fact stringlengths 4 99 | 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 0
values | line_start int64 29 1.2k | line_end int64 29 1.2k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
false = (!(p:bool). p) | false | = (!(p:bool). p) | definition | false_def | src | src/bool.ml | [] | [] | null | 48 | 48 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
$\/ = (\p1 p2. !p. (p1 ==> p) ==> (p2 ==> p) ==> p) | $\/ | = (\p1 p2. !p. (p1 ==> p) ==> (p2 ==> p) ==> p) | definition | disj_def | src | src/bool.ml | [] | [] | null | 58 | 58 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
$~ = (\p. p ==> false) | $~ | = (\p. p ==> false) | definition | not_def | src | src/bool.ml | [] | [] | null | 83 | 83 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
$?! = (\(P:'a->bool). ?x. P x /\ (!y. P y ==> y = x)) | $?! | = (\(P:'a->bool). ?x. P x /\ (!y. P y ==> y = x)) | definition | uexists_def | src | src/bool.ml | [] | [] | null | 105 | 105 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
COND =
(\p (t1:'a) t2. @x. (p = true ==> x = t1) /\ (p = false ==> x = t2)) | COND | =
(\p (t1:'a) t2. @x. (p = true ==> x = t1) /\ (p = false ==> x = t2)) | definition | cond_def | src | src/bool.ml | [] | [] | null | 128 | 128 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
true | true | theorem | truth_thm | src | src/bool.ml | [] | [] | null | 166 | 166 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A1 u A2 |- p /\ q | A1 u A2 |- p /\ q | theorem | conj_lemma0 | src | src/bool.ml | [] | [] | null | 776 | 776 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
p1_ /\ p2_ <=> (!p. (p1_ ==> p2_ ==> p) ==> p) | p1_ /\ p2_ <=> (!p. (p1_ ==> p2_ ==> p) ==> p) | theorem | conj_lemma | src | src/bool.ml | [] | [] | null | 780 | 780 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A |- p | A |- p | theorem | conjunct1_lemma | src | src/bool.ml | [] | [] | null | 812 | 812 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A |- q | A |- q | theorem | conjunct2_lemma | src | src/bool.ml | [] | [] | null | 841 | 841 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A u A1\{p} u A2\{q} |- r | A u A1\{p} u A2\{q} |- r | theorem | disj_lemma0 | src | src/bool.ml | [] | [] | null | 876 | 876 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
p1_ \/ p2_ <=> (!p. (p1_ ==> p) ==> (p2_ ==> p) ==> p) | p1_ \/ p2_ <=> (!p. (p1_ ==> p) ==> (p2_ ==> p) ==> p) | theorem | disj_cases_lemma | src | src/bool.ml | [] | [] | null | 880 | 880 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A |- p \/ q | A |- p \/ q | theorem | disj1_lemma | src | src/bool.ml | [] | [] | null | 914 | 914 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A |- p \/ q | A |- p \/ q | theorem | disj2_lemma | src | src/bool.ml | [] | [] | null | 946 | 946 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!f g. f = g <=> (!x. f x = g x) | !f g. f = g <=> (!x. f x = g x) | theorem | fun_eq_thm | src | src/bool.ml | [] | [] | null | 1,032 | 1,032 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
$==> = (\p q. p /\ q <=> p) | $==> = (\p q. p /\ q <=> p) | theorem | imp_alt_def_thm | src | src/bool.ml | [] | [] | null | 1,056 | 1,056 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
~ true <=> false | ~ true <=> false | theorem | not_true_thm | src | src/boolalg.ml | [] | [] | null | 35 | 35 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
~ false <=> true | ~ false <=> true | theorem | not_false_thm | src | src/boolalg.ml | [] | [] | null | 52 | 52 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
