formal/hol/Examples/lagrange_lemma.ml

(* ========================================================================= *)
(* Nice test for ring procedure and ordered rewriting: Lagrange lemma.       *)
(* ========================================================================= *)

prioritize_real();;

(* ------------------------------------------------------------------------- *)
(* Do the problems the (relatively) efficient way using the normalizer.      *)
(* ------------------------------------------------------------------------- *)

let LAGRANGE_4 = time prove
 (`((x1 pow 2) + (x2 pow 2) + (x3 pow 2) + (x4 pow 2)) *
   ((y1 pow 2) + (y2 pow 2) + (y3 pow 2) + (y4 pow 2)) =
     (((((x1*y1) - (x2*y2)) - (x3*y3)) - (x4*y4)) pow 2)  +
     (((((x1*y2) + (x2*y1)) + (x3*y4)) - (x4*y3)) pow 2)  +
     (((((x1*y3) - (x2*y4)) + (x3*y1)) + (x4*y2)) pow 2)  +
     (((((x1*y4) + (x2*y3)) - (x3*y2)) + (x4*y1)) pow 2)`,
  CONV_TAC REAL_RING);;

let LAGRANGE_8 = time prove
 (`(p1 pow 2 + q1 pow 2 + r1 pow 2 + s1 pow 2 + t1 pow 2 + u1 pow 2 + v1 pow 2 + w1 pow 2) *
   (p2 pow 2 + q2 pow 2 + r2 pow 2 + s2 pow 2 + t2 pow 2 + u2 pow 2 + v2 pow 2 + w2 pow 2)
   = (p1 * p2 - q1 * q2 - r1 * r2 - s1 * s2 - t1 * t2 - u1 * u2 - v1 * v2 - w1 * w2) pow 2 +
     (p1 * q2 + q1 * p2 + r1 * s2 - s1 * r2 + t1 * u2 - u1 * t2 - v1 * w2 + w1 * v2) pow 2 +
     (p1 * r2 - q1 * s2 + r1 * p2 + s1 * q2 + t1 * v2 + u1 * w2 - v1 * t2 - w1 * u2) pow 2 +
     (p1 * s2 + q1 * r2 - r1 * q2 + s1 * p2 + t1 * w2 - u1 * v2 + v1 * u2 - w1 * t2) pow 2 +
     (p1 * t2 - q1 * u2 - r1 * v2 - s1 * w2 + t1 * p2 + u1 * q2 + v1 * r2 + w1 * s2) pow 2 +
     (p1 * u2 + q1 * t2 - r1 * w2 + s1 * v2 - t1 * q2 + u1 * p2 - v1 * s2 + w1 * r2) pow 2 +
     (p1 * v2 + q1 * w2 + r1 * t2 - s1 * u2 - t1 * r2 + u1 * s2 + v1 * p2 - w1 * q2) pow 2 +
     (p1 * w2 - q1 * v2 + r1 * u2 + s1 * t2 - t1 * s2 - u1 * r2 + v1 * q2 + w1 * p2) pow 2`,
  CONV_TAC REAL_RING);;

(* ------------------------------------------------------------------------- *)
(* Or we can just use REAL_ARITH, which is also reasonably fast.             *)
(* ------------------------------------------------------------------------- *)

let LAGRANGE_4 = time prove
 (`((x1 pow 2) + (x2 pow 2) + (x3 pow 2) + (x4 pow 2)) *
   ((y1 pow 2) + (y2 pow 2) + (y3 pow 2) + (y4 pow 2)) =
     (((((x1*y1) - (x2*y2)) - (x3*y3)) - (x4*y4)) pow 2)  +
     (((((x1*y2) + (x2*y1)) + (x3*y4)) - (x4*y3)) pow 2)  +
     (((((x1*y3) - (x2*y4)) + (x3*y1)) + (x4*y2)) pow 2)  +
     (((((x1*y4) + (x2*y3)) - (x3*y2)) + (x4*y1)) pow 2)`,
  REAL_ARITH_TAC);;

let LAGRANGE_8 = time prove
 (`(p1 pow 2 + q1 pow 2 + r1 pow 2 + s1 pow 2 + t1 pow 2 + u1 pow 2 + v1 pow 2 + w1 pow 2) *
   (p2 pow 2 + q2 pow 2 + r2 pow 2 + s2 pow 2 + t2 pow 2 + u2 pow 2 + v2 pow 2 + w2 pow 2)
   = (p1 * p2 - q1 * q2 - r1 * r2 - s1 * s2 - t1 * t2 - u1 * u2 - v1 * v2 - w1 * w2) pow 2 +
     (p1 * q2 + q1 * p2 + r1 * s2 - s1 * r2 + t1 * u2 - u1 * t2 - v1 * w2 + w1 * v2) pow 2 +
     (p1 * r2 - q1 * s2 + r1 * p2 + s1 * q2 + t1 * v2 + u1 * w2 - v1 * t2 - w1 * u2) pow 2 +
     (p1 * s2 + q1 * r2 - r1 * q2 + s1 * p2 + t1 * w2 - u1 * v2 + v1 * u2 - w1 * t2) pow 2 +
     (p1 * t2 - q1 * u2 - r1 * v2 - s1 * w2 + t1 * p2 + u1 * q2 + v1 * r2 + w1 * s2) pow 2 +
     (p1 * u2 + q1 * t2 - r1 * w2 + s1 * v2 - t1 * q2 + u1 * p2 - v1 * s2 + w1 * r2) pow 2 +
     (p1 * v2 + q1 * w2 + r1 * t2 - s1 * u2 - t1 * r2 + u1 * s2 + v1 * p2 - w1 * q2) pow 2 +
     (p1 * w2 - q1 * v2 + r1 * u2 + s1 * t2 - t1 * s2 - u1 * r2 + v1 * q2 + w1 * p2) pow 2`,
  REAL_ARITH_TAC);;

