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fact
string | imports
string | filename
string | symbolic_name
string | __index_level_0__
int64 |
|---|---|---|---|---|
CoFixpoint repeat (x:A) : ∞A := x ::: repeat x.
|
MathClasses.theory.CoqStreams MathClasses.implementations.peano_naturals MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/streams
|
coq-community-math-classes
| 958 |
CoInductive ForAllIf (PA : ∞A → Prop) (PB : ∞B → Prop) : ∞A → ∞B → Prop := ext_if : ∀ s1 s2, (PA s1 → PB s2) → ForAllIf PA PB (tl s1) (tl s2) → ForAllIf PA PB s1 s2.
|
MathClasses.theory.CoqStreams MathClasses.implementations.peano_naturals MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/streams
|
coq-community-math-classes
| 959 |
Definition univwit (c : C) (d : D): (c ⟶ G d) → (F c ⟶ d) := φ⁻¹.
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.categories MathClasses.misc.workaround_tactics MathClasses.theory.jections
|
coq-community-math-classes/theory/adjunctions
|
coq-community-math-classes
| 960 |
Definition stdlib_field_theory F `{DecField F} : Field_theory.field_theory 0 1 (+) (.*.) (λ x y, x - y) (-) (λ x y, x / y) (/) (=).
|
Coq.setoid_ring.Field Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.vectorspace MathClasses.theory.fields MathClasses.theory.strong_setoids MathClasses.theory.rings
|
coq-community-math-classes/theory/dec_fields
|
coq-community-math-classes
| 961 |
Definition from_stdlib_field_theory: @DecField F Fe Fplus Fmult Fzero Fone Fnegate Frecip.
|
Coq.setoid_ring.Field Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.vectorspace MathClasses.theory.fields MathClasses.theory.strong_setoids MathClasses.theory.rings
|
coq-community-math-classes/theory/dec_fields
|
coq-community-math-classes
| 962 |
Definition id_nat_trans `{Arrows D} `{!CatId D} `(F: C → D): F ⇛ F := λ _, cat_id.
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.jections
|
coq-community-math-classes/theory/categories
|
coq-community-math-classes
| 963 |
Definition iso_arrows {x y: X} (a: x ⟶ y) (b: y ⟶ x): Prop := a ◎ b = cat_id ∧ b ◎ a = cat_id.
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.jections
|
coq-community-math-classes/theory/categories
|
coq-community-math-classes
| 964 |
Definition is_iso {x y: X} (a: x ⟶ y): Prop := ex (iso_arrows a).
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.jections
|
coq-community-math-classes/theory/categories
|
coq-community-math-classes
| 965 |
Definition isos_unique (x y: X) (a: x ⟶ y) (b b': y ⟶ x): iso_arrows a b → iso_arrows a b' → b = b'.
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.jections
|
coq-community-math-classes/theory/categories
|
coq-community-math-classes
| 966 |
Definition iso: Equiv X := λ x y, ex (uncurry (@iso_arrows x y)).
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.jections
|
coq-community-math-classes/theory/categories
|
coq-community-math-classes
| 967 |
Definition isoT: X → X → Type := λ x y, sig (uncurry (@iso_arrows x y)).
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.jections
|
coq-community-math-classes/theory/categories
|
coq-community-math-classes
| 968 |
Definition initial (x: X): Type := ∀ y: X, sig (λ a: x ⟶ y, ∀ a': x ⟶ y, a = a').
|
MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.jections
|
coq-community-math-classes/theory/categories
|
coq-community-math-classes
| 969 |
Definition liftM2 `(f: A → B → C) (m : M A) (n : M B) : M C := x ← m ; y ← n ; ret (f x y).
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.monads MathClasses.theory.functors
|
coq-community-math-classes/theory/monads
|
coq-community-math-classes
| 970 |
Definition dnn_Str_nth (s : ∞A) : DecreasingNonNegative s ↔ ∀ n, 0 ≤ Str_nth (S n) s ≤ Str_nth n s.
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 971 |
CoFixpoint everyOther (s : ∞A) : ∞A := Cons (hd s) (everyOther (tl (tl s))).
