Theorem karatsuba 13192

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 9824. (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 9558	. . . . 5 |- A e. CC
3		10nn0 9777	. . . . . . 7 |- ; 1 0 e. NN0
4	3	nn0cni 9558	. . . . . 6 |- ; 1 0 e. CC
5		karatsuba.m	. . . . . 6 |- M e. NN0
6		expcl 10977	. . . . . 6 |- ( (; 1 0 e. CC /\ M e. NN0 ) -> (; 1 0 ^ M ) e. CC )
7	4, 5, 6	mp2an 430	. . . . 5 |- (; 1 0 ^ M ) e. CC
8	2, 7	mulcli 8325	. . . 4 |- ( A x. (; 1 0 ^ M ) ) e. CC
9		karatsuba.b	. . . . 5 |- B e. NN0
10	9	nn0cni 9558	. . . 4 |- B e. CC
11		karatsuba.c	. . . . . 6 |- C e. NN0
12	11	nn0cni 9558	. . . . 5 |- C e. CC
13	12, 7	mulcli 8325	. . . 4 |- ( C x. (; 1 0 ^ M ) ) e. CC
14		karatsuba.d	. . . . 5 |- D e. NN0
15	14	nn0cni 9558	. . . 4 |- D e. CC
16	8, 10, 13, 15	muladdi 8730	. . 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 8325	. . . 4 |- ( ( A x. (; 1 0 ^ M ) ) x. ( C x. (; 1 0 ^ M ) ) ) e. CC
18	15, 10	mulcli 8325	. . . 4 |- ( D x. B ) e. CC
19	8, 15	mulcli 8325	. . . . 5 |- ( ( A x. (; 1 0 ^ M ) ) x. D ) e. CC
20	13, 10	mulcli 8325	. . . . 5 |- ( ( C x. (; 1 0 ^ M ) ) x. B ) e. CC
21	19, 20	addcli 8324	. . . 4 |- ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) ) e. CC
22	17, 18, 21	add32i 8484	. . 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 8325	. . . . . 6 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) e. CC
24		karatsuba.s	. . . . . . 7 |- S e. NN0
25	24	nn0cni 9558	. . . . . 6 |- S e. CC
26	23, 25, 7	adddiri 8331	. . . . 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 8467	. . . . . . . . 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 6089	. . . . . . . . 9 |- ( ( A x. C ) x. (; 1 0 ^ M ) ) = ( R x. (; 1 0 ^ M ) )
30	27, 29	eqtri 2259	. . . . . . . 8 |- ( ( A x. (; 1 0 ^ M ) ) x. C ) = ( R x. (; 1 0 ^ M ) )
31	30	oveq1i 6089	. . . . . . 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 2259	. . . . . 6 |- ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) = W
34	33	oveq1i 6089	. . . . 5 |- ( ( ( ( A x. (; 1 0 ^ M ) ) x. C ) + S ) x. (; 1 0 ^ M ) ) = ( W x. (; 1 0 ^ M ) )
35	8, 12, 7	mulassi 8329	. . . . . 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 8325	. . . . . . . . . . . 12 |- ( A x. C ) e. CC
37	36, 18, 25	add32i 8484	. . . . . . . . . . 11 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + S ) + ( D x. B ) )
38	28	oveq1i 6089	. . . . . . . . . . . 12 |- ( ( A x. C ) + S ) = ( R + S )
39		karatsuba.t	. . . . . . . . . . . . 13 |- ( B x. D ) = T
40	10, 15, 39	mulcomli 8327	. . . . . . . . . . . 12 |- ( D x. B ) = T
41	38, 40	oveq12i 6091	. . . . . . . . . . 11 |- ( ( ( A x. C ) + S ) + ( D x. B ) ) = ( ( R + S ) + T )
42	37, 41	eqtri 2259	. . . . . . . . . 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 8730	. . . . . . . . . 10 |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
45	42, 43, 44	3eqtr2i 2265	. . . . . . . . 9 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
46	36, 18	addcli 8324	. . . . . . . . . 10 |- ( ( A x. C ) + ( D x. B ) ) e. CC
47	2, 15	mulcli 8325	. . . . . . . . . . 11 |- ( A x. D ) e. CC
48	12, 10	mulcli 8325	. . . . . . . . . . 11 |- ( C x. B ) e. CC
49	47, 48	addcli 8324	. . . . . . . . . 10 |- ( ( A x. D ) + ( C x. B ) ) e. CC
50	46, 25, 49	addcani 8502	. . . . . . . . 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 6089	. . . . . . 7 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. D ) + ( C x. B ) ) x. (; 1 0 ^ M ) )
53	47, 48, 7	adddiri 8331	. . . . . . 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 8467	. . . . . . . 8 |- ( ( A x. D ) x. (; 1 0 ^ M ) ) = ( ( A x. (; 1 0 ^ M ) ) x. D )
55	12, 10, 7	mul32i 8467	. . . . . . . 8 |- ( ( C x. B ) x. (; 1 0 ^ M ) ) = ( ( C x. (; 1 0 ^ M ) ) x. B )
56	54, 55	oveq12i 6091	. . . . . . 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 2263	. . . . . 6 |- ( S x. (; 1 0 ^ M ) ) = ( ( ( A x. (; 1 0 ^ M ) ) x. D ) + ( ( C x. (; 1 0 ^ M ) ) x. B ) )
58	35, 57	oveq12i 6091	. . . . 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 2268	. . . 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 6091	. . 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 2263	. 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 6091	. 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 2267	1 |- ( X x. Y ) = Z

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   NN0cn0 9546  ;cdc 9760   ^cexp 10958

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-seqfrec 10868  df-exp 10959

This theorem is referenced by: (None)