(** A small interval & valuation library *)

  Definition interval := (nat * nat)%type.(* (a,b) <~~~> [a,b[ *)

  Implicit Types (i j : interval).Definition in_interval i x := let (a,b) := i in a <= x < b.Definition out_interval i x := let (a,b) := i in x < a \/ b <= x.Definition interval_disjoint i j := forall x, in_interval i x -> in_interval j x -> False.Definition interval_union (i j : interval) :=
    match i, j with (a1,b1),(a2,b2) => (min a1 a2, max b1 b2) end.Fact in_out_interval i x : in_interval i x -> out_interval i x -> False.Fact in_out_interval_dec i x : { in_interval i x } + { out_interval i x }.Fact interval_union_left i j x : in_interval i x -> in_interval (interval_union i j) x.Fact interval_union_right i j x : in_interval j x -> in_interval (interval_union i j) x.Definition valuation_union i1 (g1 : nat -> nat) i2 g2 : 
               interval_disjoint i1 i2 
            -> { g | (forall x, in_interval i1 x -> g x = g1 x)
                  /\ (forall x, in_interval i2 x -> g x = g2 x) }.Definition valuation_one_union k v i1 (g1 : nat -> nat) i2 g2 : 
               ~ in_interval (interval_union i1 i2) k 
            -> interval_disjoint i1 i2 
            -> { g | g k = v /\ (forall x, in_interval i1 x -> g x = g1 x)
                             /\ (forall x, in_interval i2 x -> g x = g2 x) }.(* v : cst = nat      constant (natural number)
     p q : par = nat    parameter 
     x y z : var = nat  existentially quantified variable 
   
     equations are of 4 types 
       1) x = v 
       2) x = p 
       3) x = y + z 
       4) x = y * z 

     We represent a relation between parameters R by a list of equations l and one
     variable x : var such that
       a) for any value f : nat -> nat of the params, l[f] has a solution
       b) for any value f : nat -> nat of the params,

                R f <-> ((x=0)::l)[f] has a solution

     How do we simulate poly1 = poly2 linearly ?

         poly1 = (eqn1,x) where x is output
         poly2 = (eqn2,y) where y is output

         example poly1 = p*(q+r)
         gives eqn1:= (0=1*2, 1=p, 2=3+4, 3=q, 4=r) and x:=0

         then we simulate poly1 = poly2 with
 
               poly1 = poly2 <-> (eqn1 U eqn2 U { w=y+u, w=x+v, z=u+v }, z)

         where u v z w are fresh

     We implement conjunction (cap), disjunction (cup) and exists so that we get a linear encoding

      cap) R by (lR,xR) and S by (lS,xS) 
           we assume (lR,xR) and (lS,xS) do not share existential variables
           (always possible by renaming those of S)

           R cap S = (lR U lS U { z = x+y },z) where z is fresh
 
      cup) R cup S = (lR U lS U { z = x*y },z)

      exists) ∃n.R f{p<-n} = (lR[p<-0], ...)

   *)

  Inductive dio_elem_expr : Set :=
    | dee_nat  : nat -> dio_elem_expr   (* c : constant *)
    | dee_var  : nat -> dio_elem_expr   (* v : existentially quant. var *)
    | dee_par  : nat -> dio_elem_expr   (* p : parameter *)
    | dee_comp : dio_op -> nat -> nat -> dio_elem_expr.(* v : cst = nat      constant (natural number)
     p q : par = nat    parameter 
     x y z : var = nat  existentially quantified variable 
   
     equations are of 4 types 
       1) x = v 
       2) x = p 
       3) x = y + z 
       4) x = y * z 

     We represent a relation between parameters R by a list of equations l and one
     variable x : var such that
       a) for any value f : nat -> nat of the params, l[f] has a solution
       b) for any value f : nat -> nat of the params,

                R f <-> ((x=0)::l)[f] has a solution

     How do we simulate poly1 = poly2 linearly ?

