Prove necessary lemma to get multiplication in structure similar to Coq, start working on proof

hol/Infra/RealSimps.hl

+
+let REAL_MUL_LE_COMPAT_NEG_L =
+  prove (`!(a:real) b c.
+           a <= &0 /\ b <= c ==>
+           a * c <= a * b`,
+         intros "!a b c; a_leq_0 b_leq_c"
+           THEN ONCE_REWRITE_TAC [SYM (SPEC `a:real` REAL_NEG_NEG)]
+           THEN ONCE_REWRITE_TAC [SYM (SPECL [`a:real`; `c:real`] REAL_MUL_LNEG)]
+           THEN ONCE_REWRITE_TAC [REAL_LE_NEG]
+           THEN MATCH_MP_TAC REAL_LE_LMUL
+           THEN split
+           THENL [
+             ONCE_REWRITE_TAC [SYM (SPEC `-- (a:real)` REAL_NEG_LE0)]
+               THEN ONCE_REWRITE_TAC [REAL_NEG_NEG]
+               THEN ASM_SIMP_TAC[];
+             ASM_SIMP_TAC[] ]);;

hol/IntervalArith.hl

@@ -3,6 +3,7 @@
 *)
 needs "Infra/tactics.hl";;
 needs "Infra/Abbrevs.hl";;
+needs "Infra/RealSimps.hl";;
 needs "Expressions.hl";;
 
 let valid = define
@@ -203,16 +204,31 @@ g `!(iv1:interval) (iv2:interval) (a:real) (b:real).
 e (SIMP_TAC[contained; multInterval; absIntvUpd; IVlo; IVhi]);;
 e (intros "!a b; contained_a lo_leq_b b_leq_hi");;
 e (destruct "contained_a" "lo_leq_a a_leq_hi");;
-e (LABEL_TAC "min4_correct" (SPECL [`(FST I1 * FST I2):real`; `(FST I1 * SND I2):real`; `(SND I1 * FST I2):real`; `(SND I1 * SND I2):real`] min4_correct));;
-e (LABEL_TAC "max4_correct" (SPECL [`(FST I1 * FST I2):real`; `(FST I1 * SND I2):real`; `(SND I1 * FST I2):real`; `(SND I1 * SND I2):real`] max4_correct));;
+e (LABEL_TAC "min4_correct"
+     (SPECL
+        [`FST (I1:real#real) * FST (I2:real#real)`; `FST (I1:real#real) * SND (I2:real#real)`; `SND (I1:real#real) * FST (I2:real#real)`; `SND (I1:real#real) * SND (I2:real#real)`] min4_correct));;
+e (LABEL_TAC "max4_correct" (SPECL [`FST (I1:real#real) * FST (I2:real#real)`; `FST (I1:real#real) * SND (I2:real#real)`; `SND (I1:real#real) * FST (I2:real#real)`; `SND (I1:real#real) * SND (I2:real#real)`] max4_correct));;
 e (destruct "min4_correct" "leq_lolo leq_lohi leq_hilo leq_hihi");;
 e (destruct "max4_correct" "lolo_leq lohi_leq hilo_leq hihi_leq");;
 e (split);;
 (* Lower Bound *) (* REAL_LE_NEGTOTAL or REAL_LE_NEG??? *)
 e (LABEL_TAC "cases_a" (SPECL [`a:real`; `&0:real`] REAL_LTE_TOTAL));;
 e (destruct "cases_a" "a_lt_0 | 0_leq_a");;
-e (LABEL_TAC "cases_hi2" (SPECL [`(SND I2):real`; `&0:real`] REAL_LTE_TOTAL));;
+e (LABEL_TAC "cases_hi2" (SPECL [`(SND (I2:real#real))`; `&0:real`] REAL_LTE_TOTAL));;
 e (destruct "cases_hi2" "hi2_lt_0 | 0_leq_hi2");;
+e (MATCH_MP_TAC REAL_LE_TRANS);;
+e (EXISTS_TAC `SND (I1:real#real) * SND (I2:real#real)`);;
+e (split);;
+e (ASM_SIMP_TAC[]);;
+e (ONCE_REWRITE_TAC [REAL_MUL_SYM]);;
+e (MATCH_MP_TAC REAL_LE_TRANS);;
+e (EXISTS_TAC `(a:real) * SND (I2:real#real)`);;
+e (split);;
+(** TODO!!! **)
+e (MATCH_MP_TAC (SPECL [`a:real`; `b:real`; `SND (I2:real#real)`] REAL_MUL_LE_COMPAT_NEG_L));;
+SPECL [`a:real`; `b:real`; `SND (I2:real#real)`] REAL_MUL_LE_COMPAT_NEG_L;;
+REAL_MUL_LE_COMPAT_NEG_L;;
+e
 (*
   TODO FIXME!
  e (MATCH_MP_TAC REAL_MULT_LE);;