Theorem karatsuba 12939

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 9630. (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 9369	. . . . 5 |- A e. CC
3		10nn0 9583	. . . . . . 7 |- ; 1 0 e. NN0
4	3	nn0cni 9369	. . . . . 6 |- ; 1 0 e. CC
5		karatsuba.m	. . . . . 6 |- M e. NN0
6		expcl 10766	. . . . . 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 8139	. . . 4 |- ( A x. (; 1 0 ^ M ) ) e. CC
9		karatsuba.b	. . . . 5 |- B e. NN0
10	9	nn0cni 9369	. . . 4 |- B e. CC
11		karatsuba.c	. . . . . 6 |- C e. NN0
12	11	nn0cni 9369	. . . . 5 |- C e. CC
13	12, 7	mulcli 8139	. . . 4 |- ( C x. (; 1 0 ^ M ) ) e. CC
14		karatsuba.d	. . . . 5 |- D e. NN0
15	14	nn0cni 9369	. . . 4 |- D e. CC
16	8, 10, 13, 15	muladdi 8543	. . 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 8139	. . . 4 |- ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) e. CC
18	15, 10	mulcli 8139	. . . 4 |- ( D x. B ) e. CC
19	8, 15	mulcli 8139	. . . . 5 |- ( ( A x. (; 1 0 ^ M ) ) x. D ) e. CC
20	13, 10	mulcli 8139	. . . . 5 |- ( ( C x. (; 1 0 ^ M ) ) x. B ) e. CC
21	19, 20	addcli 8138	. . . 4 |- ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) e. CC
22	17, 18, 21	add32i 8298	. . 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 8139	. . . . . 6 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) e. CC
24		karatsuba.s	. . . . . . 7 |- S e. NN0
25	24	nn0cni 9369	. . . . . 6 |- S e. CC
26	23, 25, 7	adddiri 8145	. . . . 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 8281	. . . . . . . . 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 6004	. . . . . . . . 9 |- ( ( A x. C ) x. (; 1 0 ^ M ) ) = ( R x. (; 1 0 ^ M ) )
30	27, 29	eqtri 2250	. . . . . . . 8 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) = ( R x. (; 1 0 ^ M ) )
31	30	oveq1i 6004	. . . . . . 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 2250	. . . . . 6 |- ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) = W
34	33	oveq1i 6004	. . . . 5 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) x. (; 1 0 ^ M ) ) = ( W x. (; 1 0 ^ M ) )
35	8, 12, 7	mulassi 8143	. . . . . 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 8139	. . . . . . . . . . . 12 |- ( A x. C ) e. CC
37	36, 18, 25	add32i 8298	. . . . . . . . . . 11 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + S ) + ( D x. B ) )
38	28	oveq1i 6004	. . . . . . . . . . . 12 |- ( ( A x. C ) + S ) = ( R + S )
39		karatsuba.t	. . . . . . . . . . . . 13 |- ( B x. D ) = T
40	10, 15, 39	mulcomli 8141	. . . . . . . . . . . 12 |- ( D x. B ) = T
41	38, 40	oveq12i 6006	. . . . . . . . . . 11 |- ( ( ( A x. C ) + S ) + ( D x. B ) ) = ( ( R + S ) + T )
42	37, 41	eqtri 2250	. . . . . . . . . 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 8543	. . . . . . . . . 10 |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
45	42, 43, 44	3eqtr2i 2256	. . . . . . . . 9 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
46	36, 18	addcli 8138	. . . . . . . . . 10 |- ( ( A x. C ) + ( D x. B ) ) e. CC
47	2, 15	mulcli 8139	. . . . . . . . . . 11 |- ( A x. D ) e. CC
48	12, 10	mulcli 8139	. . . . . . . . . . 11 |- ( C x. B ) e. CC
49	47, 48	addcli 8138	. . . . . . . . . 10 |- ( ( A x. D ) + ( C x. B ) ) e. CC
50	46, 25, 49	addcani 8316	. . . . . . . . 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 6004	. . . . . . 7 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. D ) + ( C x. B ) ) x. (; 1 0 ^ M ) )
53	47, 48, 7	adddiri 8145	. . . . . . 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 8281	. . . . . . . 8 |- ( ( A x. D ) x. (; 1 0 ^ M ) ) = ( ( A x. (; 1 0 ^ M ) ) x. D )
55	12, 10, 7	mul32i 8281	. . . . . . . 8 |- ( ( C x. B ) x. (; 1 0 ^ M ) ) = ( ( C x. (; 1 0 ^ M ) ) x. B )
56	54, 55	oveq12i 6006	. . . . . . 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 2254	. . . . . 6 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) )
58	35, 57	oveq12i 6006	. . . . 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 2259	. . . 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 6006	. . 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 2254	. 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 6006	. 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 2258	1 |- ( X x. Y ) = Z

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 5994   CCcc 7985   0cc0 7987   1c1 7988   + caddc 7990   x. cmul 7992   NN0cn0 9357  ;cdc 9566   ^cexp 10747

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-z 9435  df-dec 9567  df-uz 9711  df-seqfrec 10657  df-exp 10748

This theorem is referenced by: (None)