fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
disc_ne_zero_iff_roots_ne (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ x ≠ y ∧ x ≠ z ∧ y ≠ z :=
begin
rw [←_root_.map_ne_zero φ, disc_eq_prod_three_roots ha h3, pow_two],
simp_rw [mul_ne_zero_iff, sub_ne_zero, _root_.map_ne_zero, and_self, and_iff_right ha, and_assoc],
end | disc_ne_zero_iff_roots_ne (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ x ≠ y ∧ x ≠ z ∧ y ≠ z | begin
rw [←_root_.map_ne_zero φ, disc_eq_prod_three_roots ha h3, pow_two],
simp_rw [mul_ne_zero_iff, sub_ne_zero, _root_.map_ne_zero, and_self, and_iff_right ha, and_assoc],
end | theorem | cubic.disc_ne_zero_iff_roots_ne | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"mul_ne_zero_iff",
"pow_two"
] | null | 433 | 438 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disc_ne_zero_iff_roots_nodup (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ (map φ P).roots.nodup :=
begin
rw [disc_ne_zero_iff_roots_ne ha h3, h3],
change _ ↔ (x ::ₘ y ::ₘ {z}).nodup,
rw [nodup_cons, nodup_cons, mem_cons, mem_singleton, mem_singleton],
simp only [nodup_singleton],
tautolo... | disc_ne_zero_iff_roots_nodup (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ (map φ P).roots.nodup | begin
rw [disc_ne_zero_iff_roots_ne ha h3, h3],
change _ ↔ (x ::ₘ y ::ₘ {z}).nodup,
rw [nodup_cons, nodup_cons, mem_cons, mem_singleton, mem_singleton],
simp only [nodup_singleton],
tautology
end | theorem | cubic.disc_ne_zero_iff_roots_nodup | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"mem_cons"
] | null | 440 | 448 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_roots_of_disc_ne_zero [decidable_eq K] (ha : P.a ≠ 0)
(h3 : (map φ P).roots = {x, y, z}) (hd : P.disc ≠ 0) : (map φ P).roots.to_finset.card = 3 :=
begin
rw [to_finset_card_of_nodup $ (disc_ne_zero_iff_roots_nodup ha h3).mp hd,
← splits_iff_card_roots ha, splits_iff_roots_eq_three ha],
exact ⟨x, ⟨y, ⟨z,... | card_roots_of_disc_ne_zero [decidable_eq K] (ha : P.a ≠ 0)
(h3 : (map φ P).roots = {x, y, z}) (hd : P.disc ≠ 0) : (map φ P).roots.to_finset.card = 3 | begin
rw [to_finset_card_of_nodup $ (disc_ne_zero_iff_roots_nodup ha h3).mp hd,
← splits_iff_card_roots ha, splits_iff_roots_eq_three ha],
exact ⟨x, ⟨y, ⟨z, h3⟩⟩⟩
end | theorem | cubic.card_roots_of_disc_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | null | 450 | 456 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
directed_system (f : Π i j, i ≤ j → G i → G j) : Prop :=
(map_self [] : ∀ i x h, f i i h x = x)
(map_map [] : ∀ {i j k} hij hjk x, f j k hjk (f i j hij x) = f i k (le_trans hij hjk) x) | directed_system (f : Π i j, i ≤ j → G i → G j) : Prop | (map_self [] : ∀ i x h, f i i h x = x)
(map_map [] : ∀ {i j k} hij hjk x, f j k hjk (f i j hij x) = f i k (le_trans hij hjk) x) | class | directed_system | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | A directed system is a functor from a category (directed poset) to another category. | 43 | 45 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
directed_system.map_self [directed_system G (λ i j h, f i j h)] (i x h) :
f i i h x = x :=
directed_system.map_self (λ i j h, f i j h) i x h | directed_system.map_self [directed_system G (λ i j h, f i j h)] (i x h) :
f i i h x = x | directed_system.map_self (λ i j h, f i j h) i x h | lemma | module.directed_system.map_self | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system"
] | A copy of `directed_system.map_self` specialized to linear maps, as otherwise the
`λ i j h, f i j h` can confuse the simplifier. | 55 | 57 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
directed_system.map_map [directed_system G (λ i j h, f i j h)] {i j k} (hij hjk x) :
f j k hjk (f i j hij x) = f i k (le_trans hij hjk) x :=
directed_system.map_map (λ i j h, f i j h) hij hjk x | directed_system.map_map [directed_system G (λ i j h, f i j h)] {i j k} (hij hjk x) :
f j k hjk (f i j hij x) = f i k (le_trans hij hjk) x | directed_system.map_map (λ i j h, f i j h) hij hjk x | lemma | module.directed_system.map_map | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system"
] | A copy of `directed_system.map_map` specialized to linear maps, as otherwise the
`λ i j h, f i j h` can confuse the simplifier. | 61 | 63 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_limit : Type (max v w) :=
direct_sum ι G ⧸ (span R $ { a | ∃ (i j) (H : i ≤ j) x,
direct_sum.lof R ι G i x - direct_sum.lof R ι G j (f i j H x) = a }) | direct_limit : Type (max v w) | direct_sum ι G ⧸ (span R $ { a | ∃ (i j) (H : i ≤ j) x,
direct_sum.lof R ι G i x - direct_sum.lof R ι G j (f i j H x) = a }) | def | module.direct_limit | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum",
"direct_sum.lof"
] | The direct limit of a directed system is the modules glued together along the maps. | 70 | 72 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: add_comm_group (direct_limit G f) := quotient.add_comm_group _ | : add_comm_group (direct_limit G f) | quotient.add_comm_group _ | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"add_comm_group"
] | null | 76 | 76 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: module R (direct_limit G f) := quotient.module _ | : module R (direct_limit G f) | quotient.module _ | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"module"
] | null | 77 | 77 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: inhabited (direct_limit G f) := ⟨0⟩ | : inhabited (direct_limit G f) | ⟨0⟩ | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | null | 79 | 79 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (i) : G i →ₗ[R] direct_limit G f :=
(mkq _).comp $ direct_sum.lof R ι G i | of (i) : G i →ₗ[R] direct_limit G f | (mkq _).comp $ direct_sum.lof R ι G i | def | module.direct_limit.of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.lof"
] | The canonical map from a component to the direct limit. | 83 | 84 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_f {i j hij x} : (of R ι G f j (f i j hij x)) = of R ι G f i x :=
