Require Import Definitions.
Require Import LemPrelim.
Require Import AdderProof.



Lemma Invariant_t_O :
 BV_to_nat (appbv (stripbv (R1 0) size) (R2 0)) =
 BV_to_nat V2 × BV_to_nat (truncbv V1 0).
intros. simpl in |- ×.
rewrite BV_to_nat_app2. rewrite (trunc_O bool). rewrite <- length_V1_size.
rewrite (strip_all bool). rewrite length_V1_size.
simpl in |- ×. elim plus_n_O. elim mult_n_O. rewrite BV_null_nat. trivial with v62.
Qed.


Lemma inv_ind_false :
 ∀ 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 false (abit (BV_full_adder_sum (R2 n) V2 false) 0)
               (consbv (lowbit (R2 n)) nilbv))) (size - S n))
      (appbv (highs (Mux false (BV_full_adder_sum (R2 n) V2 false) (R2 n)))
         (Mux false (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 false nilbv)).
intros.
simpl in |- ×.
do 3 rewrite BV_to_nat_app2.
simpl in |- ×.
elim mult_n_O.
elim mult_n_O.
elim plus_n_O.
elim plus_n_O.
rewrite (length_strip bool).
rewrite <- H.
clear H.
rewrite (length_app bool).
rewrite highs_is_strip.
rewrite (length_strip bool).
rewrite length_R1.
simpl in |- ×.
replace (size - 1 + 1) with size.
rewrite minus_minus_lem2.
rewrite (strip_app bool).
rewrite BV_to_nat_app2.
rewrite (strip_strip bool).
rewrite (length_strip bool).
rewrite length_R1.
rewrite (length_strip bool).
rewrite BV_to_nat_app2.
rewrite (length_strip bool).
rewrite length_R1.
rewrite minus_minus_lem2.
replace (1 + (size - S n)) with (size - n).
rewrite minus_minus_lem2.
rewrite (minus_le_O (size - S n) (size - 1)).
rewrite (strip_O bool).
rewrite plus_assoc_reverse.
replace
 (power2 n × BV_to_nat (consbv (lowbit (R2 n)) nilbv) +
  power2 (S n) × BV_to_nat (highs (R2 n))) with (power2 n × BV_to_nat (R2 n)).
trivial with v62. symmetry in |- ×. apply BV_nat_lem1.
rewrite length_R2. exact size_not_O. exact H1. apply le_minus_minus; auto with v62.
exact H1. simpl in |- ×. rewrite minus_Sn_m. auto with v62.
exact H0. exact H1. exact H1. rewrite length_R1. auto with v62. exact H1.
rewrite length_R1. auto with v62. exact H1. exact H1. rewrite length_R1.
simpl in |- ×. rewrite minus_Sn_m. simpl in |- ×. auto with v62. exact H0. exact H1.
exact H0. rewrite plus_comm. simpl in |- ×. rewrite minus_Sn_m; auto with v62.
 exact H1. rewrite length_R1. auto with v62. exact H1. exact H1.
rewrite (length_app bool). rewrite length_highs. rewrite length_R1.
simpl in |- ×. rewrite plus_n_SO. replace (pred size) with (size - 1).
rewrite minus_Sn_m. simpl in |- ×. elim minus_n_O. apply minus_le_lem2; auto with v62. auto with v62.
apply minus_n_SO. exact H1. apply (not_nil bool). rewrite length_R1; auto with v62.
Qed.


Lemma inv_ind_true :
 ∀ 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)).