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 972 |
Definition mult_Streams := zipWith (.*.).
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 973 |
Definition powers_help : A → ∞A := iterate (.*a).
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 974 |
Definition powers : ∞A := powers_help 1.
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 975 |
Definition increments (x : A) : ∞A := iterate (+d) x.
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 976 |
Definition positives_help : A → ∞A := increments 1.
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 977 |
Definition positives : ∞A := positives_help 1.
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 978 |
CoFixpoint factorials_help (n c : A) : ∞A := c ::: factorials_help (1 + n) (n * c).
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 979 |
Fixpoint fac_help (n : nat) (m c : A) : A := match n with | O => c | S n => (m + naturals_to_semiring nat A n) * fac_help n m c end.
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 980 |
Definition factorials := factorials_help 1 1.
|
Coq.setoid_ring.Ring Coq.Arith.Factorial MathClasses.misc.workaround_tactics MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.interfaces.naturals MathClasses.interfaces.integers MathClasses.theory.nat_pow MathClasses.theory.int_pow MathClasses.theory.streams
|
coq-community-math-classes/theory/series
|
coq-community-math-classes
| 981 |
Fixpoint to_ua (e: Term): Applied := match e with | Var v => ua_packed.AppliedVar msig v tt | Unit => ua_packed.AppliedOp msig monoids.one (ua_packed.NoMoreArguments msig tt) | Comp x y => ua_packed.AppliedOp msig monoids.mult (MoreArguments msig tt _ (to_ua x) (MoreArguments msig tt _ (to_ua y) (NoMoreArguments msig tt))) end.
|
Coq(Lia) MathClasses.interfaces.abstract_algebra MathClasses.theory.ua_packed
|
coq-community-math-classes/theory/monoid_normalization
|
coq-community-math-classes
| 982 |
Definition from_ua (t: Applied): { r: Term | to_ua r ≡ t }.
|
Coq(Lia) MathClasses.interfaces.abstract_algebra MathClasses.theory.ua_packed
|
coq-community-math-classes/theory/monoid_normalization
|
coq-community-math-classes
| 983 |
Fixpoint measure (e: Term): nat := match e with | Var v => 0%nat | Unit => 1%nat | Comp x y => S (2 * measure x + measure y) end.
|
Coq(Lia) MathClasses.interfaces.abstract_algebra MathClasses.theory.ua_packed
|
coq-community-math-classes/theory/monoid_normalization
|
coq-community-math-classes
| 984 |
Definition is_ne_0 `(x : R) `{Equiv R} `{Zero R} `{p : PropHolds (x ≠ 0)} : x ≠ 0 := p.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/rings
|
coq-community-math-classes
| 985 |
Definition is_nonneg `(x : R) `{Equiv R} `{Le R} `{Zero R} `{p : PropHolds (0 ≤ x)} : 0 ≤ x := p.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/rings
|
coq-community-math-classes
| 986 |
Definition is_pos `(x : R) `{Equiv R} `{Lt R} `{Zero R} `{p : PropHolds (0 < x)} : 0 < x := p.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/rings
|
coq-community-math-classes
| 987 |
Definition negate_involutive x : - - x = x := groups.negate_involutive x.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/rings
|
coq-community-math-classes
| 988 |
Definition preserves_negate x : f (-x) = -f x := groups.preserves_negate x.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/rings
|
coq-community-math-classes
| 989 |
Definition from_stdlib_semiring_theory: @SemiRing R Re Rplus Rmult Rzero Rone.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/rings
|
coq-community-math-classes
| 990 |
Definition from_stdlib_ring_theory: @Ring R Re Rplus Rmult Rzero Rone Rnegate.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/rings
|
coq-community-math-classes
| 991 |
Definition kernel : A → Prop := (= 0) ∘ f.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.theory.rings MathClasses.theory.ring_congruence
|
coq-community-math-classes/theory/ring_ideals
|
coq-community-math-classes
| 992 |
Definition Vars V := V → M.
|
MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/quote_monoid
|
coq-community-math-classes
| 993 |
Definition novars: Vars False := False_rect _.
