(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below    *)
Require Import BuiltIn.
Require Reals.Rbasic_fun.
Require Reals.R_sqrt.
Require Reals.Rtrigo_def.
Require Reals.Rtrigo1.
Require Reals.Ratan.
Require BuiltIn.
Require HighOrd.
Require bool.Bool.
Require int.Int.
Require int.Abs.
Require int.ComputerDivision.
Require real.Real.
Require real.RealInfix.
Require real.Abs.
Require real.FromInt.
Require real.Square.
Require real.Trigonometry.
Require map.Map.

Parameter eqb:
  forall {a:Type} {a_WT:WhyType a}, a -> a -> Init.Datatypes.bool.

Axiom eqb1 :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (y:a), ((eqb x y) = Init.Datatypes.true) <-> (x = y).

Axiom eqb_false :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (y:a), ((eqb x y) = Init.Datatypes.false) <-> ~ (x = y).

Parameter neqb:
  forall {a:Type} {a_WT:WhyType a}, a -> a -> Init.Datatypes.bool.

Axiom neqb1 :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (y:a), ((neqb x y) = Init.Datatypes.true) <-> ~ (x = y).

Parameter zlt: Numbers.BinNums.Z -> Numbers.BinNums.Z -> Init.Datatypes.bool.

Parameter zleq:
  Numbers.BinNums.Z -> Numbers.BinNums.Z -> Init.Datatypes.bool.

Axiom zlt1 :
  forall (x:Numbers.BinNums.Z) (y:Numbers.BinNums.Z),
  ((zlt x y) = Init.Datatypes.true) <-> (x < y)%Z.

Axiom zleq1 :
  forall (x:Numbers.BinNums.Z) (y:Numbers.BinNums.Z),
  ((zleq x y) = Init.Datatypes.true) <-> (x <= y)%Z.

Parameter rlt:
  Reals.Rdefinitions.R -> Reals.Rdefinitions.R -> Init.Datatypes.bool.

Parameter rleq:
  Reals.Rdefinitions.R -> Reals.Rdefinitions.R -> Init.Datatypes.bool.

Axiom rlt1 :
  forall (x:Reals.Rdefinitions.R) (y:Reals.Rdefinitions.R),
  ((rlt x y) = Init.Datatypes.true) <-> (x < y)%R.

Axiom rleq1 :
  forall (x:Reals.Rdefinitions.R) (y:Reals.Rdefinitions.R),
  ((rleq x y) = Init.Datatypes.true) <-> (x <= y)%R.

(* Why3 assumption *)
Definition real_of_int (x:Numbers.BinNums.Z) : Reals.Rdefinitions.R :=
  BuiltIn.IZR x.

Axiom c_euclidian :
  forall (n:Numbers.BinNums.Z) (d:Numbers.BinNums.Z), ~ (d = 0%Z) ->
  (n = (((ZArith.BinInt.Z.quot n d) * d)%Z + (ZArith.BinInt.Z.rem n d))%Z).

Axiom cmod_remainder :
  forall (n:Numbers.BinNums.Z) (d:Numbers.BinNums.Z),
  ((0%Z <= n)%Z -> (0%Z < d)%Z ->
   (0%Z <= (ZArith.BinInt.Z.rem n d))%Z /\ ((ZArith.BinInt.Z.rem n d) < d)%Z) /\
  ((n <= 0%Z)%Z -> (0%Z < d)%Z ->
   ((-d)%Z < (ZArith.BinInt.Z.rem n d))%Z /\
   ((ZArith.BinInt.Z.rem n d) <= 0%Z)%Z) /\
  ((0%Z <= n)%Z -> (d < 0%Z)%Z ->
   (0%Z <= (ZArith.BinInt.Z.rem n d))%Z /\
   ((ZArith.BinInt.Z.rem n d) < (-d)%Z)%Z) /\
  ((n <= 0%Z)%Z -> (d < 0%Z)%Z ->
   (d < (ZArith.BinInt.Z.rem n d))%Z /\ ((ZArith.BinInt.Z.rem n d) <= 0%Z)%Z).

Axiom cdiv_neutral :
  forall (a:Numbers.BinNums.Z), ((ZArith.BinInt.Z.quot a 1%Z) = a).

Axiom cdiv_inv :
  forall (a:Numbers.BinNums.Z), ~ (a = 0%Z) ->
  ((ZArith.BinInt.Z.quot a a) = 1%Z).

