Datasets:

phanerozoic
/

Abella

statement stringlengths 13 379	proof stringlengths 5 4.7k	type stringclasses 3 values	symbolic_name stringlengths 1 33	library stringclasses 19 values	filename stringclasses 67 values	imports listlengths 0 0	deps listlengths 0 18	source_url stringclasses 1 value	commit stringclasses 1 value
$form : form -> prop	by ; $form (atom A) ; $form (tens F G) := $form F /\ $form G ; $form one ; $form (par F G) := $form F /\ $form G ; $form bot.	Define	$form	examples/compute	examples/compute/mll.thm	[]	[ "one" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
$list : list form -> prop	by ; $list nil ; $list (F :: L) := $form F /\ $list L.	Define	$list	examples/compute	examples/compute/mll.thm	[]	[ "$form" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
adj : list form -> form -> list form -> prop	by ; adj L X (X :: L) := $form X /\ $list L ; adj (Y :: L) X (Y :: K) := $form Y /\ adj L X K.	Define	adj	examples/compute	examples/compute/mll.thm	[]	[ "$form", "$list" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
$adj : forall K E L, adj K E L -> $list K /\ $form E /\ $list L.	induction on 1. intros. case H1. search. apply IH to *H3. search.	Theorem	$adj	examples/compute	examples/compute/mll.thm	[]	[ "$form", "$list", "adj" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm : list form -> list form -> prop	by ; perm nil nil ; perm L1 L2 := exists E K1 K2, adj K1 E L1 /\ adj K2 E L2 /\ perm K1 K2.	Define	perm	examples/compute	examples/compute/mll.thm	[]	[ "adj" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
$perm : forall K L, perm K L -> $list K /\ $list L.	induction on 1. intros. case H1. search. apply $adj to H2. apply $adj to H3. apply IH to *H4. search.	Theorem	$perm	examples/compute	examples/compute/mll.thm	[]	[ "$adj", "$list", "perm" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_refl : forall L, $list L -> perm L L.	induction on 1. intros. case H1. search. apply IH to *H3. apply $perm to H4. search.	Theorem	perm_refl	examples/compute	examples/compute/mll.thm	[]	[ "$list", "$perm", "perm" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_sym : forall K L, perm K L -> perm L K.	induction on 1. intros. case H1. search. apply IH to H4. search.	Theorem	perm_sym	examples/compute	examples/compute/mll.thm	[]	[ "perm" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
adj_swap : forall E F J K L, adj J E K -> adj K F L -> exists U, adj J F U /\ adj U E L.	induction on 2. intros. case H2. case H1. search. apply $adj to H6. search. case H1. apply $adj to H4. search. apply IH to H6 H4. search.	Theorem	adj_swap	examples/compute	examples/compute/mll.thm	[]	[ "$adj", "adj" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_cons_1 : forall E J L, perm (E :: J) L -> exists K, adj K E L /\ perm J K.	induction on 1. intros. case H1. case H2. search. apply IH to H4. apply adj_swap to H7 *H3. search.	Theorem	perm_cons_1	examples/compute	examples/compute/mll.thm	[]	[ "adj", "adj_swap", "perm" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_cons_2 : forall E L K, perm L (E :: K) -> exists J, adj J E L /\ perm J K.	induction on 1. intros. case H1. case H3. search. apply IH to H4. apply adj_swap to H7 *H2. search.	Theorem	perm_cons_2	examples/compute	examples/compute/mll.thm	[]	[ "adj", "adj_swap", "perm" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_trans_lem : forall J K L, $list K -> perm J K -> perm K L -> perm J L.	induction on 1. intros. case H1. compute H2 H3. search. case H6. apply perm_cons_2 to H2. apply perm_cons_1 to H3. apply IH to H5 H7 H9. search.	Theorem	perm_trans_lem	examples/compute	examples/compute/mll.thm	[]	[ "$list", "perm", "perm_cons_1", "perm_cons_2" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_trans : forall J K L, perm J K -> perm K L -> perm J L.	intros. apply $perm to H1. backchain perm_trans_lem with K = K.	Theorem	perm_trans	examples/compute	examples/compute/mll.thm	[]	[ "$perm", "perm", "perm_trans_lem" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
