Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
statement stringlengths 3 176 | proof stringlengths 0 1.55k | type stringclasses 5
values | symbolic_name stringlengths 1 27 | library stringclasses 7
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values | imports listlengths 0 0 | deps listlengths 0 13 | docstring stringclasses 1
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7 "~" := ceq. | notation | ceq | example | example/approx.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
7 "≼" := approx. | notation | approx | example | example/approx.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
ceq-refl-thru-approx : [ {x:base} x ~ x ] | {
intro; aux {auto};
capprox; areflexivity
}. | theorem | ceq-refl-thru-approx | example | example/approx.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
11 "⇓" := has-value. | notation | has-value | example | example/approx.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
10 "¬" := not. | notation | not | example | example/approx.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
bottom-diverges : [ ¬ bot ⇓ ] | {
unfold <not implies>;
intro; aux {auto; unfold <has-value>; auto};
bot-div #1
}. | theorem | bottom-diverges | example | example/approx.jonprl | [] | [
"has-value",
"implies",
"not"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
trivial-assertion : [unit] | {
assert [unit -> unit];
auto
}. | theorem | trivial-assertion | example | example/assert.jonprl | [] | [
"assert"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
100 "∈" := member. | notation | member | example | example/atoms.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
100 "~" := ceq. | notation | ceq | example | example/atoms.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
decode : (0). | operator | decode | example | example/atoms.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[decode(A)] | =def= [
match A {
"unit" => unit
| "void" => void
| _ => bot
}
]. | definition | decode | example | example/atoms.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
code : (). | operator | code | example | example/atoms.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[code] | =def= [{A : atom | decode(A) ⇓}]. | definition | code | example | example/atoms.jonprl | [] | [
"decode"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
code-wf : [code ∈ U{i}] | {
unfold <code has-value>; auto;
}. | theorem | code-wf | example | example/atoms.jonprl | [] | [
"code",
"has-value"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
decode-wf : [
{s:code} decode(s) ∈ U{i}
] | {
intro @i; aux {lemma <code-wf>};
unfold <code decode unit>; auto;
elim #1;
assert [bot ⇓] <bot-has-value>;
aux { chyp-subst <- #4 [h. h ⇓]; auto };
bot-div <bot-has-value>
}. | theorem | decode-wf | example | example/atoms.jonprl | [] | [
"assert",
"code",
"code-wf",
"decode"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
instantiate-hyps | {
@{ [H : A -> B, H' : A |- _] => elim <H> [H']
| [H : A => B, H' : A |- _] => elim <H> [H']
}
}. | tactic | instantiate-hyps | example | example/choice.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
destruct-hyps | {
*{ @{ [H : A * B |- _] => elim <H>; thin <H>} }
}. | tactic | destruct-hyps | example | example/choice.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
axiom-of-choice : [{A:U{i}} {B:U{i}} {Q : A -> B -> U{i}} (φ : (c:A) -> (b:B) * Q c b) (f : A -> B) * (a : A) Q a (f a)] | {
auto; intro [lam(w. spread(φ w; x.y.x))]; auto;
instantiate-hyps; auto; reduce;
@{ [H : =(y; φ a; _) |- _] => hyp-subst <- <H> [h. _ _ spread(h; x.y.x)] };
auto; destruct-hyps; reduce; auto
}. | theorem | axiom-of-choice | example | example/choice.jonprl | [] | [
"destruct-hyps",
"instantiate-hyps"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
test : [lam(x.x) ~ lam(x.x)] | {
auto
}. | theorem | test | example | example/computational-equality.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
test2 : [spread(lam(x.x); x.y.x) ~ spread(lam(x.x); x.y.y)] | {
auto
}. | theorem | test2 | example | example/computational-equality.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
test3 : [lam(x.x) ~ lam(x.spread(<x,x>; x._.x))] | {
auto; csymmetry; step; auto
}. | theorem | test3 | example | example/computational-equality.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
computational-eta : [{A:U{i}} {g : A -> A} g ~ lam(x. g x)] | {
auto;
assert [=(g; lam(x.g x); A -> A)];
aux { ext; auto; reduce; auto };
hyp-subst -> #3 [h. h ~ _]; auto
}. | theorem | computational-eta | example | example/computational-equality.jonprl | [] | [
"assert"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
computational-eta-tac | {
@{ [f : A -> A |- ceq(f; lam(x. f x))] =>
cut-lemma <computational-eta>;
elim <computational-eta> [A]; auto;
@{ [H : {g : A -> A} g ~ lam(x. g x) |- _] =>
elim <H> [f]; auto
}
}
}. | tactic | computational-eta-tac | example | example/computational-equality.jonprl | [] | [
"ceq",
"computational-eta"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
fix-is-fix-point : [{A:U{i}} {f : A -> A} fix(f) ~ f fix(f)] | {
auto;
csubst [f ~ lam(x.f x)] [h. fix(h) ~ _];
aux {computational-eta-tac};