Axiom cdiv_closed_remainder :
  forall (a:Numbers.BinNums.Z) (b:Numbers.BinNums.Z) (n:Numbers.BinNums.Z),
  (0%Z <= a)%Z -> (0%Z <= b)%Z ->
  (0%Z <= (b - a)%Z)%Z /\ ((b - a)%Z < n)%Z ->
  ((ZArith.BinInt.Z.rem a n) = (ZArith.BinInt.Z.rem b n)) -> (a = b).

Axiom abs_def :
  forall (x:Numbers.BinNums.Z),
  ((0%Z <= x)%Z -> ((ZArith.BinInt.Z.abs x) = x)) /\
  (~ (0%Z <= x)%Z -> ((ZArith.BinInt.Z.abs x) = (-x)%Z)).

Axiom sqrt_lin1 :
  forall (x:Reals.Rdefinitions.R), (1%R < x)%R ->
  ((Reals.R_sqrt.sqrt x) < x)%R.

Axiom sqrt_lin0 :
  forall (x:Reals.Rdefinitions.R), (0%R < x)%R /\ (x < 1%R)%R ->
  (x < (Reals.R_sqrt.sqrt x))%R.

Axiom sqrt_0 : ((Reals.R_sqrt.sqrt 0%R) = 0%R).

Axiom sqrt_1 : ((Reals.R_sqrt.sqrt 1%R) = 1%R).

Axiom Q_float_rmat_of_eulers_321_1 :
  forall (a:Reals.Rdefinitions.R) (b:Reals.Rdefinitions.R)
    (c:Reals.Rdefinitions.R),
  let r := Reals.Rtrigo_def.sin a in
  let r1 := Reals.Rtrigo_def.cos b in
  let r2 := Reals.Rtrigo_def.sin c in
  let r3 := Reals.Rtrigo_def.cos a in
  let r4 := Reals.Rtrigo_def.sin b in
  let r5 := Reals.Rtrigo_def.cos c in
  let r6 := (((-1%R)%R * (r2 * r3)%R)%R + ((r * r4)%R * r5)%R)%R in
  let r7 := ((r3 * r5)%R + ((r * r4)%R * r2)%R)%R in
  ((((((r * r)%R * r1)%R * r1)%R + (r6 * r6)%R)%R + (r7 * r7)%R)%R = 1%R).

Axiom Q_cos_sin_square :
  forall (a:Reals.Rdefinitions.R),
  let r := Reals.Rtrigo_def.sin a in
  let r1 := Reals.Rtrigo_def.cos a in (((r * r)%R + (r1 * r1)%R)%R = 1%R).

(* Why3 assumption *)
Inductive S12_RealRMat_s :=
  | S12_RealRMat_s1 : Reals.Rdefinitions.R -> Reals.Rdefinitions.R ->
      Reals.Rdefinitions.R -> Reals.Rdefinitions.R -> Reals.Rdefinitions.R ->
      Reals.Rdefinitions.R -> Reals.Rdefinitions.R -> Reals.Rdefinitions.R ->
      Reals.Rdefinitions.R -> S12_RealRMat_s.
Axiom S12_RealRMat_s_WhyType : WhyType S12_RealRMat_s.
Existing Instance S12_RealRMat_s_WhyType.

(* Why3 assumption *)
Definition F12_RealRMat_s_a22 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x8
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a21 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x7
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a20 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x6
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a12 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x5
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a11 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x4
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a10 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x3
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a02 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x2
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a01 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x1
  end.

(* Why3 assumption *)
Definition F12_RealRMat_s_a00 (v:S12_RealRMat_s) : Reals.Rdefinitions.R :=
  match v with
  | S12_RealRMat_s1 x x1 x2 x3 x4 x5 x6 x7 x8 => x
  end.