|
MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/quote_monoid
|
coq-community-math-classes
| 994 |
Definition singlevar (x: M): Vars unit := fun _ => x.
|
MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/quote_monoid
|
coq-community-math-classes
| 995 |
Definition merge {A B} (a: Vars A) (b: Vars B): Vars (A+B) := fun i => match i with inl j => a j | inr j => b j end.
|
MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/quote_monoid
|
coq-community-math-classes
| 996 |
Definition quote': ∀ x {V'} {v: Vars V'} {d: Quote novars x v}, Term _ := @quote _ _.
|
MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/quote_monoid
|
coq-community-math-classes
| 997 |
Definition eval_quote': ∀ x {V'} {v: Vars V'} {d: Quote novars x v}, eval _ (merge novars v) quote ≡ x := @eval_quote _ _.
|
MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/quote_monoid
|
coq-community-math-classes
| 998 |
Example ex x y z: x & (y & z) & mon_unit = mon_unit & (x & y) & z.
|
MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/quote_monoid
|
coq-community-math-classes
| 999 |
Variable sign: Signature.
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.interfaces.functors MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.categories.setoids MathClasses.categories.product MathClasses.categories.algebras
|
coq-community-math-classes/theory/forget_algebra
|
coq-community-math-classes
| 1,000 |
Definition object (v: algebras.Object sign): TargetObject := λ i, @setoids.object (v i) (algebras.algebra_equiv sign v i) _.
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.interfaces.functors MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.categories.setoids MathClasses.categories.product MathClasses.categories.algebras
|
coq-community-math-classes/theory/forget_algebra
|
coq-community-math-classes
| 1,001 |
Variable A : Type.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,002 |
|
CoInductive Stream : Type := Cons : A -> Stream -> Stream.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,003 |
|
Definition hd (x:Stream) := match x with | Cons a _ => a end.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,004 |
|
Definition tl (x:Stream) := match x with | Cons _ s => s end.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,005 |
|
Fixpoint Str_nth_tl (n:nat) (s:Stream) : Stream := match n with | O => s | S m => Str_nth_tl m (tl s) end.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,006 |
|
Definition Str_nth (n:nat) (s:Stream) : A := hd (Str_nth_tl n s).
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,007 |
|
CoInductive EqSt (s1 s2: Stream) : Prop := eqst : hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,008 |
|
Variable P : Stream -> Prop.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,009 |
|
CoInductive ForAll (x: Stream) : Prop := HereAndFurther : P x -> ForAll (tl x) -> ForAll x.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,010 |
|
Variable Inv : Stream -> Prop.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,011 |
|
Hypothesis InvThenP : forall x:Stream, Inv x -> P x.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,012 |
|
Hypothesis InvIsStable : forall x:Stream, Inv x -> Inv (tl x).
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,013 |
|
Variables A B : Type.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,014 |
|
Variable f : A -> B.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,015 |
|
CoFixpoint map (s:Stream A) : Stream B := Cons (f (hd s)) (map (tl s)).
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,016 |
|
Variable a : A.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,018 |
|
CoFixpoint const : Stream A := Cons a const.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,019 |
|
Variable A B C : Type.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,020 |
|
Variable f: A -> B -> C.
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,021 |
|
CoFixpoint zipWith (a:Stream A) (b:Stream B) : Stream C := Cons (f (hd a) (hd b)) (zipWith (tl a) (tl b)).
|
coq-community-math-classes/theory/CoqStreams
|
coq-community-math-classes
| 1,022 |
|
Fixpoint op_closed {o: OpType (sorts sign)}: op_type A o → Type := match o with | ne_list.one x => P x | ne_list.cons x y => λ d, ∀ z, P _ z → op_closed (d z) end.
|
Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/ua_subalgebraT
|
coq-community-math-classes
| 1,023 |
Definition op_closed_proper: ∀ (Pproper: ∀ s x x', x = x' → iffT (P s x) (P s x')) o, ∀ x x', x = x' → iffT (@op_closed o x) (@op_closed o x').
|
Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/ua_subalgebraT
|
coq-community-math-classes
| 1,024 |
Definition carrier s := sigT (P s).
|
Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/ua_subalgebraT
|
coq-community-math-classes
| 1,025 |
Fixpoint close_op {d}: ∀ (o: op_type A d), op_closed o → op_type carrier d := match d with | ne_list.one _ => λ o c, existT _ o (c) | ne_list.cons _ _ => λ o c X, close_op (o (projT1 X)) (c (projT1 X) (projT2 X)) end.
|
Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/ua_subalgebraT
|
coq-community-math-classes
| 1,026 |
Definition carrier: sorts sig → Type := λ sort, ∀ i: I, carriers i sort.