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 v62. 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 v62. unfold BV_full_adder in |- ×. trivial with v62.
simpl in |- ×. elim plus_n_O. trivial with v62. rewrite length_BV_full_adder_sum; auto with v62.
rewrite length_R2; auto with v62. rewrite length_BV_full_adder_sum; auto with v62.
rewrite length_R2; auto with v62. rewrite length_R2; auto with v62.
apply (not_nil bool); auto with v62. rewrite length_BV_full_adder_sum; auto with v62.
 rewrite length_R2; auto with v62. rewrite length_R2; auto with v62.
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 v62.
auto with v62. rewrite plus_comm. simpl in |- ×. rewrite plus_comm. trivial with v62.
simpl in |- ×. trivial with v62. rewrite mult_sym. simpl in |- ×. elim plus_n_O.
trivial with v62. auto with v62. exact H1. exact H1. rewrite length_R1; auto with v62.
 exact H1. apply le_minus_minus. auto with v62. exact H0. exact H1.
rewrite length_R1. auto with v62. exact H1. rewrite length_R1; auto with v62.
exact H1.
rewrite length_R2; auto with v62. exact H0. simpl in |- ×. rewrite minus_Sn_m.
auto with v62. auto with v62. exact H1. rewrite length_BV_full_adder_sum; auto with v62.
unfold lt in |- ×. rewrite length_R2. auto with v62. exact H1. transitivity size; auto with v62.
rewrite length_BV_full_adder_sum; auto with v62. rewrite length_R2; auto with v62.
 rewrite length_R2; auto with v62. rewrite length_R1; auto with v62.
rewrite (length_app bool). rewrite (length_strip bool).
rewrite length_R1; auto with v62. rewrite length_abit. rewrite plus_comm.
simpl in |- ×. rewrite minus_Sn_m. simpl in |- ×. elim minus_n_O. apply minus_le_lem2. auto with v62.
rewrite length_BV_full_adder_sum; auto with v62. rewrite length_R2; auto with v62.
 rewrite length_R2; auto with v62. rewrite length_R1; auto with v62.
replace (length V1) with (lengthbv V1); auto with v62.
rewrite length_V1_size. exact H1. exact H1.

Qed.


Lemma Invariant :
 ∀ t : nat,
 t ≤ size →
 BV_to_nat (appbv (stripbv (R1 t) (size - t)) (R2 t)) =
 BV_to_nat V2 × BV_to_nat (truncbv V1 t).

simple induction t. intros. elim minus_n_O. apply Invariant_t_O.
intros. rewrite <- (app_trunc_elemlist bool).
replace (elemlist bool V1 n) with (abit V1 n).
rewrite <- (R1_lem3 n). rewrite R1_eq2. rewrite R2_eq2.
rewrite <- lowbit_is_abit. case (lowbit (R1 n)).
replace (consbv (lowbit (BV_full_adder_sum (R2 n) V2 false)) nilbv) with
 (abit (BV_full_adder_sum (R2 n) V2 false) 0).
apply inv_ind_true. auto with v62.
intros. apply H. exact H1. exact H0.
apply le_Sn_le; exact H0. unfold abit in |- ×. rewrite lowbit_is_trunc.
unfold elemlist in |- ×. rewrite (strip_O bool). trivial with v62.
apply (not_nil bool). rewrite length_BV_full_adder_sum. rewrite length_R2.
exact size_not_O. apply le_Sn_le; exact H0.
rewrite length_R2. rewrite length_V2_size. trivial with v62.
apply le_Sn_le; exact H0.
replace (consbv (lowbit (BV_full_adder_sum (R2 n) V2 false)) nilbv) with
 (abit (BV_full_adder_sum (R2 n) V2 false) 0).
apply inv_ind_false. trivial with v62.
exact H. exact H0. apply le_Sn_le; exact H0. rewrite lowbit_is_trunc.
unfold abit in |- ×. unfold elemlist in |- ×. rewrite (strip_O bool). trivial with v62.
apply (not_nil bool).
rewrite length_BV_full_adder_sum. rewrite length_R2; auto with v62.
transitivity size.
rewrite length_R2; trivial with v62. apply le_Sn_le; exact H0. auto with v62.
apply (not_nil bool). rewrite length_R1; auto with v62. auto with v62. auto with v62.
replace (length V1) with (lengthbv V1). rewrite length_V1_size. exact H0.
auto with v62.

Qed.


Theorem Correct :
 BV_to_nat (appbv (R1 size) (R2 size)) = BV_to_nat V2 × BV_to_nat V1.

intros. rewrite <- (strip_O bool (R1 size)).
replace (BV_to_nat V1) with (BV_to_nat (truncbv V1 size)).
rewrite (minus_n_n size). apply Invariant. auto with v62.
rewrite <- length_V1_size. rewrite (trunc_all bool). trivial with v62.
Qed.