         poly1 = (eqn1,x) where x is output
         poly2 = (eqn2,y) where y is output

         example poly1 = p*(q+r)
         gives eqn1:= (0=1*2, 1=p, 2=3+4, 3=q, 4=r) and x:=0

         then we simulate poly1 = poly2 with
 
               poly1 = poly2 <-> (eqn1 U eqn2 U { w=y+u, w=x+v, z=u+v }, z)

         where u v z w are fresh

     We implement conjunction (cap), disjunction (cup) and exists so that we get a linear encoding

      cap) R by (lR,xR) and S by (lS,xS) 
           we assume (lR,xR) and (lS,xS) do not share existential variables
           (always possible by renaming those of S)

           R cap S = (lR U lS U { z = x+y },z) where z is fresh
 
      cup) R cup S = (lR U lS U { z = x*y },z)

      exists) ∃n.R f{p<-n} = (lR[p<-0], ...)

   *)

  Inductive dio_elem_expr : Set :=
    | dee_nat  : nat -> dio_elem_expr   (* c : constant *)
    | dee_var  : nat -> dio_elem_expr   (* v : existentially quant. var *)
    | dee_par  : nat -> dio_elem_expr   (* p : parameter *)
    | dee_comp : dio_op -> nat -> nat -> dio_elem_expr.(* v : cst = nat      constant (natural number)
     p q : par = nat    parameter 
     x y z : var = nat  existentially quantified variable 
   
     equations are of 4 types 
       1) x = v 
       2) x = p 
       3) x = y + z 
       4) x = y * z 

     We represent a relation between parameters R by a list of equations l and one
     variable x : var such that
       a) for any value f : nat -> nat of the params, l[f] has a solution
       b) for any value f : nat -> nat of the params,

                R f <-> ((x=0)::l)[f] has a solution

     How do we simulate poly1 = poly2 linearly ?

         poly1 = (eqn1,x) where x is output
         poly2 = (eqn2,y) where y is output

         example poly1 = p*(q+r)
         gives eqn1:= (0=1*2, 1=p, 2=3+4, 3=q, 4=r) and x:=0

         then we simulate poly1 = poly2 with
 
               poly1 = poly2 <-> (eqn1 U eqn2 U { w=y+u, w=x+v, z=u+v }, z)

         where u v z w are fresh

     We implement conjunction (cap), disjunction (cup) and exists so that we get a linear encoding

      cap) R by (lR,xR) and S by (lS,xS) 
           we assume (lR,xR) and (lS,xS) do not share existential variables
           (always possible by renaming those of S)

           R cap S = (lR U lS U { z = x+y },z) where z is fresh
 
      cup) R cup S = (lR U lS U { z = x*y },z)

      exists) ∃n.R f{p<-n} = (lR[p<-0], ...)

   *)

  Inductive dio_elem_expr : Set :=
    | dee_nat  : nat -> dio_elem_expr   (* c : constant *)
    | dee_var  : nat -> dio_elem_expr   (* v : existentially quant. var *)
    | dee_par  : nat -> dio_elem_expr   (* p : parameter *)
    | dee_comp : dio_op -> nat -> nat -> dio_elem_expr.(* v : cst = nat      constant (natural number)
     p q : par = nat    parameter 
     x y z : var = nat  existentially quantified variable 
   
     equations are of 4 types 
       1) x = v 
       2) x = p 
       3) x = y + z 
       4) x = y * z 

     We represent a relation between parameters R by a list of equations l and one
     variable x : var such that
       a) for any value f : nat -> nat of the params, l[f] has a solution
       b) for any value f : nat -> nat of the params,

                R f <-> ((x=0)::l)[f] has a solution

     How do we simulate poly1 = poly2 linearly ?

         poly1 = (eqn1,x) where x is output
         poly2 = (eqn2,y) where y is output

         example poly1 = p*(q+r)
         gives eqn1:= (0=1*2, 1=p, 2=3+4, 3=q, 4=r) and x:=0

         then we simulate poly1 = poly2 with
 
               poly1 = poly2 <-> (eqn1 U eqn2 U { w=y+u, w=x+v, z=u+v }, z)

         where u v z w are fresh

     We implement conjunction (cap), disjunction (cup) and exists so that we get a linear encoding

      cap) R by (lR,xR) and S by (lS,xS) 
           we assume (lR,xR) and (lS,xS) do not share existential variables
           (always possible by renaming those of S)

           R cap S = (lR U lS U { z = x+y },z) where z is fresh
 
      cup) R cup S = (lR U lS U { z = x*y },z)

      exists) ∃n.R f{p<-n} = (lR[p<-0], ...)