eq.symm $ (submodule.quotient.eq _).2 $ subset_span ⟨i, j, hij, x, rfl⟩ | of_f {i j hij x} : (of R ι G f j (f i j hij x)) = of R ι G f i x | eq.symm $ (submodule.quotient.eq _).2 $ subset_span ⟨i, j, hij, x, rfl⟩ | lemma | module.direct_limit.of_f | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"submodule.quotient.eq"
] | null | 87 | 88 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_of [nonempty ι] [is_directed ι (≤)] (z : direct_limit G f) :
∃ i x, of R ι G f i x = z :=
nonempty.elim (by apply_instance) $ assume ind : ι,
quotient.induction_on' z $ λ z, direct_sum.induction_on z
⟨ind, 0, linear_map.map_zero _⟩
(λ i x, ⟨i, x, rfl⟩)
(λ p q ⟨i, x, ihx⟩ ⟨j, y, ihy⟩, let ⟨k, hik, hjk⟩ :=... | exists_of [nonempty ι] [is_directed ι (≤)] (z : direct_limit G f) :
∃ i x, of R ι G f i x = z | nonempty.elim (by apply_instance) $ assume ind : ι,
quotient.induction_on' z $ λ z, direct_sum.induction_on z
⟨ind, 0, linear_map.map_zero _⟩
(λ i x, ⟨i, x, rfl⟩)
(λ p q ⟨i, x, ihx⟩ ⟨j, y, ihy⟩, let ⟨k, hik, hjk⟩ := exists_ge_ge i j in
⟨k, f i k hik x + f j k hjk y, by rw [linear_map.map_add, of_f, of_f, ihx,... | theorem | module.direct_limit.exists_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.induction_on",
"exists_ge_ge",
"is_directed",
"linear_map.map_add",
"linear_map.map_zero",
"quotient.induction_on'"
] | Every element of the direct limit corresponds to some element in
some component of the directed system. | 92 | 99 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction_on [nonempty ι] [is_directed ι (≤)] {C : direct_limit G f → Prop}
(z : direct_limit G f)
(ih : ∀ i x, C (of R ι G f i x)) : C z :=
let ⟨i, x, h⟩ := exists_of z in h ▸ ih i x | induction_on [nonempty ι] [is_directed ι (≤)] {C : direct_limit G f → Prop}
(z : direct_limit G f)
(ih : ∀ i x, C (of R ι G f i x)) : C z | let ⟨i, x, h⟩ := exists_of z in h ▸ ih i x | theorem | module.direct_limit.induction_on | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ih",
"is_directed"
] | null | 101 | 105 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : direct_limit G f →ₗ[R] P :=
liftq _ (direct_sum.to_module R ι P g)
(span_le.2 $ λ a ⟨i, j, hij, x, hx⟩, by rw [← hx, set_like.mem_coe, linear_map.sub_mem_ker_iff,
direct_sum.to_module_lof, direct_sum.to_module_lof, Hg]) | lift : direct_limit G f →ₗ[R] P | liftq _ (direct_sum.to_module R ι P g)
(span_le.2 $ λ a ⟨i, j, hij, x, hx⟩, by rw [← hx, set_like.mem_coe, linear_map.sub_mem_ker_iff,
direct_sum.to_module_lof, direct_sum.to_module_lof, Hg]) | def | module.direct_limit.lift | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.to_module",
"direct_sum.to_module_lof",
"lift",
"linear_map.sub_mem_ker_iff",
"set_like.mem_coe"
] | The universal property of the direct limit: maps from the components to another module
that respect the directed system structure (i.e. make some diagram commute) give rise
to a unique map out of the direct limit. | 115 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of {i} (x) : lift R ι G f g Hg (of R ι G f i x) = g i x :=
direct_sum.to_module_lof R _ _ | lift_of {i} (x) : lift R ι G f g Hg (of R ι G f i x) = g i x | direct_sum.to_module_lof R _ _ | lemma | module.direct_limit.lift_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.to_module_lof",
"lift"
] | null | 122 | 123 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_unique [nonempty ι] [is_directed ι (≤)] (F : direct_limit G f →ₗ[R] P) (x) :
F x = lift R ι G f (λ i, F.comp $ of R ι G f i)
(λ i j hij x, by rw [linear_map.comp_apply, of_f]; refl) x :=
direct_limit.induction_on x $ λ i x, by rw lift_of; refl | lift_unique [nonempty ι] [is_directed ι (≤)] (F : direct_limit G f →ₗ[R] P) (x) :
F x = lift R ι G f (λ i, F.comp $ of R ι G f i)
(λ i j hij x, by rw [linear_map.comp_apply, of_f]; refl) x | direct_limit.induction_on x $ λ i x, by rw lift_of; refl | theorem | module.direct_limit.lift_unique | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"is_directed",
"lift",
"lift_unique",
"linear_map.comp_apply"
] | null | 125 | 128 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totalize (i j) : G i →ₗ[R] G j :=
if h : i ≤ j then f i j h else 0 | totalize (i j) : G i →ₗ[R] G j | if h : i ≤ j then f i j h else 0 | def | module.direct_limit.totalize | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | `totalize G f i j` is a linear map from `G i` to `G j`, for *every* `i` and `j`.
If `i ≤ j`, then it is the map `f i j` that comes with the directed system `G`,
and otherwise it is the zero map. | 138 | 139 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totalize_of_le {i j} (h : i ≤ j) : totalize G f i j = f i j h := dif_pos h | totalize_of_le {i j} (h : i ≤ j) : totalize G f i j = f i j h | dif_pos h | lemma | module.direct_limit.totalize_of_le | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | null | 142 | 142 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totalize_of_not_le {i j} (h : ¬(i ≤ j)) : totalize G f i j = 0 := dif_neg h | totalize_of_not_le {i j} (h : ¬(i ≤ j)) : totalize G f i j = 0 | dif_neg h | lemma | module.direct_limit.totalize_of_not_le | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | null | 144 | 144 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module_totalize_of_le {x : direct_sum ι G} {i j : ι}
(hij : i ≤ j) (hx : ∀ k ∈ x.support, k ≤ i) :
direct_sum.to_module R ι (G j) (λ k, totalize G f k j) x =
f i j hij (direct_sum.to_module R ι (G i) (λ k, totalize G f k i) x) :=
begin
rw [← @dfinsupp.sum_single ι G _ _ _ x],
unfold dfinsupp.sum,
simp on... | to_module_totalize_of_le {x : direct_sum ι G} {i j : ι}
(hij : i ≤ j) (hx : ∀ k ∈ x.support, k ≤ i) :
direct_sum.to_module R ι (G j) (λ k, totalize G f k j) x =
f i j hij (direct_sum.to_module R ι (G i) (λ k, totalize G f k i) x) | begin
rw [← @dfinsupp.sum_single ι G _ _ _ x],
unfold dfinsupp.sum,
simp only [linear_map.map_sum],
refine finset.sum_congr rfl (λ k hk, _),
rw [direct_sum.single_eq_lof R k (x k), direct_sum.to_module_lof, direct_sum.to_module_lof,
totalize_of_le (hx k hk), totalize_of_le (le_trans (hx k hk) hij), direct... | lemma | module.direct_limit.to_module_totalize_of_le | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"dfinsupp.sum_single",
"direct_sum",
"direct_sum.single_eq_lof",
"direct_sum.to_module",
"direct_sum.to_module_lof",
"linear_map.map_sum"
