Theorem karatsuba 13132

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 9776. (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 9510	. . . . 5 |- A e. CC
3		10nn0 9729	. . . . . . 7 |- ; 1 0 e. NN0
4	3	nn0cni 9510	. . . . . 6 |- ; 1 0 e. CC
5		karatsuba.m	. . . . . 6 |- M e. NN0
6		expcl 10923	. . . . . 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 8281	. . . 4 |- ( A x. (; 1 0 ^ M ) ) e. CC
9		karatsuba.b	. . . . 5 |- B e. NN0
10	9	nn0cni 9510	. . . 4 |- B e. CC
11		karatsuba.c	. . . . . 6 |- C e. NN0
12	11	nn0cni 9510	. . . . 5 |- C e. CC
13	12, 7	mulcli 8281	. . . 4 |- ( C x. (; 1 0 ^ M ) ) e. CC
14		karatsuba.d	. . . . 5 |- D e. NN0
15	14	nn0cni 9510	. . . 4 |- D e. CC
16	8, 10, 13, 15	muladdi 8684	. . 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 8281	. . . 4 |- ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) e. CC
18	15, 10	mulcli 8281	. . . 4 |- ( D x. B ) e. CC
19	8, 15	mulcli 8281	. . . . 5 |- ( ( A x. (; 1 0 ^ M ) ) x. D ) e. CC
20	13, 10	mulcli 8281	. . . . 5 |- ( ( C x. (; 1 0 ^ M ) ) x. B ) e. CC
21	19, 20	addcli 8280	. . . 4 |- ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) e. CC
22	17, 18, 21	add32i 8439	. . 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 8281	. . . . . 6 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) e. CC
24		karatsuba.s	. . . . . . 7 |- S e. NN0
25	24	nn0cni 9510	. . . . . 6 |- S e. CC
26	23, 25, 7	adddiri 8287	. . . . 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 8422	. . . . . . . . 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 6062	. . . . . . . . 9 |- ( ( A x. C ) x. (; 1 0 ^ M ) ) = ( R x. (; 1 0 ^ M ) )
30	27, 29	eqtri 2255	. . . . . . . 8 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) = ( R x. (; 1 0 ^ M ) )
31	30	oveq1i 6062	. . . . . . 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 2255	. . . . . 6 |- ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) = W
34	33	oveq1i 6062	. . . . 5 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) x. (; 1 0 ^ M ) ) = ( W x. (; 1 0 ^ M ) )
35	8, 12, 7	mulassi 8285	. . . . . 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 8281	. . . . . . . . . . . 12 |- ( A x. C ) e. CC
37	36, 18, 25	add32i 8439	. . . . . . . . . . 11 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + S ) + ( D x. B ) )
38	28	oveq1i 6062	. . . . . . . . . . . 12 |- ( ( A x. C ) + S ) = ( R + S )
39		karatsuba.t	. . . . . . . . . . . . 13 |- ( B x. D ) = T
40	10, 15, 39	mulcomli 8283	. . . . . . . . . . . 12 |- ( D x. B ) = T
41	38, 40	oveq12i 6064	. . . . . . . . . . 11 |- ( ( ( A x. C ) + S ) + ( D x. B ) ) = ( ( R + S ) + T )
42	37, 41	eqtri 2255	. . . . . . . . . 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 8684	. . . . . . . . . 10 |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
45	42, 43, 44	3eqtr2i 2261	. . . . . . . . 9 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
46	36, 18	addcli 8280	. . . . . . . . . 10 |- ( ( A x. C ) + ( D x. B ) ) e. CC
47	2, 15	mulcli 8281	. . . . . . . . . . 11 |- ( A x. D ) e. CC
48	12, 10	mulcli 8281	. . . . . . . . . . 11 |- ( C x. B ) e. CC
49	47, 48	addcli 8280	. . . . . . . . . 10 |- ( ( A x. D ) + ( C x. B ) ) e. CC
50	46, 25, 49	addcani 8457	. . . . . . . . 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 6062	. . . . . . 7 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. D ) + ( C x. B ) ) x. (; 1 0 ^ M ) )
53	47, 48, 7	adddiri 8287	. . . . . . 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 8422	. . . . . . . 8 |- ( ( A x. D ) x. (; 1 0 ^ M ) ) = ( ( A x. (; 1 0 ^ M ) ) x. D )
55	12, 10, 7	mul32i 8422	. . . . . . . 8 |- ( ( C x. B ) x. (; 1 0 ^ M ) ) = ( ( C x. (; 1 0 ^ M ) ) x. B )
56	54, 55	oveq12i 6064	. . . . . . 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 2259	. . . . . 6 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) )
58	35, 57	oveq12i 6064	. . . . 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 2264	. . . 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 6064	. . 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 2259	. 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 6064	. 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 2263	1 |- ( X x. Y ) = Z

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   NN0cn0 9498  ;cdc 9712   ^cexp 10904

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-seqfrec 10814  df-exp 10905

This theorem is referenced by: (None)