$nat : nat -> prop	by ; $nat z ; $nat (s X) := $nat X.	Define	$nat	examples/compute	examples/compute/nats.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
ten : forall X, $nat (s^10 X) -> $nat X.	intros. compute H1. search.	Theorem	ten	examples/compute	examples/compute/nats.thm	[]	[ "$nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
leq : nat -> nat -> prop	by ; leq z X ; leq (s X) (s Y) := leq X Y.	Define	leq	examples/compute	examples/compute/nats.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
filter : forall X L, member X (s^10 z :: s^9 z :: s^8 z :: L) -> leq X (s^5 z) -> member X L.	/* Built-in: Suspend member E L := L. */ intros. compute H1 H2. search.	Theorem	filter	examples/compute	examples/compute/nats.thm	[]	[ "leq" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
filter_alt : forall X L, member X (s^10 z :: s^9 z :: s^8 z :: L) -> leq X (s^5 z) -> member X L.	% the "compute (all)" form tries compute on all hypotheses intros. compute (*all). search.	Theorem	filter_alt	examples/compute	examples/compute/nats.thm	[]	[ "leq" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
plus : nat -> nat -> nat -> prop	by ; plus z X X := $nat X ; plus (s X) Y (s Z) := plus X Y Z.	Define	plus	examples/compute	examples/compute/nats.thm	[]	[ "$nat", "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
eight : forall X, plus (s^3 z) X (s^8 z) -> X = s^5 z.	intros. compute (*all). search.	Theorem	eight	examples/compute	examples/compute/nats.thm	[]	[ "plus" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
nine : forall X Y, plus (s^3 z) X (s^8 z) -> plus Y (s^4 z) (s^9 z) -> X = Y.	intros. compute (all). search.	Theorem	nine	examples/compute	examples/compute/nats.thm	[]	[ "plus" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
plus_invert : forall M N K, plus M N K -> $nat M /\ $nat N /\ $nat K.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	plus_invert	examples/compute	examples/compute/nats.thm	[]	[ "$nat", "plus" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
plus_zero : forall N, $nat N -> plus N z N.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	plus_zero	examples/compute	examples/compute/nats.thm	[]	[ "$nat", "plus" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
plus_succ : forall M N K, plus M N K -> plus M (s N) (s K).	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	plus_succ	examples/compute	examples/compute/nats.thm	[]	[ "plus" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
plus_comm : forall M N K, $nat K -> plus M N K -> plus N M K.	induction on 2. intros. case H2. apply plus_zero to H1. search. compute *H1. apply IH to H4 H3. apply plus_succ to H5. search.	Theorem	plus_comm	examples/compute	examples/compute/nats.thm	[]	[ "$nat", "plus", "plus_succ", "plus_zero" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
plus_assoc : forall A B C AB ABC, plus A B AB -> plus AB C ABC -> exists BC, plus B C BC /\ plus A BC ABC.	induction on 1. intros. case H1. apply plus_invert to H2. search. compute *H2. apply IH to H3 H4. search.	Theorem	plus_assoc	examples/compute	examples/compute/nats.thm	[]	[ "plus", "plus_invert" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
plus_det : forall M N J K, plus M N J -> plus M N K -> J = K.	induction on 1. intros. case H1. case H2. search. case H2. apply IH to H3 H4. search.	Theorem	plus_det	examples/compute	examples/compute/nats.thm	[]	[ "plus" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fib : nat -> nat -> prop	by ; fib z z ; fib (s z) (s z) ; fib (s (s X)) F := exists A B, fib (s X) A /\ fib X B /\ plus A B F.	Define	fib	examples/compute	examples/compute/nats.thm	[]	[ "nat", "plus" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fib_three : forall F, fib (s^3 z) F -> F = s^2 z.	intros. compute *H1. search.	Theorem	fib_three	examples/compute	examples/compute/nats.thm	[]	[ "fib" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fib_four : forall F, fib (s^4 z) F -> F = s^3 z.	intros. compute *H1. search.	Theorem	fib_four	examples/compute	examples/compute/nats.thm	[]	[ "fib" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fib_five : forall F, fib (s^5 z) F -> F = s^5 z.	intros. compute *H1. search.	