~ (true <=> false) | ~ (true <=> false) | theorem | true_not_eq_false_thm | src | src/boolalg.ml | [] | [] | null | 66 | 66 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q. ~ (p \/ q) <=> ~ p /\ ~ q | !p q. ~ (p \/ q) <=> ~ p /\ ~ q | theorem | not_dist_disj_thm | src | src/boolalg.ml | [] | [] | null | 80 | 80 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p /\ true <=> p | !p. p /\ true <=> p | theorem | conj_id_thm | src | src/boolalg.ml | [] | [] | null | 110 | 110 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p /\ false <=> false | !p. p /\ false <=> false | theorem | conj_zero_thm | src | src/boolalg.ml | [] | [] | null | 122 | 122 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p /\ p <=> p | !p. p /\ p <=> p | theorem | conj_idem_thm | src | src/boolalg.ml | [] | [] | null | 134 | 134 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q. p /\ q <=> q /\ p | !p q. p /\ q <=> q /\ p | theorem | conj_comm_thm | src | src/boolalg.ml | [] | [] | null | 146 | 146 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. p /\ (q /\ r) <=> (p /\ q) /\ r | !p q r. p /\ (q /\ r) <=> (p /\ q) /\ r | theorem | conj_assoc_thm | src | src/boolalg.ml | [] | [] | null | 160 | 160 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q. p /\ (p \/ q) <=> p | !p q. p /\ (p \/ q) <=> p | theorem | conj_absorb_disj_thm | src | src/boolalg.ml | [] | [] | null | 184 | 184 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. p /\ (q \/ r) <=> (p /\ q) \/ (p /\ r) | !p q r. p /\ (q \/ r) <=> (p /\ q) \/ (p /\ r) | theorem | conj_dist_right_disj_thm | src | src/boolalg.ml | [] | [] | null | 197 | 197 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. (p \/ q) /\ r <=> (p /\ r) \/ (q /\ r) | !p q r. (p \/ q) /\ r <=> (p /\ r) \/ (q /\ r) | theorem | conj_dist_left_disj_thm | src | src/boolalg.ml | [] | [] | null | 227 | 227 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p /\ ~ p <=> false | !p. p /\ ~ p <=> false | theorem | conj_contr_thm | src | src/boolalg.ml | [] | [] | null | 242 | 242 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p \/ false <=> p | !p. p \/ false <=> p | theorem | disj_id_thm | src | src/boolalg.ml | [] | [] | null | 259 | 259 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p \/ true <=> true | !p. p \/ true <=> true | theorem | disj_zero_thm | src | src/boolalg.ml | [] | [] | null | 275 | 275 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p \/ p <=> p | !p. p \/ p <=> p | theorem | disj_idem_thm | src | src/boolalg.ml | [] | [] | null | 287 | 287 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q. p \/ q <=> q \/ p | !p q. p \/ q <=> q \/ p | theorem | disj_comm_thm | src | src/boolalg.ml | [] | [] | null | 300 | 300 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. p \/ (q \/ r) <=> (p \/ q) \/ r | !p q r. p \/ (q \/ r) <=> (p \/ q) \/ r | theorem | disj_assoc_thm | src | src/boolalg.ml | [] | [] | null | 315 | 315 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q. p \/ (p /\ q) <=> p | !p q. p \/ (p /\ q) <=> p | theorem | disj_absorb_conj_thm | src | src/boolalg.ml | [] | [] | null | 341 | 341 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. p \/ (q /\ r) <=> (p \/ q) /\ (p \/ r) | !p q r. p \/ (q /\ r) <=> (p \/ q) /\ (p \/ r) | theorem | disj_dist_right_conj_thm | src | src/boolalg.ml | [] | [] | null | 357 | 357 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. (p /\ q) \/ r <=> (p \/ r) /\ (q \/ r) | !p q r. (p /\ q) \/ r <=> (p \/ r) /\ (q \/ r) | theorem | disj_dist_left_conj_thm | src | src/boolalg.ml | [] | [] | null | 388 | 388 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p ==> true | !p. p ==> true | theorem | imp_right_zero_thm | src | src/boolalg.ml | [] | [] | null | 403 | 403 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. (true ==> p) <=> p | !p. (true ==> p) <=> p | theorem | imp_left_id_thm | src | src/boolalg.ml | [] | [] | null | 412 | 412 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. false ==> p | !p. false ==> p | theorem | imp_left_zero_thm | src | src/boolalg.ml | [] | [] | null | 424 | 424 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. p ==> p | !p. p ==> p | theorem | imp_refl_thm | src | src/boolalg.ml | [] | [] | null | 433 | 433 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. (p \/ q ==> r) <=> (p ==> r) /\ (q ==> r) | !p q r. (p \/ q ==> r) <=> (p ==> r) /\ (q ==> r) | theorem | imp_dist_left_disj_thm | src | src/boolalg.ml | [] | [] | null | 442 | 442 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. (p ==> q /\ r) <=> (p ==> q) /\ (p ==> r) | !p q r. (p ==> q /\ r) <=> (p ==> q) /\ (p ==> r) | theorem | imp_dist_right_conj_thm | src | src/boolalg.ml | [] | [] | null | 469 | 469 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q r. (p ==> q ==> r) <=> (p /\ q ==> r) | !p q r. (p ==> q ==> r) <=> (p /\ q ==> r) | theorem | imp_imp_thm | src | src/boolalg.ml | [] | [] | null | 490 | 490 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P Q. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x) | !P Q. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x) | theorem | forall_dist_conj_thm | src | src/boolalg.ml | [] | [] | null | 513 | 513 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P a. (!x. x = a ==> P x) <=> P a | !P a. (!x. x = a ==> P x) <=> P a | theorem | forall_one_point_thm | src | src/boolalg.ml | [] | [] | null | 534 | 534 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!t. (!(x:'a). t) <=> t | !t. (!(x:'a). t) <=> t | theorem | forall_null_thm | src | src/boolalg.ml | [] | [] | null | 553 | 553 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!(a:'a). (@x. x = a) = a | !(a:'a). (@x. x = a) = a | theorem | select_eq_thm | src | src/boolclass.ml | [] | [] | null | 101 | 101 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P Q. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x) | !P Q. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x) | theorem | exists_dist_disj_thm | src | src/boolclass.ml | [] | [] | null | 114 | 114 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P a. (?x. x = a /\ P x) <=> P a | !P a. (?x. x = a /\ P x) <=> P a | theorem | exists_one_point_thm | src | src/boolclass.ml | [] | [] | null | 144 | 144 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!x. (?y. y = x) | !x. (?y. y = x) | theorem | exists_value_thm | src | src/boolclass.ml | [] | [] | null | 162 | 162 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!t. (?x. t) <=> t | !t. (?x. t) <=> t | theorem | exists_null_thm | src | src/boolclass.ml | [] | [] | null | 173 | 173 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. (?!x. P x) <=> (?x. P x /\ (!y. P y ==> y = x)) | !P. (?!x. P x) <=> (?x. P x /\ (!y. P y ==> y = x)) | theorem | uexists_thm1 | src | src/boolclass.ml | [] | [] | null | 186 | 186 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. (?!x. P x) <=> (?x. !y. P y <=> x = y) | !P. (?!x. P x) <=> (?x. !y. P y <=> x = y) | theorem | uexists_thm2 | src | src/boolclass.ml | [] | [] | null | 198 | 198 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. (?!x. P x) <=> (?x. P x) /\ (!x x'. P x /\ P x' ==> x = x') | !P. (?!x. P x) <=> (?x. P x) /\ (!x x'. P x /\ P x' ==> x = x') | theorem | uexists_thm3 | src | src/boolclass.ml | [] | [] | null | 228 | 228 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P a. (?!x. x = a /\ P x) <=> P a | !P a. (?!x. x = a /\ P x) <=> P a | theorem | uexists_one_point_thm | src | src/boolclass.ml | [] | [] | null | 269 | 269 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. (!x. ?y. P x y) <=> (?f. !x. P x (f x)) | !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x)) | theorem | skolem_thm | src | src/boolclass.ml | [] | [] | null | 293 | 293 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. (!x. ?!y. P x y) <=> (?f. !x y. P x y <=> f x = y) | !P. (!x. ?!y. P x y) <=> (?f. !x y. P x y <=> f x = y) | theorem | unique_skolem_thm | src | src/boolclass.ml | [] | [] | null | 315 | 315 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. ~ (?x. P x) <=> (!x. ~ P x) | !P. ~ (?x. P x) <=> (!x. ~ P x) | theorem | not_dist_exists_thm | src | src/boolclass.ml | [] | [] | null | 335 | 335 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. (p <=> true) \/ (p <=> false) | !p. (p <=> true) \/ (p <=> false) | theorem | bool_cases_thm | src | src/boolclass.ml | [] | [] | null | 407 | 407 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A\{~ p} |- p | A\{~ p} |- p | theorem | ccontr_lemma | src | src/boolclass.ml | [] | [] | null | 437 | 437 