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option_mul {T : Mul.type} (o1 o2 : option T) : option T
:= match o1, o2 with | Some n, Some m => Some (mul n m) | _, _ => None end.
Definition
option_mul
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "type" ]
We need a functorial construction (a container) which preserves both structures. The simplest one is the option type.
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
option_square {T : Sq.type} (o : option T) : option T
:= match o with | Some n => Some (sq n) | None => None end.
Definition
option_square
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
sq_mul (V : Mul.type) (v : V) : sq v = mul v v.
Proof. by reflexivity. Qed.
Lemma
sq_mul
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "type" ]
As we expect we can proved this (by reflexivity)
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
problem (W : Mul.type) (w : option W) : sq w = mul w w.
Proof. Fail reflexivity. (* What? It used to work! *) Fail rewrite sq_mul. (* Lemmas don't cross the container either! *) (* Let's investigate *) rewrite /mul/= /sq/=. (* As we expect, we are on the option type. In the LHS it is the Sq built using the NFI instance option_square w = option_mul w w *) rewrite /...
Lemma
problem
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "option_mul", "option_square", "sq_mul", "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
option_sq_mul {T : Sq.type} (o : option T) : option_square o = mul o o.
Proof. by rewrite /option_square; case: o => [x|//]; rewrite sq_mul. Qed.
Lemma
option_sq_mul
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "option_square", "sq_mul", "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
problem (W : Sq.type) (w : option W) : sq w = mul w w.
Proof. by rewrite sq_mul. Qed.
Lemma
problem
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "sq_mul", "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
link {xT T : Type} {f : xT -> T} {g : T -> xT} (canfg : forall x, f (g x) = x)
:= T.
Definition
link
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "canfg" ]
This is the type which is used as a feather factory. - xT plays the role of a rich type, - T is a new type linked to xT by some lemma. In this case a very strong cancellation lemma canfg
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
link_eqtest (x y : T) : bool
:= eqtest (g x) (g y).
Definition
link_eqtest
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
link_eqOK (x y : T) : reflect (x = y) (link_eqtest x y).
Proof. rewrite /link_eqtest; case: (eqOK (g x) (g y)) => [E|abs]. by constructor; rewrite -[x]canfg -[y]canfg E canfg. by constructor=> /(f_equal g)/abs. Qed.
Lemma
link_eqOK
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "canfg", "link_eqtest" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
link_def : link canfg
:= f def.
Definition
link_def
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "canfg", "link" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
link_all_def x : eqtest x link_def = true.
Proof. rewrite /link_def; have /eqOK <- := all_def (g x). by rewrite canfg; case: (eqOK x x). Qed.
Lemma
link_all_def
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "canfg", "link_def" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
B : Type.
Axiom
B
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
We assume a known type B which is both an Equality and a Singleton
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
testB : B -> B -> bool.
Axiom
testB
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
testOKB : forall x y, reflect (x = y) (testB x y).
Axiom
testOKB
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "testB" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
defB : B.
Axiom
defB
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
all_defB : forall x, eqtest x defB = true.
Axiom
all_defB
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "defB" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
A : Type.
Axiom
A
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
Now we copy all instances from B to A via link
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
f : B -> A.
Axiom
f
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
g : A -> B.
Axiom
g
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
canfg : forall x, f (g x) = x.
Axiom
canfg
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"n ^ m"
:= (Nat.pow n m) : nat_scope.
Notation
n ^ m
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
*******************************************************************
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
new_concept
:= 999999.
Definition
new_concept
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[]
not a very clever construction, but a large one. Bare with us.
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
test x : new_concept ^ x ^ new_concept = x ^ new_concept ^ new_concept.
Proof. (* this goal is not trivial, and maybe even false, but you may call some automation on it anyway *) Time Fail reflexivity. (* takes 7s, note that both by and // call reflexivity *) Abort.
Lemma
test
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "new_concept" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
test x : new_concept ^ x ^ new_concept = x ^ new_concept ^ new_concept.
Time Fail reflexivity. (* takes 0s *) rewrite new_concept.unlock. Time Fail reflexivity. (* takes 7s, the original body is restored *) Abort.
Lemma
test
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "new_concept" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
unlock_new_concept
:= Unlockable new_concept.unlock.
Canonical
unlock_new_concept
examples
examples/hulk.v
[ "HB", "structures", "ssreflect", "ssrfun", "ssrbool" ]
[ "new_concept" ]
Notation new_concept := new_concept.body
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"0"