(* Why3 assumption *)
Definition EqS12_RealRMat_s (S:S12_RealRMat_s) (S1:S12_RealRMat_s) : Prop :=
  (((((((((F12_RealRMat_s_a00 S1) = (F12_RealRMat_s_a00 S)) /\
         ((F12_RealRMat_s_a01 S1) = (F12_RealRMat_s_a01 S))) /\
        ((F12_RealRMat_s_a02 S1) = (F12_RealRMat_s_a02 S))) /\
       ((F12_RealRMat_s_a10 S1) = (F12_RealRMat_s_a10 S))) /\
      ((F12_RealRMat_s_a11 S1) = (F12_RealRMat_s_a11 S))) /\
     ((F12_RealRMat_s_a12 S1) = (F12_RealRMat_s_a12 S))) /\
    ((F12_RealRMat_s_a20 S1) = (F12_RealRMat_s_a20 S))) /\
   ((F12_RealRMat_s_a21 S1) = (F12_RealRMat_s_a21 S))) /\
  ((F12_RealRMat_s_a22 S1) = (F12_RealRMat_s_a22 S)).

(* Why3 assumption *)
Definition L_mult_RealRMat (rmat1:S12_RealRMat_s) (rmat2:S12_RealRMat_s) :
    S12_RealRMat_s :=
  let r := F12_RealRMat_s_a00 rmat1 in
  let r1 := F12_RealRMat_s_a00 rmat2 in
  let r2 := F12_RealRMat_s_a01 rmat1 in
  let r3 := F12_RealRMat_s_a10 rmat2 in
  let r4 := F12_RealRMat_s_a02 rmat1 in
  let r5 := F12_RealRMat_s_a20 rmat2 in
  let r6 := F12_RealRMat_s_a01 rmat2 in
  let r7 := F12_RealRMat_s_a11 rmat2 in
  let r8 := F12_RealRMat_s_a21 rmat2 in
  let r9 := F12_RealRMat_s_a02 rmat2 in
  let r10 := F12_RealRMat_s_a12 rmat2 in
  let r11 := F12_RealRMat_s_a22 rmat2 in
  let r12 := F12_RealRMat_s_a10 rmat1 in
  let r13 := F12_RealRMat_s_a11 rmat1 in
  let r14 := F12_RealRMat_s_a12 rmat1 in
  let r15 := F12_RealRMat_s_a20 rmat1 in
  let r16 := F12_RealRMat_s_a21 rmat1 in
  let r17 := F12_RealRMat_s_a22 rmat1 in
  S12_RealRMat_s1 (((r * r1)%R + (r2 * r3)%R)%R + (r4 * r5)%R)%R
  (((r * r6)%R + (r2 * r7)%R)%R + (r4 * r8)%R)%R
  (((r * r9)%R + (r2 * r10)%R)%R + (r4 * r11)%R)%R
  (((r12 * r1)%R + (r13 * r3)%R)%R + (r14 * r5)%R)%R
  (((r12 * r6)%R + (r13 * r7)%R)%R + (r14 * r8)%R)%R
  (((r12 * r9)%R + (r13 * r10)%R)%R + (r14 * r11)%R)%R
  (((r15 * r1)%R + (r16 * r3)%R)%R + (r17 * r5)%R)%R
  (((r15 * r6)%R + (r16 * r7)%R)%R + (r17 * r8)%R)%R
  (((r15 * r9)%R + (r16 * r10)%R)%R + (r17 * r11)%R)%R.

(* Why3 assumption *)
Inductive addr :=
  | addr'mk : Numbers.BinNums.Z -> Numbers.BinNums.Z -> addr.
Axiom addr_WhyType : WhyType addr.
Existing Instance addr_WhyType.

(* Why3 assumption *)
Definition offset (v:addr) : Numbers.BinNums.Z :=
  match v with
  | addr'mk x x1 => x1
  end.

(* Why3 assumption *)
Definition base (v:addr) : Numbers.BinNums.Z :=
  match v with
  | addr'mk x x1 => x
  end.

Parameter addr_le: addr -> addr -> Prop.

Parameter addr_lt: addr -> addr -> Prop.

Parameter addr_le_bool: addr -> addr -> Init.Datatypes.bool.

Parameter addr_lt_bool: addr -> addr -> Init.Datatypes.bool.

Axiom addr_le_def :
  forall (p:addr) (q:addr), ((base p) = (base q)) ->
  addr_le p q <-> ((offset p) <= (offset q))%Z.

Axiom addr_lt_def :
  forall (p:addr) (q:addr), ((base p) = (base q)) ->
  addr_lt p q <-> ((offset p) < (offset q))%Z.

Axiom addr_le_bool_def :
  forall (p:addr) (q:addr),
  addr_le p q <-> ((addr_le_bool p q) = Init.Datatypes.true).

