Section TransferEqType.
Variables (T : Type) (eT : eqType) (f : T -> eT).
Admitted.
End SubType.
Arguments SubType [T P].
Arguments Sub [T P s].
Arguments vrefl [T P].
Arguments vrefl_rect [T P].
Arguments clone_subType [T P] U v [sT] _ [c Urec cK].
Arguments insub [T P sT].
Arguments insubT [T] P [sT x].
Arguments val_inj [T P sT].
Prenex Implicits val Sub vrefl vrefl_rect insub insubd val_inj.
Local Notation inlined_sub_rect :=
(fun K K_S u => let (x, Px) as u return K u := u in K_S x Px).
Local Notation inlined_new_rect :=
(fun K K_S u => let (x) as u return K u := u in K_S x).
Notation "[ 'subType' 'for' v ]" := (SubType _ v _ inlined_sub_rect vrefl_rect)
(at level 0, only parsing) : form_scope.
Notation "[ 'sub' 'Type' 'for' v ]" := (SubType _ v _ _ vrefl_rect)
(at level 0, format "[ 'sub' 'Type'  'for'  v ]") : form_scope.
Notation "[ 'subType' 'for' v 'by' rec ]" := (SubType _ v _ rec vrefl)
(at level 0, format "[ 'subType'  'for'  v  'by'  rec ]") : form_scope.
Notation "[ 'subType' 'of' U 'for' v ]" := (clone_subType U v id idfun)
(at level 0, format "[ 'subType'  'of'  U  'for'  v ]") : form_scope.
(* Notation "[ 'subType' 'for' v ]" := (clone_subType _ v id idfun) (at level 0, format "[ 'subType'  'for'  v ]") : form_scope. *)
Notation "[ 'subType' 'of' U ]" := (clone_subType U _ id id)
(at level 0, format "[ 'subType'  'of'  U ]") : form_scope.
Definition NewType T U v c Urec :=
let Urec' P IH := Urec P (fun x : T => IH x isT : P _) in
SubType U v (fun x _ => c x) Urec'.
Arguments NewType [T U].
Notation "[ 'newType' 'for' v ]" := (NewType v _ inlined_new_rect vrefl_rect)
(at level 0, only parsing) : form_scope.
Notation "[ 'new' 'Type' 'for' v ]" := (NewType v _ _ vrefl_rect)
(at level 0, format "[ 'new' 'Type'  'for'  v ]") : form_scope.
Notation "[ 'newType' 'for' v 'by' rec ]" := (NewType v _ rec vrefl)
(at level 0, format "[ 'newType'  'for'  v  'by'  rec ]") : form_scope.
Definition innew T nT x := @Sub T predT nT x (erefl true).
Arguments innew [T nT].
Prenex Implicits innew.
Canonical nat_eqMixin := EqMixin eqnP.
Canonical nat_eqType := Eval hnf in EqType nat nat_eqMixin.
Arguments eqnP [x y].
Prenex Implicits eqnP.
Coercion nat_of_bool (b : bool) := if b then 1 else 0.
Fixpoint odd n := if n is n'.+1 then ~~ odd n' else false.
Definition subn_rec := minus.
Notation "m - n" := (subn_rec m n) : nat_rec_scope.
Definition subn := nosimpl subn_rec.
Notation "m - n" := (subn m n) : nat_scope.
Definition leq m n := m - n == 0.
Notation "m <= n" := (leq m n) : nat_scope.
Notation "m < n"  := (m.+1 <= n) : nat_scope.
Notation "m >= n" := (n <= m) (only parsing) : nat_scope.
Notation "m > n"  := (n < m) (only parsing)  : nat_scope.
Notation "m <= n <= p" := ((m <= n) && (n <= p)) : nat_scope.
Notation "m < n <= p" := ((m < n) && (n <= p)) : nat_scope.
Notation "m <= n < p" := ((m <= n) && (n < p)) : nat_scope.
Notation "m < n < p" := ((m < n) && (n < p)) : nat_scope.
Open Scope nat_scope.
Hint Resolve leqnn.
Admitted.
Admitted.
Admitted.
Hint Resolve leqnSn.
Definition muln_rec := mult.
Notation "m * n" := (muln_rec m n) : nat_rec_scope.
Definition muln := nosimpl muln_rec.
Notation "m * n" := (muln m n) : nat_scope.

Lemma mulnS m n : m * n.+1 = m + m * n.