Theorem	fib_five	examples/compute	examples/compute/nats.thm	[]	[ "fib" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fib_six : forall F, fib (s^6 z) F -> F = s^8 z.	intros. compute *H1. search.	Theorem	fib_six	examples/compute	examples/compute/nats.thm	[]	[ "fib" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fib_seven : forall F, fib (s^7 z) F -> F = s^13 z.	intros. compute *H1. search.	Theorem	fib_seven	examples/compute	examples/compute/nats.thm	[]	[ "fib" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fib_det : forall A B C, fib A B -> fib A C -> B = C.	induction on 1. intros. case H1. compute H2. search. compute H2. search. compute *H2. apply IH to H4 H7. apply IH to H3 H8. apply plus_det to H5 H6. search.	Theorem	fib_det	examples/compute	examples/compute/nats.thm	[]	[ "fib", "plus_det" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
three_solutions : forall N, (N = z \/ N = s z \/ N = s^5 z) -> fib N N.	intros. compute H1. search. search. search 10.	Theorem	three_solutions	examples/compute	examples/compute/nats.thm	[]	[ "fib" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
p_cases : forall X Y Z, {p X, p Y \|- p Z} -> Z = X \/ Z = Y.	intros. compute H1. search. search.	Theorem	p_cases	examples/compute	examples/compute/nats.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
spec_ten : forall X, {is_nat (s^10 X)} -> {is_nat X}.	intros. compute H1. search.	Theorem	spec_ten	examples/compute	examples/compute/nats.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
no_progress : forall X Y, {is_nat X \|- is_nat Y} -> false.	intros. compute H1. % no progress is possible because is_nat Y is suspended abort.	Theorem	no_progress	examples/compute	examples/compute/nats.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
no_progress_nabla : nabla X Y, {is_nat X \|- is_nat Y} -> false.	intros. compute H1.	Theorem	no_progress_nabla	examples/compute	examples/compute/nats.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
total : (A -> prop) -> prop	by total P := exists X, P X.	Define	total	examples/compute	examples/compute/rels.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
deterministic : (A -> prop) -> prop	by deterministic P := forall X Y, P X -> P Y -> X = Y.	Define	deterministic	examples/compute	examples/compute/rels.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
singleton : (A -> prop) -> prop	by singleton P := total P /\ deterministic P.	Define	singleton	examples/compute	examples/compute/rels.thm	[]	[ "deterministic", "total" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
deterministic_upto : (A -> prop) -> (A -> A -> prop) -> prop	by deterministic_upto P Eq := forall X Y, P X -> P Y -> Eq X Y.	Define	deterministic_upto	examples/compute	examples/compute/rels.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
singleton_upto : (A -> prop) -> (A -> A -> prop) -> prop	by singleton_upto P Eq := total P /\ deterministic_upto P Eq.	Define	singleton_upto	examples/compute	examples/compute/rels.thm	[]	[ "deterministic_upto", "total" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
reflexive : (A -> A -> prop) -> prop	by reflexive P := forall X, P X X.	Define	reflexive	examples/compute	examples/compute/rels.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
symmetric : (A -> A -> prop) -> prop	by symmetric P := forall X Y, P X Y -> P Y X.	Define	symmetric	examples/compute	examples/compute/rels.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
transitive : (A -> A -> prop) -> prop	by transitive P := forall X Y Z, P X Y -> P Y Z -> P X Z.	Define	transitive	examples/compute	examples/compute/rels.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
equivalence : (A -> A -> prop) -> prop	by equivalence P := reflexive P /\ symmetric P /\ transitive P.	Define	equivalence	examples/compute	examples/compute/rels.thm	[]	[ "reflexive", "symmetric", "transitive" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