] | null | 151 | 162 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of.zero_exact_aux [nonempty ι] [is_directed ι (≤)] {x : direct_sum ι G}
(H : submodule.quotient.mk x = (0 : direct_limit G f)) :
∃ j, (∀ k ∈ x.support, k ≤ j) ∧
direct_sum.to_module R ι (G j) (λ i, totalize G f i j) x = (0 : G j) :=
nonempty.elim (by apply_instance) $ assume ind : ι,
span_induction ((quotient.m... | of.zero_exact_aux [nonempty ι] [is_directed ι (≤)] {x : direct_sum ι G}
(H : submodule.quotient.mk x = (0 : direct_limit G f)) :
∃ j, (∀ k ∈ x.support, k ≤ j) ∧
direct_sum.to_module R ι (G j) (λ i, totalize G f i j) x = (0 : G j) | nonempty.elim (by apply_instance) $ assume ind : ι,
span_induction ((quotient.mk_eq_zero _).1 H)
(λ x ⟨i, j, hij, y, hxy⟩, let ⟨k, hik, hjk⟩ := exists_ge_ge i j in
⟨k, begin
clear_,
subst hxy,
split,
{ intros i0 hi0,
rw [dfinsupp.mem_support_iff, direct_sum.sub_apply, ← direct_sum.... | lemma | module.direct_limit.of.zero_exact_aux | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"dfinsupp.mem_support_iff",
"dfinsupp.single_apply",
"dfinsupp.support_add",
"direct_sum",
"direct_sum.apply_eq_component",
"direct_sum.component.of",
"direct_sum.single_eq_lof",
"direct_sum.sub_apply",
"direct_sum.support_smul",
"direct_sum.to_module",
"exists_ge_ge",
"finset.not_mem_empty",
... | null | 164 | 196 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of.zero_exact [is_directed ι (≤)] {i x} (H : of R ι G f i x = 0) :
∃ j hij, f i j hij x = (0 : G j) :=
by haveI : nonempty ι := ⟨i⟩; exact
let ⟨j, hj, hxj⟩ := of.zero_exact_aux H in
if hx0 : x = 0 then ⟨i, le_rfl, by simp [hx0]⟩
else
have hij : i ≤ j, from hj _ $
by simp [direct_sum.apply_eq_component, hx0],
... | of.zero_exact [is_directed ι (≤)] {i x} (H : of R ι G f i x = 0) :
∃ j hij, f i j hij x = (0 : G j) | by haveI : nonempty ι := ⟨i⟩; exact
let ⟨j, hj, hxj⟩ := of.zero_exact_aux H in
if hx0 : x = 0 then ⟨i, le_rfl, by simp [hx0]⟩
else
have hij : i ≤ j, from hj _ $
by simp [direct_sum.apply_eq_component, hx0],
⟨j, hij, by simpa [totalize_of_le hij] using hxj⟩ | theorem | module.direct_limit.of.zero_exact | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.apply_eq_component",
"is_directed",
"le_rfl"
] | A component that corresponds to zero in the direct limit is already zero in some
bigger module in the directed system. | 200 | 208 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_limit (f : Π i j, i ≤ j → G i →+ G j) : Type* :=
@module.direct_limit ℤ _ ι _ _ G _ _
(λ i j hij, (f i j hij).to_int_linear_map) | direct_limit (f : Π i j, i ≤ j → G i →+ G j) : Type* | @module.direct_limit ℤ _ ι _ _ G _ _
(λ i j hij, (f i j hij).to_int_linear_map) | def | add_comm_group.direct_limit | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"module.direct_limit"
] | The direct limit of a directed system is the abelian groups glued together along the maps. | 221 | 223 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
directed_system [h : directed_system G (λ i j h, f i j h)] :
directed_system G (λ i j hij, (f i j hij).to_int_linear_map) :=
h | directed_system [h : directed_system G (λ i j h, f i j h)] :
directed_system G (λ i j hij, (f i j hij).to_int_linear_map) | h | lemma | add_comm_group.direct_limit.directed_system | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system"
] | null | 231 | 233 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: add_comm_group (direct_limit G f) :=
module.direct_limit.add_comm_group G (λ i j hij, (f i j hij).to_int_linear_map) | : add_comm_group (direct_limit G f) | module.direct_limit.add_comm_group G (λ i j hij, (f i j hij).to_int_linear_map) | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"add_comm_group"
] | null | 239 | 240 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: inhabited (direct_limit G f) := ⟨0⟩ | : inhabited (direct_limit G f) | ⟨0⟩ | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | null | 242 | 242 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (i) : G i →ₗ[ℤ] direct_limit G f :=
module.direct_limit.of ℤ ι G (λ i j hij, (f i j hij).to_int_linear_map) i | of (i) : G i →ₗ[ℤ] direct_limit G f | module.direct_limit.of ℤ ι G (λ i j hij, (f i j hij).to_int_linear_map) i | def | add_comm_group.direct_limit.of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"module.direct_limit.of"
] | The canonical map from a component to the direct limit. | 245 | 246 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_f {i j} (hij) (x) : of G f j (f i j hij x) = of G f i x :=
module.direct_limit.of_f | of_f {i j} (hij) (x) : of G f j (f i j hij x) = of G f i x | module.direct_limit.of_f | lemma | add_comm_group.direct_limit.of_f | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"module.direct_limit.of_f"
] | null | 249 | 250 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction_on [nonempty ι] [is_directed ι (≤)] {C : direct_limit G f → Prop}
(z : direct_limit G f) (ih : ∀ i x, C (of G f i x)) : C z :=
module.direct_limit.induction_on z ih | induction_on [nonempty ι] [is_directed ι (≤)] {C : direct_limit G f → Prop}
(z : direct_limit G f) (ih : ∀ i x, C (of G f i x)) : C z | module.direct_limit.induction_on z ih | theorem | add_comm_group.direct_limit.induction_on | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ih",
"is_directed",
"module.direct_limit.induction_on"
] | null | 252 | 255 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of.zero_exact [is_directed ι (≤)] [directed_system G (λ i j h, f i j h)] (i x)
(h : of G f i x = 0) :
∃ j hij, f i j hij x = 0 :=
module.direct_limit.of.zero_exact h | of.zero_exact [is_directed ι (≤)] [directed_system G (λ i j h, f i j h)] (i x)
(h : of G f i x = 0) :
∃ j hij, f i j hij x = 0 | module.direct_limit.of.zero_exact h | theorem | add_comm_group.direct_limit.of.zero_exact | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system",
"is_directed",
"module.direct_limit.of.zero_exact"
] | A component that corresponds to zero in the direct limit is already zero in some
bigger module in the directed system. | 259 | 262 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : direct_limit G f →ₗ[ℤ] P :=
module.direct_limit.lift ℤ ι G (λ i j hij, (f i j hij).to_int_linear_map)