(* ------------------------------------------------------------------------- *)
(* But they can be done (slowly) simply by ordered rewriting.                *)
(* ------------------------------------------------------------------------- *)

let LAGRANGE_4 = time prove
 (`((x1 pow 2) + (x2 pow 2) + (x3 pow 2) + (x4 pow 2)) *
   ((y1 pow 2) + (y2 pow 2) + (y3 pow 2) + (y4 pow 2)) =
     (((((x1*y1) - (x2*y2)) - (x3*y3)) - (x4*y4)) pow 2)  +
     (((((x1*y2) + (x2*y1)) + (x3*y4)) - (x4*y3)) pow 2)  +
     (((((x1*y3) - (x2*y4)) + (x3*y1)) + (x4*y2)) pow 2)  +
     (((((x1*y4) + (x2*y3)) - (x3*y2)) + (x4*y1)) pow 2)`,
  REWRITE_TAC[REAL_POW_2; REAL_ADD_LDISTRIB; REAL_ADD_RDISTRIB;
              REAL_SUB_LDISTRIB; REAL_SUB_RDISTRIB;
              REAL_ARITH `a + (b - c) = (a + b) - c`;
              REAL_ARITH `a - (b - c) = a + (c - b)`;
              REAL_ARITH `(a - b) + c = (a + c) - b`;
              REAL_ARITH `(a - b) - c = a - (b + c)`;
              REAL_ARITH `(a - b = c) = (a = b + c)`;
              REAL_ARITH `(a = b - c) = (a + c = b)`;
              REAL_ADD_AC; REAL_MUL_AC]);;

let LAGRANGE_8 = time prove
 (`(p1 pow 2 + q1 pow 2 + r1 pow 2 + s1 pow 2 + t1 pow 2 + u1 pow 2 + v1 pow 2 + w1 pow 2) *
   (p2 pow 2 + q2 pow 2 + r2 pow 2 + s2 pow 2 + t2 pow 2 + u2 pow 2 + v2 pow 2 + w2 pow 2)
   = (p1 * p2 - q1 * q2 - r1 * r2 - s1 * s2 - t1 * t2 - u1 * u2 - v1 * v2 - w1 * w2) pow 2 +
     (p1 * q2 + q1 * p2 + r1 * s2 - s1 * r2 + t1 * u2 - u1 * t2 - v1 * w2 + w1 * v2) pow 2 +
     (p1 * r2 - q1 * s2 + r1 * p2 + s1 * q2 + t1 * v2 + u1 * w2 - v1 * t2 - w1 * u2) pow 2 +
     (p1 * s2 + q1 * r2 - r1 * q2 + s1 * p2 + t1 * w2 - u1 * v2 + v1 * u2 - w1 * t2) pow 2 +
     (p1 * t2 - q1 * u2 - r1 * v2 - s1 * w2 + t1 * p2 + u1 * q2 + v1 * r2 + w1 * s2) pow 2 +
     (p1 * u2 + q1 * t2 - r1 * w2 + s1 * v2 - t1 * q2 + u1 * p2 - v1 * s2 + w1 * r2) pow 2 +
     (p1 * v2 + q1 * w2 + r1 * t2 - s1 * u2 - t1 * r2 + u1 * s2 + v1 * p2 - w1 * q2) pow 2 +
     (p1 * w2 - q1 * v2 + r1 * u2 + s1 * t2 - t1 * s2 - u1 * r2 + v1 * q2 + w1 * p2) pow 2`,
  REWRITE_TAC[REAL_POW_2; REAL_ADD_LDISTRIB; REAL_ADD_RDISTRIB;
              REAL_SUB_LDISTRIB; REAL_SUB_RDISTRIB;
              REAL_ARITH `a + (b - c) = (a + b) - c`;
              REAL_ARITH `a - (b - c) = a + (c - b)`;
              REAL_ARITH `(a - b) + c = (a + c) - b`;
              REAL_ARITH `(a - b) - c = a - (b + c)`;
              REAL_ARITH `(a - b = c) = (a = b + c)`;
              REAL_ARITH `(a = b - c) = (a + c = b)`;
              REAL_ADD_AC; REAL_MUL_AC]);;