preorder : (A -> A -> prop) -> prop	by preorder P := reflexive P /\ transitive P.	Define	preorder	examples/compute	examples/compute/rels.thm	[]	[ "reflexive", "transitive" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
similar : (A -> A -> prop) -> prop	by similar P := reflexive P /\ symmetric P.	Define	similar	examples/compute	examples/compute/rels.thm	[]	[ "reflexive", "symmetric" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
upto : (A -> prop) -> (A -> A -> prop) -> A -> prop	by upto P Eq X := exists Y, Eq X Y /\ P Y.	Define	upto	examples/compute	examples/compute/rels.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
change_forall_exists : total p -> (forall x, p x -> q x) -> (exists x, p x /\ q x).	intros. compute H1. apply H2 to H3. search.	Theorem	change_forall_exists	examples/compute	examples/compute/rels.thm	[]	[ "total" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
change_exists_forall : deterministic p -> (exists x, p x /\ q x) -> (forall x, p x -> q x).	intros. compute H1 H2. apply H6 to H3 H5. search.	Theorem	change_exists_forall	examples/compute	examples/compute/rels.thm	[]	[ "deterministic" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
change_forall_exists_upto : total p -> (forall x, p x -> upto q eqv x) -> (exists x, p x /\ upto q eqv x).	/* minor variant of change_forall_exists / intros. compute H1. apply H2 to H3. compute H4. search.	Theorem	change_forall_exists_upto	examples/compute	examples/compute/rels.thm	[]	[ "total", "upto" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
change_exists_forall_upto : deterministic_upto p eqv -> transitive eqv -> (exists x, p x /\ upto q eqv x) -> (forall x, p x -> upto q eqv x).	intros. compute H1 H2 *H3. apply H9 to H4 H7. apply H8 to H10 H6. search.	Theorem	change_exists_forall_upto	examples/compute	examples/compute/rels.thm	[]	[ "deterministic_upto", "transitive", "upto" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
nat : nat -> prop	by nat z ; nat (s X) := nat X.	Define	nat	examples/first-order	examples/first-order/ackermann.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
ack : nat -> nat -> nat -> prop	by ack z N (s N) ; ack (s M) z R := ack M (s z) R ; ack (s M) (s N) R := exists R', ack (s M) N R' /\ ack M R' R.	Define	ack	examples/first-order	examples/first-order/ackermann.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
ack_total : forall M N, nat M -> nat N -> exists R, nat R /\ ack M N R.	induction on 1. induction on 2. intros. case H1 (keep). search. case H2. apply IH to H3 _ with N = s z. search. apply IH1 to H1 H4. apply IH to H3 H5. search.	Theorem	ack_total	examples/first-order	examples/first-order/ackermann.thm	[]	[ "ack", "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
add_base : forall N, {nat N} -> {add N z N}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	add_base	examples/first-order	examples/first-order/add.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
add_step : forall A B C, {add A B C} -> {add A (s B) (s C)}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	add_step	examples/first-order	examples/first-order/add.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
add_comm : forall A B C, {nat B} -> {add A B C} -> {add B A C}.	induction on 2. intros. case H2. apply add_base to H1. search. apply IH to H1 H3. apply add_step to H4. search.	Theorem	add_comm	examples/first-order	examples/first-order/add.thm	[]	[ "add_base", "add_step", "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
add_det : forall A B C D, {add A B C} -> {add A B D} -> C = D.	induction on 1. intros. case H1. case H2. search. case H2. apply IH to H3 H4. search.	Theorem	add_det	examples/first-order	examples/first-order/add.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
add_assoc : forall A B C AB ABC, {add A B AB} -> {add AB C ABC} -> exists BC, {add B C BC} /\ {add A BC ABC}.	induction on 1. intros. case H1. search. case H2. apply IH to H3 H4. search.	Theorem	add_assoc	examples/first-order	examples/first-order/add.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
nat : nat -> prop	by nat z ; nat (s N) := nat N.	Define	nat	examples/first-order	examples/first-order/even-odd.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
