fact stringlengths 3 2.59k | type stringclasses 20
values | library stringclasses 4
values | imports listlengths 0 18 | filename stringclasses 207
values | symbolic_name stringlengths 1 36 | docstring stringclasses 269
values |
|---|---|---|---|---|---|---|
clos_rt_t : forall x y z,
clos_refl_trans R x y -> trans_clos R y z ->
trans_clos R x z. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rt_t | null |
clos_rst_is_equiv : Equivalence (clos_refl_sym_trans R). | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst_is_equiv | Correctness of the reflexive-symmetric-transitive closure |
clos_rst_idempotent :
inclusion (clos_refl_sym_trans (clos_refl_sym_trans R))
(clos_refl_sym_trans R). | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst_idempotent | Idempotency of the reflexive-symmetric-transitive closure operator |
clos_t1n_trans : forall x y, trans_clos_1n R x y -> trans_clos R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_t1n_trans | Direct transitive closure vs left-step extension |
trans_clos_t1n : forall x y, trans_clos R x y -> trans_clos_1n R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | trans_clos_t1n | null |
trans_clos_t1n_iff : forall x y,
trans_clos R x y <-> trans_clos_1n R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | trans_clos_t1n_iff | null |
clos_tn1_trans : forall x y, trans_clos_n1 R x y -> trans_clos R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_tn1_trans | Direct transitive closure vs right-step extension |
trans_clos_tn1 : forall x y, trans_clos R x y -> trans_clos_n1 R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | trans_clos_tn1 | null |
trans_clos_tn1_iff : forall x y,
trans_clos R x y <-> trans_clos_n1 R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | trans_clos_tn1_iff | null |
clos_rt1n_step : forall x y, R x y -> clos_refl_trans_1n R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rt1n_step | Direct reflexive-transitive closure is equivalent to
transitivity by left-step extension |
clos_rtn1_step : forall x y, R x y -> clos_refl_trans_n1 R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rtn1_step | null |
clos_rt1n_rt : forall x y,
clos_refl_trans_1n R x y -> clos_refl_trans R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rt1n_rt | null |
clos_rt_rt1n : forall x y,
clos_refl_trans R x y -> clos_refl_trans_1n R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rt_rt1n | null |
clos_rt_rt1n_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_1n R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rt_rt1n_iff | null |
clos_rtn1_rt : forall x y,
clos_refl_trans_n1 R x y -> clos_refl_trans R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rtn1_rt | Direct reflexive-transitive closure is equivalent to
transitivity by right-step extension |
clos_rt_rtn1 : forall x y,
clos_refl_trans R x y -> clos_refl_trans_n1 R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rt_rtn1 | null |
clos_rt_rtn1_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_n1 R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rt_rtn1_iff | null |
clos_refl_trans_ind_left :
forall (x:A) (P:A -> Type), P x ->
(forall y z:A, clos_refl_trans R x y -> P y -> R y z -> P z) ->
forall z:A, clos_refl_trans R x z -> P z. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_refl_trans_ind_left | Induction on the left transitive step |
rt1n_ind_right : forall (P : A -> Type) (z:A),
P z ->
(forall x y, R x y -> clos_refl_trans_1n R y z -> P y -> P x) ->
forall x, clos_refl_trans_1n R x z -> P x. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | rt1n_ind_right | Induction on the right transitive step |
clos_refl_trans_ind_right : forall (P : A -> Type) (z:A),
P z ->
(forall x y, R x y -> P y -> clos_refl_trans R y z -> P x) ->
forall x, clos_refl_trans R x z -> P x. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_refl_trans_ind_right | null |
clos_rst1n_rst : forall x y,
clos_refl_sym_trans_1n R x y -> clos_refl_sym_trans R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst1n_rst | Direct reflexive-symmetric-transitive closure is equivalent to
transitivity by symmetric left-step extension |
