Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
| prioritize_num();; | |
| let EGCD_INVARIANT = thm `; | |
| !m n d. d divides egcd(m,n) <=> d divides m /\ d divides n | |
| proof | |
| let m n be num; | |
| (!m'' n'. | |
| m'' + n' < m + n | |
| ==> (!d. d divides egcd (m'',n') <=> | |
| d divides m'' /\ d divides n')) | |
| ==> (!d. d divides egcd (m,n) <=> d divides m /\ d divides n) [1] | |
| proof | |
| assume !m'' n'. | |
| m'' + n' < m + n | |
| ==> (!d. d divides egcd (m'',n') <=> | |
| d divides m'' /\ d divides n') [2]; | |
| !d. d divides | |
| (if m = 0 | |
| then n | |
| else | |
| if n = 0 | |
| then m | |
| else if m <= n then egcd (m,n - m) else egcd (m - n,n)) <=> | |
| d divides m /\ d divides n [3] | |
| proof | |
| let d be num; | |
| m = 0 ==> (d divides n <=> d divides m /\ d divides n) [4] | |
| by DIVIDES_0; | |
| ~(m = 0) | |
| ==> (d divides | |
| (if n = 0 | |
| then m | |
| else | |
| if m <= n then egcd (m,n - m) else egcd (m - n,n)) <=> | |
| d divides m /\ d divides n) [5] | |
| proof | |
| assume ~(m = 0) [6]; | |
| n = 0 ==> (d divides m <=> d divides m /\ d divides n) [7] | |
| by DIVIDES_0; | |
| ~(n = 0) | |
| ==> (d divides | |
| (if m <= n then egcd (m,n - m) else egcd (m - n,n)) <=> | |
| d divides m /\ d divides n) [8] | |
| proof | |
| assume ~(n = 0) [9]; | |
| m <= n | |
| ==> (d divides egcd (m,n - m) <=> | |
| d divides m /\ d divides n) [10] | |
| proof | |
| assume m <= n; | |
| m + (n - m) < m + n by ARITH_TAC,6; | |
| qed by #; | |
| ~(m <= n) | |
| ==> (d divides egcd (m - n,n) <=> | |
| d divides m /\ d divides n) [11] | |
| proof | |
| assume ~(m <= n); | |
| (m - n) + n < m + n by ARITH_TAC,9; | |
| d divides egcd (m - n,n) <=> d divides m - n /\ d divides n | |
| by 2; | |
| ... <=> d divides (m - n) + n /\ d divides n by DIVIDES_ADD; | |
| :: #1 | |
| :: 1: inference error | |
| qed by 2,DIVIDES_SUB; | |
| :: #1 | |
| qed by COND_CASES_TAC from 10,11; | |
| qed by COND_CASES_TAC from 7,8; | |
| qed by COND_CASES_TAC from 4,5; | |
| qed by ONCE_REWRITE_TAC[egcd] from 3; | |
| qed by WF_INDUCT_TAC (parse_term "m + n") from 1; | |
| :: #1 | |
| `;; | |