statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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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"
] | 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⟩) = 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. | 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 $ 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. | 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⟩ | 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"
] | [] | 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 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. | 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, 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"
] | 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 | 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"
] | 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... | 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"
] | 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
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"
] | 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, 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. | 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).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. | 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_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. | 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 _ _ | 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"
] | 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 | 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"
] | 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.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"
] | 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 _ _ _).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"
] | 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) | 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. | 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)] | 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"
] | 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] | 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"
] | 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.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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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 $ε^2 = 0$. | 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 | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | [] | 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 _ _ | 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 `•`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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"
] | 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 | 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"
] | 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 | 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`. | 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 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`. | 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 _ _ | 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"
] | 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 _ | lemma | dual_number.lift_eps | algebra | src/algebra/dual_number.lean | [
"algebra.triv_sq_zero_ext"
] | [
"alg_hom.id",
"lift"
] | 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_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. | 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 | lemma | quaternion.re_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.im_i_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.im_j_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.im_k_fst_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.re_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.im_i_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.im_j_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.im_k_snd_dual_number_equiv | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.fst_re_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.fst_im_i_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.fst_im_j_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.fst_im_k_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.snd_re_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.snd_im_i_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.snd_im_j_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 | lemma | quaternion.snd_im_k_dual_number_equiv_symm | algebra | src/algebra/dual_quaternion.lean | [
"algebra.dual_number",
"algebra.quaternion"
] | [
"dual_number",
"quaternion"
] | 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 }) | 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. | 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 }) | 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. | 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 }) | 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. | 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 }) | 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. | 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 | 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. | 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 | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | 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 | 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`. | 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, *] } | 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"
] | 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 | 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`. | 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 | 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`. | 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 } } | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of' (f : free_magma α →ₙ* β) : lift (f ∘ of) = f | lift.apply_symm_apply f | 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"
] | [
"free_magma",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) : free_magma α →ₙ* free_magma β | lift (of ∘ f) | def | free_magma.map | 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",
"lift"
] | The unique magma homomorphism `free_magma α →ₙ* free_magma β` that sends
each `of x` to `of (f x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_of (x) : map f (of x) = of (f x) | rfl | lemma | free_magma.map_of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec_on_pure {C : free_magma α → Sort l} (x)
(ih1 : ∀ x, C (pure x)) (ih2 : ∀ x y, C x → C y → C (x * y)) :
C x | free_magma.rec_on_mul x ih1 ih2 | def | free_magma.rec_on_pure | 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",
"free_magma.rec_on_mul"
] | Recursor on `free_magma` using `pure` instead of `of`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pure (f : α → β) (x) : (f <$> pure x : free_magma β) = pure (f x) | rfl | lemma | free_magma.map_pure | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul' (f : α → β) (x y : free_magma α) : (f <$> (x * y)) = (f <$> x * f <$> y) | rfl | lemma | free_magma.map_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_bind (f : α → free_magma β) (x) : (pure x >>= f) = f x | rfl | lemma | free_magma.pure_bind | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bind (f : α → free_magma β) (x y : free_magma α) :
(x * y >>= f) = ((x >>= f) * (y >>= f)) | rfl | lemma | free_magma.mul_bind | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_seq {α β : Type u} {f : α → β} {x : free_magma α} : pure f <*> x = f <$> x | rfl | lemma | free_magma.pure_seq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_seq {α β : Type u} {f g : free_magma (α → β)} {x : free_magma α} :