clos_rst1n_trans : forall x y z, clos_refl_sym_trans_1n R x y ->
clos_refl_sym_trans_1n R y z -> clos_refl_sym_trans_1n R x z. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst1n_trans | null |
clos_rst1n_sym : forall x y, clos_refl_sym_trans_1n R x y ->
clos_refl_sym_trans_1n R y x. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst1n_sym | null |
clos_rst_rst1n : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_1n R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst_rst1n | null |
clos_rst_rst1n_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_1n R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst_rst1n_iff | null |
clos_rstn1_rst : forall x y,
clos_refl_sym_trans_n1 R x y -> clos_refl_sym_trans R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rstn1_rst | Direct reflexive-symmetric-transitive closure is equivalent to
transitivity by symmetric right-step extension |
clos_rstn1_trans : forall x y z, clos_refl_sym_trans_n1 R x y ->
clos_refl_sym_trans_n1 R y z -> clos_refl_sym_trans_n1 R x z. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rstn1_trans | null |
clos_rstn1_sym : forall x y, clos_refl_sym_trans_n1 R x y ->
clos_refl_sym_trans_n1 R y x. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rstn1_sym | null |
clos_rst_rstn1 : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_n1 R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst_rstn1 | null |
clos_rst_rstn1_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_n1 R x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | clos_rst_rstn1_iff | null |
trans_clos_transp_permute : forall x y,
transp (trans_clos R) x y <-> trans_clos (transp R) x y. | Lemma | theories | [
"CRelationClasses",
"Equations",
"Equations",
"Equations"
] | theories/Type/Relation_Properties.v | trans_clos_transp_permute | null |
FixWf `{WF:WellFounded A R} (P : A -> Type)
(step : forall x : A, (forall y : A, R y x -> P y) -> P x) : forall x : A, P x :=
Fix wellfounded P step. | Definition | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | FixWf | The fixpoint combinator associated to a well-founded relation,
just reusing the [WellFounded.Fix] combinator. |
step_fn_ext {A} {R} (P : A -> Type) :=
fun step : forall x : A, (forall y : A, R y x -> P y) -> P x =>
forall x (f g : forall y (H : R y x), P y),
(forall y H, f y H = g y H) ->
step x f = step x g. | Definition | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | step_fn_ext | null |
FixWf_unfold `{WF : WellFounded A R} (P : A -> Type)
(step : forall x : A, (forall y : A, R y x -> P y) -> P x)
(step_ext : step_fn_ext P step) (x : A) :
FixWf P step x = step x (fun y _ => FixWf P step y). | Lemma | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | FixWf_unfold | null |
FixWf_unfold_step :
forall (A : Type) (R : relation A) (WF : WellFounded R) (P : A -> Type)
(step : forall x : A, (forall y : A, R y x -> P y) -> P x) (x : A)
(step_ext : step_fn_ext P step)
(step' : forall y : A, R y x -> P y),
step' = (fun (y : A) (_ : R y x) => FixWf P step y) ->
FixWf P step x... | Lemma | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | FixWf_unfold_step | The fixpoint combinator associated to a well-founded relation,
just reusing the [WellFounded.Fix] combinator. |
FixWf_unfold_ext `{WF : WellFounded A R} (P : A -> Type)
(step : forall x : A, (forall y : A, R y x -> P y) -> P x) (x : A) :
FixWf P step x = step x (fun y _ => FixWf P step y). | Lemma | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | FixWf_unfold_ext | null |
FixWf_unfold_ext_step :
forall (A : Type) (R : relation A) (WF : WellFounded R) (P : A -> Type)
(step : forall x : A, (forall y : A, R y x -> P y) -> P x) (x : A)
(step' : forall y : A, R y x -> P y),
step' = (fun (y : A) (_ : R y x) => FixWf P step y) ->
FixWf P step x = step x step'. | Lemma | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | FixWf_unfold_ext_step | null |
WellFounded_trans_clos `(WF : WellFounded A R) : WellFounded (trans_clos R). | Lemma | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | WellFounded_trans_clos | We can automatically use the well-foundedness of a relation to get
the well-foundedness of its transitive closure.