(λ i, (g i).to_int_linear_map) Hg | lift : direct_limit G f →ₗ[ℤ] P | module.direct_limit.lift ℤ ι G (λ i j hij, (f i j hij).to_int_linear_map)
(λ i, (g i).to_int_linear_map) Hg | def | add_comm_group.direct_limit.lift | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"lift",
"module.direct_limit.lift"
] | The universal property of the direct limit: maps from the components to another abelian group
that respect the directed system structure (i.e. make some diagram commute) give rise
to a unique map out of the direct limit. | 272 | 274 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of (i x) : lift G f P g Hg (of G f i x) = g i x :=
module.direct_limit.lift_of _ _ _ | lift_of (i x) : lift G f P g Hg (of G f i x) = g i x | module.direct_limit.lift_of _ _ _ | lemma | add_comm_group.direct_limit.lift_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"lift",
"module.direct_limit.lift_of"
] | null | 277 | 278 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_unique [nonempty ι] [is_directed ι (≤)] (F : direct_limit G f →+ P) (x) :
F x = lift G f P (λ i, F.comp (of G f i).to_add_monoid_hom)
(λ i j hij x, by simp) x :=
direct_limit.induction_on x $ λ i x, by simp | lift_unique [nonempty ι] [is_directed ι (≤)] (F : direct_limit G f →+ P) (x) :
F x = lift G f P (λ i, F.comp (of G f i).to_add_monoid_hom)
(λ i j hij x, by simp) x | direct_limit.induction_on x $ λ i x, by simp | lemma | add_comm_group.direct_limit.lift_unique | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"is_directed",
"lift",
"lift_unique"
] | null | 280 | 283 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_limit : Type (max v w) :=
free_comm_ring (Σ i, G i) ⧸ (ideal.span { a |
(∃ i j H x, of (⟨j, f i j H x⟩ : Σ i, G i) - of ⟨i, x⟩ = a) ∨
(∃ i, of (⟨i, 1⟩ : Σ i, G i) - 1 = a) ∨
(∃ i x y, of (⟨i, x + y⟩ : Σ i, G i) - (of ⟨i, x⟩ + of ⟨i, y⟩) = a) ∨
(∃ i x y, of (⟨i, x * y⟩ : Σ i, G i) - (of ⟨i, x⟩ * of ⟨i, y⟩... | direct_limit : Type (max v w) | free_comm_ring (Σ i, G i) ⧸ (ideal.span { a |
(∃ i j H x, of (⟨j, f i j H x⟩ : Σ i, G i) - of ⟨i, x⟩ = a) ∨
(∃ i, of (⟨i, 1⟩ : Σ i, G i) - 1 = a) ∨
(∃ i x y, of (⟨i, x + y⟩ : Σ i, G i) - (of ⟨i, x⟩ + of ⟨i, y⟩) = a) ∨
(∃ i x y, of (⟨i, x * y⟩ : Σ i, G i) - (of ⟨i, x⟩ * of ⟨i, y⟩) = a) }) | def | ring.direct_limit | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"free_comm_ring",
"ideal.span"
] | The direct limit of a directed system is the rings glued together along the maps. | 300 | 305 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: comm_ring (direct_limit G f) :=
ideal.quotient.comm_ring _ | : comm_ring (direct_limit G f) | ideal.quotient.comm_ring _ | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"comm_ring",
"ideal.quotient.comm_ring"
] | null | 309 | 310 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: ring (direct_limit G f) :=
comm_ring.to_ring _ | : ring (direct_limit G f) | comm_ring.to_ring _ | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ring"
] | null | 312 | 313 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: inhabited (direct_limit G f) := ⟨0⟩ | : inhabited (direct_limit G f) | ⟨0⟩ | instance | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | null | 315 | 315 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (i) : G i →+* direct_limit G f :=
ring_hom.mk'
{ to_fun := λ x, ideal.quotient.mk _ (of (⟨i, x⟩ : Σ i, G i)),
map_one' := ideal.quotient.eq.2 $ subset_span $ or.inr $ or.inl ⟨i, rfl⟩,
map_mul' := λ x y, ideal.quotient.eq.2 $ subset_span $ or.inr $ or.inr $ or.inr ⟨i, x, y, rfl⟩, }
(λ x y, ideal.quotient.eq.2 $ s... | of (i) : G i →+* direct_limit G f | ring_hom.mk'
{ to_fun := λ x, ideal.quotient.mk _ (of (⟨i, x⟩ : Σ i, G i)),
map_one' := ideal.quotient.eq.2 $ subset_span $ or.inr $ or.inl ⟨i, rfl⟩,
map_mul' := λ x y, ideal.quotient.eq.2 $ subset_span $ or.inr $ or.inr $ or.inr ⟨i, x, y, rfl⟩, }
(λ x y, ideal.quotient.eq.2 $ subset_span $ or.inr $ or.inr $ or.inl... | def | ring.direct_limit.of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk",
"ring_hom.mk'"
] | The canonical map from a component to the direct limit. | 318 | 323 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_f {i j} (hij) (x) : of G f j (f i j hij x) = of G f i x :=
ideal.quotient.eq.2 $ subset_span $ or.inl ⟨i, j, hij, x, rfl⟩ | of_f {i j} (hij) (x) : of G f j (f i j hij x) = of G f i x | ideal.quotient.eq.2 $ subset_span $ or.inl ⟨i, j, hij, x, rfl⟩ | lemma | ring.direct_limit.of_f | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | null | 327 | 328 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_of [nonempty ι] [is_directed ι (≤)] (z : direct_limit G f) :
∃ i x, of G f i x = z :=
nonempty.elim (by apply_instance) $ assume ind : ι,
quotient.induction_on' z $ λ x, free_abelian_group.induction_on x
⟨ind, 0, (of _ _ ind).map_zero⟩
(λ s, multiset.induction_on s
⟨ind, 1, (of _ _ ind).map_one⟩
(λ... | exists_of [nonempty ι] [is_directed ι (≤)] (z : direct_limit G f) :
∃ i x, of G f i x = z | nonempty.elim (by apply_instance) $ assume ind : ι,
quotient.induction_on' z $ λ x, free_abelian_group.induction_on x
⟨ind, 0, (of _ _ ind).map_zero⟩
(λ s, multiset.induction_on s
⟨ind, 1, (of _ _ ind).map_one⟩
(λ a s ih, let ⟨i, x⟩ := a, ⟨j, y, hs⟩ := ih, ⟨k, hik, hjk⟩ := exists_ge_ge i j in
⟨k, f i ... | theorem | ring.direct_limit.exists_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"exists_ge_ge",
"free_abelian_group.induction_on",
"ih",
"is_directed",
"map_mul",
"multiset.induction_on",
"quotient.induction_on'"
] | Every element of the direct limit corresponds to some element in
some component of the directed system. | 332 | 343 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial.exists_of [nonempty ι] [is_directed ι (≤)]
(q : polynomial (direct_limit G (λ i j h, f' i j h))) :
∃ i p, polynomial.map (of G (λ i j h, f' i j h) i) p = q :=
polynomial.induction_on q
(λ z, let ⟨i, x, h⟩ := exists_of z in ⟨i, C x, by rw [map_C, h]⟩)
(λ q₁ q₂ ⟨i₁, p₁, ih₁⟩ ⟨i₂, p₂, ih₂⟩, let ⟨i, h1, ... | polynomial.exists_of [nonempty ι] [is_directed ι (≤)]
(q : polynomial (direct_limit G (λ i j h, f' i j h))) :
∃ i p, polynomial.map (of G (λ i j h, f' i j h) i) p = q | polynomial.induction_on q