:= zero.
Notation
0
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "zero" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"- x"
:= (opp x).
Notation
- x
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "opp" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
example (G : AbelianGrp.type) (x : G) : x + (- x) = - 0.
Proof. by rewrite addrC addNr -[LHS](addNr zero) addrC add0r. Qed.
Lemma
example
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "add0r", "addNr", "addrC", "type", "zero" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
Z_add_assoc : forall x y z, Z.add x (Z.add y z) = Z.add (Z.add x y) z.
Axiom
Z_add_assoc
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "add" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
Z_add_comm : forall x y, Z.add x y = Z.add y x.
Axiom
Z_add_comm
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "add" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
Z_add_0_l : forall x, Z.add Z0 x = x.
Axiom
Z_add_0_l
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "add" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
Z_add_opp_diag_l : forall x, Z.add (Z.opp x) x = Z0.
Axiom
Z_add_opp_diag_l
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "add", "opp" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
example2 (x : Z) : x + (- x) = - 0.
Proof. by rewrite example. Qed.
Lemma
example2
examples
examples/readme.v
[ "HB", "structures", "Corelib", "ssreflect", "BinNums", "IntDef" ]
[ "example" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
funext : forall {T : Type} {U : T -> Type} [f g : forall t, U t], (forall t, f t = g t) -> f = g.
Axiom
funext
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
we assume a few axioms to make life easier
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
propext : forall P Q : Prop, P <-> Q -> P = Q.
Axiom
propext
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
Prop_irrelevance : forall (P : Prop) (x y : P), x = y.
Axiom
Prop_irrelevance
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
U
:= Type.
Notation
U
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
Shortcut
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
homs T
:= (@hom T _ _).
Notation
homs
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"a ~> b"
:= (hom a b) (at level 99, b at level 200, format "a ~> b") : cat_scope.
Notation
a ~> b
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"a ~>_ C b"
:= (@hom C a b) (at level 99, C at level 0, only parsing) : cat_scope.
Notation
a ~>_ C b
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"f \o g"
:= (comp g f) : cat_scope.
Notation
f \o g
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"f \; g :> T"
:= (@comp T _ _ _ f g) (at level 60, g, T at level 60, format "f \; g :> T", only parsing) : cat_scope.
Notation
f \; g :> T
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"f \; g"
:= (comp f g) : cat_scope.
Notation
f \; g
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"\idmap_ a"
:= (@idmap _ a) (only parsing, at level 0) : cat_scope.
Notation
\idmap_ a
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
discrete (T : U)
:= T.
Definition
discrete
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
the discrete category on a type cannot be the default, we make an alias
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
etransA T (a b c d : discrete T) (f : a ~> b) (g : b ~> c) (h : c ~> d) : f \; (g \; h) = (f \; g) \; h.
Proof. by rewrite /idmap/comp/=; case: _ / h; case: _ / g. Qed.
Lemma
etransA
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "discrete" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
Ucomp (X Y Z : U) (f : X ~> Y) (g : Y ~> Z) : f \; g = (g \o f)%function.
Proof. by []. Qed.
Lemma
Ucomp
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
Ucompx (X Y Z : U) (f : X ~> Y) (g : Y ~> Z) x : (f \; g) x = g (f x).
Proof. by []. Qed.
Lemma
Ucompx
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
U1 (X : U) : \idmap_X = idfun.
Proof. by []. Qed.
Lemma
U1
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
U1x (X : U) x : \idmap_X x = x.
Proof. by []. Qed.
Lemma
U1x
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"F ^$"
:= (@Fhom _ _ F _ _) (at level 1, format "F ^$") : cat_scope.
Notation
F ^$
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"F <$> f"
:= (@Fhom _ _ F _ _ f) (at level 58, format "F <$> f", right associativity) : cat_scope.
Notation
F <$> f
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
prefunctorP (C D : quiver) (F G : C ~> D) (eqFG : F =1 G) : let homF a b F := F a ~> F b in (forall a b f, eq_rect _ (homF a b) (F <$> f) _ (funext eqFG) = G <$> f) -> F = G.
Proof. move: F G => [F [[/= Fhom]]] [G [[/= Ghom]]] in eqFG *. case: _ / (funext eqFG) => /= in Ghom * => eqFGhom. congr PreFunctor.Pack; congr PreFunctor.Class; congr IsPreFunctor.Axioms_. by do 3!apply: funext=> ?. Qed.
Lemma
prefunctorP
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "funext", "homF" ]
prefunctors are equal if their object and hom part are respectively equal
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
functorP (C D : precat) (F G : C ~> D) (eqFG : F =1 G) : let homF a b F := F a ~> F b in (forall a b f, eq_rect _ (homF a b) (F^$ f) _ (funext eqFG) = G^$ f) -> F = G.
Proof. move=> /= /prefunctorP {eqFG}. case: F G => [F [/= Fm Fm']] [G [/= Gm Gm']]//=. move=> [_] eqFG; have /= := eq_sigT_rect_existT _ _ eqFG; apply => {}eqFG. case: _ / eqFG => /= in Gm Gm' * => eqFG. case: _ / eqFG in Gm' *. congr Functor.Pack; congr Functor.Class. case: Fm' Gm' => [F1 Fc] [G1 Gc]. by congr PreFunc...
Lemma
functorP
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "Prop_irrelevance", "funext", "homF", "prefunctorP" ]
functor equality is the same as prefunctor because of PI
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
