_zsmul_near_of_pos' ha (b - c) | lemma | exists_unique_sub_zsmul_mem_Ico | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_unique_zsmul_near_of_pos'",
"set.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique_add_zsmul_mem_Ico {a : α} (ha : 0 < a) (b c : α) :
∃! m : ℤ, b + m • a ∈ set.Ico c (c + a) | (equiv.neg ℤ).bijective.exists_unique_iff.2 $
by simpa only [equiv.neg_apply, neg_zsmul, ← sub_eq_add_neg]
using exists_unique_sub_zsmul_mem_Ico ha b c | lemma | exists_unique_add_zsmul_mem_Ico | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_unique_sub_zsmul_mem_Ico",
"set.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique_add_zsmul_mem_Ioc {a : α} (ha : 0 < a) (b c : α) :
∃! m : ℤ, b + m • a ∈ set.Ioc c (c + a) | (equiv.add_right (1 : ℤ)).bijective.exists_unique_iff.2 $
by simpa only [add_one_zsmul, sub_lt_iff_lt_add', le_sub_iff_add_le', ← add_assoc, and.comm,
mem_Ioc, equiv.coe_add_right, add_le_add_iff_right]
using exists_unique_zsmul_near_of_pos ha (c - b) | lemma | exists_unique_add_zsmul_mem_Ioc | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_unique_zsmul_near_of_pos",
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unique_sub_zsmul_mem_Ioc {a : α} (ha : 0 < a) (b c : α) :
∃! m : ℤ, b - m • a ∈ set.Ioc c (c + a) | (equiv.neg ℤ).bijective.exists_unique_iff.2 $
by simpa only [equiv.neg_apply, neg_zsmul, sub_neg_eq_add]
using exists_unique_add_zsmul_mem_Ioc ha b c | lemma | exists_unique_sub_zsmul_mem_Ioc | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_unique_add_zsmul_mem_Ioc",
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nat_gt [strict_ordered_semiring α] [archimedean α] (x : α) : ∃ n : ℕ, x < n | let ⟨n, h⟩ := archimedean.arch x zero_lt_one in
⟨n+1, lt_of_le_of_lt (by rwa ← nsmul_one)
(nat.cast_lt.2 (nat.lt_succ_self _))⟩ | theorem | exists_nat_gt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"nsmul_one",
"strict_ordered_semiring",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nat_ge [strict_ordered_semiring α] [archimedean α] (x : α) :
∃ n : ℕ, x ≤ n | begin
nontriviality α,
exact (exists_nat_gt x).imp (λ n, le_of_lt)
end | theorem | exists_nat_ge | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"exists_nat_gt",
"strict_ordered_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_pow_unbounded_of_pos [strict_ordered_semiring α] [archimedean α] (x : α) {y : α}
(hy : 0 < y) :
∃ n : ℕ, x < (y + 1) ^ n | have 0 ≤ 1 + y, from add_nonneg zero_le_one hy.le,
let ⟨n, h⟩ := archimedean.arch x hy in
⟨n, calc x ≤ n • y : h
... = n * y : nsmul_eq_mul _ _
... < 1 + n * y : lt_one_add _
... ≤ (1 + y) ^ n : one_add_mul_le_pow' (mul_nonneg hy.le hy.le) (mul_nonneg this this)
(add_no... | lemma | add_one_pow_unbounded_of_pos | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"lt_one_add",
"nsmul_eq_mul",
"one_add_mul_le_pow'",
"strict_ordered_semiring",
"zero_le_one",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_unbounded_of_one_lt (x : α) {y : α} (hy1 : 1 < y) :
∃ n : ℕ, x < y ^ n | sub_add_cancel y 1 ▸ add_one_pow_unbounded_of_pos _ (sub_pos.2 hy1) | lemma | pow_unbounded_of_one_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"add_one_pow_unbounded_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_int_gt (x : α) : ∃ n : ℤ, x < n | let ⟨n, h⟩ := exists_nat_gt x in ⟨n, by rwa int.cast_coe_nat⟩ | theorem | exists_int_gt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_nat_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_int_lt (x : α) : ∃ n : ℤ, (n : α) < x | let ⟨n, h⟩ := exists_int_gt (-x) in ⟨-n, by rw int.cast_neg; exact neg_lt.1 h⟩ | theorem | exists_int_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_int_gt",
"int.cast_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_floor (x : α) :
∃ (fl : ℤ), ∀ (z : ℤ), z ≤ fl ↔ (z : α) ≤ x | begin
haveI := classical.prop_decidable,
have : ∃ (ub : ℤ), (ub:α) ≤ x ∧ ∀ (z : ℤ), (z:α) ≤ x → z ≤ ub :=