(f * g) <*> x = (f <*> x) * (g <*> x) | rfl | lemma | free_magma.mul_seq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_magma.traverse {m : Type u → Type u} [applicative m] {α β : Type u}
(F : α → m β) :
free_magma α → m (free_magma β) | | (free_magma.of x) := free_magma.of <$> F x
| (x * y) := (*) <$> x.traverse <*> y.traverse | def | free_magma.traverse | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | `free_magma` is traversable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_add_magma.traverse {m : Type u → Type u} [applicative m] {α β : Type u}
(F : α → m β) :
free_add_magma α → m (free_add_magma β) | | (free_add_magma.of x) := free_add_magma.of <$> F x
| (x + y) := (+) <$> x.traverse <*> y.traverse | def | free_add_magma.traverse | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_add_magma"
] | `free_add_magma` is traversable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
traverse_pure (x) : traverse F (pure x : free_magma α) = pure <$> F x | rfl | lemma | free_magma.traverse_pure | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_pure' : traverse F ∘ pure = λ x, (pure <$> F x : m (free_magma β)) | rfl | lemma | free_magma.traverse_pure' | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_mul (x y : free_magma α) :
traverse F (x * y) = (*) <$> traverse F x <*> traverse F y | rfl | lemma | free_magma.traverse_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_mul' :
function.comp (traverse F) ∘ @has_mul.mul (free_magma α) _ =
λ x y, (*) <$> traverse F x <*> traverse F y | rfl | lemma | free_magma.traverse_mul' | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
traverse_eq (x) : free_magma.traverse F x = traverse F x | rfl | lemma | free_magma.traverse_eq | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"free_magma.traverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_map_seq (x y : free_magma α) :
((*) <$> x <*> y : id (free_magma α)) = (x * y : free_magma α) | rfl | lemma | free_magma.mul_map_seq | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
free_magma.repr {α : Type u} [has_repr α] : free_magma α → string | | (free_magma.of x) := repr x
| (x * y) := "( " ++ x.repr ++ " * " ++ y.repr ++ " )" | def | free_magma.repr | 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"
] | Representation of an element of a free magma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_add_magma.repr {α : Type u} [has_repr α] : free_add_magma α → string | | (free_add_magma.of x) := repr x
| (x + y) := "( " ++ x.repr ++ " + " ++ y.repr ++ " )" | def | free_add_magma.repr | 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"
] | Representation of an element of a free additive magma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_magma.length {α : Type u} : free_magma α → ℕ | | (free_magma.of x) := 1
| (x * y) := x.length + y.length | def | free_magma.length | 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"
] | Length of an element of a free magma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_add_magma.length {α : Type u} : free_add_magma α → ℕ | | (free_add_magma.of x) := 1
| (x + y) := x.length + y.length | def | free_add_magma.length | 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"
] | Length of an element of a free additive magma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_magma.assoc_rel (α : Type u) [has_add α] : α → α → Prop
| intro : ∀ x y z, add_magma.assoc_rel ((x + y) + z) (x + (y + z))
| left : ∀ w x y z, add_magma.assoc_rel (w + ((x + y) + z)) (w + (x + (y + z))) | inductive | add_magma.assoc_rel | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Associativity relations for an additive magma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
magma.assoc_rel (α : Type u) [has_mul α] : α → α → Prop
| intro : ∀ x y z, magma.assoc_rel ((x * y) * z) (x * (y * z))
| left : ∀ w x y z, magma.assoc_rel (w * ((x * y) * z)) (w * (x * (y * z))) | inductive | magma.assoc_rel | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Associativity relations for a magma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
assoc_quotient (α : Type u) [has_mul α] : Type u | quot $ assoc_rel α | def | magma.assoc_quotient | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Semigroup quotient of a magma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mk_assoc (x y z : α) : quot.mk (assoc_rel α) (x * y * z) = quot.mk _ (x * (y * z)) | quot.sound (assoc_rel.intro _ _ _) | lemma | magma.assoc_quotient.quot_mk_assoc | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_mk_assoc_left (x y z w : α) :
quot.mk (assoc_rel α) (x * (y * z * w)) = quot.mk _ (x * (y * (z * w))) | quot.sound (assoc_rel.left _ _ _ _) | lemma | magma.assoc_quotient.quot_mk_assoc_left | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of : α →ₙ* assoc_quotient α | ⟨quot.mk _, λ x y, rfl⟩ | def | magma.assoc_quotient.of | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Embedding from magma to its free semigroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction_on {C : assoc_quotient α → Prop} (x : assoc_quotient α)
(ih : ∀ x, C (of x)) : C x | quot.induction_on x ih | lemma | magma.assoc_quotient.induction_on | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {f g : assoc_quotient α →ₙ* β} (h : f.comp of = g.comp of) : f = g | fun_like.ext _ _ $ λ x, assoc_quotient.induction_on x $ fun_like.congr_fun h | lemma | magma.assoc_quotient.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"
] | [
"fun_like.congr_fun",
"fun_like.ext",
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : (α →ₙ* β) ≃ (assoc_quotient α →ₙ* β) | { to_fun := λ f,
{ to_fun := λ x, quot.lift_on x f $
by rintros a b (⟨c, d, e⟩ | ⟨c, d, e, f⟩); simp only [map_mul, mul_assoc],
map_mul' := λ x y, quot.induction_on₂ x y (map_mul f) },
inv_fun := λ f, f.comp of,
left_inv := λ f, fun_like.ext _ _ $ λ x, rfl,
right_inv := λ f, hom_ext $ fun_like.e... | def | magma.assoc_quotient.lift | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"fun_like.ext",
"hom_ext",
"inv_fun",
"lift",
"map_mul",
"mul_assoc",
"quot.induction_on₂"
] | Lifts a magma homomorphism `α → β` to a semigroup homomorphism `magma.assoc_quotient α → β`
given a semigroup `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of (x : α) : lift f (of x) = f x | rfl | lemma | magma.assoc_quotient.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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of : (lift f).comp of = f | lift.symm_apply_apply f | lemma | magma.assoc_quotient.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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of' (f : assoc_quotient α →ₙ* β) :
lift (f.comp of) = f | lift.apply_symm_apply f | lemma | magma.assoc_quotient.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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map : assoc_quotient α →ₙ* assoc_quotient β | lift (of.comp f) | def | magma.assoc_quotient.map | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [
"lift"
] | From a magma homomorphism `α →ₙ* β` to a semigroup homomorphism
`magma.assoc_quotient α →ₙ* magma.assoc_quotient β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_add_semigroup (α : Type u) | (head : α) (tail : list α) | structure | free_add_semigroup | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Free additive semigroup over a given alphabet. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_semigroup (α : Type u) | (head : α) (tail : list α) | structure | free_semigroup | algebra | src/algebra/free.lean | [
"algebra.hom.group",
"algebra.hom.equiv.basic",
"control.applicative",
"control.traversable.basic",
"logic.equiv.defs",
"data.list.basic"
] | [] | Free semigroup over a given alphabet. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
head_mul (x y : free_semigroup α) : (x * y).1 = x.1 | rfl | lemma | free_semigroup.head_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_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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