Axiom addr_lt_bool_def :
  forall (p:addr) (q:addr),
  addr_lt p q <-> ((addr_lt_bool p q) = Init.Datatypes.true).

(* Why3 assumption *)
Definition null : addr := addr'mk 0%Z 0%Z.

(* Why3 assumption *)
Definition global (b:Numbers.BinNums.Z) : addr := addr'mk b 0%Z.

(* Why3 assumption *)
Definition shift (p:addr) (k:Numbers.BinNums.Z) : addr :=
  addr'mk (base p) ((offset p) + k)%Z.

(* Why3 assumption *)
Definition included (p:addr) (a:Numbers.BinNums.Z) (q:addr)
    (b:Numbers.BinNums.Z) : Prop :=
  (0%Z < a)%Z ->
  (0%Z <= b)%Z /\
  ((base p) = (base q)) /\
  ((offset q) <= (offset p))%Z /\
  (((offset p) + a)%Z <= ((offset q) + b)%Z)%Z.

(* Why3 assumption *)
Definition separated (p:addr) (a:Numbers.BinNums.Z) (q:addr)
    (b:Numbers.BinNums.Z) : Prop :=
  (a <= 0%Z)%Z \/
  (b <= 0%Z)%Z \/
  ~ ((base p) = (base q)) \/
  (((offset q) + b)%Z <= (offset p))%Z \/
  (((offset p) + a)%Z <= (offset q))%Z.

(* Why3 assumption *)
Definition eqmem {a:Type} {a_WT:WhyType a} (m1:addr -> a) (m2:addr -> a)
    (p:addr) (a1:Numbers.BinNums.Z) : Prop :=
  forall (q:addr), included q 1%Z p a1 -> ((m1 q) = (m2 q)).

Parameter havoc:
  forall {a:Type} {a_WT:WhyType a}, (addr -> a) -> (addr -> a) -> addr ->
  Numbers.BinNums.Z -> addr -> a.

(* Why3 assumption *)
Definition valid_rw (m:Numbers.BinNums.Z -> Numbers.BinNums.Z) (p:addr)
    (n:Numbers.BinNums.Z) : Prop :=
  (0%Z < n)%Z ->
  (0%Z < (base p))%Z /\
  (0%Z <= (offset p))%Z /\ (((offset p) + n)%Z <= (m (base p)))%Z.

(* Why3 assumption *)
Definition valid_rd (m:Numbers.BinNums.Z -> Numbers.BinNums.Z) (p:addr)
    (n:Numbers.BinNums.Z) : Prop :=
  (0%Z < n)%Z ->
  ~ (0%Z = (base p)) /\
  (0%Z <= (offset p))%Z /\ (((offset p) + n)%Z <= (m (base p)))%Z.

(* Why3 assumption *)
Definition valid_obj (m:Numbers.BinNums.Z -> Numbers.BinNums.Z) (p:addr)
    (n:Numbers.BinNums.Z) : Prop :=
  (0%Z < n)%Z ->
  (p = null) \/
  ~ (0%Z = (base p)) /\
  (0%Z <= (offset p))%Z /\ (((offset p) + n)%Z <= (1%Z + (m (base p)))%Z)%Z.

(* Why3 assumption *)
Definition invalid (m:Numbers.BinNums.Z -> Numbers.BinNums.Z) (p:addr)
    (n:Numbers.BinNums.Z) : Prop :=
  (n <= 0%Z)%Z \/
  ((base p) = 0%Z) \/
  ((m (base p)) <= (offset p))%Z \/ (((offset p) + n)%Z <= 0%Z)%Z.

Axiom valid_rw_rd :
  forall (m:Numbers.BinNums.Z -> Numbers.BinNums.Z), forall (p:addr),
  forall (n:Numbers.BinNums.Z), valid_rw m p n -> valid_rd m p n.

Axiom valid_string :
  forall (m:Numbers.BinNums.Z -> Numbers.BinNums.Z), forall (p:addr),
  ((base p) < 0%Z)%Z ->
  (0%Z <= (offset p))%Z /\ ((offset p) < (m (base p)))%Z ->
  valid_rd m p 1%Z /\ ~ valid_rw m p 1%Z.

