Datasets:

phanerozoic
/

HOL-Light

statement stringlengths 2 4.33k	proof stringlengths 0 87.5k	type stringclasses 13 values	symbolic_name stringlengths 1 70	library stringclasses 42 values	filename stringclasses 393 values	imports listlengths 0 0	deps listlengths 0 64	source_url stringclasses 1 value	commit stringclasses 1 value
(PRE 0 = 0) /\ (!n. PRE (SUC n) = n)		new_recursive_definition	PRE	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!n. 0 + n = n) /\ (!m n. (SUC m) + n = SUC(m + n))		new_recursive_definition	ADD	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m. m + 0 = m	INDUCT_TAC THEN ASM_REWRITE_TAC[ADD]	prove	ADD_0	Root	arith.ml	[]	[ "ADD" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m + (SUC n) = SUC(m + n)	INDUCT_TAC THEN ASM_REWRITE_TAC[ADD]	prove	ADD_SUC	Root	arith.ml	[]	[ "ADD" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!n. 0 + n = n) /\ (!m. m + 0 = m) /\ (!m n. (SUC m) + n = SUC(m + n)) /\ (!m n. m + (SUC n) = SUC(m + n))	REWRITE_TAC[ADD; ADD_0; ADD_SUC]	prove	ADD_CLAUSES	Root	arith.ml	[]	[ "ADD", "ADD_0", "ADD_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m + n = n + m	INDUCT_TAC THEN ASM_REWRITE_TAC[ADD_CLAUSES]	prove	ADD_SYM	Root	arith.ml	[]	[ "ADD_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m + (n + p) = (m + n) + p	INDUCT_TAC THEN ASM_REWRITE_TAC[ADD_CLAUSES]	prove	ADD_ASSOC	Root	arith.ml	[]	[ "ADD_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(m + n = n + m) /\ ((m + n) + p = m + (n + p)) /\ (m + (n + p) = n + (m + p))	MESON_TAC[ADD_ASSOC; ADD_SYM]	prove	ADD_AC	Root	arith.ml	[]	[ "ADD_ASSOC", "ADD_SYM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m + n = 0) <=> (m = 0) /\ (n = 0)	REPEAT INDUCT_TAC THEN REWRITE_TAC[ADD_CLAUSES; NOT_SUC]	prove	ADD_EQ_0	Root	arith.ml	[]	[ "ADD_CLAUSES", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m + n = m + p) <=> (n = p)	INDUCT_TAC THEN ASM_REWRITE_TAC[ADD_CLAUSES; SUC_INJ]	prove	EQ_ADD_LCANCEL	Root	arith.ml	[]	[ "ADD_CLAUSES", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m + p = n + p) <=> (m = n)	ONCE_REWRITE_TAC[ADD_SYM] THEN MATCH_ACCEPT_TAC EQ_ADD_LCANCEL	prove	EQ_ADD_RCANCEL	Root	arith.ml	[]	[ "ADD_SYM", "EQ_ADD_LCANCEL" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m + n = m) <=> (n = 0)	INDUCT_TAC THEN ASM_REWRITE_TAC[ADD_CLAUSES; SUC_INJ]	prove	EQ_ADD_LCANCEL_0	Root	arith.ml	[]	[ "ADD_CLAUSES", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m + n = n) <=> (m = 0)	ONCE_REWRITE_TAC[ADD_SYM] THEN MATCH_ACCEPT_TAC EQ_ADD_LCANCEL_0	prove	EQ_ADD_RCANCEL_0	Root	arith.ml	[]	[ "ADD_SYM", "EQ_ADD_LCANCEL_0" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. BIT0 n = n + n	INDUCT_TAC THEN ASM_REWRITE_TAC[BIT0_DEF; ADD_CLAUSES]	prove	BIT0	Root	arith.ml	[]	[ "ADD_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. BIT1 n = SUC(n + n)	REWRITE_TAC[BIT1_DEF; BIT0]	prove	BIT1	Root	arith.ml	[]	[ "BIT0" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. NUMERAL (BIT0 n) = NUMERAL n + NUMERAL n	REWRITE_TAC[NUMERAL; BIT0]	prove	BIT0_THM	Root	arith.ml	[]	[ "BIT0" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. NUMERAL (BIT1 n) = SUC(NUMERAL n + NUMERAL n)	REWRITE_TAC[NUMERAL; BIT1]	prove	BIT1_THM	Root	arith.ml	[]	[ "BIT1" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
