Theorem karatsuba 12624

Description: The Karatsuba multiplication algorithm. If X and Y are decomposed into two groups of digits of length M (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then X Y can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log₂ 3) instead of n^2 for the "naive" algorithm decmul1c 9538. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.)

Hypotheses

Ref	Expression
karatsuba.a	|- A e. NN0
karatsuba.b	|- B e. NN0
karatsuba.c	|- C e. NN0
karatsuba.d	|- D e. NN0
karatsuba.s	|- S e. NN0
karatsuba.m	|- M e. NN0
karatsuba.r	|- ( A x. C ) = R
karatsuba.t	|- ( B x. D ) = T
karatsuba.e	|- ( ( A + B ) x. ( C + D ) ) = ( ( R + S ) + T )
karatsuba.x	|- ( ( A x. (; 1 0 ^ M ) ) + B ) = X
karatsuba.y	|- ( ( C x. (; 1 0 ^ M ) ) + D ) = Y
karatsuba.w	|- ( ( R x. (; 1 0 ^ M ) ) + S ) = W
karatsuba.z	|- ( ( W x. (; 1 0 ^ M ) ) + T ) = Z

Assertion

Ref	Expression
karatsuba	|- ( X x. Y ) = Z

Proof of Theorem karatsuba

Step	Hyp	Ref	Expression
1		karatsuba.a	. . . . . 6 |- A e. NN0
2	1	nn0cni 9278	. . . . 5 |- A e. CC
3		10nn0 9491	. . . . . . 7 |- ; 1 0 e. NN0
4	3	nn0cni 9278	. . . . . 6 |- ; 1 0 e. CC
5		karatsuba.m	. . . . . 6 |- M e. NN0
6		expcl 10666	. . . . . 6 |- ( (; 1 0 e. CC /\ M e. NN0 ) -> (; 1 0 ^ M ) e. CC )
7	4, 5, 6	mp2an 426	. . . . 5 |- (; 1 0 ^ M ) e. CC
8	2, 7	mulcli 8048	. . . 4 |- ( A x. (; 1 0 ^ M ) ) e. CC
9		karatsuba.b	. . . . 5 |- B e. NN0
10	9	nn0cni 9278	. . . 4 |- B e. CC
11		karatsuba.c	. . . . . 6 |- C e. NN0
12	11	nn0cni 9278	. . . . 5 |- C e. CC
13	12, 7	mulcli 8048	. . . 4 |- ( C x. (; 1 0 ^ M ) ) e. CC
14		karatsuba.d	. . . . 5 |- D e. NN0
15	14	nn0cni 9278	. . . 4 |- D e. CC
16	8, 10, 13, 15	muladdi 8452	. . 3 |- ( ( ( A x. (; 1 0 ^ M ) ) + B ) x. ( ( C x. (; 1 0 ^ M ) ) + D ) ) = ( ( ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) + ( D x. B ) ) + ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) )
17	8, 13	mulcli 8048	. . . 4 |- ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) e. CC
18	15, 10	mulcli 8048	. . . 4 |- ( D x. B ) e. CC
19	8, 15	mulcli 8048	. . . . 5 |- ( ( A x. (; 1 0 ^ M ) ) x. D ) e. CC
20	13, 10	mulcli 8048	. . . . 5 |- ( ( C x. (; 1 0 ^ M ) ) x. B ) e. CC
21	19, 20	addcli 8047	. . . 4 |- ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) e. CC
22	17, 18, 21	add32i 8207	. . 3 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) + ( D x. B ) ) + ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) ) = ( ( ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) + ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) ) + ( D x. B ) )
23	8, 12	mulcli 8048	. . . . . 6 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) e. CC
24		karatsuba.s	. . . . . . 7 |- S e. NN0
25	24	nn0cni 9278	. . . . . 6 |- S e. CC
26	23, 25, 7	adddiri 8054	. . . . 5 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) x. (; 1 0 ^ M ) ) = ( ( ( ( A x. (; 1 0 ^ M ) ) x. C ) x. (; 1 0 ^ M ) ) + ( S x. (; 1 0 ^ M ) ) )
27	2, 7, 12	mul32i 8190	. . . . . . . . 9 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) = ( ( A x. C ) x. (; 1 0 ^ M ) )
28		karatsuba.r	. . . . . . . . . 10 |- ( A x. C ) = R
29	28	oveq1i 5935	. . . . . . . . 9 |- ( ( A x. C ) x. (; 1 0 ^ M ) ) = ( R x. (; 1 0 ^ M ) )
30	27, 29	eqtri 2217	. . . . . . . 8 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) = ( R x. (; 1 0 ^ M ) )
31	30	oveq1i 5935	. . . . . . 7 |- ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) = ( ( R x. (; 1 0 ^ M ) ) + S )
32		karatsuba.w	. . . . . . 7 |- ( ( R x. (; 1 0 ^ M ) ) + S ) = W
33	31, 32	eqtri 2217	. . . . . 6 |- ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) = W
34	33	oveq1i 5935	. . . . 5 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) x. (; 1 0 ^ M ) ) = ( W x. (; 1 0 ^ M ) )