Axiom separated_1 :
  forall (p:addr) (q:addr),
  forall (a:Numbers.BinNums.Z) (b:Numbers.BinNums.Z) (i:Numbers.BinNums.Z)
    (j:Numbers.BinNums.Z),
  separated p a q b -> ((offset p) <= i)%Z /\ (i < ((offset p) + a)%Z)%Z ->
  ((offset q) <= j)%Z /\ (j < ((offset q) + b)%Z)%Z ->
  ~ ((addr'mk (base p) i) = (addr'mk (base q) j)).

Parameter region: Numbers.BinNums.Z -> Numbers.BinNums.Z.

Parameter linked: (Numbers.BinNums.Z -> Numbers.BinNums.Z) -> Prop.

Parameter sconst: (addr -> Numbers.BinNums.Z) -> Prop.

(* Why3 assumption *)
Definition framed (m:addr -> addr) : Prop :=
  forall (p:addr), ((region (base (m p))) <= 0%Z)%Z.

Axiom separated_included :
  forall (p:addr) (q:addr),
  forall (a:Numbers.BinNums.Z) (b:Numbers.BinNums.Z), (0%Z < a)%Z ->
  (0%Z < b)%Z -> separated p a q b -> ~ included p a q b.

Axiom included_trans :
  forall (p:addr) (q:addr) (r:addr),
  forall (a:Numbers.BinNums.Z) (b:Numbers.BinNums.Z) (c:Numbers.BinNums.Z),
  included p a q b -> included q b r c -> included p a r c.

Axiom separated_trans :
  forall (p:addr) (q:addr) (r:addr),
  forall (a:Numbers.BinNums.Z) (b:Numbers.BinNums.Z) (c:Numbers.BinNums.Z),
  included p a q b -> separated q b r c -> separated p a r c.

Axiom separated_sym :
  forall (p:addr) (q:addr),
  forall (a:Numbers.BinNums.Z) (b:Numbers.BinNums.Z),
  separated p a q b <-> separated q b p a.

Axiom eqmem_included :
  forall {a:Type} {a_WT:WhyType a},
  forall (m1:addr -> a) (m2:addr -> a), forall (p:addr) (q:addr),
  forall (a1:Numbers.BinNums.Z) (b:Numbers.BinNums.Z), included p a1 q b ->
  eqmem m1 m2 q b -> eqmem m1 m2 p a1.

Axiom eqmem_sym :
  forall {a:Type} {a_WT:WhyType a},
  forall (m1:addr -> a) (m2:addr -> a), forall (p:addr),
  forall (a1:Numbers.BinNums.Z), eqmem m1 m2 p a1 -> eqmem m2 m1 p a1.

Axiom havoc_access :
  forall {a:Type} {a_WT:WhyType a},
  forall (m0:addr -> a) (m1:addr -> a), forall (q:addr) (p:addr),
  forall (a1:Numbers.BinNums.Z),
  (separated q 1%Z p a1 -> ((havoc m0 m1 p a1 q) = (m1 q))) /\
  (~ separated q 1%Z p a1 -> ((havoc m0 m1 p a1 q) = (m0 q))).

Parameter cinits: (addr -> Init.Datatypes.bool) -> Prop.

(* Why3 assumption *)
Definition is_init_range (m:addr -> Init.Datatypes.bool) (p:addr)
    (l:Numbers.BinNums.Z) : Prop :=
  forall (i:Numbers.BinNums.Z), (0%Z <= i)%Z /\ (i < l)%Z ->
  ((m (shift p i)) = Init.Datatypes.true).

Parameter set_init:
  (addr -> Init.Datatypes.bool) -> addr -> Numbers.BinNums.Z ->
  addr -> Init.Datatypes.bool.

Axiom set_init_access :
  forall (m:addr -> Init.Datatypes.bool), forall (q:addr) (p:addr),
  forall (a:Numbers.BinNums.Z),
  (separated q 1%Z p a -> ((set_init m p a q) = (m q))) /\
  (~ separated q 1%Z p a -> ((set_init m p a q) = Init.Datatypes.true)).

(* Why3 assumption *)
Definition monotonic_init (m1:addr -> Init.Datatypes.bool)
    (m2:addr -> Init.Datatypes.bool) : Prop :=
  forall (p:addr), ((m1 p) = Init.Datatypes.true) ->
  ((m2 p) = Init.Datatypes.true).

Parameter int_of_addr: addr -> Numbers.BinNums.Z.

Parameter addr_of_int: Numbers.BinNums.Z -> addr.

