Simplify a coq proof

coq/IntervalValidation.v

@@ -24,33 +24,31 @@ Fixpoint validIntervalbounds (e:exp Q) (absenv:analysisResult) (P:precond):=
   let (intv, _) := absenv e in
     match e with
     | Var _ v => false
-    | Param _ v =>
-      isSupersetIntv (P v) intv
-    | Const n =>
-      isSupersetIntv (n,n) intv
-    | Unop o f1 =>
-    let rec := validIntervalbounds f1 absenv P in
-    let (iv1, _) := absenv f1 in
+    | Param _ v => isSupersetIntv (P v) intv
+    | Const n => isSupersetIntv (n,n) intv
+    | Unop o f =>
+    let rec := validIntervalbounds f absenv P in
+    let (iv, _) := absenv f in
     let opres :=
         match o with
         | Neg =>
-          let new_iv := negateIntv iv1 in
+          let new_iv := negateIntv iv in
           isSupersetIntv new_iv intv
         | Inv =>
           let nodiv0 := orb
-                          (andb (Qleb (ivhi iv1) 0) (negb (Qeq_bool (ivhi iv1) 0)))
-                          (andb (Qleb 0 (ivlo iv1)) (negb (Qeq_bool (ivlo iv1) 0))) in
-          let new_iv := invertIntv iv1 in
+                          (andb (Qleb (ivhi iv) 0) (negb (Qeq_bool (ivhi iv) 0)))
+                          (andb (Qleb 0 (ivlo iv)) (negb (Qeq_bool (ivlo iv) 0))) in
+          let new_iv := invertIntv iv in
           andb (isSupersetIntv new_iv intv) nodiv0
         end
     in
     andb rec opres
-    | Binop b e1 e2 =>
-      let rec := andb (validIntervalbounds e1 absenv P) (validIntervalbounds e2 absenv P) in
-      let (iv1,_) := absenv e1 in
-      let (iv2,_) := absenv e2 in
+    | Binop op f1 f2 =>
+      let rec := andb (validIntervalbounds f1 absenv P) (validIntervalbounds f2 absenv P) in
+      let (iv1,_) := absenv f1 in
+      let (iv2,_) := absenv f2 in
       let opres :=
-          match b with
+          match op with
           | Plus =>
             let new_iv := addIntv iv1 iv2 in
             isSupersetIntv new_iv intv
@@ -79,11 +77,9 @@ Proof.
   induction f; unfold validIntervalbounds.
   - intros approx_true v v_in_fV; simpl in *; contradiction.
   - intros approx_true v v_in_fV; simpl in *.
-    unfold isSupersetIntv.
     destruct v_in_fV; try contradiction.
     subst.
-    destruct (P v); destruct (absenv (Param Q v)); simpl in *.
-    destruct i; simpl in *.
+    destruct (absenv (Param Q v)); simpl in *.
     apply Is_true_eq_left in approx_true; auto.
   - intros approx_true v0 v_in_fV; simpl in *; contradiction.
   - intros approx_unary_true v v_in_fV.
@@ -152,7 +148,6 @@ Theorem validIntervalbounds_sound (f:exp Q) (absenv:analysisResult) (P:precond)
 Proof.
   induction f.
   - intros vR valid_bounds eval_f.
-    pose proof (ivbounds_approximatesPrecond_sound (Var Q n) absenv P valid_bounds) as env_approx_p.
     unfold validIntervalbounds in valid_bounds.
     destruct (absenv (Var Q n)); inversion valid_bounds.
   - intros vR valid_bounds eval_f.
@@ -181,7 +176,6 @@ Proof.
     apply Qle_Rle in abslo_le_ivlo; apply Qle_Rle in ivhi_le_abshi.
     split; lra.
   - intros vR valid_bounds eval_f.
-    pose proof (ivbounds_approximatesPrecond_sound (Const v) absenv P valid_bounds) as env_approx_p.
     unfold validIntervalbounds in valid_bounds.
     destruct (absenv (Const v)) as [intv err]; simpl.
     apply Is_true_eq_left in valid_bounds.
@@ -199,8 +193,7 @@ Proof.
       unfold Qleb in *.
       apply Qle_bool_iff in valid_hi.
       apply Qle_Rle in valid_hi; auto.
-  - intros vR valid_bounds eval_f;
-      pose proof (ivbounds_approximatesPrecond_sound (Unop u f) absenv P valid_bounds) as env_approx_p.
+  - intros vR valid_bounds eval_f.
     case_eq (absenv (Unop u f)); intros intv err absenv_unop.
     destruct intv as [unop_lo unop_hi]; simpl.
     unfold validIntervalbounds in valid_bounds.
@@ -299,7 +292,6 @@ Proof.
              apply Rlt_Qlt in nodiv0_pos; apply Rle_Qle in H2; lra.
          }
   - intros vR valid_bounds eval_f; inversion eval_f; subst.
-    pose proof (ivbounds_approximatesPrecond_sound (Binop b f1 f2) absenv P valid_bounds) as env_approx_p.
     rewrite delta_0_deterministic in eval_f; auto.
     rewrite delta_0_deterministic; auto.
     simpl in valid_bounds.