even : nat -> prop	by even z ; even (s (s N)) := even N.	Define	even	examples/first-order	examples/first-order/even-odd.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
odd : nat -> prop	by odd (s z) ; odd (s (s N)) := odd N.	Define	odd	examples/first-order	examples/first-order/even-odd.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
even_or_odd : forall N, nat N -> even N \/ odd N.	induction on 1. intros. case H1. search. case H2. search. apply IH to H3. case H4. search. search.	Theorem	even_or_odd	examples/first-order	examples/first-order/even-odd.thm	[]	[ "even", "nat", "odd" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
even' : nat -> prop, odd' : nat -> prop	by even' z ; odd' (s N) := even' N ; even' (s N) := odd' N.	Define	even'	examples/first-order	examples/first-order/even-odd.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
even'_nat : forall N, even' N -> nat N.	induction on 1. intros. case H1. search. case H2. apply IH to H3. search.	Theorem	even'_nat	examples/first-order	examples/first-order/even-odd.thm	[]	[ "even'", "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
odd'_nat : forall N, odd' N -> nat N.	induction on 1. intros. case H1. case H2. search. apply IH to H3. search.	Theorem	odd'_nat	examples/first-order	examples/first-order/even-odd.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
sub_total_tt : forall X Y, {less X Y tt} -> exists Z, {sub Y X Z}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	sub_total_tt	examples/first-order	examples/first-order/gcd.thm	[]	[ "sub" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
sub_total_ff : forall X Y, {less X Y ff} -> exists Z, {sub X Y Z}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	sub_total_ff	examples/first-order	examples/first-order/gcd.thm	[]	[ "sub" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
less_total : forall X Y, {nat X} -> {nat Y} -> exists B, {bool B} /\ {less X Y B}.	induction on 1. intros. case H1. case H2. search. search. case H2. search. apply IH to H3 H4. search.	Theorem	less_total	examples/first-order	examples/first-order/gcd.thm	[]	[ "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
lt_trans : forall X Y Z, {lt X (s Y)} -> {lt Y Z} -> {lt X Z}.	induction on 1. intros. case H1. case H2. search. search. case H2. case H3. apply IH to H3 H4. search.	Theorem	lt_trans	examples/first-order	examples/first-order/gcd.thm	[]	[ "lt" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
lt_nat : forall X Y, {lt X Y} -> {nat X}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	lt_nat	examples/first-order	examples/first-order/gcd.thm	[]	[ "lt", "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
lt_x_sx : forall X, {nat X} -> {lt X (s X)}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	lt_x_sx	examples/first-order	examples/first-order/gcd.thm	[]	[ "lt", "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
lt_s : forall X Y, {lt X Y} -> {lt X (s Y)}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	lt_s	examples/first-order	examples/first-order/gcd.thm	[]	[ "lt" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
sub_lt : forall X Y Z, {nat X} -> {sub X Y Z} -> {lt Z (s X)}.	induction on 2. intros. case H2. apply lt_x_sx to H1. search. case H1. apply IH to H4 H3. apply lt_s to H5. search.	Theorem	sub_lt	examples/first-order	examples/first-order/gcd.thm	[]	[ "lt", "lt_s", "lt_x_sx", "nat", "sub" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
gcd_total_strong : forall A B, {nat A} -> {nat B} -> (forall X Y, {lt X A} -> {lt Y B} -> exists Z, {nat Z} /\ {gcd X Y Z}).	induction on 1. induction on 2. intros. apply lt_nat to H3. apply lt_nat to H4. % Trivial base cases case H1 (keep). case H3. case H2 (keep). case H4. % A = s N, B = s N1 apply IH to H7 H2. apply IH1 to H1 H8. clear IH IH1 H1 H2 H7 H8. % X = z case H3. search. % Y = z case H4. search. ...	Theorem	gcd_total_strong	examples/first-order	examples/first-order/gcd.thm	[]	[ "gcd", "less_total", "lt", "lt_nat", "lt_trans", "nat", "sub_lt", "sub_total_ff", "sub_total_tt" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