(λ z, let ⟨i, x, h⟩ := exists_of z in ⟨i, C x, by rw [map_C, h]⟩)
(λ q₁ q₂ ⟨i₁, p₁, ih₁⟩ ⟨i₂, p₂, ih₂⟩, let ⟨i, h1, h2⟩ := exists_ge_ge i₁ i₂ in
⟨i, p₁.map (f' i₁ i h1) + p₂.map (f' i₂ i h2),
by { rw [polynomial.map_add, map_map, map_map, ← ih₁, ← ih₂],
congr' 2; ext x; simp... | theorem | ring.direct_limit.polynomial.exists_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"exists_ge_ge",
"ih",
"is_directed",
"polynomial",
"polynomial.induction_on",
"polynomial.map",
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_pow",
"ring_hom.comp_apply"
] | null | 352 | 362 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction_on [nonempty ι] [is_directed ι (≤)]
{C : direct_limit G f → Prop}
(z : direct_limit G f) (ih : ∀ i x, C (of G f i x)) : C z :=
let ⟨i, x, hx⟩ := exists_of z in hx ▸ ih i x | induction_on [nonempty ι] [is_directed ι (≤)]
{C : direct_limit G f → Prop}
(z : direct_limit G f) (ih : ∀ i x, C (of G f i x)) : C z | let ⟨i, x, hx⟩ := exists_of z in hx ▸ ih i x | theorem | ring.direct_limit.induction_on | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ih",
"is_directed"
] | null | 366 | 369 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of.zero_exact_aux2 {x : free_comm_ring Σ i, G i} {s t} (hxs : is_supported x s) {j k}
(hj : ∀ z : Σ i, G i, z ∈ s → z.1 ≤ j) (hk : ∀ z : Σ i, G i, z ∈ t → z.1 ≤ k)
(hjk : j ≤ k) (hst : s ⊆ t) :
f' j k hjk (lift (λ ix : s, f' ix.1.1 j (hj ix ix.2) ix.1.2) (restriction s x)) =
lift (λ ix : t, f' ix.1.1 k (hk ix i... | of.zero_exact_aux2 {x : free_comm_ring Σ i, G i} {s t} (hxs : is_supported x s) {j k}
(hj : ∀ z : Σ i, G i, z ∈ s → z.1 ≤ j) (hk : ∀ z : Σ i, G i, z ∈ t → z.1 ≤ k)
(hjk : j ≤ k) (hst : s ⊆ t) :
f' j k hjk (lift (λ ix : s, f' ix.1.1 j (hj ix ix.2) ix.1.2) (restriction s x)) =
lift (λ ix : t, f' ix.1.1 k (hk ix i... | begin
refine subring.in_closure.rec_on hxs _ _ _ _,
{ rw [(restriction _).map_one, (free_comm_ring.lift _).map_one, (f' j k hjk).map_one,
(restriction _).map_one, (free_comm_ring.lift _).map_one] },
{ rw [(restriction _).map_neg, (restriction _).map_one,
(free_comm_ring.lift _).map_neg, (free_comm... | lemma | ring.direct_limit.of.zero_exact_aux2 | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"free_comm_ring",
"free_comm_ring.lift",
"ih",
"lift",
"map_mul",
"map_one",
"subring.in_closure.rec_on"
] | null | 378 | 405 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of.zero_exact_aux [nonempty ι] [is_directed ι (≤)] {x : free_comm_ring Σ i, G i}
(H : ideal.quotient.mk _ x = (0 : direct_limit G (λ i j h, f' i j h))) :
∃ j s, ∃ H : (∀ k : Σ i, G i, k ∈ s → k.1 ≤ j), is_supported x s ∧
lift (λ ix : s, f' ix.1.1 j (H ix ix.2) ix.1.2) (restriction s x) = (0 : G j) :=
begin
re... | of.zero_exact_aux [nonempty ι] [is_directed ι (≤)] {x : free_comm_ring Σ i, G i}
(H : ideal.quotient.mk _ x = (0 : direct_limit G (λ i j h, f' i j h))) :
∃ j s, ∃ H : (∀ k : Σ i, G i, k ∈ s → k.1 ≤ j), is_supported x s ∧
lift (λ ix : s, f' ix.1.1 j (H ix ix.2) ix.1.2) (restriction s x) = (0 : G j) | begin
refine span_induction (ideal.quotient.eq_zero_iff_mem.1 H) _ _ _ _,
{ rintros x (⟨i, j, hij, x, rfl⟩ | ⟨i, rfl⟩ | ⟨i, x, y, rfl⟩ | ⟨i, x, y, rfl⟩),
{ refine ⟨j, {⟨i, x⟩, ⟨j, f' i j hij x⟩}, _,
is_supported_sub (is_supported_of.2 $ or.inr rfl) (is_supported_of.2 $ or.inl rfl), _⟩,
{ rintros k... | lemma | ring.direct_limit.of.zero_exact_aux | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"exists_ge_ge",
"free_comm_ring",
"free_comm_ring.lift",
"ideal.quotient.mk",
"is_directed",
"lift",
"map_mul",
"map_one",
"mul_zero",
"set.mem_singleton",
"set.subset_union_left",
"set.subset_union_right",
"smul_eq_mul"
] | null | 408 | 482 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of.zero_exact [is_directed ι (≤)] {i x} (hix : of G (λ i j h, f' i j h) i x = 0) :
∃ j (hij : i ≤ j), f' i j hij x = 0 :=
by haveI : nonempty ι := ⟨i⟩; exact
let ⟨j, s, H, hxs, hx⟩ := of.zero_exact_aux hix in
have hixs : (⟨i, x⟩ : Σ i, G i) ∈ s, from is_supported_of.1 hxs,
⟨j, H ⟨i, x⟩ hixs, by rw [restriction_of, di... | of.zero_exact [is_directed ι (≤)] {i x} (hix : of G (λ i j h, f' i j h) i x = 0) :
∃ j (hij : i ≤ j), f' i j hij x = 0 | by haveI : nonempty ι := ⟨i⟩; exact
let ⟨j, s, H, hxs, hx⟩ := of.zero_exact_aux hix in
have hixs : (⟨i, x⟩ : Σ i, G i) ∈ s, from is_supported_of.1 hxs,
⟨j, H ⟨i, x⟩ hixs, by rw [restriction_of, dif_pos hixs, lift_of] at hx; exact hx⟩ | lemma | ring.direct_limit.of.zero_exact | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"is_directed"
] | A component that corresponds to zero in the direct limit is already zero in some
bigger module in the directed system. | 486 | 491 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_injective [is_directed ι (≤)] [directed_system G (λ i j h, f' i j h)]
(hf : ∀ i j hij, function.injective (f' i j hij)) (i) :
function.injective (of G (λ i j h, f' i j h) i) :=
begin
suffices : ∀ x, of G (λ i j h, f' i j h) i x = 0 → x = 0,
{ intros x y hxy, rw ← sub_eq_zero, apply this,
rw [(of G _ i).m... | of_injective [is_directed ι (≤)] [directed_system G (λ i j h, f' i j h)]
(hf : ∀ i j hij, function.injective (f' i j hij)) (i) :
function.injective (of G (λ i j h, f' i j h) i) | begin
suffices : ∀ x, of G (λ i j h, f' i j h) i x = 0 → x = 0,
{ intros x y hxy, rw ← sub_eq_zero, apply this,
rw [(of G _ i).map_sub, hxy, sub_self] },
intros x hx, rcases of.zero_exact hx with ⟨j, hij, hfx⟩,
apply hf i j hij, rw [hfx, (f' i j hij).map_zero]
end | theorem | ring.direct_limit.of_injective | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system",
"is_directed"
] | If the maps in the directed system are injective, then the canonical maps
from the components to the direct limits are injective. | 498 | 507 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : direct_limit G f →+* P :=
ideal.quotient.lift _ (free_comm_ring.lift $ λ (x : Σ i, G i), g x.1 x.2) begin
suffices : ideal.span _ ≤
ideal.comap (free_comm_ring.lift (λ (x : Σ (i : ι), G i), g (x.fst) (x.snd))) ⊥,
{ intros x hx, exact (mem_bot P).1 (this hx) },
rw ideal.span_le, intros x hx,
rw [set_l... | lift : direct_limit G f →+* P | ideal.quotient.lift _ (free_comm_ring.lift $ λ (x : Σ i, G i), g x.1 x.2) begin
