Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 7,326 Bytes
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horizon := 0;;
let SUC_INJ_1 = thm `;
now
now [1]
let m n be num;
now [2]
assume mk_num (IND_SUC (dest_num m)) =
mk_num (IND_SUC (dest_num n)) [3];
now [4]
let p be num;
NUM_REP (dest_num p) [5]
by REWRITE_TAC[fst num_tydef; snd num_tydef] ;
thus NUM_REP (IND_SUC (dest_num p))
by MATCH_MP_TAC (CONJUNCT2 NUM_REP_RULES) from 5;
end;
!p. NUM_REP (IND_SUC (dest_num p)) [6] by GEN_TAC from 4;
now [7]
assume !p. dest_num (mk_num (IND_SUC (dest_num p))) =
IND_SUC (dest_num p) [8];
mk_num (dest_num m) = mk_num (dest_num n) ==> m = n [9]
by REWRITE_TAC[fst num_tydef];
dest_num m = dest_num n ==> m = n [10]
by DISCH_THEN(MP_TAC o AP_TERM (parse_term "mk_num")) from 9;
thus dest_num (mk_num (IND_SUC (dest_num m))) =
dest_num (mk_num (IND_SUC (dest_num n)))
==> m = n by ASM_REWRITE_TAC[IND_SUC_INJ],8 from 10;
end;
(!p. dest_num (mk_num (IND_SUC (dest_num p))) =
IND_SUC (dest_num p))
==> dest_num (mk_num (IND_SUC (dest_num m))) =
dest_num (mk_num (IND_SUC (dest_num n)))
==> m = n [11] by DISCH_TAC from 7;
(!p. NUM_REP (IND_SUC (dest_num p)))
==> dest_num (mk_num (IND_SUC (dest_num m))) =
dest_num (mk_num (IND_SUC (dest_num n)))
==> m = n [12] by REWRITE_TAC[fst num_tydef; snd num_tydef] from 11;
dest_num (mk_num (IND_SUC (dest_num m))) =
dest_num (mk_num (IND_SUC (dest_num n)))
==> m = n [13]
by SUBGOAL_THEN (parse_term "!p. NUM_REP (IND_SUC (dest_num p))")
MP_TAC from 6,12;
thus m = n
by POP_ASSUM(MP_TAC o AP_TERM (parse_term "dest_num")),3 from 13;
end;
mk_num (IND_SUC (dest_num m)) = mk_num (IND_SUC (dest_num n))
==> m = n [14] by DISCH_TAC from 2;
now [15]
assume m = n [16];
thus mk_num (IND_SUC (dest_num m)) =
mk_num (IND_SUC (dest_num n)) by ASM_REWRITE_TAC[],16;
end;
m = n
==> mk_num (IND_SUC (dest_num m)) =
mk_num (IND_SUC (dest_num n)) [17] by DISCH_TAC from 15;
mk_num (IND_SUC (dest_num m)) = mk_num (IND_SUC (dest_num n)) <=>
m = n [18] by EQ_TAC from 14,17;
thus SUC m = SUC n <=> m = n by REWRITE_TAC[SUC_DEF] from 18;
end;
thus !m n. SUC m = SUC n <=> m = n by REPEAT GEN_TAC from 1;
end;
`;;
let SUC_INJ_2 = thm `;
!m n. SUC m = SUC n <=> m = n [1]
proof
let m n be num;
mk_num (IND_SUC (dest_num m)) = mk_num (IND_SUC (dest_num n))
==> m = n [2]
proof
assume mk_num (IND_SUC (dest_num m)) =
mk_num (IND_SUC (dest_num n)) [3];
!p. NUM_REP (IND_SUC (dest_num p)) [4]
proof
let p be num;
NUM_REP (dest_num p) [5]
by REWRITE_TAC[fst num_tydef; snd num_tydef];
qed by MATCH_MP_TAC (CONJUNCT2 NUM_REP_RULES) from 5;
(!p. dest_num (mk_num (IND_SUC (dest_num p))) =
IND_SUC (dest_num p))
==> dest_num (mk_num (IND_SUC (dest_num m))) =
dest_num (mk_num (IND_SUC (dest_num n)))
==> m = n [6]
proof
assume !p. dest_num (mk_num (IND_SUC (dest_num p))) =
IND_SUC (dest_num p) [7];
mk_num (dest_num m) = mk_num (dest_num n) ==> m = n [8]
by REWRITE_TAC[fst num_tydef];
dest_num m = dest_num n ==> m = n [9]
by DISCH_THEN(MP_TAC o AP_TERM (parse_term "mk_num")) from 8;
qed by ASM_REWRITE_TAC[IND_SUC_INJ],* from 9;
(!p. NUM_REP (IND_SUC (dest_num p)))
==> dest_num (mk_num (IND_SUC (dest_num m))) =