Axiom table : Type.
Parameter table_WhyType : WhyType table.
Existing Instance table_WhyType.

Parameter table_of_base: Numbers.BinNums.Z -> table.

Parameter table_to_offset: table -> Numbers.BinNums.Z -> Numbers.BinNums.Z.

Axiom table_to_offset_zero :
  forall (t:table), ((table_to_offset t 0%Z) = 0%Z).

Axiom table_to_offset_monotonic :
  forall (t:table), forall (o1:Numbers.BinNums.Z) (o2:Numbers.BinNums.Z),
  (o1 <= o2)%Z <-> ((table_to_offset t o1) <= (table_to_offset t o2))%Z.

Axiom int_of_addr_bijection :
  forall (a:Numbers.BinNums.Z), ((int_of_addr (addr_of_int a)) = a).

Axiom addr_of_int_bijection :
  forall (p:addr), ((addr_of_int (int_of_addr p)) = p).

Axiom addr_of_null : ((int_of_addr null) = 0%Z).

(* Why3 assumption *)
Definition L_l_RMat_of_FloatQuat (Mf32:addr -> Reals.Rdefinitions.R)
    (q:addr) : S12_RealRMat_s :=
  let r := Mf32 (shift q 2%Z) in
  let r1 := (r * r)%R in
  let r2 := ((-1%R)%R * r1)%R in
  let r3 := Mf32 (shift q 3%Z) in
  let r4 := (r3 * r3)%R in
  let r5 := ((-1%R)%R * r4)%R in
  let r6 := Mf32 (shift q 0%Z) in
  let r7 := (r6 * r6)%R in
  let r8 := Mf32 (shift q 1%Z) in
  let r9 := (r8 * r8)%R in
  let r10 := (r6 * r3)%R in
  let r11 := (r8 * r)%R in
  let r12 := (r6 * r)%R in
  let r13 := (r8 * r3)%R in
  let r14 := ((-1%R)%R * r9)%R in
  let r15 := (r6 * r8)%R in
  let r16 := (r * r3)%R in
  S12_RealRMat_s1 (((r2 + r5)%R + r7)%R + r9)%R
  (2%R * (((-1%R)%R * r10)%R + r11)%R)%R (2%R * (r12 + r13)%R)%R
  (2%R * (r10 + r11)%R)%R (((r14 + r5)%R + r7)%R + r1)%R
  (2%R * (((-1%R)%R * r15)%R + r16)%R)%R
  (2%R * (((-1%R)%R * r12)%R + r13)%R)%R (2%R * (r15 + r16)%R)%R
  (((r14 + r2)%R + r7)%R + r4)%R.

(* Why3 assumption *)
Definition L_l_RMat_of_FloatQuat_bis_1 (Mf32:addr -> Reals.Rdefinitions.R)
    (q:addr) : S12_RealRMat_s :=
  let r := Mf32 (shift q 2%Z) in
  let r1 := (r * r)%R in
  let r2 := Mf32 (shift q 3%Z) in
  let r3 := (r2 * r2)%R in
  let r4 := Mf32 (shift q 0%Z) in
  let r5 := (r4 * r2)%R in
  let r6 := Mf32 (shift q 1%Z) in
  let r7 := (r6 * r)%R in
  let r8 := (r4 * r)%R in
  let r9 := (r6 * r2)%R in
  let r10 := (r6 * r6)%R in
  let r11 := (r4 * r6)%R in
  let r12 := (r * r2)%R in
  S12_RealRMat_s1 (1%R + ((-1%R)%R * (2%R * (r1 + r3)%R)%R)%R)%R
  (2%R * (((-1%R)%R * r5)%R + r7)%R)%R (2%R * (r8 + r9)%R)%R
  (2%R * (r5 + r7)%R)%R (1%R + ((-1%R)%R * (2%R * (r10 + r3)%R)%R)%R)%R
  (2%R * (((-1%R)%R * r11)%R + r12)%R)%R (2%R * (((-1%R)%R * r8)%R + r9)%R)%R
  (2%R * (r11 + r12)%R)%R (1%R + ((-1%R)%R * (2%R * (r10 + r1)%R)%R)%R)%R.

