statement stringlengths 2 99 | proof stringclasses 25
values | type stringclasses 3
values | symbolic_name stringlengths 6 25 | library stringclasses 1
value | filename stringclasses 12
values | imports listlengths 0 0 | deps listlengths 0 0 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
!n. n < SUC n | theorem | lt_suc_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!n. 0 < SUC n | theorem | zero_lt_suc_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!n. n = 0 \/ 0 < n | theorem | lt_zero_cases_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. SUC m < SUC n <=> m < n | theorem | suc_lt_cancel_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!n. ~ (n < n) | theorem | lt_irrefl_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!l m n. l < m /\ m < n ==> l < n | theorem | lt_trans_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. ~ (m < n /\ n < m) | theorem | lt_asym_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!n. 0 < n <=> ~ (n = 0) | theorem | zero_lt_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. ~ (m < n) <=> n = m \/ n < m | theorem | not_lt_cases_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!l m n. l + n < m + n <=> l < m | theorem | add_lt_cancel_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!l m n. ~ (l = 0) ==> l * m < l * n <=> m < n | theorem | mult_lt_cancel_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. m < | = n <=> m < n \/ m = n | definition | le_def | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!n. n <= n | theorem | le_refl_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. m <= n /\ n <= m ==> m = n | theorem | le_antisym_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!l m n. l <= m /\ m <= n ==> l <= n | theorem | le_trans_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!n. n <= 0 <=> n = 0 | theorem | le_zero_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!n. 0 <= n | theorem | zero_le_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. SUC m <= SUC n <=> m <= n | theorem | suc_le_cancel_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!l m n. l + n <= m + n <=> l <= m | theorem | add_le_cancel_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!l m n. ~ (l = 0) ==> l * m <= l * n <=> m <= n | theorem | mult_le_cancel_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. m < SUC n <=> m <= n | theorem | lt_suc_le_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. SUC m <= n <=> m < n | theorem | suc_le_lt_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. m <= n ==> m - n = 0 | theorem | sub_floor_thm | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. m > n <=> n < m | definition | gt_def | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!m n. m > | = n <=> n <= m | definition | ge_def | src | src/natrel.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
MkPairRep | = \(x:'a) (y:'b). \a b. a = x /\ b = y | definition | mk_pair_rep_def | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
IsPairRep | = \(r:'a->'b->bool). ?a b. r = MkPairRep a b | definition | is_pair_rep_def | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!c d. ?a b. MkPairRep c d = MkPairRep a b | theorem | mk_pair_rep_lemma | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
PAIR | = \(x:'a) (y:'b). PairAbs (MkPairRep x y) | definition | pair_def | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!x y. PairRep (PairAbs (MkPairRep x y)) = MkPairRep x y | theorem | rep_abs_pair_lemma | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!x y u v. (x,y) = (u,v) <=> x = u /\ y = v | theorem | pair_eq_thm | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!p. ?x y. p = (x,y) | theorem | pair_surjective_thm | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
FST | = \(p:'a#'b). @x. ?y. p = (x,y) | definition | fst_def | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
SND | = \(p:'a#'b). @y. ?x. p = (x,y) | definition | snd_def | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!x y. FST (x,y) = x | theorem | fst_thm | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!x y. SND (x,y) = y | theorem | snd_thm | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 | ||
!p. (FST p, SND p) = p | theorem | fst_snd_thm | src | src/pair.ml | [] | [] | http://www.proof-technologies.com/holzero/ | 0.6.3 |
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