gcd_total : forall X Y, {nat X} -> {nat Y} -> exists Z, {nat Z} /\ {gcd X Y Z}.	intros. assert {nat (s X)}. assert {nat (s Y)}. apply gcd_total_strong to H3 H4. apply lt_x_sx to H1. apply lt_x_sx to H2. apply H5 to H6 H7. search.	Theorem	gcd_total	examples/first-order	examples/first-order/gcd.thm	[]	[ "gcd", "gcd_total_strong", "lt_x_sx", "nat" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
app_total : forall A B, {list A} -> {list B} -> exists C, {append A B C}.	induction on 1. intros. case H1. search. apply IH to H3 H2. search.	Theorem	app_total	examples/first-order	examples/first-order/lists.thm	[]	[ "append" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
app_det : forall A B C C', {append A B C} -> {append A B C'} -> C = C'.	induction on 1. intros. case H1. case H2. search. case H2. apply IH to H3 H4. search.	Theorem	app_det	examples/first-order	examples/first-order/lists.thm	[]	[ "append" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
app_assoc : forall A B C AB ABC, {append A B AB} -> {append AB C ABC} -> exists BC, {append B C BC} /\ {append A BC ABC}.	induction on 1. intros. case H1. search. case H2. apply IH to H3 H4. search.	Theorem	app_assoc	examples/first-order	examples/first-order/lists.thm	[]	[ "append" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
rev_lemma : forall A A' B X, {rev A A'} -> {append A (cons X nl) B} -> {rev B (cons X A')}.	induction on 1. intros. case H1. case H2. search. case H2. apply IH to H3 H5. search.	Theorem	rev_lemma	examples/first-order	examples/first-order/lists.thm	[]	[ "append" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
rev_rev : forall A B, {rev A B} -> {rev B A}.	induction on 1. intros. case H1. search. apply IH to H2. apply rev_lemma to H4 H3. search.	Theorem	rev_rev	examples/first-order	examples/first-order/lists.thm	[]	[ "rev_lemma" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_lemma : forall A B B' X, {perm B' A} -> {select B X B'} -> {perm B (cons X A)}.	induction on 2. intros. case H2. search. case H1. apply IH to H5 H3. search.	Theorem	perm_lemma	examples/first-order	examples/first-order/lists.thm	[]	[ "perm" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
perm_perm : forall A B, {perm A B} -> {perm B A}.	induction on 1. intros. case H1. search. apply IH to H3. apply perm_lemma to H4 H2. search.	Theorem	perm_perm	examples/first-order	examples/first-order/lists.thm	[]	[ "perm", "perm_lemma" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
app : lst -> lst -> lst -> prop	by app nl C C ; app (cons X A) B (cons X C) := app A B C.	Define	app	examples/first-order	examples/first-order/lists.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
meta_implies_obj : forall A B C, app A B C -> {append A B C}.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	meta_implies_obj	examples/first-order	examples/first-order/lists.thm	[]	[ "app", "append" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
obj_implies_meta : forall A B C, {append A B C} -> app A B C.	induction on 1. intros. case H1. search. apply IH to H2. search.	Theorem	obj_implies_meta	examples/first-order	examples/first-order/lists.thm	[]	[ "app", "append" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
ctx2 : olist -> olist -> prop	by ctx2 nil nil ; nabla x p, ctx2 (bred x x :: G) (path x p :: D) := ctx2 G D ; nabla x, ctx2 ((pi u\ bred x u <= bred N u) :: G) ((pi q\ path x q <= path N q) :: D) := ctx2 G D.	Define	ctx2	examples/higher-order	examples/higher-order/breduce.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
name : tm -> prop	by nabla n, name n.	Define	name	examples/higher-order	examples/higher-order/breduce.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
fresh : tm -> tm -> prop	by nabla n, fresh n X.	Define	fresh	examples/higher-order	examples/higher-order/breduce.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
pname : p -> prop	by nabla p, pname p.	Define	pname	examples/higher-order	examples/higher-order/breduce.thm	[]	[]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