(* Why3 assumption *)
Definition P_unary_quaterion (Mf32:addr -> Reals.Rdefinitions.R) (q:addr) :
    Prop :=
  let r := Mf32 (shift q 0%Z) in
  let r1 := Mf32 (shift q 1%Z) in
  let r2 := Mf32 (shift q 2%Z) in
  let r3 := Mf32 (shift q 3%Z) in
  (((((r * r)%R + (r1 * r1)%R)%R + (r2 * r2)%R)%R + (r3 * r3)%R)%R = 1%R).

(* Why3 assumption *)
Definition L_l_RMat_of_FloatQuat_bis_2 (Mf32:addr -> Reals.Rdefinitions.R)
    (q:addr) : S12_RealRMat_s :=
  let r := Mf32 (shift q 0%Z) in
  let r1 := (r * r)%R in
  let r2 := Mf32 (shift q 1%Z) in
  let r3 := Mf32 (shift q 3%Z) in
  let r4 := (r * r3)%R in
  let r5 := Mf32 (shift q 2%Z) in
  let r6 := (r2 * r5)%R in
  let r7 := (r * r5)%R in
  let r8 := (r2 * r3)%R in
  let r9 := (r * r2)%R in
  let r10 := (r5 * r3)%R in
  S12_RealRMat_s1 ((-1%R)%R + (2%R * (r1 + (r2 * r2)%R)%R)%R)%R
  (2%R * (((-1%R)%R * r4)%R + r6)%R)%R (2%R * (r7 + r8)%R)%R
  (2%R * (r4 + r6)%R)%R ((-1%R)%R + (2%R * (r1 + (r5 * r5)%R)%R)%R)%R
  (2%R * (((-1%R)%R * r9)%R + r10)%R)%R (2%R * (((-1%R)%R * r7)%R + r8)%R)%R
  (2%R * (r9 + r10)%R)%R ((-1%R)%R + (2%R * (r1 + (r3 * r3)%R)%R)%R)%R.

(* Why3 assumption *)
Definition L_mult_scalar (rmat:S12_RealRMat_s) (a:Numbers.BinNums.Z) :
    S12_RealRMat_s :=
  let r := real_of_int a in
  S12_RealRMat_s1 (r * (F12_RealRMat_s_a00 rmat))%R
  (r * (F12_RealRMat_s_a01 rmat))%R (r * (F12_RealRMat_s_a02 rmat))%R
  (r * (F12_RealRMat_s_a10 rmat))%R (r * (F12_RealRMat_s_a11 rmat))%R
  (r * (F12_RealRMat_s_a12 rmat))%R (r * (F12_RealRMat_s_a20 rmat))%R
  (r * (F12_RealRMat_s_a21 rmat))%R (r * (F12_RealRMat_s_a22 rmat))%R.

(* Why3 assumption *)
Definition L_transpose (rmat:S12_RealRMat_s) : S12_RealRMat_s :=
  S12_RealRMat_s1 (F12_RealRMat_s_a00 rmat) (F12_RealRMat_s_a10 rmat)
  (F12_RealRMat_s_a20 rmat) (F12_RealRMat_s_a01 rmat)
  (F12_RealRMat_s_a11 rmat) (F12_RealRMat_s_a21 rmat)
  (F12_RealRMat_s_a02 rmat) (F12_RealRMat_s_a12 rmat)
  (F12_RealRMat_s_a22 rmat).

Axiom Q_mult_id_rmat_neutral :
  forall (rmat:S12_RealRMat_s),
  EqS12_RealRMat_s
  (L_mult_RealRMat rmat
   (S12_RealRMat_s1 1%R 0%R 0%R 0%R 1%R 0%R 0%R 0%R 1%R))
  rmat.

Axiom Q_mult_rmat_id_neutral :
  forall (rmat:S12_RealRMat_s),
  EqS12_RealRMat_s
  (L_mult_RealRMat (S12_RealRMat_s1 1%R 0%R 0%R 0%R 1%R 0%R 0%R 0%R 1%R)
   rmat)
  rmat.

Axiom Q_mutliple_def_rmat_of_quat_1 :
  forall (Malloc:Numbers.BinNums.Z -> Numbers.BinNums.Z)
    (Mf32:addr -> Reals.Rdefinitions.R) (q:addr),
  P_unary_quaterion Mf32 q -> valid_rd Malloc q 4%Z ->
  EqS12_RealRMat_s (L_l_RMat_of_FloatQuat Mf32 q)
  (L_l_RMat_of_FloatQuat_bis_1 Mf32 q).