   *)

  Inductive dio_elem_expr : Set :=
    | dee_nat  : nat -> dio_elem_expr   (* c : constant *)
    | dee_var  : nat -> dio_elem_expr   (* v : existentially quant. var *)
    | dee_par  : nat -> dio_elem_expr   (* p : parameter *)
    | dee_comp : dio_op -> nat -> nat -> dio_elem_expr.(* v : cst = nat      constant (natural number)
     p q : par = nat    parameter 
     x y z : var = nat  existentially quantified variable 
   
     equations are of 4 types 
       1) x = v 
       2) x = p 
       3) x = y + z 
       4) x = y * z 

     We represent a relation between parameters R by a list of equations l and one
     variable x : var such that
       a) for any value f : nat -> nat of the params, l[f] has a solution
       b) for any value f : nat -> nat of the params,

                R f <-> ((x=0)::l)[f] has a solution

     How do we simulate poly1 = poly2 linearly ?

         poly1 = (eqn1,x) where x is output
         poly2 = (eqn2,y) where y is output

         example poly1 = p*(q+r)
         gives eqn1:= (0=1*2, 1=p, 2=3+4, 3=q, 4=r) and x:=0

         then we simulate poly1 = poly2 with
 
               poly1 = poly2 <-> (eqn1 U eqn2 U { w=y+u, w=x+v, z=u+v }, z)

         where u v z w are fresh

     We implement conjunction (cap), disjunction (cup) and exists so that we get a linear encoding

      cap) R by (lR,xR) and S by (lS,xS) 
           we assume (lR,xR) and (lS,xS) do not share existential variables
           (always possible by renaming those of S)

           R cap S = (lR U lS U { z = x+y },z) where z is fresh
 
      cup) R cup S = (lR U lS U { z = x*y },z)

      exists) ∃n.R f{p<-n} = (lR[p<-0], ...)

   *)

  Inductive dio_elem_expr : Set :=
    | dee_nat  : nat -> dio_elem_expr   (* c : constant *)
    | dee_var  : nat -> dio_elem_expr   (* v : existentially quant. var *)
    | dee_par  : nat -> dio_elem_expr   (* p : parameter *)
    | dee_comp : dio_op -> nat -> nat -> dio_elem_expr.(* v1 op v2 *)

  Notation dee_add := (dee_comp do_add).Notation dee_mul := (dee_comp do_mul).(* ρ σ ν φ *)

  Definition dee_eval φ ν e := 
    match e with
      | dee_nat n => n
      | dee_var v => φ v
      | dee_par i => ν i
      | dee_comp o v w => do_eval o (φ v) (φ w) 
    end.Definition dee_vars e x  :=
    match e with
      | dee_nat _ => False
      | dee_var v => x = v
      | dee_par _ => False
      | dee_comp _ v w => x = v \/ x = w
    end.Fact dee_eval_ext e φ1 ν1 φ2 ν2  : 
        (forall x, dee_vars e x -> φ1 x = φ2 x) 
     -> (forall x, ν1 x = ν2 x)
     -> dee_eval φ1 ν1 e = dee_eval φ2 ν2 e.(* ρ σ ν *)