comp_Fun (a b : C) (f : a ~> b) : (G \o F)%function <$> f = G <$> (F <$> f).
Proof. by []. Qed.
Lemma
comp_Fun
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
comp_F1 (a : C) : (G \o F)%function <$> \idmap_a = idmap.
Proof. by rewrite !comp_Fun !F1. Qed.
Lemma
comp_F1
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "comp_Fun" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
comp_Fcomp (a b c : C) (f : a ~> b) (g : b ~> c) : (G \o F)%function <$> (f \; g) = (G \o F)%function <$> f \; (G \o F)%function <$> g.
Proof. by rewrite !comp_Fun !Fcomp. Qed.
Lemma
comp_Fcomp
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "comp_Fun" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
funext_frefl A B (f : A -> B) : funext (frefl f) = erefl.
Proof. exact: Prop_irrelevance. Qed.
Lemma
funext_frefl
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "Prop_irrelevance", "funext" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
precat_cat : PreCat_IsCat precat.
Proof. by split=> [C D F|C D F|C D C' D' F G H]; apply/functorP => a b f /=; rewrite funext_frefl. Qed.
Definition
precat_cat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "functorP", "funext_frefl" ]
precategories and categories form a category
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
cat_cat : PreCat_IsCat cat.
Proof. by split=> [C D F|C D F|C D C' D' F G H]; apply/functorP => a b f /=; rewrite funext_frefl. Qed.
Definition
cat_cat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "functorP", "funext_frefl" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
concrete : PreConcreteQuiver.sort >-> Sortclass.
Coercion
concrete
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "sort" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
quiver_concrete_subproof : PreConcrete_IsConcrete quiver.
Proof. constructor=> C D F G FG; apply: prefunctorP. by move=> x; congr (_ x); apply: FG. by move=> *; admit. Admitted.
Lemma
quiver_concrete_subproof
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "prefunctorP" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
precat_concrete_subproof : PreConcrete_IsConcrete precat.
Proof. constructor=> C D F G FG; apply: functorP. by move=> x; congr (_ x); apply: FG. by move=> *; admit. Admitted.
Lemma
precat_concrete_subproof
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "functorP" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
cat_concrete_subproof : PreConcrete_IsConcrete cat.
Proof. constructor=> C D F G FG; apply: functorP. by move=> x; congr (_ x); apply: FG. by move=> *; admit. Admitted.
Lemma
cat_concrete_subproof
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "functorP" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
cst (C D : quiver) (c : C)
:= fun & D => c.
Definition
cst
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
constant functor
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
catop (C : U) : U
:= C.
Definition
catop
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
opposite category
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"C ^op"
:= (catop C) (at level 2, format "C ^op") : cat_scope.
Notation
C ^op
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "catop" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
hom_Fhom_subproof (C : cat) (x : C) : PreFunctor_IsFunctor _ _ (hom x).
Proof. by split=> *; apply/funext => h; [apply: compo1 | apply: compoA]. Qed.
Lemma
hom_Fhom_subproof
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "funext" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
hom_op {C : quiver} (c : C^op) : hom c = (@hom C)^~ c.
Proof. reflexivity. Qed.
Lemma
hom_op
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
homFhomx {C : precat} (a b c : C) (f : a ~> b) (g : c ~> a) : (hom c <$> f) g = g \; f.
Proof. reflexivity. Qed.
Lemma
homFhomx
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
dprod {I : U} (C : I -> U)
:= forall i, C i.
Definition
dprod
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
nary product of categories
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
dprod_hom_subdef (a b : dprod C)
:= forall i, a i ~> b i.
Definition
dprod_hom_subdef
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "dprod" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
dprod_idmap_subdef (a : dprod C) : a ~> a
:= fun=> idmap.
Definition
dprod_idmap_subdef
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "dprod" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
dprod_comp_subdef (a b c : dprod C) (f : a ~> b) (g : b ~> c) : a ~> c
:= fun i => f i \; g i.
Definition
dprod_comp_subdef
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "dprod" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
type
:= (dprod C).
Notation
type
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "dprod" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
dprod_is_cat : PreCat_IsCat type.
Proof. split=> [a b f|a b f|a b c d f g h]; apply/funext => i; [exact: comp1o | exact: compo1 | exact: compoA]. Qed.
Lemma
dprod_is_cat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "funext", "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
prod_hom_subdef (a b : C * D)
:= ((a.1 ~> b.1) * (a.2 ~> b.2))%type.
Definition
prod_hom_subdef
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
type
:= (C * D)%type.
Notation
type
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
prod_is_cat : PreCat_IsCat type.
Proof. split=> [[a1 a2] [b1 b2] [f1 f2]|[a1 a2] [b1 b2] [f1 f2]| [a1 a2] [b1 b2] [c1 c2] [d1 d2] [f1 f2] [g1 g2] [h1 h2]]; congr (_, _) => //=; by [exact: comp1o | exact: compo1 | exact: compoA]. Qed.
Lemma
prod_is_cat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
naturalx (C : precat) (D : concrete_precat) (F G : C ~>_quiver D) (n : F ~> G) (a b : C) (f : a ~> b) g : (concrete <$> n b) ((concrete <$> F <$> f) g) = (concrete <$> G <$> f) ((concrete <$> n a) g).