1 = SUC 0	REWRITE_TAC[BIT1; REWRITE_RULE[NUMERAL] ADD_CLAUSES; NUMERAL]	prove	ONE	Root	arith.ml	[]	[ "ADD_CLAUSES", "BIT1" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
2 = SUC 1	REWRITE_TAC[BIT0; BIT1; REWRITE_RULE[NUMERAL] ADD_CLAUSES; NUMERAL]	prove	TWO	Root	arith.ml	[]	[ "ADD_CLAUSES", "BIT0", "BIT1" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m. SUC m = m + 1	REWRITE_TAC[BIT1_THM; ADD_CLAUSES]	prove	ADD1	Root	arith.ml	[]	[ "ADD_CLAUSES", "BIT1_THM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!n. 0 * n = 0) /\ (!m n. (SUC m) * n = (m * n) + n)		new_recursive_definition	MULT	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m. m * 0 = 0	INDUCT_TAC THEN ASM_REWRITE_TAC[MULT; ADD_CLAUSES]	prove	MULT_0	Root	arith.ml	[]	[ "ADD_CLAUSES", "MULT" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m * (SUC n) = m + (m * n)	INDUCT_TAC THEN ASM_REWRITE_TAC[MULT; ADD_CLAUSES; ADD_ASSOC]	prove	MULT_SUC	Root	arith.ml	[]	[ "ADD_ASSOC", "ADD_CLAUSES", "MULT" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!n. 0 * n = 0) /\ (!m. m * 0 = 0) /\ (!n. 1 * n = n) /\ (!m. m * 1 = m) /\ (!m n. (SUC m) * n = (m * n) + n) /\ (!m n. m * (SUC n) = m + (m * n))	REWRITE_TAC[BIT1_THM; MULT; MULT_0; MULT_SUC; ADD_CLAUSES]	prove	MULT_CLAUSES	Root	arith.ml	[]	[ "ADD_CLAUSES", "BIT1_THM", "MULT", "MULT_0", "MULT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m * n = n * m	INDUCT_TAC THEN ASM_REWRITE_TAC[MULT_CLAUSES; EQT_INTRO(SPEC_ALL ADD_SYM)]	prove	MULT_SYM	Root	arith.ml	[]	[ "ADD_SYM", "MULT_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m * (n + p) = (m * n) + (m * p)	GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[ADD; MULT_CLAUSES; ADD_ASSOC]	prove	LEFT_ADD_DISTRIB	Root	arith.ml	[]	[ "ADD", "ADD_ASSOC", "MULT_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m + n) * p = (m * p) + (n * p)	ONCE_REWRITE_TAC[MULT_SYM] THEN MATCH_ACCEPT_TAC LEFT_ADD_DISTRIB	prove	RIGHT_ADD_DISTRIB	Root	arith.ml	[]	[ "LEFT_ADD_DISTRIB", "MULT_SYM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m * (n * p) = (m * n) * p	INDUCT_TAC THEN ASM_REWRITE_TAC[MULT_CLAUSES; RIGHT_ADD_DISTRIB]	prove	MULT_ASSOC	Root	arith.ml	[]	[ "MULT_CLAUSES", "RIGHT_ADD_DISTRIB" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(m * n = n * m) /\ ((m * n) * p = m * (n * p)) /\ (m * (n * p) = n * (m * p))	MESON_TAC[MULT_ASSOC; MULT_SYM]	prove	MULT_AC	Root	arith.ml	[]	[ "MULT_ASSOC", "MULT_SYM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m * n = 0) <=> (m = 0) \/ (n = 0)	REPEAT INDUCT_TAC THEN REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES; NOT_SUC]	prove	MULT_EQ_0	Root	arith.ml	[]	[ "ADD_CLAUSES", "MULT_CLAUSES", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m * n = m * p) <=> (m = 0) \/ (n = p)	INDUCT_TAC THEN REWRITE_TAC[MULT_CLAUSES; NOT_SUC] THEN REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES; GSYM NOT_SUC; NOT_SUC] THEN ASM_REWRITE_TAC[SUC_INJ; GSYM ADD_ASSOC; EQ_ADD_LCANCEL]	prove	EQ_MULT_LCANCEL	Root	arith.ml	[]	[ "ADD_ASSOC", "ADD_CLAUSES", "EQ_ADD_LCANCEL", "MULT_CLAUSES", "NOT_SUC", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m * p = n * p) <=> (m = n) \/ (p = 0)	ONCE_REWRITE_TAC[MULT_SYM; DISJ_SYM] THEN MATCH_ACCEPT_TAC EQ_MULT_LCANCEL	prove	EQ_MULT_RCANCEL	Root	arith.ml	[]	[ "DISJ_SYM", "EQ_MULT_LCANCEL", "MULT_SYM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. 