int.exists_greatest_of_bdd
(let ⟨n, hn⟩ := exists_int_gt x in ⟨n, λ z h',
int.cast_le.1 $ le_trans h' $ le_of_lt hn⟩)
(let ⟨n, hn⟩ := exists_int_lt x in ⟨n, le_of_lt hn⟩),
refine this.imp (λ f... | theorem | exists_floor | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_int_gt",
"exists_int_lt",
"int.exists_greatest_of_bdd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nat_pow_near {x : α} {y : α} (hx : 1 ≤ x) (hy : 1 < y) :
∃ n : ℕ, y ^ n ≤ x ∧ x < y ^ (n + 1) | have h : ∃ n : ℕ, x < y ^ n, from pow_unbounded_of_one_lt _ hy,
by classical; exact let n := nat.find h in
have hn : x < y ^ n, from nat.find_spec h,
have hnp : 0 < n, from pos_iff_ne_zero.2 (λ hn0,
by rw [hn0, pow_zero] at hn; exact (not_le_of_gt hn hx)),
have hnsp : nat.pred n + 1 = n, from nat.suc... | lemma | exists_nat_pow_near | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"pow_unbounded_of_one_lt",
"pow_zero"
] | Every x greater than or equal to 1 is between two successive
natural-number powers of every y greater than one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_mem_Ico_zpow (hx : 0 < x) (hy : 1 < y) : ∃ n : ℤ, x ∈ Ico (y ^ n) (y ^ (n + 1)) | by classical; exact
let ⟨N, hN⟩ := pow_unbounded_of_one_lt x⁻¹ hy in
have he: ∃ m : ℤ, y ^ m ≤ x, from
⟨-N, le_of_lt (by { rw [zpow_neg y (↑N), zpow_coe_nat],
exact (inv_lt hx (lt_trans (inv_pos.2 hx) hN)).1 hN })⟩,
let ⟨M, hM⟩ := pow_unbounded_of_one_lt x hy in
have hb: ∃ b : ℤ, ∀ m, y ^ m ≤ x → m ≤ b, fro... | lemma | exists_mem_Ico_zpow | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"int.exists_greatest_of_bdd",
"inv_lt",
"pow_unbounded_of_one_lt",
"zpow_coe_nat",
"zpow_le_of_le",
"zpow_neg"
] | Every positive `x` is between two successive integer powers of
another `y` greater than one. This is the same as `exists_mem_Ioc_zpow`,
but with ≤ and < the other way around. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_mem_Ioc_zpow (hx : 0 < x) (hy : 1 < y) : ∃ n : ℤ, x ∈ Ioc (y ^ n) (y ^ (n + 1)) | let ⟨m, hle, hlt⟩ := exists_mem_Ico_zpow (inv_pos.2 hx) hy in
have hyp : 0 < y, from lt_trans zero_lt_one hy,
⟨-(m+1),
by rwa [zpow_neg, inv_lt (zpow_pos_of_pos hyp _) hx],
by rwa [neg_add, neg_add_cancel_right, zpow_neg,
le_inv hx (zpow_pos_of_pos hyp _)]⟩ | lemma | exists_mem_Ioc_zpow | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_mem_Ico_zpow",
"inv_lt",
"le_inv",
"zero_lt_one",
"zpow_neg",
"zpow_pos_of_pos"
] | Every positive `x` is between two successive integer powers of
another `y` greater than one. This is the same as `exists_mem_Ico_zpow`,
but with ≤ and < the other way around. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_pow_lt_of_lt_one (hx : 0 < x) (hy : y < 1) : ∃ n : ℕ, y ^ n < x | begin
by_cases y_pos : y ≤ 0,
{ use 1, simp only [pow_one], linarith, },
rw [not_le] at y_pos,
rcases pow_unbounded_of_one_lt (x⁻¹) (one_lt_inv y_pos hy) with ⟨q, hq⟩,
exact ⟨q, by rwa [inv_pow, inv_lt_inv hx (pow_pos y_pos _)] at hq⟩
end | lemma | exists_pow_lt_of_lt_one | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"inv_lt_inv",
"inv_pow",
"one_lt_inv",
"pow_one",
"pow_pos",
"pow_unbounded_of_one_lt"
] | For any `y < 1` and any positive `x`, there exists `n : ℕ` with `y ^ n < x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nat_pow_near_of_lt_one (xpos : 0 < x) (hx : x ≤ 1) (ypos : 0 < y) (hy : y < 1) :
∃ n : ℕ, y ^ (n + 1) < x ∧ x ≤ y ^ n | begin
rcases exists_nat_pow_near (one_le_inv_iff.2 ⟨xpos, hx⟩) (one_lt_inv_iff.2 ⟨ypos, hy⟩)
with ⟨n, hn, h'n⟩,
refine ⟨n, _, _⟩,
{ rwa [inv_pow, inv_lt_inv xpos (pow_pos ypos _)] at h'n },
{ rwa [inv_pow, inv_le_inv (pow_pos ypos _) xpos] at hn }
end | lemma | exists_nat_pow_near_of_lt_one | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_nat_pow_near",
"inv_le_inv",
"inv_lt_inv",
"inv_pow",
"pow_pos"
] | Given `x` and `y` between `0` and `1`, `x` is between two successive powers of `y`.
