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finish all adder proof for mpn_add_n

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ZhuangYumin
2025-06-22 07:37:10 +00:00
parent 0fdf4fc328
commit fd26d9669e
2 changed files with 289 additions and 4 deletions
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projects/lib/gmp_proof_manual.v
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@ -932,13 +932,298 @@ Proof.
Qed.
Lemma proof_of_mpn_add_n_entail_wit_3_2 : mpn_add_n_entail_wit_3_2.
Proof. Admitted.
Proof.
pre_process.
rewrite replace_Znth_app_r.
assert (l_a_3 = l_a_2). {
pose proof (list_store_Z_compact_reverse_injection l_a_3 l_a_2 val_a val_a).
specialize (H37 H13 H28).
apply H37.
reflexivity.
}
subst l_a_3.
assert (l_b_3 = l_b_2). {
pose proof (list_store_Z_compact_reverse_injection l_b_3 l_b_2 val_b val_b).
specialize (H37 H14 H24).
apply H37.
reflexivity.
}
subst l_b_3.
- Exists l_r_suffix'.
rewrite H29.
rewrite H18.
assert (i - i = 0) by lia.
rewrite H37; clear H37.
set (partial_result_1 := (unsigned_last_nbits (Znth i l_a_2 0 + cy) 32)).
set (partial_result_2 := (unsigned_last_nbits (partial_result_1 + Znth i l_b_2 0) 32)).
rewrite replace_Znth_nothing; try lia.
assert ((replace_Znth 0 partial_result_2 (a :: nil)) = partial_result_2 :: nil). {
unfold replace_Znth.
simpl.
reflexivity.
}
rewrite H37; clear H37.
Exists (l_r_prefix_2 ++ partial_result_2 :: nil).
Exists (val_r_prefix_2 + partial_result_2 * (UINT_MOD ^ i)).
Exists (val_b_prefix_2 + (Znth i l_b_2 0) * (UINT_MOD ^ i)).
Exists (val_a_prefix_2 + (Znth i l_a_2 0) * (UINT_MOD ^ i)).
Exists l_b_2 l_a_2.
entailer!.
+ assert ( (val_a_prefix_2 + Znth i l_a_2 0 * 4294967296 ^ i +(val_b_prefix_2 + Znth i l_b_2 0 * 4294967296 ^ i)) = (val_a_prefix_2 + val_b_prefix_2) + Znth i l_a_2 0 * 4294967296 ^ i + Znth i l_b_2 0 * 4294967296 ^ i).
{
lia.
}
rewrite H37; clear H37.
rewrite <- H19.
assert ( (Znth i l_a_2 0) + (Znth i l_b_2 0) + cy = partial_result_2 + UINT_MOD * 2). {
unfold unsigned_last_nbits in H4, H3.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H37 in H4, H3; clear H37.
apply Z_mod_3add_carry11; try lia; try tauto;
try unfold list_store_Z_compact in H13, H14;
try apply list_within_bound_Znth;
try lia;
try tauto.
}
assert ( partial_result_2 * 4294967296 ^ i + (1 + 1) * 4294967296 ^ (i + 1) = cy * 4294967296 ^ i + Znth i l_a_2 0 * 4294967296 ^ i + Znth i l_b_2 0 * 4294967296 ^ i). {
rewrite <- Z.mul_add_distr_r.
rewrite (Zpow_add_1 4294967296 i); try lia.
}
lia.
+ pose proof (Zlength_app l_r_prefix_2 (partial_result_2 :: nil)).
assert (Zlength (partial_result_2 :: nil) = 1). {
unfold Zlength.
simpl.
reflexivity.
}
rewrite H38 in H37; clear H38.
rewrite H18 in H37.
apply H37.
+ pose proof (list_store_Z_concat l_r_prefix_2 (partial_result_2 :: nil) val_r_prefix_2 partial_result_2).
rewrite H18 in H37.
apply H37.
tauto.
unfold list_store_Z.
simpl.
split.
reflexivity.
split.
unfold partial_result_2.
unfold unsigned_last_nbits.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H38; clear H38.
apply Z.mod_pos_bound.
lia.
tauto.
+ pose proof (list_store_Z_list_append l_b_2 i val_b_prefix_2 val_b).
apply H37.
lia.
tauto.
tauto.