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.categories.varieties
|
coq-community-math-classes/theory/ua_products
|
coq-community-math-classes
| 1,028 |
Fixpoint rec_impl o: (∀ i, op_type (carriers i) o) → op_type carrier o := match o return (∀ i, op_type (carriers i) o) → op_type carrier o with | ne_list.one _ => id | ne_list.cons _ g => λ X X0, rec_impl g (λ i, X i (X0 i)) end.
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.categories.varieties
|
coq-community-math-classes/theory/ua_products
|
coq-community-math-classes
| 1,029 |
Fixpoint nqe {t}: op_type carrier t → (∀ i, op_type (carriers i) t) → Prop := match t with | ne_list.one _ => λ f g, ∀ i, f i = g i | ne_list.cons _ _ => λ f g, ∀ tuple, nqe (f tuple) (λ i, g i (tuple i)) end.
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.theory.categories MathClasses.categories.varieties
|
coq-community-math-classes/theory/ua_products
|
coq-community-math-classes
| 1,030 |
Definition sum: sq R → R := @fold sq _ _ (0:R) (+).
|
MathClasses.theory.categories MathClasses.interfaces.functors MathClasses.interfaces.abstract_algebra MathClasses.interfaces.sequences
|
coq-community-math-classes/theory/sequences
|
coq-community-math-classes
| 1,031 |
Definition product: sq R → R := @fold sq _ _ (1:R) mult.
|
MathClasses.theory.categories MathClasses.interfaces.functors MathClasses.interfaces.abstract_algebra MathClasses.interfaces.sequences
|
coq-community-math-classes/theory/sequences
|
coq-community-math-classes
| 1,032 |
Definition head_arg x y (a: Arguments (ne_list.cons x y)): Applied x := match a with | MoreArguments k l m n => m end.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,033 |
Definition tail_args x y (a: Arguments (ne_list.cons x y)): Arguments y := match a with | MoreArguments k l m n => n end.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,034 |
Fixpoint forallArgs {o} (a: Arguments o): Type := match a with | NoMoreArguments _ => True | MoreArguments _ _ z v => prod (P z) (forallArgs v) end.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,035 |
Definition PofSplit {o}: Arguments o → Type := match o with | ne_list.one _ => λ _, unit | ne_list.cons x y => λ a, prod (P (head_arg x y a)) (forallArgs (tail_args _ _ a)) end.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,036 |
Definition forallSplit `(a: Arguments (ne_list.cons y z)): forallArgs a → prod (P (head_arg y z a)) (forallArgs (tail_args _ _ a)).
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,037 |
Fixpoint better_Applied_rect {o} (a: Applied o): P a := match a with | AppliedOp x y => Pop x y (better_args y) | AppliedVar x v => Pvar x v end with better_args {o} (a: Arguments o): forallArgs a := match a with | NoMoreArguments _ => I | MoreArguments _ _ x y => (better_Applied_rect x, better_args y) end.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,038 |
Fixpoint curry {σ} {V} {o} (a: Applied σ o): Term σ V (ne_list.one o) := match a in (Applied _ s) with | AppliedOp _ op y => apply_args y (app_tree σ (Op σ V op)) | AppliedVar _ v x => Var σ V v x end with apply_args {σ} {V} {o} (a: @Arguments σ V o): op_type (Term0 σ V) o → Term0 σ V (result _ o) := match a with | NoMoreArguments _ y => id | MoreArguments _ x y u q => λ z, apply_args q (z (curry u)) end.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,039 |
Fixpoint decode `(t: Term σ V o): Arguments σ o → Applied σ (result _ o) := match t with | Var _ _ x y => λ z, AppliedVar σ x y | Op _ _ x => AppliedOp σ x | App _ _ x y z w => λ p, decode z (MoreArguments σ y x (decode w (NoMoreArguments σ _)) p) end.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,040 |
Definition curry_decode `(t: Term σ V o) (a: Arguments σ o): curry (decode t a) ≡ apply_args a (app_tree σ t).