This is the same as `exists_nat_pow_near`, but for elements between `0` and `1` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_rat_gt (x : α) : ∃ q : ℚ, x < q | let ⟨n, h⟩ := exists_nat_gt x in ⟨n, by rwa rat.cast_coe_nat⟩ | lemma | exists_rat_gt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_nat_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_rat_lt (x : α) : ∃ q : ℚ, (q : α) < x | let ⟨n, h⟩ := exists_int_lt x in ⟨n, by rwa rat.cast_coe_int⟩ | theorem | exists_rat_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_int_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_rat_btwn {x y : α} (h : x < y) : ∃ q : ℚ, x < q ∧ (q:α) < y | begin
cases exists_nat_gt (y - x)⁻¹ with n nh,
cases exists_floor (x * n) with z zh,
refine ⟨(z + 1 : ℤ) / n, _⟩,
have n0' := (inv_pos.2 (sub_pos.2 h)).trans nh,
have n0 := nat.cast_pos.1 n0',
rw [rat.cast_div_of_ne_zero, rat.cast_coe_nat, rat.cast_coe_int, div_lt_iff n0'],
refine ⟨(lt_div_iff n0').2 $
... | theorem | exists_rat_btwn | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"div_lt_iff",
"div_lt_iff'",
"exists_floor",
"exists_nat_gt",
"int.cast_add",
"int.cast_coe_nat",
"int.cast_one",
"le_rfl",
"lt_add_one",
"lt_div_iff",
"lt_iff_lt_of_le_iff_le",
"nat.cast_eq_zero",
"nat.cast_one",
"one_div",
"one_ne_zero",
"rat.cast_coe_int",
"rat.cast_coe_nat",
"r... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_rat_lt_imp_le (h : ∀ q : ℚ, (q : α) < x → (q : α) ≤ y) : x ≤ y | le_of_not_lt $ λ hyx, let ⟨q, hy, hx⟩ := exists_rat_btwn hyx in hy.not_le $ h _ hx | lemma | le_of_forall_rat_lt_imp_le | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_rat_btwn"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_forall_lt_rat_imp_le (h : ∀ q : ℚ, y < q → x ≤ q) : x ≤ y | le_of_not_lt $ λ hyx, let ⟨q, hy, hx⟩ := exists_rat_btwn hyx in hx.not_le $ h _ hy | lemma | le_of_forall_lt_rat_imp_le | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_rat_btwn"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_forall_rat_lt_iff_lt (h : ∀ q : ℚ, (q : α) < x ↔ (q : α) < y) : x = y | (le_of_forall_rat_lt_imp_le $ λ q hq, ((h q).1 hq).le).antisymm $ le_of_forall_rat_lt_imp_le $
λ q hq, ((h q).2 hq).le | lemma | eq_of_forall_rat_lt_iff_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"le_of_forall_rat_lt_imp_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_forall_lt_rat_iff_lt (h : ∀ q : ℚ, x < q ↔ y < q) : x = y | (le_of_forall_lt_rat_imp_le $ λ q hq, ((h q).2 hq).le).antisymm $ le_of_forall_lt_rat_imp_le $
λ q hq, ((h q).1 hq).le | lemma | eq_of_forall_lt_rat_iff_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"le_of_forall_lt_rat_imp_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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