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p. ~ ~ p <=> p | !p. ~ ~ p <=> p | theorem | not_dneg_thm | src | src/boolclass.ml | [] | [] | null | 467 | 467 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q. (p ==> q) <=> (~ p \/ q) | !p q. (p ==> q) <=> (~ p \/ q) | theorem | imp_disj_thm | src | src/boolclass.ml | [] | [] | null | 490 | 490 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p q. ~ (p /\ q) <=> ~ p \/ ~ q | !p q. ~ (p /\ q) <=> ~ p \/ ~ q | theorem | not_dist_conj_thm | src | src/boolclass.ml | [] | [] | null | 514 | 514 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. ~ (!x. P x) <=> (?x. ~ P x) | !P. ~ (!x. P x) <=> (?x. ~ P x) | theorem | not_dist_forall_thm | src | src/boolclass.ml | [] | [] | null | 542 | 542 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!t1 t2. (if true then t1 else t2) = t1 | !t1 t2. (if true then t1 else t2) = t1 | theorem | cond_true_thm | src | src/boolclass.ml | [] | [] | null | 567 | 567 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!t1 t2. (if false then t1 else t2) = t2 | !t1 t2. (if false then t1 else t2) = t2 | theorem | cond_false_thm | src | src/boolclass.ml | [] | [] | null | 598 | 598 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p t. (if p then t else t) = t | !p t. (if p then t else t) = t | theorem | cond_idem_thm | src | src/boolclass.ml | [] | [] | null | 633 | 633 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!p t1 t2. (if ~ p then t1 else t2) = (if p then t2 else t1) | !p t1 t2. (if ~ p then t1 else t2) = (if p then t2 else t1) | theorem | cond_not_thm | src | src/boolclass.ml | [] | [] | null | 653 | 653 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A |- ~ p1 <=> ~ p2 | A |- ~ p1 <=> ~ p2 | theorem | not_fn | src | src/eqcong.ml | [] | [] | null | 198 | 198 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A1 u A2 |- p1 /\ q1 <=> p2 /\ q2 | A1 u A2 |- p1 /\ q1 <=> p2 /\ q2 | theorem | conj_fn | src | src/eqcong.ml | [] | [] | null | 219 | 219 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A1 u A2 |- p1 \/ q1 <=> p2 \/ q2 | A1 u A2 |- p1 \/ q1 <=> p2 \/ q2 | theorem | disj_fn | src | src/eqcong.ml | [] | [] | null | 281 | 281 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
A1 u A2 |- p1 ==> q1 <=> p2 ==> q2 | A1 u A2 |- p1 ==> q1 <=> p2 ==> q2 | theorem | imp_fn | src | src/eqcong.ml | [] | [] | null | 343 | 343 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
LET = (\(f:'a->'b) (x:'a). f x) | LET | = (\(f:'a->'b) (x:'a). f x) | definition | let_def | src | src/equal.ml | [] | [] | null | 29 | 29 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
ONTO = (\(f:'a->'b). !y. ?x. y = f x) | ONTO | = (\(f:'a->'b). !y. ?x. y = f x) | definition | onto_def | src | src/equal.ml | [] | [] | null | 63 | 63 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
?(f:ind->ind). ONE_ONE f /\ ~ ONTO f | ?(f:ind->ind). ONE_ONE f /\ ~ ONTO f | axiom | infinity_ax | src | src/ind.ml | [] | [] | null | 45 | 45 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!f. ~ ONTO f <=> ?y. !x. ~(f x = y) | !f. ~ ONTO f <=> ?y. !x. ~(f x = y) | theorem | not_onto_lemma | src | src/ind.ml | [] | [] | null | 69 | 69 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
?(s:ind->ind) z. ONE_ONE s /\ (!i. ~(s i = z)) | ?(s:ind->ind) z. ONE_ONE s /\ (!i. ~(s i = z)) | theorem | ind_suc_zero_exists_lemma | src | src/ind.ml | [] | [] | null | 90 | 90 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!i j. IND_SUC i = IND_SUC j <=> i = j | !i j. IND_SUC i = IND_SUC j <=> i = j | theorem | ind_suc_injective_thm | src | src/ind.ml | [] | [] | null | 129 | 129 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!i. ~(IND_SUC i = IND_ZERO) | !i. ~(IND_SUC i = IND_ZERO) | theorem | ind_suc_not_zero_thm | src | src/ind.ml | [] | [] | null | 155 | 155 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!(i:ind). IsNatRep i <=>
(!P. P IND_ZERO /\ (!j. P j ==> P (IND_SUC j)) ==> P i) | !(i:ind). IsNatRep i <=>