  Definition dee_move k p :=
    match p with
      | dee_nat n      => dee_nat n
      | dee_var v      => dee_var (k+v)
      | dee_par i      => dee_par i
      | dee_comp o v w => dee_comp o (k+v) (k+w)
    end.Fact dee_eval_move k φ ν e : dee_eval φ ν (dee_move k e) = dee_eval (fun x => φ (k+x)) ν e.Fact dee_vars_move k e x : dee_vars (dee_move k e) x <-> exists y, dee_vars e y /\ x = k+y.Definition dee_dec k e :=
    match e with
      | dee_nat n      => dee_nat n
      | dee_var v      => dee_var v
      | dee_par 0      => dee_var k 
      | dee_par (S i)  => dee_par i
      | dee_comp o v w => dee_comp o v w
    end.Fact dee_eval_dec φ ν k e : dee_eval φ ν (dee_dec k e) = dee_eval φ (fun x => match x with 0 => φ k | S x => ν x end) e.Fact dee_vars_dec k e x : dee_vars (dee_dec k e) x -> x = k \/ dee_vars e x.Definition dio_constraint := (nat * dio_elem_expr)%type.Implicit Type (c : dio_constraint).Definition dc_eval φ ν c := φ (fst c) = dee_eval φ ν (snd c).Definition dc_vars c x := x = fst c \/ dee_vars (snd c) x.Fact dc_eval_ext c φ1 ν1 φ2 ν2  : 
        (forall x, dc_vars c x -> φ1 x = φ2 x) 
     -> (forall x, ν1 x = ν2 x)
     -> dc_eval φ1 ν1 c <-> dc_eval φ2 ν2 c.Definition dc_move k c := (k+fst c, dee_move k (snd c)).Fact dc_eval_move k φ ν c : dc_eval φ ν (dc_move k c) <-> dc_eval (fun x => φ (k+x)) ν c.Fact dc_vars_move k c x : dc_vars (dc_move k c) x <-> exists y, x = k + y /\ dc_vars c y.Definition dc_dec k c := (fst c, dee_dec k (snd c)).Fact dc_eval_dec φ ν k c : dc_eval φ ν (dc_dec k c) <-> dc_eval φ  (fun x => match x with 0 => φ k | S x => ν x end) c.Fact dc_vars_dec k c x : dc_vars (dc_dec k c) x -> x = k \/ dc_vars c x.Implicit Type (R : (nat -> nat) -> Prop).(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A diophantine system for R is an interval i, a list l of constraints, a reference variable x
     such that
       1) the variables in l U {x} and belong to i
       2) for any valuation, the equations of l are satisfiable
       3) for any valuation ν, R ν holds iff the equations of { x=0 } U l are satisfiable
   *)

  Record dio_repr_at R a n l := {
    ds_eqns : list dio_constraint;
    ds_ref  : nat;
    ds_H0   : length ds_eqns = l;
    ds_H1   : forall x c, In c ds_eqns -> dc_vars c x -> a <= x < a+n;
    ds_H2   : a <= ds_ref < a+n;
    ds_H3   : forall ν, exists φ, Forall (dc_eval φ ν) ds_eqns;
    ds_H4   : forall ν, R ν <-> exists φ, Forall (dc_eval φ ν) ds_eqns /\ φ ds_ref = 0;
  }.(* A compiler from expressions to lists of small expressions *)

    Fixpoint de_eqns e x :=
      match e with
        | de_cst n      => (x,dee_nat n)::nil 
        | de_var p      => (x,dee_par p)::nil
        | de_comp o p q => (x,dee_comp o (x+1) (x+1+de_size p)) :: de_eqns p (x+1) ++ de_eqns q (x+1+de_size p)
      end.Fact de_eqns_length e x : length (de_eqns e x) = de_size e.Fact de_size_ge_1 e : 1 <= de_size e.Fact de_eqns_vars e x : forall c, In c (de_eqns e x) -> forall y, dc_vars c y -> x <= y < x+de_size e.(* If equations in de_eqns e x are satisfied in φ, then φ x must be equal to de_eval ν e *)

    Fact dc_Forall_eval φ ν e x : Forall (dc_eval φ ν) (de_eqns e x) -> de_eval ν e = φ x.(* Converselly, equations in de_eqns e x are satisfiable *)

    Fact dc_eval_exists_Forall ν e x : { φ | Forall (dc_eval φ ν) (de_eqns e x) }.

Lemma dio_repr_at_exst R a n m p : dio_repr_at R a n p -> m = n+1 -> dio_repr_at (fun ν => exists n, R (dv_lift ν n)) a m p.