Note that this definition is transparent as well as [wf_clos_trans],
to allow computations with functions defined by well-founded recursion. |
#[export]
Instance wf_inverse_image {A R} `(WellFounded A R) {B} (f : B -> A) :
WellFounded (inverse_image R f) | (WellFounded (inverse_image _ _)). | Instance | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | wf_inverse_image | null |
trans_clos_stepr A (R : relation A) (x y z : A) :
R y z -> trans_clos R x y -> trans_clos R x z. | Lemma | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | trans_clos_stepr | null |
NoCycle_WellFounded {A} (R : relation A) (wfR : WellFounded R) : NoCyclePackage A :=
{| NoCycle := R;
noCycle := well_founded_irreflexive (wfR:=wfR) |}. | Definition | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | NoCycle_WellFounded | We specialize the tactic for [x] of type [A], first packing
[x] with its indices into a sigma type and finding the declared
relation on this type. |
#[export]
Existing Instance NoCycle_WellFounded. | Existing | theories | [
"Equations.Init",
"Equations.Signature",
"Equations",
"Equations"
] | theories/Type/Subterm.v | Instance | null |
tele_sigma (t : tele@{i}) : Type@{i} :=
| tip A := A
| ext A B := @sigma A (fun x => tele_sigma (B x)). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_sigma | null |
tele_val : tele@{i} -> Type@{i+1} :=
| tip_val {A} (a : A) : tele_val (tip A)
| ext_val {A B} (a : A) (b : tele_val (B a)) : tele_val (ext A B). | Inductive | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_val | null |
tele_pred : tele -> Type :=
| tip A := A -> Type
| ext A B := forall x : A, tele_pred (B x). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_pred | null |
tele_rel : tele -> tele -> Type :=
| tip A | tip B := A -> B -> Type
| ext A B | ext A' B' := forall (x : A) (y : A'), tele_rel (B x) (B' y)
| _ | _ := False. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_rel | null |
tele_rel_app (T U : tele) (P : tele_rel T U) (x : tele_sigma T) (y : tele_sigma U) : Type :=
| tip A, tip A', P, a, a' := P a a'
| ext A B, ext A' B', P, (a, b), (a', b') := tele_rel_app (B a) (B' a') (P a a') b b'. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_rel_app | null |
tele_fn : tele@{i} -> Type@{j} -> Type@{k} :=
| tip A, concl := A -> concl
| ext A B, concl := forall x : A, tele_fn (B x) concl. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_fn | null |
tele_MR (T : tele@{i}) (A : Type@{j}) (f : tele_fn T A) : T -> A :=
tele_MR (tip A) C f => f;
tele_MR (ext A B) C f => fun x => tele_MR (B x.1) C (f x.1) x.2. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_MR | null |
tele_measure (T : tele@{i}) (A : Type@{j}) (f : tele_fn T A) (R : A -> A -> Type@{k}) :
T -> T -> Type@{k} :=
tele_measure T C f R := fun x y => R (tele_MR T C f x) (tele_MR T C f y). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_measure | null |
tele_type : tele@{i} -> Type@{k} :=
| tip A := A -> Type@{j};
| ext A B := forall x : A, tele_type (B x). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_type | null |
tele_type_app (T : tele@{i}) (P : tele_type T) (x : tele_sigma T) : Type@{k} :=
tele_type_app (tip A) P a := P a;
tele_type_app (ext A B) P (a, b) := tele_type_app (B a) (P a) b. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_type_app | null |
tele_forall (T : tele@{i}) (P : tele_type T) : Type@{k} :=
| tip A, P := forall x : A, P x;
| ext A B, P := forall x : A, tele_forall (B x) (P x). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall | null |
tele_forall_impl (T : tele@{i}) (P : tele_type T) (Q : tele_type T) : Type :=
| tip A, P, Q := forall x : A, P x -> Q x;
| ext A B, P, Q := forall x : A, tele_forall_impl (B x) (P x) (Q x). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_impl | null |
tele_forall_app (T : tele@{i}) (P : tele_type T) (f : tele_forall T P) (x : T) : tele_type_app T P x :=
tele_forall_app (tip A) P f x := f x;
tele_forall_app (ext A B) P f x := tele_forall_app (B x.1) (P x.1) (f x.1) x.2. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_app | null |