ctx2_mem_G : forall G D F, ctx2 G D -> member F G -> ( (exists x, F = bred x x /\ name x) \/ (exists x N, F = (pi u\ bred x u <= bred N u) /\ fresh x N)).	induction on 1. intros. case H1. case H2. case H2. search. apply IH to H3 H4. case H5. search. search. case H2. search. apply IH to H3 H4. case H5. search. search.	Theorem	ctx2_mem_G	examples/higher-order	examples/higher-order/breduce.thm	[]	[ "ctx2", "fresh", "name" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
ctx2_mem_D : forall G D F, ctx2 G D -> member F D -> ( (exists x p, F = path x p /\ name x /\ pname p) \/ (exists x N, F = (pi q\ path x q <= path N q) /\ fresh x N) ).	induction on 1. intros. case H1. case H2. case H2. search. apply IH to H3 H4. case H5. search. search. case H2. search. apply IH to H3 H4. case H5. search. search.	Theorem	ctx2_mem_D	examples/higher-order	examples/higher-order/breduce.thm	[]	[ "ctx2", "fresh", "name", "pname" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
ctx2_uniform : forall G D X, nabla n, ctx2 (G n) (D n) -> member (pi u\ bred n u <= bred X u) (G n) -> member (pi q\ path n q <= path X q) (D n).	induction on 1. intros. case H1. case H2. case H2. apply IH to H3 H4. search. case H2. apply IH to H3 H4. search. case H2. apply IH to H3 H4. search. case H2. search. apply IH to H3 H4. search.	Theorem	ctx2_uniform	examples/higher-order	examples/higher-order/breduce.thm	[]	[ "ctx2" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
member_prune_D : forall G D E, nabla (n:tm), ctx2 G D -> member (E n) D -> exists F, E = x\ F.	induction on 1. intros. case H1. case H2. case H2. search. apply IH to H3 H4. apply IH to H3 H4. search. case H2. search. apply IH to H3 H4. apply IH to H3 H4. search.	Theorem	member_prune_D	examples/higher-order	examples/higher-order/breduce.thm	[]	[ "ctx2" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
member_D_determinate : forall G D X Y, nabla n, ctx2 (G n) (D n) -> member (pi q\ path n q <= path X q) (D n) -> member (pi q\ path n q <= path Y q) (D n) -> X = Y.	induction on 1. intros. case H1. case H2. case H2. case H3. apply IH to H4 H5 H6. search. case H2. case H3. apply IH to H4 H5 H6. search. case H2. case H3. apply IH to H4 H5 H6. search. case H2. case H3. search. apply member_prune_D to H4 H5. apply member_prune_D to H4 H5.	Theorem	member_D_determinate	examples/higher-order	examples/higher-order/breduce.thm	[]	[ "ctx2", "member_prune_D" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39
member_D_discrim : forall G D X P, nabla n, ctx2 (G n) (D n) -> member (pi q\ path n q <= path X q) (D n) -> member (path n P) (D n) -> false.	induction on 1. intros. case H1. case H2. case H2. case H3. apply IH to H4 H5 H6. case H2. apply member_prune_D to H4 H5. case H2. case H3. apply IH to H4 H5 H6. case H3. apply member_prune_D to H4 H5.	Theorem	member_D_discrim	examples/higher-order	examples/higher-order/breduce.thm	[]	[ "ctx2", "member_prune_D" ]	https://github.com/abella-prover/abella	6332b52c67922ef326fa780e5b6a8b76dcbe8a39

End of preview. Expand in Data Studio

Abella

Declarations from Abella.

Source

Repository: https://github.com/abella-prover/abella
Commit: 6332b52c67922ef326fa780e5b6a8b76dcbe8a39
Files: 67
License: other

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: 769
With proof: 769 (100.0%)
With docstring: 0 (0.0%)
Libraries: 19

By type

Type	Count
Theorem	604
Define	147
CoDefine	18

Example

$form : form -> prop
by
; $form (atom A)
; $form (tens F G) := $form F /\ $form G
; $form one
; $form (par F G) := $form F /\ $form G
; $form bot.

type: Define | symbolic_name: $form | examples/compute/mll.thm

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{abella_dataset,
  title  = {Abella},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/abella-prover/abella, commit 6332b52c6792},
  url    = {https://huggingface.co/datasets/phanerozoic/Abella}
}

Downloads last month: 59

Collection including phanerozoic/Abella

Proof Assistant Projects

Collection

Digesting proof assistant libraries for AI ingestion. • 103 items • Updated 19 days ago • 3