2 * n = n + n	GEN_TAC THEN REWRITE_TAC[BIT0_THM; MULT_CLAUSES; RIGHT_ADD_DISTRIB]	prove	MULT_2	Root	arith.ml	[]	[ "BIT0_THM", "MULT_CLAUSES", "RIGHT_ADD_DISTRIB" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m * n = 1) <=> (m = 1) /\ (n = 1)	INDUCT_TAC THEN INDUCT_TAC THEN REWRITE_TAC [MULT_CLAUSES; ADD_CLAUSES; BIT0_THM; BIT1_THM; GSYM NOT_SUC] THEN REWRITE_TAC[SUC_INJ; ADD_EQ_0; MULT_EQ_0] THEN CONV_TAC TAUT	prove	MULT_EQ_1	Root	arith.ml	[]	[ "ADD_CLAUSES", "ADD_EQ_0", "BIT0_THM", "BIT1_THM", "MULT_CLAUSES", "MULT_EQ_0", "NOT_SUC", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!m. m EXP 0 = 1) /\ (!m n. m EXP (SUC n) = m * (m EXP n))		new_recursive_definition	EXP	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m EXP n = 0) <=> (m = 0) /\ ~(n = 0)	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC [BIT1_THM; NOT_SUC; NOT_SUC; EXP; MULT_CLAUSES; ADD_CLAUSES; ADD_EQ_0]	prove	EXP_EQ_0	Root	arith.ml	[]	[ "ADD_CLAUSES", "ADD_EQ_0", "BIT1_THM", "EXP", "MULT_CLAUSES", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!x n. x EXP n = 1 <=> x = 1 \/ n = 0	GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[EXP; MULT_EQ_1; NOT_SUC] THEN CONV_TAC TAUT	prove	EXP_EQ_1	Root	arith.ml	[]	[ "EXP", "MULT_EQ_1", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. 0 EXP n = if n = 0 then 1 else 0	GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[EXP_EQ_0; EXP_EQ_1]	prove	EXP_ZERO	Root	arith.ml	[]	[ "EXP", "EXP_EQ_0", "EXP_EQ_1" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m EXP (n + p) = (m EXP n) * (m EXP p)	GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[EXP; ADD_CLAUSES; MULT_CLAUSES; MULT_AC]	prove	EXP_ADD	Root	arith.ml	[]	[ "ADD_CLAUSES", "EXP", "MULT_AC", "MULT_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. 1 EXP n = 1	INDUCT_TAC THEN ASM_REWRITE_TAC[EXP; MULT_CLAUSES]	prove	EXP_ONE	Root	arith.ml	[]	[ "EXP", "MULT_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. n EXP 1 = n	REWRITE_TAC[ONE; EXP; MULT_CLAUSES; ADD_CLAUSES]	prove	EXP_1	Root	arith.ml	[]	[ "ADD_CLAUSES", "EXP", "MULT_CLAUSES", "ONE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. n EXP 2 = n * n	REWRITE_TAC[BIT0_THM; BIT1_THM; EXP; EXP_ADD; MULT_CLAUSES; ADD_CLAUSES]	prove	EXP_2	Root	arith.ml	[]	[ "ADD_CLAUSES", "BIT0_THM", "BIT1_THM", "EXP", "EXP_ADD", "MULT_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!p m n. (m * n) EXP p = m EXP p * n EXP p	INDUCT_TAC THEN ASM_REWRITE_TAC[EXP; MULT_CLAUSES; MULT_AC]	prove	MULT_EXP	Root	arith.ml	[]	[ "EXP", "MULT_AC", "MULT_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m EXP (n * p) = (m EXP n) EXP p	GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[EXP_ADD; EXP; MULT_CLAUSES] THENL [CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[EXP; MULT_CLAUSES]; REWRITE_TAC[MULT_EXP] THEN MATCH_ACCEPT_TAC MULT_SYM]	prove	EXP_MULT	Root	arith.ml	[]	[ "EXP", "EXP_ADD", "MULT_CLAUSES", "MULT_EXP", "MULT_SYM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!x m n. (x EXP m) EXP n = x EXP (m * n)	REWRITE_TAC[EXP_MULT]	prove	EXP_EXP	Root	arith.ml	[]	[ "EXP", "EXP_MULT" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!m. (m <= 0) <=> (m = 0)) /\ (!m n. (m <= SUC n) <=> (m = SUC n) \/ (m <= n))		new_recursive_definition	LE	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!m. (m < 0) <=> F) /\ (!m n. (m < SUC n) <=> (m = n) \/ (m < n))		new_recursive_definition	LT	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
