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Text Generation
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language-modeling
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English
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File size: 2,107 Bytes
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horizon := -1;;
let FOO = thm `;
!x n. x pow n = &1 <=> abs x = &1 /\ (x < &0 ==> EVEN n) \/ n = 0 [1]
proof
let x be real;
let n be num;
n = 0
==> (x pow n = &1 <=> abs x = &1 /\ (x < &0 ==> EVEN n) \/ n = 0) [2]
proof
assume n = 0 [3];
qed by ASM_REWRITE_TAC[real_pow],3;
~(n = 0)
==> (x pow n = &1 <=> abs x = &1 /\ (x < &0 ==> EVEN n) \/ n = 0) [4]
proof
assume ~(n = 0) [5];
abs x = &1 ==> (x pow n = &1 <=> abs x = &1 /\ (x < &0 ==> EVEN n)) [6]
proof
assume abs x = &1 [7];
&1 < &0 ==> EVEN n [8] by REAL_ARITH_TAC,5;
&1 pow n = &1 <=> &1 < &0 ==> EVEN n [9]
by ASM_REWRITE_TAC[REAL_POW_NEG; REAL_POW_ONE],5 from 8;
EVEN n ==> (&1 = &1 <=> -- &1 < &0 ==> T) [10]
proof
assume EVEN n [11];
qed by ASM_REWRITE_TAC[],5,11;
~EVEN n ==> (-- &1 = &1 <=> -- &1 < &0 ==> F) [12]
proof
assume ~EVEN n [13];
-- &1 = &1 <=> ~(-- &1 < &0) [14] by REAL_ARITH_TAC,5,13;
qed by ASM_REWRITE_TAC[],5,13 from 14;
(if EVEN n then &1 else -- &1) = &1 <=> -- &1 < &0 ==> EVEN n [15]
by REPEAT COND_CASES_TAC,5 from 10,12;
-- &1 pow n = &1 <=> -- &1 < &0 ==> EVEN n [16]
by ASM_REWRITE_TAC[REAL_POW_NEG; REAL_POW_ONE],5 from 15;
x pow n = &1 <=> x < &0 ==> EVEN n [17]
by FIRST_X_ASSUM(DISJ_CASES_THEN SUBST1_TAC o MATCH_MP (REAL_ARITH (parse_term "abs x = a ==> x = a \\/ x = --a"))),5,7
from 9,16;
x pow n = &1 <=> abs x = &1 /\ (x < &0 ==> EVEN n) [18]
by ASM_REWRITE_TAC[],5,7 from 17;
qed by ALL_TAC,5,7 from 18;
~(abs x = &1)
==> (x pow n = &1 <=> abs x = &1 /\ (x < &0 ==> EVEN n)) [19]
proof
assume ~(abs x = &1) [20];
qed by ASM_MESON_TAC[REAL_POW_EQ_1_IMP],5,20;
:: #2
:: 2: inference time-out
qed by ASM_REWRITE_TAC[real_pow],5 from 18;
:: #4
:: 4: unknown label
qed by ASM_CASES_TAC (parse_term "n = 0") from 2,4;`;;
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