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,041 |
Definition decode0 `(t: Term0 σ V s): Applied σ s := decode t (NoMoreArguments σ _).
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,042 |
Definition curry_decode0 `(t: Term0 σ V o): curry (decode0 t) ≡ t.
|
MathClasses.interfaces.canonical_names MathClasses.interfaces.universal_algebra Coq.Program.Program
|
coq-community-math-classes/theory/ua_packed
|
coq-community-math-classes
| 1,043 |
Definition M (T: Type): Type := Term sign T (ne_list.one tt).
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.interfaces.monads
|
coq-community-math-classes/theory/ua_term_monad
|
coq-community-math-classes
| 1,044 |
Fixpoint geneq {s s'} (x: Term sign A s) (y: Term sign A s'): Prop := match x, y with | Var _ _ v _, Var _ _ w _ => v = w | App _ _ _ z t t', App _ _ _ z' t'' t''' => geneq t t'' ∧ geneq t' t''' | Op _ _ o, Op _ _ o' => o ≡ o' | _, _ => False end.
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.interfaces.monads
|
coq-community-math-classes/theory/ua_term_monad
|
coq-community-math-classes
| 1,045 |
Definition gen_bind_aux {A B: Type} (f: A → M B): ∀ {s}, Term sign A s → Term sign B s := fix F {s} (t: Term sign A s): Term sign B s := match t with | Var _ _ v tt => f v | App _ _ o z x y => App _ _ _ z (F x) (F y) | Op _ _ o => Op _ _ o end.
|
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.interfaces.monads
|
coq-community-math-classes/theory/ua_term_monad
|
coq-community-math-classes
| 1,046 |
Definition retract_is_nat_to_sr : NaturalsToSemiRing SR := λ R _ _ _ _ , naturals_to_semiring N R ∘ f⁻¹.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.implementations.peano_naturals MathClasses.theory.rings Coq.Arith.PeanoNat MathClasses.categories.varieties MathClasses.theory.ua_transference MathClasses.interfaces.naturals
|
coq-community-math-classes/theory/naturals
|
coq-community-math-classes
| 1,047 |
Definition retract_is_int_to_ring : IntegersToRing Z2 := λ Z2 _ _ _ _ _, integers_to_ring Z Z2 ∘ f⁻¹.
|
Coq.setoid_ring.Ring MathClasses.interfaces.naturals MathClasses.interfaces.abstract_algebra MathClasses.implementations.natpair_integers MathClasses.interfaces.integers
|
coq-community-math-classes/theory/integers
|
coq-community-math-classes
| 1,048 |
Definition inverse_inverse `{Inverse A B f} : Inverse (f⁻¹) := f.
|
MathClasses.theory.setoids MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/theory/jections
|
coq-community-math-classes
| 1,049 |
Definition poly := list R.
|
Coq.Lists.List MathClasses.interfaces.abstract_algebra MathClasses.interfaces.vectorspace MathClasses.theory.rings Ring
|
coq-community-math-classes/implementations/polynomials
|
coq-community-math-classes
| 1,050 |
Definition all (l: list Prop): Prop := fold_left and l True.
|
Coq.Lists.List MathClasses.interfaces.abstract_algebra MathClasses.interfaces.vectorspace MathClasses.theory.rings Ring
|
coq-community-math-classes/implementations/polynomials
|
coq-community-math-classes
| 1,051 |
Definition poly_eq_zero: poly → Prop := all ∘ map ((=) 0).