m >= n <=> n <= m		new_definition	GE	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
m > n <=> n < m		new_definition	GT	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. MAX m n = if m <= n then n else m		new_definition	MAX	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. MIN m n = if m <= n then m else n		new_definition	MIN	Root	arith.ml	[]	[]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (SUC m <= n) <=> (m < n)	GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[LE; LT; NOT_SUC; SUC_INJ]	prove	LE_SUC_LT	Root	arith.ml	[]	[ "LE", "LT", "NOT_SUC", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m < SUC n) <=> (m <= n)	GEN_TAC THEN INDUCT_TAC THEN ONCE_REWRITE_TAC[LT; LE] THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[LT]	prove	LT_SUC_LE	Root	arith.ml	[]	[ "LE", "LT" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (SUC m <= SUC n) <=> (m <= n)	REWRITE_TAC[LE_SUC_LT; LT_SUC_LE]	prove	LE_SUC	Root	arith.ml	[]	[ "LE_SUC_LT", "LT_SUC_LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (SUC m < SUC n) <=> (m < n)	REWRITE_TAC[LT_SUC_LE; LE_SUC_LT]	prove	LT_SUC	Root	arith.ml	[]	[ "LE_SUC_LT", "LT_SUC_LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. 0 <= n	INDUCT_TAC THEN ASM_REWRITE_TAC[LE]	prove	LE_0	Root	arith.ml	[]	[ "LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. 0 < SUC n	REWRITE_TAC[LT_SUC_LE; LE_0]	prove	LT_0	Root	arith.ml	[]	[ "LE_0", "LT_SUC_LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. n <= n	INDUCT_TAC THEN REWRITE_TAC[LE]	prove	LE_REFL	Root	arith.ml	[]	[ "LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. ~(n < n)	INDUCT_TAC THEN ASM_REWRITE_TAC[LT_SUC] THEN REWRITE_TAC[LT]	prove	LT_REFL	Root	arith.ml	[]	[ "LT", "LT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n:num. m < n ==> ~(m = n)	MESON_TAC[LT_REFL]	prove	LT_IMP_NE	Root	arith.ml	[]	[ "LT_REFL" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m <= n /\ n <= m) <=> (m = n)	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; SUC_INJ] THEN REWRITE_TAC[LE; NOT_SUC; GSYM NOT_SUC]	prove	LE_ANTISYM	Root	arith.ml	[]	[ "LE", "LE_SUC", "NOT_SUC", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. ~(m < n /\ n < m)	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LT_SUC] THEN REWRITE_TAC[LT]	prove	LT_ANTISYM	Root	arith.ml	[]	[ "LT", "LT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. ~(m <= n /\ n < m)	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; LT_SUC] THEN REWRITE_TAC[LE; LT; NOT_SUC]	prove	LET_ANTISYM	Root	arith.ml	[]	[ "LE", "LE_SUC", "LT", "LT_SUC", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. ~(m < n /\ n <= m)	ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[LET_ANTISYM]	prove	LTE_ANTISYM	Root	arith.ml	[]	[ "CONJ_SYM", "LET_ANTISYM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m <= n /\ n <= p ==> m <= p	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; LE_0] THEN REWRITE_TAC[LE; NOT_SUC]	prove	LE_TRANS	Root	arith.ml	[]	[ "LE", "LE_0", "LE_SUC", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m < n /\ n < p ==> m < p	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LT_SUC; LT_0] THEN REWRITE_TAC[LT; NOT_SUC]	prove	LT_TRANS	Root	arith.ml	[]	[ "LT", "LT_0", "LT_SUC", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m <= n /\ n < p ==> m < p	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; LT_SUC; LT_0] THEN REWRITE_TAC[LT; LE; NOT_SUC]	prove	LET_TRANS	Root	arith.ml	[]	[ "LE", "LE_SUC", "LT", "LT_0", "LT_SUC", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. m < n /\ n <= p ==> m < p	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; LT_SUC; LT_0] THEN REWRITE_TAC[LT; LE; NOT_SUC]	prove	LTE_TRANS	Root	arith.ml	[]	[ "LE", "LE_SUC", "LT", "LT_0", "LT_SUC", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m <= n \/ n <= m	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_0; LE_SUC]	prove	LE_CASES	Root	arith.ml	[]	[ "LE_0", "LE_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m < n) \/ (n < m) \/ (m = n)	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LT_SUC; SUC_INJ] THEN REWRITE_TAC[LT; NOT_SUC; GSYM NOT_SUC] THEN W(W (curry SPEC_TAC) o hd o frees o snd) THEN INDUCT_TAC THEN REWRITE_TAC[LT_0]	prove	LT_CASES	Root	arith.ml	[]	[ "LT", "LT_0", "LT_SUC", "NOT_SUC", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m <= n \/ n < m	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC_LT; LT_SUC_LE; LE_0]	prove	LET_CASES	Root	arith.ml	[]	[ "LE_0", "LE_SUC_LT", "LT_SUC_LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m < n \/ n <= m	ONCE_REWRITE_TAC[DISJ_SYM] THEN MATCH_ACCEPT_TAC LET_CASES	prove	LTE_CASES	Root	arith.ml	[]	[ "DISJ_SYM", "LET_CASES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m <= n) <=> (m < n) \/ (m = n)	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; LT_SUC; SUC_INJ; LE_0; LT_0] THEN REWRITE_TAC[LE; LT]	prove	LE_LT	Root	arith.ml	[]	[ "LE", "LE_0", "LE_SUC", "LT", "LT_0", "LT_SUC", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m < n) <=> (m <= n) /\ ~(m = n)	REWRITE_TAC[LE_LT] THEN REPEAT GEN_TAC THEN EQ_TAC THENL [DISCH_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST_ALL_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[LT_REFL]; DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[]]	prove	LT_LE	Root	arith.ml	[]	[ "LE_LT", "LT_REFL" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. ~(m <= n) <=> (n < m)	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; LT_SUC] THEN REWRITE_TAC[LE; LT; NOT_SUC; GSYM NOT_SUC; LE_0] THEN W(W (curry SPEC_TAC) o hd o frees o snd) THEN INDUCT_TAC THEN REWRITE_TAC[LT_0]	prove	NOT_LE	Root	arith.ml	[]	[ "LE", "LE_0", "LE_SUC", "LT", "LT_0", "LT_SUC", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. ~(m < n) <=> n <= m	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[LE_SUC; LT_SUC] THEN REWRITE_TAC[LE; LT; NOT_SUC; GSYM NOT_SUC; LE_0] THEN W(W (curry SPEC_TAC) o hd o frees o snd) THEN INDUCT_TAC THEN REWRITE_TAC[LT_0]	prove	NOT_LT	Root	arith.ml	[]	[ "LE", "LE_0", "LE_SUC", "LT", "LT_0", "LT_SUC", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m < n ==> m <= n	REWRITE_TAC[LT_LE] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]	prove	LT_IMP_LE	Root	arith.ml	[]	[ "LT_LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m = n) ==> m <= n	REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[LE_REFL]	prove	EQ_IMP_LE	Root	arith.ml	[]	[ "LE_REFL" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!n. 0 < n <=> ~(n = 0)	INDUCT_TAC THEN ASM_REWRITE_TAC[NOT_SUC; LT; EQ_SYM_EQ] THEN CONV_TAC TAUT	prove	LT_NZ	Root	arith.ml	[]	[ "EQ_SYM_EQ", "LT", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