suffices : ideal.span _ ≤
ideal.comap (free_comm_ring.lift (λ (x : Σ (i : ι), G i), g (x.fst) (x.snd))) ⊥,
{ intros x hx, exact (mem_bot P).1 (this hx) },
rw ideal.span_le, intros x hx,
rw [set_like.mem_coe, ideal.mem_comap, mem... | def | ring.direct_limit.lift | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"free_comm_ring.lift",
"ideal.comap",
"ideal.mem_comap",
"ideal.quotient.lift",
"ideal.span",
"ideal.span_le",
"lift",
"map_mul",
"map_one",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_sub",
"set_like.mem_coe"
] | The universal property of the direct limit: maps from the components to another ring
that respect the directed system structure (i.e. make some diagram commute) give rise
to a unique map out of the direct limit. | 521 | 531 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of (i x) : lift G f P g Hg (of G f i x) = g i x := free_comm_ring.lift_of _ _ | lift_of (i x) : lift G f P g Hg (of G f i x) = g i x | free_comm_ring.lift_of _ _ | lemma | ring.direct_limit.lift_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"free_comm_ring.lift_of",
"lift"
] | null | 536 | 536 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_unique [nonempty ι] [is_directed ι (≤)] (F : direct_limit G f →+* P) (x) :
F x = lift G f P (λ i, F.comp $ of G f i) (λ i j hij x, by simp) x :=
direct_limit.induction_on x $ λ i x, by simp | lift_unique [nonempty ι] [is_directed ι (≤)] (F : direct_limit G f →+* P) (x) :
F x = lift G f P (λ i, F.comp $ of G f i) (λ i j hij x, by simp) x | direct_limit.induction_on x $ λ i x, by simp | theorem | ring.direct_limit.lift_unique | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"is_directed",
"lift",
"lift_unique"
] | null | 538 | 540 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nontrivial [directed_system G (λ i j h, f' i j h)] :
nontrivial (ring.direct_limit G (λ i j h, f' i j h)) :=
⟨⟨0, 1, nonempty.elim (by apply_instance) $ assume i : ι, begin
change (0 : ring.direct_limit G (λ i j h, f' i j h)) ≠ 1,
rw ← (ring.direct_limit.of _ _ _).map_one,
intros H, rcases ring.direct_limit.of.... | nontrivial [directed_system G (λ i j h, f' i j h)] :
nontrivial (ring.direct_limit G (λ i j h, f' i j h)) | ⟨⟨0, 1, nonempty.elim (by apply_instance) $ assume i : ι, begin
change (0 : ring.direct_limit G (λ i j h, f' i j h)) ≠ 1,
rw ← (ring.direct_limit.of _ _ _).map_one,
intros H, rcases ring.direct_limit.of.zero_exact H.symm with ⟨j, hij, hf⟩,
rw (f' i j hij).map_one at hf,
exact one_ne_zero hf
end ⟩⟩ | instance | field.direct_limit.nontrivial | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system",
"map_one",
"nontrivial",
"one_ne_zero",
"ring.direct_limit",
"ring.direct_limit.of",
"ring.direct_limit.of.zero_exact"
] | null | 557 | 565 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_inv {p : ring.direct_limit G f} : p ≠ 0 → ∃ y, p * y = 1 :=
ring.direct_limit.induction_on p $ λ i x H,
⟨ring.direct_limit.of G f i (x⁻¹), by erw [← (ring.direct_limit.of _ _ _).map_mul,
mul_inv_cancel (assume h : x = 0, H $ by rw [h, (ring.direct_limit.of _ _ _).map_zero]),
(ring.direct_limit.of _ _ _).... | exists_inv {p : ring.direct_limit G f} : p ≠ 0 → ∃ y, p * y = 1 | ring.direct_limit.induction_on p $ λ i x H,
⟨ring.direct_limit.of G f i (x⁻¹), by erw [← (ring.direct_limit.of _ _ _).map_mul,
mul_inv_cancel (assume h : x = 0, H $ by rw [h, (ring.direct_limit.of _ _ _).map_zero]),
(ring.direct_limit.of _ _ _).map_one]⟩ | theorem | field.direct_limit.exists_inv | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"map_mul",
"map_one",
"mul_inv_cancel",
"ring.direct_limit",
"ring.direct_limit.induction_on",
"ring.direct_limit.of"
] | null | 567 | 571 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv (p : ring.direct_limit G f) : ring.direct_limit G f :=
if H : p = 0 then 0 else classical.some (direct_limit.exists_inv G f H) | inv (p : ring.direct_limit G f) : ring.direct_limit G f | if H : p = 0 then 0 else classical.some (direct_limit.exists_inv G f H) | def | field.direct_limit.inv | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ring.direct_limit"
] | Noncomputable multiplicative inverse in a direct limit of fields. | 577 | 578 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_cancel {p : ring.direct_limit G f} (hp : p ≠ 0) : p * inv G f p = 1 :=
by rw [inv, dif_neg hp, classical.some_spec (direct_limit.exists_inv G f hp)] | mul_inv_cancel {p : ring.direct_limit G f} (hp : p ≠ 0) : p * inv G f p = 1 | by rw [inv, dif_neg hp, classical.some_spec (direct_limit.exists_inv G f hp)] | theorem | field.direct_limit.mul_inv_cancel | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"mul_inv_cancel",
"ring.direct_limit"
] | null | 580 | 581 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_mul_cancel {p : ring.direct_limit G f} (hp : p ≠ 0) : inv G f p * p = 1 :=
by rw [_root_.mul_comm, direct_limit.mul_inv_cancel G f hp] | inv_mul_cancel {p : ring.direct_limit G f} (hp : p ≠ 0) : inv G f p * p = 1 | by rw [_root_.mul_comm, direct_limit.mul_inv_cancel G f hp] | theorem | field.direct_limit.inv_mul_cancel | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"inv_mul_cancel",
"ring.direct_limit"
] | null | 583 | 584 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
field [directed_system G (λ i j h, f' i j h)] :
field (ring.direct_limit G (λ i j h, f' i j h)) :=
{ inv := inv G (λ i j h, f' i j h),
mul_inv_cancel := λ p, direct_limit.mul_inv_cancel G (λ i j h, f' i j h),
inv_zero := dif_pos rfl,
.. ring.direct_limit.comm_ring G (λ i j h, f' i j h),
.. direct_limit.nontri... | field [directed_system G (λ i j h, f' i j h)] :
field (ring.direct_limit G (λ i j h, f' i j h)) | { inv := inv G (λ i j h, f' i j h),
mul_inv_cancel := λ p, direct_limit.mul_inv_cancel G (λ i j h, f' i j h),
inv_zero := dif_pos rfl,
.. ring.direct_limit.comm_ring G (λ i j h, f' i j h),
.. direct_limit.nontrivial G (λ i j h, f' i j h) } | def | field.direct_limit.field | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system",
"field",
"inv_zero",
"mul_inv_cancel",
"ring.direct_limit"
] | Noncomputable field structure on the direct limit of fields.