35	8, 12, 7	mulassi 8052	. . . . . 6 |- ( ( ( A x. (; 1 0 ^ M ) ) x. C ) x. (; 1 0 ^ M ) ) = ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) )
36	2, 12	mulcli 8048	. . . . . . . . . . . 12 |- ( A x. C ) e. CC
37	36, 18, 25	add32i 8207	. . . . . . . . . . 11 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + S ) + ( D x. B ) )
38	28	oveq1i 5935	. . . . . . . . . . . 12 |- ( ( A x. C ) + S ) = ( R + S )
39		karatsuba.t	. . . . . . . . . . . . 13 |- ( B x. D ) = T
40	10, 15, 39	mulcomli 8050	. . . . . . . . . . . 12 |- ( D x. B ) = T
41	38, 40	oveq12i 5937	. . . . . . . . . . 11 |- ( ( ( A x. C ) + S ) + ( D x. B ) ) = ( ( R + S ) + T )
42	37, 41	eqtri 2217	. . . . . . . . . 10 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( R + S ) + T )
43		karatsuba.e	. . . . . . . . . 10 |- ( ( A + B ) x. ( C + D ) ) = ( ( R + S ) + T )
44	2, 10, 12, 15	muladdi 8452	. . . . . . . . . 10 |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
45	42, 43, 44	3eqtr2i 2223	. . . . . . . . 9 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
46	36, 18	addcli 8047	. . . . . . . . . 10 |- ( ( A x. C ) + ( D x. B ) ) e. CC
47	2, 15	mulcli 8048	. . . . . . . . . . 11 |- ( A x. D ) e. CC
48	12, 10	mulcli 8048	. . . . . . . . . . 11 |- ( C x. B ) e. CC
49	47, 48	addcli 8047	. . . . . . . . . 10 |- ( ( A x. D ) + ( C x. B ) ) e. CC
50	46, 25, 49	addcani 8225	. . . . . . . . 9 |- ( ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) <-> S = ( ( A x. D ) + ( C x. B ) ) )
51	45, 50	mpbi 145	. . . . . . . 8 |- S = ( ( A x. D ) + ( C x. B ) )
52	51	oveq1i 5935	. . . . . . 7 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. D ) + ( C x. B ) ) x. (; 1 0 ^ M ) )
53	47, 48, 7	adddiri 8054	. . . . . . 7 |- ( ( ( A x. D ) + ( C x. B ) ) x. (; 1 0 ^ M ) ) = ( ( ( A x. D ) x. (; 1 0 ^ M ) ) + ( ( C x. B ) x. (; 1 0 ^ M ) ) )
54	2, 15, 7	mul32i 8190	. . . . . . . 8 |- ( ( A x. D ) x. (; 1 0 ^ M ) ) = ( ( A x. (; 1 0 ^ M ) ) x. D )
55	12, 10, 7	mul32i 8190	. . . . . . . 8 |- ( ( C x. B ) x. (; 1 0 ^ M ) ) = ( ( C x. (; 1 0 ^ M ) ) x. B )
56	54, 55	oveq12i 5937	. . . . . . 7 |- ( ( ( A x. D ) x. (; 1 0 ^ M ) ) + ( ( C x. B ) x. (; 1 0 ^ M ) ) ) = ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) )
57	52, 53, 56	3eqtri 2221	. . . . . 6 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) )
58	35, 57	oveq12i 5937	. . . . 5 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. C ) x. (; 1 0 ^ M ) ) + ( S x. (; 1 0 ^ M ) ) ) = ( ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) + ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) )
59	26, 34, 58	3eqtr3ri 2226	. . . 4 |- ( ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) + ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) ) = ( W x. (; 1 0 ^ M ) )
60	59, 40	oveq12i 5937	. . 3 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) + ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) ) + ( D x. B ) ) = ( ( W x. (; 1 0 ^ M ) ) + T )
61	16, 22, 60	3eqtri 2221	. 2 |- ( ( ( A x. (; 1 0 ^ M ) ) + B ) x. ( ( C x. (; 1 0 ^ M ) ) + D ) ) = ( ( W x. (; 1 0 ^ M ) ) + T )
62		karatsuba.x	. . 3 |- ( ( A x. (; 1 0 ^ M ) ) + B ) = X
63		karatsuba.y	. . 3 |- ( ( C x. (; 1 0 ^ M ) ) + D ) = Y
64	62, 63	oveq12i 5937	. 2 |- ( ( ( A x. (; 1 0 ^ M ) ) + B ) x. ( ( C x. (; 1 0 ^ M ) ) + D ) ) = ( X x. Y )
65		karatsuba.z	. 2 |- ( ( W x. (; 1 0 ^ M ) ) + T ) = Z
66	61, 64, 65	3eqtr3i 2225	1 |- ( X x. Y ) = Z

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   NN0cn0 9266  ;cdc 9474   ^cexp 10647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-dec 9475  df-uz 9619  df-seqfrec 10557  df-exp 10648

This theorem is referenced by: (None)