proposition: Lemma inv_ind_true : forall t n : nat, (n <= size -> BV_to_nat (appbv (stripbv (R1 n) (size - n)) (R2 n)) = BV_to_nat V2 * BV_to_nat (truncbv V1 n)) -> S n <= size -> n <= size -> BV_to_nat (appbv (stripbv (appbv (highs (R1 n)) (Mux true (abit (BV_full_adder_sum (R2 n) V2 false) 0) (consbv (lowbit (R2 n)) nilbv))) (size - S n)) (appbv (highs (Mux true (BV_full_adder_sum (R2 n) V2 false) (R2 n))) (Mux true (consbv (BV_full_adder_carry (R2 n) V2 false) nilbv) (consbv false nilbv)))) = BV_to_nat V2 * BV_to_nat (appbv (truncbv V1 n) (consbv true nilbv)).
proof: Proof. intros. simpl in |- *. do 3 rewrite BV_to_nat_app2. simpl in |- *. rewrite mult_plus_distr2. rewrite (mult_plus_distr2 (BV_to_nat V2) (BV_to_nat (truncbv V1 n)) (power2 (lengthbv (truncbv V1 n)) * 1)). rewrite <- H. clear H. elim plus_n_O. rewrite highs_is_strip. rewrite highs_is_strip. rewrite (length_trunc bool). rewrite (length_strip bool). rewrite (length_app bool). rewrite (length_strip bool). rewrite (length_strip bool). rewrite length_abit. rewrite length_R1. rewrite (plus_comm (size - 1)). replace (1 + (size - 1)) with size. rewrite minus_minus_lem2. rewrite length_BV_full_adder_sum. rewrite length_R2. rewrite (strip_app bool). rewrite BV_to_nat_app2. rewrite (strip_strip bool). rewrite (length_strip bool). rewrite (length_strip bool). rewrite length_R1. rewrite BV_to_nat_app2. simpl in |- *. rewrite minus_Sn_m. rewrite (minus_le_O (size - S n) (size - 1)). simpl in |- *. rewrite minus_minus_lem2. rewrite (strip_O bool). rewrite (length_strip bool). rewrite length_R1. rewrite minus_minus_lem2. replace (power2 n + power2 n) with (power2 (S n)). replace (power2 n * 1) with (power2 n). rewrite <- (mult_assoc_reverse (power2 (S n))). rewrite <- power2_plus. simpl in |- *. rewrite <- power2_eq2. rewrite <- power2_eq2. replace (S (n + (size - 1))) with (n + size). repeat rewrite plus_assoc_reverse. replace (power2 n * BV_to_nat (abit (BV_full_adder_sum (R2 n) V2 false) 0) + (power2 (S n) * BV_to_nat (stripbv (BV_full_adder_sum (R2 n) V2 false) 1) + power2 (n + size) * bool_to_nat (BV_full_adder_carry (R2 n) V2 false))) with (power2 n * BV_to_nat (R2 n) + BV_to_nat V2 * power2 n). trivial with arith. rewrite plus_assoc. rewrite <- highs_is_strip. unfold abit in |- *. unfold elemlist in |- *. rewrite (strip_O bool). rewrite <- lowbit_is_trunc. rewrite BV_nat_lem1. rewrite power2_plus. rewrite (mult_assoc_reverse (power2 n)). rewrite <- mult_plus_distr2. replace size with (lengthbv (BV_full_adder_sum (R2 n) V2 false)). replace (bool_to_nat (BV_full_adder_carry (R2 n) V2 false)) with (BV_to_nat (consbv (BV_full_adder_carry (R2 n) V2 false) nilbv)). rewrite <- BV_to_nat_app2. replace (appbv (BV_full_adder_sum (R2 n) V2 false) (consbv (BV_full_adder_carry (R2 n) V2 false) nilbv)) with (BV_full_adder (R2 n) V2 false). rewrite BV_full_adder_ok. rewrite mult_plus_distr2. rewrite mult_plus_distr2. simpl in |- *. elim mult_n_O. elim plus_n_O. rewrite (mult_sym (BV_to_nat V2)). trivial with arith. unfold BV_full_adder in |- *. trivial with arith. simpl in |- *. elim plus_n_O. trivial with arith. rewrite length_BV_full_adder_sum; auto with arith. rewrite length_R2; auto with arith. rewrite length_BV_full_adder_sum; auto with arith. rewrite length_R2; auto with arith. rewrite length_R2; auto with arith. apply (not_nil bool); auto with arith. rewrite length_BV_full_adder_sum; auto with arith. rewrite length_R2; auto with arith. rewrite length_R2; auto with arith. replace (S (n + (size - 1))) with (S n + (size - 1)). replace (S n + (size - 1)) with (n + S (size - 1)). rewrite minus_Sn_m. simpl in |- *. elim minus_n_O. trivial with arith. auto with arith. rewrite plus_comm. simpl in |- *. rewrite plus_comm. trivial with arith. simpl in |- *. trivial with arith. rewrite mult_sym. simpl in |- *. elim plus_n_O. trivial with arith. auto with arith. exact H1. exact H1. rewrite length_R1; auto with arith. exact H1. apply le_minus_minus. auto with arith. exact H0. exact H1. rewrite length_R1. auto with arith. exact H1. rewrite length_R1; auto with arith. exact H1. rewrite length_R2; auto with arith. exact H0. simpl in |- *. rewrite minus_Sn_m. auto with arith. auto with arith. exact H1. rewrite length_BV_full_adder_sum; auto with arith. unfold lt in |- *. rewrite length_R2. auto with arith. exact H1. transitivity size; auto with arith. rewrite length_BV_full_adder_sum; auto with arith. rewrite length_R2; auto with arith. rewrite length_R2; auto with arith. rewrite length_R1; auto with arith. rewrite (length_app bool). rewrite (length_strip bool). rewrite length_R1; auto with arith. rewrite length_abit. rewrite plus_comm. simpl in |- *. rewrite minus_Sn_m. simpl in |- *. elim minus_n_O. apply minus_le_lem2. auto with arith. rewrite length_BV_full_adder_sum; auto with arith. rewrite length_R2; auto with arith. rewrite length_R2; auto with arith. rewrite length_R1; auto with arith. replace (length V1) with (lengthbv V1); auto with arith. rewrite length_V1_size. exact H1. exact H1. Qed.
imports: Definitions LemPrelim AdderProof