See note [reducible non-instances]. | 588 | 595 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_number (R : Type*) : Type* := triv_sq_zero_ext R R | dual_number (R : Type*) : Type* | triv_sq_zero_ext R R | abbreviation | dual_number | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"triv_sq_zero_ext"
] | The type of dual numbers, numbers of the form $a + bε$ where $ε^2 = 0$. | 44 | 44 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_number.eps [has_zero R] [has_one R] : dual_number R := triv_sq_zero_ext.inr 1 | dual_number.eps [has_zero R] [has_one R] : dual_number R | triv_sq_zero_ext.inr 1 | def | dual_number.eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"dual_number",
"triv_sq_zero_ext.inr"
] | The unit element $ε$ that squares to zero. | 47 | 47 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_eps [has_zero R] [has_one R] : fst ε = (0 : R) := fst_inr _ _ | fst_eps [has_zero R] [has_one R] : fst ε = (0 : R) | fst_inr _ _ | lemma | dual_number.fst_eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [] | null | 58 | 58 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_eps [has_zero R] [has_one R] : snd ε = (1 : R) := snd_inr _ _ | snd_eps [has_zero R] [has_one R] : snd ε = (1 : R) | snd_inr _ _ | lemma | dual_number.snd_eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [] | null | 59 | 59 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_mul [semiring R] (x y : R[ε]) : snd (x * y) = fst x * snd y + snd x * fst y :=
snd_mul _ _ | snd_mul [semiring R] (x y : R[ε]) : snd (x * y) = fst x * snd y + snd x * fst y | snd_mul _ _ | lemma | dual_number.snd_mul | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"semiring"
] | A version of `triv_sq_zero_ext.snd_mul` with `*` instead of `•`. | 62 | 63 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eps_mul_eps [semiring R] : (ε * ε : R[ε]) = 0 := inr_mul_inr _ _ _ | eps_mul_eps [semiring R] : (ε * ε : R[ε]) = 0 | inr_mul_inr _ _ _ | lemma | dual_number.eps_mul_eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"semiring"
] | null | 65 | 65 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_eq_smul_eps [mul_zero_one_class R] (r : R) : inr r = (r • ε : R[ε]) :=
ext (mul_zero r).symm (mul_one r).symm | inr_eq_smul_eps [mul_zero_one_class R] (r : R) : inr r = (r • ε : R[ε]) | ext (mul_zero r).symm (mul_one r).symm | lemma | dual_number.inr_eq_smul_eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"mul_one",
"mul_zero",
"mul_zero_one_class"
] | null | 67 | 68 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom_ext {A} [comm_semiring R] [semiring A] [algebra R A]
⦃f g : R[ε] →ₐ[R] A⦄ (h : f ε = g ε) : f = g :=
alg_hom_ext' $ linear_map.ext_ring $ h | alg_hom_ext {A} [comm_semiring R] [semiring A] [algebra R A]
⦃f g : R[ε] →ₐ[R] A⦄ (h : f ε = g ε) : f = g | alg_hom_ext' $ linear_map.ext_ring $ h | lemma | dual_number.alg_hom_ext | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"algebra",
"comm_semiring",
"linear_map.ext_ring",
"semiring"
] | For two algebra morphisms out of `R[ε]` to agree, it suffices for them to agree on `ε`. | 71 | 73 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : {e : A // e * e = 0} ≃ (R[ε] →ₐ[R] A) :=
equiv.trans
(show {e : A // e * e = 0} ≃ {f : R →ₗ[R] A // ∀ x y, f x * f y = 0}, from
(linear_map.ring_lmap_equiv_self R ℕ A).symm.to_equiv.subtype_equiv $ λ a, begin
dsimp,
simp_rw smul_mul_smul,
refine ⟨λ h x y, h.symm ▸ smul_zero _, λ h, by sim... | lift : {e : A // e * e = 0} ≃ (R[ε] →ₐ[R] A) | equiv.trans
(show {e : A // e * e = 0} ≃ {f : R →ₗ[R] A // ∀ x y, f x * f y = 0}, from
(linear_map.ring_lmap_equiv_self R ℕ A).symm.to_equiv.subtype_equiv $ λ a, begin
dsimp,
simp_rw smul_mul_smul,
refine ⟨λ h x y, h.symm ▸ smul_zero _, λ h, by simpa using h 1 1⟩,
end)
triv_sq_zero_ext.lif... | def | dual_number.lift | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"equiv.trans",
"lift",
"linear_map.ring_lmap_equiv_self",
"smul_mul_smul",
"smul_zero",
"triv_sq_zero_ext.lift"
] | A universal property of the dual numbers, providing a unique `R[ε] →ₐ[R] A` for every element
of `A` which squares to `0`.
This isomorphism is named to match the very similar `complex.lift`. | 81 | 90 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_apply_eps (e : {e : A // e * e = 0}) : lift e (ε : R[ε]) = e :=
(triv_sq_zero_ext.lift_aux_apply_inr _ _ _).trans $ one_smul _ _ | lift_apply_eps (e : {e : A // e * e = 0}) : lift e (ε : R[ε]) = e | (triv_sq_zero_ext.lift_aux_apply_inr _ _ _).trans $ one_smul _ _ | lemma | dual_number.lift_apply_eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"lift",
"one_smul",
"triv_sq_zero_ext.lift_aux_apply_inr"
] | null | 93 | 95 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_eps : lift ⟨ε, by exact eps_mul_eps⟩ = alg_hom.id R R[ε] :=
alg_hom_ext $ lift_apply_eps _ | lift_eps : lift ⟨ε, by exact eps_mul_eps⟩ = alg_hom.id R R[ε] | alg_hom_ext $ lift_apply_eps _ | lemma | dual_number.lift_eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"alg_hom.id",
"lift"
] | null | 98 | 100 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_number_equiv :
quaternion (dual_number R) ≃ₐ[R] dual_number (quaternion R) :=
{ to_fun := λ q,
(⟨q.re.fst, q.im_i.fst, q.im_j.fst, q.im_k.fst⟩,
⟨q.re.snd, q.im_i.snd, q.im_j.snd, q.im_k.snd⟩),
inv_fun := λ d,
⟨(d.fst.re, d.snd.re), (d.fst.im_i, d.snd.im_i),
(d.fst.im_j, d.snd.im_j), (d.fst.im... | dual_number_equiv :
quaternion (dual_number R) ≃ₐ[R] dual_number (quaternion R) | { to_fun := λ q,
(⟨q.re.fst, q.im_i.fst, q.im_j.fst, q.im_k.fst⟩,
⟨q.re.snd, q.im_i.snd, q.im_j.snd, q.im_k.snd⟩),
inv_fun := λ d,
⟨(d.fst.re, d.snd.re), (d.fst.im_i, d.snd.im_i),
(d.fst.im_j, d.snd.im_j), (d.fst.im_k, d.snd.im_k)⟩,
left_inv := λ ⟨⟨r, rε⟩, ⟨i, iε⟩, ⟨j, jε⟩, ⟨k, kε⟩⟩, rfl,
right_... | def | quaternion.dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"inv_fun",
"quaternion",
"ring"
] | The dual quaternions can be equivalently represented as a quaternion with dual coefficients,
or as a dual number with quaternion coefficients.
See also `matrix.dual_number_equiv` for a similar result. | 36 | 59 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.re = q.re.fst := rfl | re_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.re = q.re.fst | rfl | lemma | quaternion.re_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 64 | 65 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_i_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.im_i = q.im_i.fst := rfl | im_i_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.im_i = q.im_i.fst | rfl | lemma | quaternion.im_i_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 66 | 67 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_j_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.im_j = q.im_j.fst := rfl | im_j_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.im_j = q.im_j.fst | rfl | lemma | quaternion.im_j_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 68 | 69 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_k_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.im_k = q.im_k.fst := rfl | im_k_fst_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).fst.im_k = q.im_k.fst | rfl | lemma | quaternion.im_k_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 70 | 71 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.re = q.re.snd := rfl | re_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.re = q.re.snd | rfl | lemma | quaternion.re_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 72 | 73 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_i_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.im_i = q.im_i.snd := rfl | im_i_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.im_i = q.im_i.snd | rfl | lemma | quaternion.im_i_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 74 | 75 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_j_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.im_j = q.im_j.snd := rfl | im_j_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.im_j = q.im_j.snd | rfl | lemma | quaternion.im_j_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 76 | 77 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_k_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.im_k = q.im_k.snd := rfl | im_k_snd_dual_number_equiv (q : quaternion (dual_number R)) :