Proof. have /(congr1 (fun h => (concrete <$> h) g)) := natural n f. by rewrite !Fcomp. Qed.
Lemma
naturalx
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "concrete" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
naturalU (C : precat) (F G : C ~>_quiver U) (n : F ~> G) (a b : C) (f : a ~> b) g : n b (F^$ f g) = G^$ f (n a g).
Proof. exact: (naturalx n). Qed.
Lemma
naturalU
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "naturalx" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
natP (C D : precat) (F G : C ~>_quiver D) (n m : F ~> G) : Natural.sort n = Natural.sort m -> n = m.
Proof. case: n m => [/= n nP] [/= m mP] enm. elim: _ / enm in mP *; congr Natural.Pack. case: nP mP => [[?]] [[?]]; congr Natural.Class. congr IsNatural.Axioms_. exact: Prop_irrelevance. Qed.
Lemma
natP
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "Prop_irrelevance", "sort" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"F ~~> G"
:= (F ~>_(homs quiver) G) (at level 99, G at level 200, format "F ~~> G").
Notation
F ~~> G
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "homs" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"F ~~> G :> C ~> D"
:= (F ~> G :> (C ~>_quiver D)) (at level 99, G at level 200, C, D at level 0, format "F ~~> G :> C ~> D").
Notation
F ~~> G :> C ~> D
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
natural_id {C D : precat} (F : C ~>_quiver D) (a : C)
:= \idmap_(F a).
Definition
natural_id
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
natural_id_natural (C D : cat) (F : C ~>_quiver D) : IsNatural C D F F (natural_id F).
Proof. by constructor=> a b f; rewrite /natural_id/= compo1 comp1o. Qed.
Definition
natural_id_natural
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "natural_id" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
natural_comp {C D : precat} (F G H : C ~>_quiver D) (m : F ~> G) (n : G ~> H) (a : C)
:= m a \; n a.
Definition
natural_comp
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
natural_comp_natural (C D : cat) (F G H : C ~>_quiver D) m n : IsNatural C D F H (@natural_comp C D F G H m n).
Proof. constructor=> a b f; rewrite /natural_comp/=. by rewrite compoA natural -compoA natural compoA. Qed.
Definition
natural_comp_natural
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "natural_comp" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
prefunctor_cat {C D : cat} : PreCat_IsCat (PreFunctor.type C D).
Proof. constructor => [F G f|F G f|F G H J f g h]. - by apply/natP/funext => a; rewrite /= /natural_comp comp1o. - by apply/natP/funext => a; rewrite /= /natural_comp compo1. - by apply/natP/funext => a; rewrite /= /natural_comp compoA. Qed.
Lemma
prefunctor_cat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "funext", "natP", "natural_comp", "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
functor_cat {C D : cat} : PreCat_IsCat (Functor.type C D).
Proof. constructor => [F G f|F G f|F G H J f g h]. - by apply/natP/funext => a; rewrite /= /natural_comp comp1o. - by apply/natP/funext => a; rewrite /= /natural_comp compo1. - by apply/natP/funext => a; rewrite /= /natural_comp compoA. Qed.
Lemma
functor_cat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "funext", "natP", "natural_comp", "type" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
idFmap (C : cat) (a b : C) (f : a ~> b) : idfun <$> f = f.
Proof. by []. Qed.
Lemma
idFmap
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
compFmap (C D E : cat) (F : C ~> D) (G : D ~> E) (a b : C) (f : a ~> b) : (G \o F) <$> f = G <$> F <$> f.
Proof. by []. Qed.
Lemma
compFmap
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
whiskerl_fun (eta : forall c, F c ~> G c) (H : D ~> E) : forall c, (F \; H) c ~> (G \; H) c
:= fun c => H <$> eta c.
Definition
whiskerl_fun
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "eta" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
whiskerl_is_nat (eta : F ~> G) (H : D ~> E) : IsNatural _ _ _ _ (whiskerl_fun eta H).
Proof. by constructor=> a b f; rewrite /whiskerl_fun/= -!Fcomp natural. Qed.
Lemma
whiskerl_is_nat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "eta", "whiskerl_fun" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
whiskerl (eta : F ~> G) (H : D ~> E) : (F \; H) ~> (G \; H)
:= whiskerl_fun eta H : Natural.type _ _.
Definition
whiskerl
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "eta", "type", "whiskerl_fun" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"F <o$> n"
:= (whiskerl F n) (at level 58, format "F <o$> n", right associativity) : cat_scope.
Notation
F <o$> n
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "whiskerl" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
whiskerr_fun (H : C ~> D) (eta : forall d, F d ~> G d) (c : C) : (H \; F) c ~> (H \; G) c
:= eta (H c).
Definition
whiskerr_fun
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "eta" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
whiskerr_is_nat (H : C ~> D) (eta : F ~> G) : IsNatural _ _ _ _ (whiskerr_fun H eta).
Proof. by constructor=> a b f; rewrite /whiskerr_fun/= natural. Qed.
Lemma
whiskerr_is_nat
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "eta", "whiskerr_fun" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
whiskerr (H : C ~> D) (eta : F ~> G) : (H \; F) ~> (H \; G)
:= whiskerr_fun H eta : Natural.type _ _.
Definition
whiskerr
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "eta", "type", "whiskerr_fun" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
"F <$o> n"
:= (whiskerr F n) (at level 58, format "F <$o> n", right associativity) : cat_scope.
Notation
F <$o> n
examples.cat
examples/cat/cat.v
[ "ssreflect", "ssrfun", "HB", "structures" ]
[ "whiskerr" ]
https://github.com/math-comp/hierarchy-builder
002a61eee7cf02e08e4b60abca0cf46a793497aa
End of preview. Expand in Data Studio