|
Coq.Lists.List MathClasses.interfaces.abstract_algebra MathClasses.interfaces.vectorspace MathClasses.theory.rings Ring
|
coq-community-math-classes/implementations/polynomials
|
coq-community-math-classes
| 1,052 |
Definition Npair_to_Z (x : SRpair N) : Z := ('pos x - 'neg x)%mc.
|
Coq.ZArith.BinInt Coq.setoid_ring.Ring Coq.Arith.Arith Coq.NArith.NArith Coq.ZArith.ZArith Coq.Numbers.Integer.Binary.ZBinary MathClasses.interfaces.abstract_algebra MathClasses.interfaces.integers MathClasses.implementations.natpair_integers MathClasses.implementations.stdlib_binary_naturals MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.implementations.nonneg_integers_naturals
|
coq-community-math-classes/implementations/stdlib_binary_integers
|
coq-community-math-classes
| 1,053 |
Fixpoint listset_add_raw (x : A) (l : list A) : list A := match l with | [] => [x] | y :: l => y :: if decide_rel (=) x y then l else listset_add_raw x l end.
|
Coq.Lists.List Coq.Lists.SetoidList MathClasses.implementations.list MathClasses.interfaces.abstract_algebra MathClasses.interfaces.finite_sets MathClasses.interfaces.orders MathClasses.theory.lattices MathClasses.orders.lattices
|
coq-community-math-classes/implementations/list_finite_set
|
coq-community-math-classes
| 1,054 |
Fixpoint listset_extend_raw `{Bottom B} `{Join B} (f : A → B) (l : list A) : B := match l with | [] => ⊥ | x :: l => f x ⊔ listset_extend_raw f l end.
|
Coq.Lists.List Coq.Lists.SetoidList MathClasses.implementations.list MathClasses.interfaces.abstract_algebra MathClasses.interfaces.finite_sets MathClasses.interfaces.orders MathClasses.theory.lattices MathClasses.orders.lattices
|
coq-community-math-classes/implementations/list_finite_set
|
coq-community-math-classes
| 1,055 |
Record Dyadic Z := dyadic { mant: Z; expo: Z }.
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.integers MathClasses.interfaces.naturals MathClasses.interfaces.rationals MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.orders.minmax MathClasses.orders.integers MathClasses.orders.rationals MathClasses.implementations.nonneg_integers_naturals MathClasses.implementations.stdlib_rationals MathClasses.theory.rationals MathClasses.theory.shiftl MathClasses.theory.int_pow MathClasses.theory.nat_pow MathClasses.theory.abs
|
coq-community-math-classes/implementations/dyadics
|
coq-community-math-classes
| 1,056 |
Definition DtoQ_slow (x : Dyadic) := ZtoQ (mant x) * 2 ^ (expo x).
|
Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.integers MathClasses.interfaces.naturals MathClasses.interfaces.rationals MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.orders.minmax MathClasses.orders.integers MathClasses.orders.rationals MathClasses.implementations.nonneg_integers_naturals MathClasses.implementations.stdlib_rationals MathClasses.theory.rationals MathClasses.theory.shiftl MathClasses.theory.int_pow MathClasses.theory.nat_pow MathClasses.theory.abs
|
coq-community-math-classes/implementations/dyadics
|
coq-community-math-classes
| 1,057 |
Definition to_listset (X : @set_type _ mset) : @set_type _ (listset elt) := props.to_list X↾elements_spec2w X.
|
Coq.MSets.MSetInterface Coq.MSets.MSetFacts Coq.MSets.MSetProperties MathClasses.implementations.list MathClasses.implementations.list_finite_set MathClasses.theory.finite_sets MathClasses.interfaces.finite_sets MathClasses.interfaces.orders MathClasses.interfaces.abstract_algebra
|
coq-community-math-classes/implementations/mset_finite_set
|
coq-community-math-classes
| 1,058 |
Fixpoint app (a b: L) {struct a}: L := match a with | one x => cons x b | cons x y => cons x (app y b) end.
|
Coq.Unicode.Utf8 Coq.Lists.List Coq.Setoids.Setoid Coq.Classes.Morphisms Coq.Sorting.Permutation
|
coq-community-math-classes/implementations/ne_list
|
coq-community-math-classes
| 1,059 |
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