(dual_number_equiv q).snd.im_k = q.im_k.snd | rfl | lemma | quaternion.im_k_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 78 | 79 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_re_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).re.fst = d.fst.re := rfl | fst_re_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).re.fst = d.fst.re | rfl | lemma | quaternion.fst_re_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 80 | 81 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_im_i_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_i.fst = d.fst.im_i := rfl | fst_im_i_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_i.fst = d.fst.im_i | rfl | lemma | quaternion.fst_im_i_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 82 | 83 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_im_j_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_j.fst = d.fst.im_j := rfl | fst_im_j_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_j.fst = d.fst.im_j | rfl | lemma | quaternion.fst_im_j_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 84 | 85 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_im_k_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_k.fst = d.fst.im_k := rfl | fst_im_k_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_k.fst = d.fst.im_k | rfl | lemma | quaternion.fst_im_k_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 86 | 87 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_re_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).re.snd = d.snd.re := rfl | snd_re_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).re.snd = d.snd.re | rfl | lemma | quaternion.snd_re_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 88 | 89 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_im_i_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_i.snd = d.snd.im_i := rfl | snd_im_i_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_i.snd = d.snd.im_i | rfl | lemma | quaternion.snd_im_i_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 90 | 91 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_im_j_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_j.snd = d.snd.im_j := rfl | snd_im_j_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_j.snd = d.snd.im_j | rfl | lemma | quaternion.snd_im_j_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 92 | 93 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_im_k_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_k.snd = d.snd.im_k := rfl | snd_im_k_dual_number_equiv_symm (d : dual_number (quaternion R)) :
(dual_number_equiv.symm d).im_k.snd = d.snd.im_k | rfl | lemma | quaternion.snd_im_k_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | null | 94 | 95 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_one (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_one expr) :=
do
(t, one) ← t.mk_app `has_one.one [],
pure (t, { one := one }) | has_one (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_one expr) | do
(t, one) ← t.mk_app `has_one.one [],
pure (t, { one := one }) | def | expr.has_one | algebra | src/algebra/expr.lean | [
"tactic.core"
] | [
"tactic.instance_cache"
] | Produce a `has_one` instance for the type cached by `t`, such that `1 : expr` is the one of that
type. | 21 | 25 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_zero expr) :=
do
(t, zero) ← t.mk_app `has_zero.zero [],
pure (t, { zero := zero }) | has_zero (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_zero expr) | do
(t, zero) ← t.mk_app `has_zero.zero [],
pure (t, { zero := zero }) | def | expr.has_zero | algebra | src/algebra/expr.lean | [
"tactic.core"
] | [
"tactic.instance_cache"
] | Produce a `has_zero` instance for the type cached by `t`, such that `0 : expr` is the zero of
that type. | 29 | 33 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_mul (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_mul expr) :=
do
(t, mul) ← t.mk_app `has_mul.mul [],
pure (t, { mul := λ a b, mul a b }) | has_mul (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_mul expr) | do
(t, mul) ← t.mk_app `has_mul.mul [],
pure (t, { mul := λ a b, mul a b }) | def | expr.has_mul | algebra | src/algebra/expr.lean | [
"tactic.core"
] | [
"tactic.instance_cache"
] | Produce a `has_mul` instance for the type cached by `t`, such that `(*) : expr → expr → expr` is
the multiplication of that type. | 37 | 41 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_add (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_add expr) :=
do
(t, add) ← t.mk_app `has_add.add [],
pure (t, { add := λ a b, add a b }) | has_add (t : tactic.instance_cache) :
tactic (tactic.instance_cache × has_add expr) | do
(t, add) ← t.mk_app `has_add.add [],
pure (t, { add := λ a b, add a b }) | def | expr.has_add | algebra | src/algebra/expr.lean | [
"tactic.core"
] | [
"tactic.instance_cache"
] | Produce a `has_add` instance for the type cached by `t`, such that `(+) : expr → expr → expr` is
the addition of that type. | 45 | 49 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_magma (α : Type u) : Type u
| of : α → free_magma
| mul : free_magma → free_magma → free_magma | free_magma (α : Type u) : Type u
| of : α → free_magma
| mul : free_magma → free_magma → free_magma | inductive | free_magma | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Free magma over a given alphabet. | 35 | 38 | false | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_add_magma (α : Type u) : Type u
| of : α → free_add_magma
| add : free_add_magma → free_add_magma → free_add_magma | free_add_magma (α : Type u) : Type u
| of : α → free_add_magma
| add : free_add_magma → free_add_magma → free_add_magma | inductive | free_add_magma | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Free nonabelian additive magma over a given alphabet. | 41 | 44 | false | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[inhabited α] : inhabited (free_magma α) := ⟨of default⟩ | [inhabited α] : inhabited (free_magma α) | ⟨of default⟩ | instance | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | null | 52 | 53 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
: has_mul (free_magma α) := ⟨free_magma.mul⟩ | : has_mul (free_magma α) | ⟨free_magma.mul⟩ | instance | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | null | 55 | 56 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq (x y : free_magma α) : mul x y = x * y := rfl | mul_eq (x y : free_magma α) : mul x y = x * y | rfl | theorem | free_magma.mul_eq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | null | 60 | 61 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rec_on_mul {C : free_magma α → Sort l} (x)
(ih1 : ∀ x, C (of x)) (ih2 : ∀ x y, C x → C y → C (x * y)) :
C x :=
free_magma.rec_on x ih1 ih2 | rec_on_mul {C : free_magma α → Sort l} (x)
(ih1 : ∀ x, C (of x)) (ih2 : ∀ x y, C x → C y → C (x * y)) :
C x | free_magma.rec_on x ih1 ih2 | def | free_magma.rec_on_mul | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | Recursor for `free_magma` using `x * y` instead of `free_magma.mul x y`. | 64 | 69 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_ext {β : Type v} [has_mul β] {f g : free_magma α →ₙ* β} (h : f ∘ of = g ∘ of) : f = g :=
fun_like.ext _ _ $ λ x, rec_on_mul x (congr_fun h) $ by { intros, simp only [map_mul, *] } | hom_ext {β : Type v} [has_mul β] {f g : free_magma α →ₙ* β} (h : f ∘ of = g ∘ of) : f = g | fun_like.ext _ _ $ λ x, rec_on_mul x (congr_fun h) $ by { intros, simp only [map_mul, *] } | lemma | free_magma.hom_ext | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma",
"fun_like.ext",
"hom_ext",
"map_mul"
] | null | 71 | 73 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_magma.lift_aux {α : Type u} {β : Type v} [has_mul β] (f : α → β) : free_magma α → β
| (free_magma.of x) := f x
| (x * y) := x.lift_aux * y.lift_aux | free_magma.lift_aux {α : Type u} {β : Type v} [has_mul β] (f : α → β) : free_magma α → β
| (free_magma.of x) | f x
| (x * y) := x.lift_aux * y.lift_aux | def | free_magma.lift_aux | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | Lifts a function `α → β` to a magma homomorphism `free_magma α → β` given a magma `β`. | 78 | 80 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_add_magma.lift_aux {α : Type u} {β : Type v} [has_add β] (f : α → β) : free_add_magma α → β
| (free_add_magma.of x) := f x
| (x + y) := x.lift_aux + y.lift_aux | free_add_magma.lift_aux {α : Type u} {β : Type v} [has_add β] (f : α → β) : free_add_magma α → β
| (free_add_magma.of x) | f x
| (x + y) := x.lift_aux + y.lift_aux | def | free_add_magma.lift_aux | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_add_magma"
] | Lifts a function `α → β` to an additive magma homomorphism `free_add_magma α → β` given
an additive magma `β`. | 84 | 86 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (α → β) ≃ (free_magma α →ₙ* β) :=
{ to_fun := λ f,
{ to_fun := lift_aux f,
map_mul' := λ x y, rfl, },
inv_fun := λ F, F ∘ of,
left_inv := λ f, by { ext, refl },
right_inv := λ F, by { ext, refl } } | lift : (α → β) ≃ (free_magma α →ₙ* β) | { to_fun := λ f,
{ to_fun := lift_aux f,
map_mul' := λ x y, rfl, },
inv_fun := λ F, F ∘ of,
left_inv := λ f, by { ext, refl },
right_inv := λ F, by { ext, refl } } | def | free_magma.lift | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma",
"inv_fun",
"lift"
] | The universal property of the free magma expressing its adjointness. | 97 | 105 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of (x) : lift f (of x) = f x := rfl | lift_of (x) : lift f (of x) = f x | rfl | lemma | free_magma.lift_of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"lift"
] | null | 107 | 107 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_comp_of : lift f ∘ of = f := rfl | lift_comp_of : lift f ∘ of = f | rfl | lemma | free_magma.lift_comp_of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"lift"
] | null | 108 | 108 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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