dest_num (mk_num (IND_SUC (dest_num n)))
==> m = n [10] by REWRITE_TAC[fst num_tydef; snd num_tydef] from 6;
dest_num (mk_num (IND_SUC (dest_num m))) =
dest_num (mk_num (IND_SUC (dest_num n)))
==> m = n [11]
by SUBGOAL_THEN (parse_term "!p. NUM_REP (IND_SUC (dest_num p))")
MP_TAC
from 4,10;
qed by POP_ASSUM(MP_TAC o AP_TERM (parse_term "dest_num")),3 from 11;
m = n
==> mk_num (IND_SUC (dest_num m)) = mk_num (IND_SUC (dest_num n)) [12]
proof
assume m = n [13];
qed by ASM_REWRITE_TAC[],*;
mk_num (IND_SUC (dest_num m)) = mk_num (IND_SUC (dest_num n)) <=>
m = n [14] by EQ_TAC from 2,12;
qed by REWRITE_TAC[SUC_DEF] from 14;`;;
let num_INDUCTION_ = thm `;
now [1]
let P be num->bool;
let n be num;
assume P _0;
assume !n. P n ==> P (SUC n);
now [2]
let i be ind;
assume NUM_REP i;
assume P (mk_num i);
NUM_REP i [3] by ASM_REWRITE_TAC[],*;
thus NUM_REP (IND_SUC i)
by MATCH_MP_TAC(CONJUNCT2 NUM_REP_RULES) from 3;
end;
now [4]
let i be ind;
assume NUM_REP i;
assume P (mk_num i);
NUM_REP i [5] by FIRST_ASSUM MATCH_ACCEPT_TAC,*;
dest_num (mk_num i) = i [6] by REWRITE_TAC[GSYM(snd num_tydef)] from 5;
i = dest_num (mk_num i) [7] by CONV_TAC SYM_CONV from 6;
mk_num (IND_SUC i) = mk_num (IND_SUC (dest_num (mk_num i))) [8]
by REPEAT AP_TERM_TAC from 7;
mk_num (IND_SUC i) = SUC (mk_num i) [9] by REWRITE_TAC[SUC_DEF] from 8;
P (mk_num i) [10] by FIRST_ASSUM MATCH_ACCEPT_TAC,*;
P (SUC (mk_num i)) [11] by FIRST_ASSUM MATCH_MP_TAC,* from 10;
thus P (mk_num (IND_SUC i))
by SUBGOAL_THEN (parse_term "mk_num(IND_SUC i) = SUC(mk_num i)")
SUBST1_TAC
from 9,11;
end;
!i. NUM_REP i /\ P (mk_num i)
==> NUM_REP (IND_SUC i) /\ P (mk_num (IND_SUC i)) [12]
by REPEAT STRIP_TAC from 2,4;
(NUM_REP (dest_num n)
==> NUM_REP (dest_num n) /\ P (mk_num (dest_num n)))
==> P n [13] by REWRITE_TAC[fst num_tydef; snd num_tydef];
(!a. NUM_REP a ==> NUM_REP a /\ P (mk_num a)) ==> P n [14]
by DISCH_THEN(MP_TAC o SPEC (parse_term "dest_num n")) from 13;
((!i. NUM_REP i /\ P (mk_num i)
==> NUM_REP (IND_SUC i) /\ P (mk_num (IND_SUC i)))
==> (!a. NUM_REP a ==> NUM_REP a /\ P (mk_num a)))
==> P n [15]
by W(C SUBGOAL_THEN (fun t -> REWRITE_TAC[t]) o funpow 2 lhand o snd)
from 12,14;
((\i. NUM_REP i /\ P (mk_num i)) IND_0 /\
(!i. (\i. NUM_REP i /\ P (mk_num i)) i
==> (\i. NUM_REP i /\ P (mk_num i)) (IND_SUC i))
==> (!a. NUM_REP a ==> (\i. NUM_REP i /\ P (mk_num i)) a))
==> P n [16] by ASM_REWRITE_TAC[GSYM ZERO_DEF; NUM_REP_RULES],* from 15;
thus P n by MP_TAC (SPEC (parse_term
"\\i. NUM_REP i /\\ P(mk_num i):bool") NUM_REP_INDUCT) from 16;
end;
thus !P. P(_0) /\ (!n. P(n) ==> P(SUC n)) ==> !n. P n
by REPEAT STRIP_TAC from 1;
`;;
let num_RECURSION_STD = thm `;
!e:Z f. ?fn. (fn 0 = e) /\ (!n. fn (SUC n) = f n (fn n))
proof
!e:Z f. ?fn. fn 0 = e /\ (!n. fn (SUC n) = f n (fn n)) [1]
proof
let e be Z;
let f be num->Z->Z;
(?fn. fn 0 = e /\ (!n. fn (SUC n) = (\z n. f n z) (fn n) n))
==> (?fn. fn 0 = e /\ (!n. fn (SUC n) = f n (fn n))) [2]
by REWRITE_TAC[];
qed by MP_TAC(ISPECL [(parse_term "e:Z");
(parse_term "(\\z n. (f:num->Z->Z) n z)")] num_RECURSION) from 2;
qed by REPEAT GEN_TAC from 1;
`;;
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