+ pose proof (list_store_Z_list_append l_a_2 i val_a_prefix_2 val_a).
apply H37.
lia.
tauto.
tauto.
- pose proof (Zlength_sublist0 i l_r_prefix_2).
lia.
Qed.
Lemma proof_of_mpn_add_n_entail_wit_3_3 : mpn_add_n_entail_wit_3_3.
Proof. Admitted.
Proof.
pre_process.
rewrite replace_Znth_app_r.
assert (l_a_3 = l_a_2). {
pose proof (list_store_Z_compact_reverse_injection l_a_3 l_a_2 val_a val_a).
specialize (H37 H13 H28).
apply H37.
reflexivity.
}
subst l_a_3.
assert (l_b_3 = l_b_2). {
pose proof (list_store_Z_compact_reverse_injection l_b_3 l_b_2 val_b val_b).
specialize (H37 H14 H24).
apply H37.
reflexivity.
}
subst l_b_3.
- Exists l_r_suffix'.
rewrite H29.
rewrite H18.
assert (i - i = 0) by lia.
rewrite H37; clear H37.
set (partial_result_1 := (unsigned_last_nbits (Znth i l_a_2 0 + cy) 32)).
set (partial_result_2 := (unsigned_last_nbits (partial_result_1 + Znth i l_b_2 0) 32)).
rewrite replace_Znth_nothing; try lia.
assert ((replace_Znth 0 partial_result_2 (a :: nil)) = partial_result_2 :: nil). {
unfold replace_Znth.
simpl.
reflexivity.
}
rewrite H37; clear H37.
Exists (l_r_prefix_2 ++ partial_result_2 :: nil).
Exists (val_r_prefix_2 + partial_result_2 * (UINT_MOD ^ i)).
Exists (val_b_prefix_2 + (Znth i l_b_2 0) * (UINT_MOD ^ i)).
Exists (val_a_prefix_2 + (Znth i l_a_2 0) * (UINT_MOD ^ i)).
Exists l_b_2 l_a_2.
entailer!.
+ assert ( (val_a_prefix_2 + Znth i l_a_2 0 * 4294967296 ^ i +(val_b_prefix_2 + Znth i l_b_2 0 * 4294967296 ^ i)) = (val_a_prefix_2 + val_b_prefix_2) + Znth i l_a_2 0 * 4294967296 ^ i + Znth i l_b_2 0 * 4294967296 ^ i).
{
lia.
}
rewrite H37; clear H37.
rewrite <- H19.
assert ( (Znth i l_a_2 0) + (Znth i l_b_2 0) + cy = partial_result_2). {
unfold unsigned_last_nbits in H4, H3.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H37 in H4, H3; clear H37.
apply Z_mod_3add_carry00; try lia; try tauto;
try unfold list_store_Z_compact in H13, H14;
try apply list_within_bound_Znth;
try lia;
try tauto.
}
assert ( partial_result_2 * 4294967296 ^ i + (0 + 0) * 4294967296 ^ (i + 1) = cy * 4294967296 ^ i + Znth i l_a_2 0 * 4294967296 ^ i + Znth i l_b_2 0 * 4294967296 ^ i). {
rewrite <- Z.mul_add_distr_r.
rewrite (Zpow_add_1 4294967296 i); try lia.
}
lia.
+ pose proof (Zlength_app l_r_prefix_2 (partial_result_2 :: nil)).
assert (Zlength (partial_result_2 :: nil) = 1). {
unfold Zlength.
simpl.
reflexivity.
}
rewrite H38 in H37; clear H38.
rewrite H18 in H37.
apply H37.
+ pose proof (list_store_Z_concat l_r_prefix_2 (partial_result_2 :: nil) val_r_prefix_2 partial_result_2).
rewrite H18 in H37.
apply H37.
tauto.
unfold list_store_Z.
simpl.
split.
reflexivity.
split.
unfold partial_result_2.
unfold unsigned_last_nbits.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H38; clear H38.
apply Z.mod_pos_bound.
lia.
tauto.
+ pose proof (list_store_Z_list_append l_b_2 i val_b_prefix_2 val_b).
apply H37.
lia.
tauto.
tauto.
+ pose proof (list_store_Z_list_append l_a_2 i val_a_prefix_2 val_a).
apply H37.
lia.
tauto.
tauto.
- pose proof (Zlength_sublist0 i l_r_prefix_2).
lia.
Qed.