(!n. ~(n = 0) ==> 0 < n) /\ (!n. ~(n = 0) ==> 1 <= n) /\ (!n. 0 < n ==> ~(n = 0)) /\ (!n. 0 < n ==> 1 <= n) /\ (!n. 1 <= n ==> 0 < n) /\ (!n. 1 <= n ==> ~(n = 0))	REWRITE_TAC[LT_NZ; GSYM NOT_LT; ONE; LT]	prove	LE_1	Root	arith.ml	[]	[ "LT", "LT_NZ", "NOT_LT", "ONE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m <= n) <=> (?d. n = m + d)	GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[LE] THENL [REWRITE_TAC[CONV_RULE(LAND_CONV SYM_CONV) (SPEC_ALL ADD_EQ_0)] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL]; EQ_TAC THENL [DISCH_THEN(DISJ_CASES_THEN2 SUBST1_TAC MP_TAC) THENL [EXISTS_TAC `0` THEN REWRITE_TAC[ADD_CLAUSES]; DIS...	prove	LE_EXISTS	Root	arith.ml	[]	[ "ADD_CLAUSES", "ADD_EQ_0", "EQ_ADD_LCANCEL", "EXISTS_REFL", "LE", "LEFT_IMP_EXISTS_THM", "RIGHT_EXISTS_AND_THM", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m < n) <=> (?d. n = m + SUC d)	GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[LT; ADD_CLAUSES; GSYM NOT_SUC] THEN ASM_REWRITE_TAC[SUC_INJ] THEN EQ_TAC THENL [DISCH_THEN(DISJ_CASES_THEN2 SUBST1_TAC MP_TAC) THENL [EXISTS_TAC `0` THEN REWRITE_TAC[ADD_CLAUSES]; DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST1_TAC) THEN EXISTS_TAC `SUC d` THEN REW...	prove	LT_EXISTS	Root	arith.ml	[]	[ "ADD_CLAUSES", "EQ_ADD_LCANCEL", "EXISTS_REFL", "LEFT_IMP_EXISTS_THM", "LT", "NOT_SUC", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. m <= m + n	GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[LE; ADD_CLAUSES; LE_REFL]	prove	LE_ADD	Root	arith.ml	[]	[ "ADD_CLAUSES", "LE", "LE_REFL" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. n <= m + n	ONCE_REWRITE_TAC[ADD_SYM] THEN MATCH_ACCEPT_TAC LE_ADD	prove	LE_ADDR	Root	arith.ml	[]	[ "ADD_SYM", "LE_ADD" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (m < m + n) <=> (0 < n)	INDUCT_TAC THEN ASM_REWRITE_TAC[ADD_CLAUSES; LT_SUC]	prove	LT_ADD	Root	arith.ml	[]	[ "ADD_CLAUSES", "LT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (n < m + n) <=> (0 < m)	ONCE_REWRITE_TAC[ADD_SYM] THEN MATCH_ACCEPT_TAC LT_ADD	prove	LT_ADDR	Root	arith.ml	[]	[ "ADD_SYM", "LT_ADD" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m + n) <= (m + p) <=> n <= p	REWRITE_TAC[LE_EXISTS; GSYM ADD_ASSOC; EQ_ADD_LCANCEL]	prove	LE_ADD_LCANCEL	Root	arith.ml	[]	[ "ADD_ASSOC", "EQ_ADD_LCANCEL", "LE_EXISTS" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m + p) <= (n + p) <=> (m <= n)	ONCE_REWRITE_TAC[ADD_SYM] THEN MATCH_ACCEPT_TAC LE_ADD_LCANCEL	prove	LE_ADD_RCANCEL	Root	arith.ml	[]	[ "ADD_SYM", "LE_ADD_LCANCEL" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m + n) < (m + p) <=> n < p	REWRITE_TAC[LT_EXISTS; GSYM ADD_ASSOC; EQ_ADD_LCANCEL; SUC_INJ]	prove	LT_ADD_LCANCEL	Root	arith.ml	[]	[ "ADD_ASSOC", "EQ_ADD_LCANCEL", "LT_EXISTS", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m + p) < (n + p) <=> (m < n)	ONCE_REWRITE_TAC[ADD_SYM] THEN MATCH_ACCEPT_TAC LT_ADD_LCANCEL	prove	LT_ADD_RCANCEL	Root	arith.ml	[]	[ "ADD_SYM", "LT_ADD_LCANCEL" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p q. m <= p /\ n <= q ==> m + n <= p + q	REPEAT GEN_TAC THEN REWRITE_TAC[LE_EXISTS] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `a:num`) (X_CHOOSE_TAC `b:num`)) THEN EXISTS_TAC `a + b` THEN