tele_forall_type_app (T : tele@{i}) (P : tele_type T)
(fn : forall t, tele_type_app T P t) : tele_forall T P :=
| tip A, P, fn := fn
| ext A B, P, fn := fun a : A => tele_forall_type_app (B a) (P a) (fun b => fn (a, b)). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_type_app | null |
tele_forall_app_type (T : tele@{i}) (P : tele_type T) (f : forall t, tele_type_app T P t) :
forall x, tele_forall_app T P (tele_forall_type_app T P f) x = f x. | Lemma | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_app_type | null |
tele_forall_uncurry (T : tele@{i}) (P : T -> Type@{j}) : Type@{k} :=
| tip A , P := forall x : A, P x
| ext A B , P := forall x : A, tele_forall_uncurry (B x) (fun y : tele_sigma (B x) => P (x, y)). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_uncurry | null |
tele_rel_pack (T U : tele) (x : tele_rel T U) : tele_sigma T -> tele_sigma U -> Type by struct T :=
tele_rel_pack (tip A) (tip A') P := P;
tele_rel_pack (ext A B) (ext A' B') P := fun x y => tele_rel_pack (B x.1) (B' y.1) (P _ _) x.2 y.2. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_rel_pack | null |
tele_pred_pack (T : tele) (P : tele_pred T) : tele_sigma T -> Type :=
| tip A, P := P
| ext A B, P := fun x => tele_pred_pack (B x.1) (P x.1) x.2. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_pred_pack | null |
tele_type_unpack (T : tele) (P : tele_sigma T -> Type) : tele_type T :=
tele_type_unpack (tip A) P := P;
tele_type_unpack (ext A B) P := fun x => tele_type_unpack (B x) (fun y => P (x, y)). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_type_unpack | null |
tele_pred_fn_pack (T U : tele) (P : tele_fn T (tele_pred U)) : tele_sigma T -> tele_sigma U -> Type :=
tele_pred_fn_pack (tip A) U P := fun x => tele_pred_pack U (P x);
tele_pred_fn_pack (ext A B) U P := fun x => tele_pred_fn_pack (B x.1) U (P x.1) x.2. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_pred_fn_pack | null |
tele_rel_curried T := tele_fn T (tele_pred T). | Definition | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_rel_curried | null |
tele_forall_pack (T : tele) (P : T -> Type) (f : tele_forall_uncurry T P) (t : T) : P t :=
| tip A | P | f | t := f t;
| ext A B | P | f | (a, b) := tele_forall_pack (B a) (fun b => P (a, b)) (f a) b. | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_pack | null |
tele_forall_unpack (T : tele@{i}) (P : T -> Type@{j}) (f : forall (t : T), P t) : tele_forall_uncurry T P :=
| tip A | P | f := f
| ext A B | P | f := fun a : A => tele_forall_unpack (B a) (fun b => P (a, b)) (fun b => f (a, b)). | Equations | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_unpack | null |
tele_forall_pack_unpack (T : tele) (P : T -> Type) (f : forall t, P t) :
forall x, tele_forall_pack T P (tele_forall_unpack T P f) x = f x. | Lemma | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_forall_pack_unpack | null |
#[export]
Instance wf_tele_measure@{i j k| i <= k, j <= k}
{T : tele@{i}} (A : Type@{j}) (f : tele_fn@{i j k} T A) (R : A -> A -> Type@{k}) :
WellFounded R -> WellFounded (tele_measure T A f R) | (WellFounded (tele_measure _ _ _ _)). | Instance | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | wf_tele_measure | null |
tele_fix_functional_type :=
tele_forall_uncurry@{i m m} T (fun x =>
((tele_forall_uncurry@{i m m} T (fun y =>
R y x -> tele_type_app T P y))) ->
tele_type_app T P x). | Definition | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_fix_functional_type | Telescopes: allows treating variable arity fixpoints |
tele_fix : tele_forall T P :=
tele_forall_type_app _ _
(@FixWf (tele_sigma T) _ wf (tele_type_app T P)
(fun x H => tele_forall_pack T _ fn x (tele_forall_unpack T _ H))). | Definition | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_fix | null |
tele_fix_unfold :
tele_forall_app T P (tele_fix R wf P fn) x =
tele_forall_pack T _ fn x
(tele_forall_unpack T _ (fun y _ => tele_forall_app T P (tele_fix R wf P fn) y)). | Lemma | theories | [
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations",
"Equations"
] | theories/Type/Telescopes.v | tele_fix_unfold | null |