Lemma proof_of_mpn_add_n_entail_wit_3_4 : mpn_add_n_entail_wit_3_4.
Proof. Admitted.
Proof.
pre_process.
rewrite replace_Znth_app_r.
assert (l_a_3 = l_a_2). {
pose proof (list_store_Z_compact_reverse_injection l_a_3 l_a_2 val_a val_a).
specialize (H37 H13 H28).
apply H37.
reflexivity.
}
subst l_a_3.
assert (l_b_3 = l_b_2). {
pose proof (list_store_Z_compact_reverse_injection l_b_3 l_b_2 val_b val_b).
specialize (H37 H14 H24).
apply H37.
reflexivity.
}
subst l_b_3.
- Exists l_r_suffix'.
rewrite H29.
rewrite H18.
assert (i - i = 0) by lia.
rewrite H37; clear H37.
set (partial_result_1 := (unsigned_last_nbits (Znth i l_a_2 0 + cy) 32)).
set (partial_result_2 := (unsigned_last_nbits (partial_result_1 + Znth i l_b_2 0) 32)).
rewrite replace_Znth_nothing; try lia.
assert ((replace_Znth 0 partial_result_2 (a :: nil)) = partial_result_2 :: nil). {
unfold replace_Znth.
simpl.
reflexivity.
}
rewrite H37; clear H37.
Exists (l_r_prefix_2 ++ partial_result_2 :: nil).
Exists (val_r_prefix_2 + partial_result_2 * (UINT_MOD ^ i)).
Exists (val_b_prefix_2 + (Znth i l_b_2 0) * (UINT_MOD ^ i)).
Exists (val_a_prefix_2 + (Znth i l_a_2 0) * (UINT_MOD ^ i)).
Exists l_b_2 l_a_2.
entailer!.
+ assert ( (val_a_prefix_2 + Znth i l_a_2 0 * 4294967296 ^ i +(val_b_prefix_2 + Znth i l_b_2 0 * 4294967296 ^ i)) = (val_a_prefix_2 + val_b_prefix_2) + Znth i l_a_2 0 * 4294967296 ^ i + Znth i l_b_2 0 * 4294967296 ^ i).
{
lia.
}
rewrite H37; clear H37.
rewrite <- H19.
assert ( (Znth i l_a_2 0) + (Znth i l_b_2 0) + cy = partial_result_2 + UINT_MOD). {
unfold unsigned_last_nbits in H4, H3.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H37 in H4, H3; clear H37.
apply Z_mod_3add_carry01; try lia; try tauto;
try unfold list_store_Z_compact in H13, H14;
try apply list_within_bound_Znth;
try lia;
try tauto.
}
assert ( partial_result_2 * 4294967296 ^ i + (0 + 1) * 4294967296 ^ (i + 1) = cy * 4294967296 ^ i + Znth i l_a_2 0 * 4294967296 ^ i + Znth i l_b_2 0 * 4294967296 ^ i). {
rewrite <- Z.mul_add_distr_r.
rewrite (Zpow_add_1 4294967296 i); try lia.
}
lia.
+ pose proof (Zlength_app l_r_prefix_2 (partial_result_2 :: nil)).
assert (Zlength (partial_result_2 :: nil) = 1). {
unfold Zlength.
simpl.
reflexivity.
}
rewrite H38 in H37; clear H38.
rewrite H18 in H37.
apply H37.
+ pose proof (list_store_Z_concat l_r_prefix_2 (partial_result_2 :: nil) val_r_prefix_2 partial_result_2).
rewrite H18 in H37.
apply H37.
tauto.
unfold list_store_Z.
simpl.
split.
reflexivity.
split.
unfold partial_result_2.
unfold unsigned_last_nbits.
assert (2 ^ 32 = 4294967296). { nia. }
rewrite H38; clear H38.
apply Z.mod_pos_bound.
lia.
tauto.
+ pose proof (list_store_Z_list_append l_b_2 i val_b_prefix_2 val_b).
apply H37.
lia.
tauto.
tauto.
+ pose proof (list_store_Z_list_append l_a_2 i val_a_prefix_2 val_a).
apply H37.
lia.
tauto.
tauto.
- pose proof (Zlength_sublist0 i l_r_prefix_2).
lia.
Qed.
Lemma proof_of_mpn_add_n_return_wit_1 : mpn_add_n_return_wit_1.
Proof. Admitted.