ASM_REWRITE_TAC[ADD_AC]	prove	LE_ADD2	Root	arith.ml	[]	[ "ADD_AC", "LE_EXISTS" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p q. m <= p /\ n < q ==> m + n < p + q	REPEAT GEN_TAC THEN REWRITE_TAC[LE_EXISTS; LT_EXISTS] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `a:num`) (X_CHOOSE_TAC `b:num`)) THEN EXISTS_TAC `a + b` THEN ASM_REWRITE_TAC[SUC_INJ; ADD_CLAUSES; ADD_AC]	prove	LET_ADD2	Root	arith.ml	[]	[ "ADD_AC", "ADD_CLAUSES", "LE_EXISTS", "LT_EXISTS", "SUC_INJ" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p q. m < p /\ n <= q ==> m + n < p + q	ONCE_REWRITE_TAC[ADD_SYM; CONJ_SYM] THEN MATCH_ACCEPT_TAC LET_ADD2	prove	LTE_ADD2	Root	arith.ml	[]	[ "ADD_SYM", "CONJ_SYM", "LET_ADD2" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p q. m < p /\ n < q ==> m + n < p + q	REPEAT STRIP_TAC THEN MATCH_MP_TAC LTE_ADD2 THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LT_IMP_LE THEN ASM_REWRITE_TAC[]	prove	LT_ADD2	Root	arith.ml	[]	[ "LTE_ADD2", "LT_IMP_LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n. (0 < m * n) <=> (0 < m) /\ (0 < n)	REPEAT INDUCT_TAC THEN REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES; LT_0]	prove	LT_MULT	Root	arith.ml	[]	[ "ADD_CLAUSES", "LT_0", "MULT_CLAUSES" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p q. m <= n /\ p <= q ==> m * p <= n * q	REPEAT GEN_TAC THEN REWRITE_TAC[LE_EXISTS] THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `a:num`) (X_CHOOSE_TAC `b:num`)) THEN EXISTS_TAC `a * p + m * b + a * b` THEN ASM_REWRITE_TAC[LEFT_ADD_DISTRIB] THEN REWRITE_TAC[LEFT_ADD_DISTRIB; RIGHT_ADD_DISTRIB; ADD_ASSOC]	prove	LE_MULT2	Root	arith.ml	[]	[ "ADD_ASSOC", "LEFT_ADD_DISTRIB", "LE_EXISTS", "RIGHT_ADD_DISTRIB" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. ~(m = 0) /\ n < p ==> m * n < m * p	REPEAT GEN_TAC THEN REWRITE_TAC[LT_LE] THEN STRIP_TAC THEN CONJ_TAC THENL [MATCH_MP_TAC LE_MULT2 THEN ASM_REWRITE_TAC[LE_REFL]; ASM_REWRITE_TAC[EQ_MULT_LCANCEL]]	prove	LT_LMULT	Root	arith.ml	[]	[ "EQ_MULT_LCANCEL", "LE_MULT2", "LE_REFL", "LT_LE" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m * n) <= (m * p) <=> (m = 0) \/ n <= p	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES; LE_REFL; LE_0; NOT_SUC] THEN REWRITE_TAC[LE_SUC] THEN REWRITE_TAC[LE; LE_ADD_LCANCEL; GSYM ADD_ASSOC] THEN ASM_REWRITE_TAC[GSYM(el 4(CONJUNCTS MULT_CLAUSES)); NOT_SUC]	prove	LE_MULT_LCANCEL	Root	arith.ml	[]	[ "ADD_ASSOC", "ADD_CLAUSES", "LE", "LE_0", "LE_ADD_LCANCEL", "LE_REFL", "LE_SUC", "MULT_CLAUSES", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m * p) <= (n * p) <=> (m <= n) \/ (p = 0)	ONCE_REWRITE_TAC[MULT_SYM; DISJ_SYM] THEN MATCH_ACCEPT_TAC LE_MULT_LCANCEL	prove	LE_MULT_RCANCEL	Root	arith.ml	[]	[ "DISJ_SYM", "LE_MULT_LCANCEL", "MULT_SYM" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff
!m n p. (m * n) < (m * p) <=> ~(m = 0) /\ n < p	REPEAT INDUCT_TAC THEN ASM_REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES; LT_REFL; LT_0; NOT_SUC] THEN REWRITE_TAC[LT_SUC] THEN REWRITE_TAC[LT; LT_ADD_LCANCEL; GSYM ADD_ASSOC] THEN ASM_REWRITE_TAC[GSYM(el 4(CONJUNCTS MULT_CLAUSES)); NOT_SUC]	prove	LT_MULT_LCANCEL	Root	arith.ml	[]	[ "ADD_ASSOC", "ADD_CLAUSES", "LT", "LT_0", "LT_ADD_LCANCEL", "LT_REFL", "LT_SUC", "MULT_CLAUSES", "NOT_SUC" ]	https://github.com/jrh13/hol-light	87433ef35d8404ef3e336b8bdef8eddd09c768ff