Axiom Q_mutliple_def_rmat_of_quat_2 :
  forall (Malloc:Numbers.BinNums.Z -> Numbers.BinNums.Z)
    (Mf32:addr -> Reals.Rdefinitions.R) (q:addr),
  P_unary_quaterion Mf32 q -> valid_rd Malloc q 4%Z ->
  EqS12_RealRMat_s (L_l_RMat_of_FloatQuat Mf32 q)
  (L_l_RMat_of_FloatQuat_bis_2 Mf32 q).

Axiom Q_mutliple_def_rmat_of_quat_3 :
  forall (Malloc:Numbers.BinNums.Z -> Numbers.BinNums.Z)
    (Mf32:addr -> Reals.Rdefinitions.R) (q:addr),
  P_unary_quaterion Mf32 q -> valid_rd Malloc q 4%Z ->
  EqS12_RealRMat_s (L_l_RMat_of_FloatQuat_bis_1 Mf32 q)
  (L_l_RMat_of_FloatQuat_bis_2 Mf32 q).

Axiom Q_transpose_id :
  forall (a:Numbers.BinNums.Z),
  let a1 :=
  L_mult_scalar (S12_RealRMat_s1 1%R 0%R 0%R 0%R 1%R 0%R 0%R 0%R 1%R) a in
  EqS12_RealRMat_s (L_transpose a1) a1.

Axiom Q_transpose_transpose_rmat :
  forall (rmat:S12_RealRMat_s),
  EqS12_RealRMat_s (L_transpose (L_transpose rmat)) rmat.

Axiom Q_mul_sqrt_float :
  forall (x:Reals.Rdefinitions.R),
  let r := Reals.R_sqrt.sqrt x in (0%R <= x)%R -> ((r * r)%R = x).

Lemma sin_cos_sqr:
  forall r,
  (Rpow_def.pow (Rtrigo_def.sin r) 2 + Rpow_def.pow (Rtrigo_def.cos r) 2)%R = 1%R.
Proof.
  intros r.
  simpl.
  repeat rewrite Rmult_comm with (r2:=1%R); repeat rewrite Rmult_1_l. 
  rewrite Q_cos_sin_square.
  auto.
Qed.


Lemma cos_sin_sqr:
  forall r,
  (Rpow_def.pow (Rtrigo_def.cos r) 2 + Rpow_def.pow (Rtrigo_def.sin r) 2)%R = 1%R.
Proof.
  intros r.
  rewrite Rplus_comm.
  apply sin_cos_sqr.
Qed.

Lemma sin_to_cos_sqr:
  forall r,
  Rpow_def.pow (Rtrigo_def.sin r) 2 = (1%R - Rpow_def.pow (Rtrigo_def.cos r) 2)%R.
 Proof.
  intros r.
  ring_simplify.
  rewrite <- Q_cos_sin_square with (a := r).
  ring_simplify.
  auto.
Qed.

(* Why3 goal *)
Theorem wp_goal :
  forall (r:Reals.Rdefinitions.R) (r1:Reals.Rdefinitions.R)
    (r2:Reals.Rdefinitions.R),
  let r3 := Reals.Rtrigo_def.sin r in
  let r4 := Reals.Rtrigo_def.cos r1 in
  let r5 := Reals.Rtrigo_def.sin r2 in
  let r6 := Reals.Rtrigo_def.cos r in
  let r7 := Reals.Rtrigo_def.sin r1 in
  let r8 := Reals.Rtrigo_def.cos r2 in
  let r9 := (((-1%R)%R * (r5 * r6)%R)%R + ((r3 * r7)%R * r8)%R)%R in
  let r10 := ((r6 * r8)%R + ((r3 * r7)%R * r5)%R)%R in
  ((((((r3 * r3)%R * r4)%R * r4)%R + (r9 * r9)%R)%R + (r10 * r10)%R)%R = 1%R).
(* Why3 intros r r1 r2 r3 r4 r5 r6 r7 r8 r9 r10. *)
Proof.
  intros r r1 r2 r3 r4 r5 r6 r7 r8 r9 r10.
  unfold r9, r10, r3, r4, r5, r6, r7, r8.
  ring_simplify.
  repeat rewrite sin_to_cos_sqr.
  ring_simplify.
  auto.
Qed.