Acc (x : A) : Type@{max(i,j)} :=
| Acc_intro : (forall y, R y x -> Acc y) -> Acc x. | Inductive | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Acc | null |
Acc_inv {x} (H : Acc x) : forall y, R y x -> Acc y. | Definition | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Acc_inv | null |
Acc_prop i (x y : Acc i) : x = y. | Lemma | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Acc_prop | null |
well_founded := forall x, Acc x. | Definition | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | well_founded | null |
Fix_F (x : A) (a : Acc x) : P x :=
step x (fun y r => Fix_F y (Acc_inv a y r)). | Fixpoint | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Fix_F | null |
Fix (x : A) : P x :=
Fix_F R P step x (WF x). | Definition | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Fix | null |
well_founded_irreflexive {A} {R : relation A} {wfR : well_founded R} :
forall x y : A, R x y -> x = y -> Empty. | Lemma | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | well_founded_irreflexive | null |
well_founded_antisym@{i j} {A : Type@{i}} {R : relation@{i j} A}{wfR : well_founded R} :
forall x y : A, R x y -> R y x -> Empty. | Lemma | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | well_founded_antisym | null |
incl_trans_clos : inclusion R trans_clos. | Lemma | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | incl_trans_clos | null |
Acc_trans_clos : forall x:A, Acc R x -> Acc trans_clos x. | Lemma | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Acc_trans_clos | null |
Acc_inv_trans : forall x y:A, trans_clos y x -> Acc R x -> Acc R y. | Lemma | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Acc_inv_trans | null |
wf_trans_clos : well_founded R -> well_founded trans_clos. | Theorem | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | wf_trans_clos | null |
inverse_image := fun x y => R (f x) (f y). | Definition | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | inverse_image | null |
Acc_lemma : forall y : B, Acc R y -> forall x : A, y = f x -> Acc inverse_image x. | Remark | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Acc_lemma | null |
Acc_inverse_image : forall x:A, Acc R (f x) -> Acc inverse_image x. | Lemma | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | Acc_inverse_image | null |
wf_inverse_image : well_founded R -> well_founded inverse_image. | Theorem | theories | [
"Coq.Extraction",
"Coq.CRelationClasses",
"Equations",
"Equations"
] | theories/Type/WellFounded.v | wf_inverse_image | null |
le : nat -> nat -> Set :=
| le_0 x : le 0 x
| le_S {x y} : le x y -> le (S x) (S y). | Inductive | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | le | null |
lt x y := le (S x) y. | Definition | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | lt | null |
le_eq_lt x y : le x y -> (x = y) + (lt x y). | Lemma | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | le_eq_lt | null |
lt_wf : WellFounded lt. | Instance | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | lt_wf | null |
lt_n_Sn n : lt n (S n). | Lemma | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | lt_n_Sn | null |
lexprod : A * B -> A * B -> Type :=
| left_lex :
forall {x x':A} {y:B} {y':B},
leA x x' -> lexprod (x, y) (x', y')
| right_lex :
forall {x:A} {y y':B},
leB y y' -> lexprod (x, y) (x, y'). | Inductive | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | lexprod | null |
acc_A_B_lexprod :
forall x:A, Acc leA x -> (well_founded leB) ->
forall y:B, Acc leB y -> Acc lexprod (x, y). | Lemma | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | acc_A_B_lexprod | null |
wf_lexprod :
well_founded leA ->
well_founded leB -> well_founded lexprod. | Theorem | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | wf_lexprod | null |
#[export]
Instance wellfounded_lexprod A B R S `(wfR : WellFounded A R, wfS : WellFounded B S) :
WellFounded (lexprod A B R S) := wf_lexprod A B R S wfR wfS. | Instance | theories | [
"Equations"
] | theories/Type/WellFoundedInstances.v | wellfounded_lexprod | null |
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