Coq-HierarchyBuilder

Structured dataset from Hierarchy Builder — High-level commands for packed class hierarchies.

Source

Schema

Column Type Description
statement string Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof string Verbatim proof/body, empty if the declaration has none
type string Declaration keyword
symbolic_name string Declaration identifier
library string Sub-library
filename string Repository-relative source path
imports list[string] File-level Require/Import modules
deps list[string] Intra-corpus identifiers referenced
docstring string Preceding documentation comment, empty if absent
source_url string Upstream repository
commit string Upstream commit extracted

Statistics

  • Entries: 593
  • With proof: 555 (93.6%)
  • With docstring: 43 (7.3%)
  • Libraries: 14

By type

Type Count
Lemma 238
Definition 153
Notation 135
Axiom 29
Record 7
Hypothesis 6
Coercion 4
Fact 4
Example 4
Canonical 3
Let 3
Inductive 2
Variant 2
Theorem 1
CoInductive 1
Class 1

Example

option_mul {T : Mul.type} (o1 o2 : option T) : option T
:=
  match o1, o2 with
  | Some n, Some m => Some (mul n m)
  | _, _ => None
  end.
  • type: Definition | symbolic_name: option_mul | examples/hulk.v

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{coq_hierarchybuilder_dataset,
  title  = {Coq-HierarchyBuilder},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/math-comp/hierarchy-builder, commit 002a61eee7cf},
  url    = {https://huggingface.co/datasets/phanerozoic/Coq-HierarchyBuilder}
}
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