(!P. P IND_ZERO /\ (!j. P j ==> P (IND_SUC j)) ==> P i) | definition | is_nat_rep_def | src | src/nat.ml | [] | [] | null | 48 | 48 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
IsNatRep IND_ZERO | IsNatRep IND_ZERO | theorem | ind_zero_is_nat_rep_lemma | src | src/nat.ml | [] | [] | null | 58 | 58 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!i. IsNatRep i ==> IsNatRep (IND_SUC i) | !i. IsNatRep i ==> IsNatRep (IND_SUC i) | theorem | ind_suc_is_nat_rep_lemma | src | src/nat.ml | [] | [] | null | 76 | 76 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!n. IsNatRep (NatRep n) | !n. IsNatRep (NatRep n) | theorem | is_nat_rep_lemma | src | src/nat.ml | [] | [] | null | 139 | 139 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!m n. NatRep m = NatRep n <=> m = n | !m n. NatRep m = NatRep n <=> m = n | theorem | nat_rep_injective_lemma | src | src/nat.ml | [] | [] | null | 154 | 154 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
ZERO = NatAbs IND_ZERO | ZERO | = NatAbs IND_ZERO | definition | zero_def | src | src/nat.ml | [] | [] | null | 180 | 180 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
!n. SUC n = NatAbs (IND_SUC (NatRep n)) | !n. SUC n | = NatAbs (IND_SUC (NatRep n)) | definition | suc_def | src | src/nat.ml | [] | [] | null | 184 | 184 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
NatRep ZERO = IND_ZERO | NatRep ZERO = IND_ZERO | theorem | nat_rep_zero_lemma | src | src/nat.ml | [] | [] | null | 209 | 209 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!n. NatRep (SUC n) = IND_SUC (NatRep n) | !n. NatRep (SUC n) = IND_SUC (NatRep n) | theorem | nat_rep_suc_lemma | src | src/nat.ml | [] | [] | null | 224 | 224 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!n. ~ (SUC n = ZERO) | !n. ~ (SUC n = ZERO) | theorem | suc_not_zero_thm0 | src | src/nat.ml | [] | [] | null | 245 | 245 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!m n. SUC m = SUC n <=> m = n | !m n. SUC m = SUC n <=> m = n | theorem | suc_injective_thm | src | src/nat.ml | [] | [] | null | 271 | 271 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!P. P ZERO /\ (!n. P n ==> P (SUC n)) ==> (!n. P n) | !P. P ZERO /\ (!n. P n ==> P (SUC n)) ==> (!n. P n) | theorem | nat_induction_thm0 | src | src/nat.ml | [] | [] | null | 300 | 300 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!n. n = ZERO \/ (?m. n = SUC m) | !n. n = ZERO \/ (?m. n = SUC m) | theorem | nat_cases_thm0 | src | src/nat.ml | [] | [] | null | 369 | 369 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!n. ?!y. PRG n y <PRG-functional> | !n. ?!y. PRG n y <PRG-functional> | theorem | lemma3 | src | src/nat.ml | [] | [] | null | 622 | 622 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
?fn. fn ZERO = e /\ (!n. fn (SUC n) = f (fn n) n) | ?fn. fn ZERO = e /\ (!n. fn (SUC n) = f (fn n) n) | theorem | lemma4 | src | src/nat.ml | [] | [] | null | 747 | 747 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!e f. ?fn. fn ZERO = e /\ (!n. fn (SUC n) = f (fn n) n) | !e f. ?fn. fn ZERO = e /\ (!n. fn (SUC n) = f (fn n) n) | theorem | nat_recursion_thm0 | src | src/nat.ml | [] | [] | null | 773 | 773 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
(!n. 0 + n = n) /\
(!m n. (SUC m) + n = SUC (m + n)) | (!n. 0 + n | = n) /\
(!m n. (SUC m) + n = SUC (m + n)) | definition | add_def | src | src/natarith.ml | [] | [] | null | 45 | 45 | true | http://www.proof-technologies.com/holzero/ | 0.6.3 |
!n. 0 + n = n | !n. 0 + n = n | theorem | add_left_id_lemma | src | src/natarith.ml | [] | [] | null | 66 | 66 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!m n. (SUC m) + n = SUC (m + n) | !m n. (SUC m) + n = SUC (m + n) | theorem | add_dist_left_suc_thm | src | src/natarith.ml | [] | [] | null | 73 | 73 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 | |
!m n. m + (SUC n) = SUC (m + n) | !m n. m + (SUC n) = SUC (m + n) | theorem | add_dist_right_suc_thm | src | src/natarith.ml | [] | [] | null | 82 | 82 | false | http://www.proof-technologies.com/holzero/ | 0.6.3 |
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