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HOL-Light

A structured dataset of theorems and definitions from HOL Light, an interactive theorem prover for higher-order logic written in OCaml.

Source

Repository: https://github.com/jrh13/hol-light
Commit: 87433ef35d8404ef3e336b8bdef8eddd09c768ff
Files: 666
License: bsd-2-clause

Schema

Column	Type	Description
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has none
type	string	Declaration keyword
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted

Statistics

Entries: 38,435
With proof: 35,659 (92.8%)
With docstring: 0 (0.0%)
Libraries: 42

By type

Type	Count
prove	33,354
new_definition	2,161
prove_by_refinement	2,111
define	320
PROVE	203
new_recursive_definition	156
new_axiom	47
new_infix_definition	25
new_specification	22
new_type_definition	21
new_basic_definition	13
new_basic_type_definition	1
new_inductive_definition	1

Example

(!n. 0 + n = n) /\
   (!m. m + 0 = m) /\
   (!m n. (SUC m) + n = SUC(m + n)) /\
   (!m n. m + (SUC n) = SUC(m + n))
REWRITE_TAC[ADD; ADD_0; ADD_SUC]

type: prove | symbolic_name: ADD_CLAUSES | arith.ml

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{hol_light_dataset,
  title  = {HOL-Light},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/jrh13/hol-light, commit 87433ef35d84},
  url    = {https://huggingface.co/datasets/phanerozoic/HOL-Light}
}

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Collection

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