step;
csubst [lam(x.f x) ~ f] [h. h fix(h) ~ _];
aux {csymmetry; computational-eta-tac};
auto
}. | theorem | fix-is-fix-point | example | example/computational-equality.jonprl | [] | [
"computational-eta-tac"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
transitive : [
{a:base} {b:base} {c:base}
=(a;b;base)
=> =(b;c;base)
=> =(a;b;base)
] | {
auto;
elim #4;
elim #5;
chyp-subst → #6 [h. h ~ c];
chyp-subst ← #7 [h. b ~ h];
auto
}. | theorem | transitive | example | example/computational-equality.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
complex : (1; 1; 2; 0; 1). | operator | complex | example | example/computational-equality.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
complex-ceq : [
complex(x.x; x.unit; x.y.y; <>; x.<>)
~ complex(x.spread(<x,x>; x._.x); x.unit; x.y.y; <>; x.<>)
] | {
auto; reduce; auto
}. | theorem | complex-ceq | example | example/computational-equality.jonprl | [] | [
"complex"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
top : (). | operator | top | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[top] | =def= [void => void]. | definition | top | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
top-is-top :
[{x:base} {y:base} =(x; y; top)] | {
unfold <top>; auto
}. | theorem | top-is-top | example | example/conats.jonprl | [] | [
"top"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
corec : (1). | operator | corec | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[corec(x.F[x])] | =def= [{n:nat} natrec(n; top; _.x. F[x])]. | definition | corec | example | example/conats.jonprl | [] | [
"top"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
conatF : (0). | operator | conatF | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[conatF(X)] | =def= [unit + X]. | definition | conatF | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
conat : (). | operator | conat | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[conat] | =def= [corec(R. conatF(R))]. | definition | conat | example | example/conats.jonprl | [] | [
"conatF",
"corec"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
czero : (). | operator | czero | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[czero] | =def= [inl(<>)]. | definition | czero | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
csucc : (0). | operator | csucc | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[csucc(M)] | =def= [inr(M)]. | definition | csucc | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Y : (1). | operator | Y | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Y(z.f[z])] | =def= [lam(x.f[x x]) lam(x.f [(x x)])]. | definition | Y | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
omega : (). | operator | omega | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[omega] | =def= [Y(x.csucc(x))]. | definition | omega | example | example/conats.jonprl | [] | [
"csucc"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
instantiate-hyp | {
@{ [H : P => Q, H' : P |- _] => elim <H> [H'] }
}. | tactic | instantiate-hyp | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
refl-from-trans | {
@{ [H : =(y; x; P) |- =(x;x; P)] => hyp-subst <- <H> [h.=(h;h;_)] };
}. | tactic | refl-from-trans | example | example/conats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
conat-wf-overture | {
*{ unfold <corec conat czero csucc conatF top omega unit Y> }; auto;
@{ [n:nat |- _] => elim <n>; thin <n> };
reduce 1; auto
}. | tactic | conat-wf-overture | example | example/conats.jonprl | [] | [
"conat",
"conatF",
"corec",
"csucc",
"czero",
"omega",
"top"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
zero-wf : [member(czero; conat)] | {
conat-wf-overture
}. | theorem | zero-wf | example | example/conats.jonprl | [] | [
"conat",
"conat-wf-overture",
"czero",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
succ-wf : [{x:conat} member(csucc(x); conat)] | {
conat-wf-overture;
instantiate-hyp; auto;
refl-from-trans; auto
}. | theorem | succ-wf | example | example/conats.jonprl | [] | [
"conat",
"conat-wf-overture",
"csucc",
"instantiate-hyp",
"member",
"refl-from-trans"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
omega-wf : [member(omega; conat)] | {
conat-wf-overture;
csubst [ceq(lam(x.inr(x x)) lam(x.inr(x x));
inr(lam(x.inr(x x)) lam(x.inr(x x))))]
[h.=(h;h;_)];
[step, reduce 1]; auto
}. | theorem | omega-wf | example | example/conats.jonprl | [] | [
"ceq",
"conat",
"conat-wf-overture",
"member",
"omega"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
20 "∈" := member. | notation | member | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
20 "⇓" := has-value. | notation | has-value | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
has-value-wf : [{M:base} M ⇓ ∈ U{i}] | {
unfold <has-value>; auto
}. | theorem | has-value-wf | example | example/container.jonprl | [] | [
"has-value"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat-decode : (0). | operator | nat-decode | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[nat-decode(x)] | =def= [
match x {
"ze" => void
| "su" => unit
| _ => bot
}
]. | definition | nat-decode | example | example/container.jonprl | [] | [
"su",
"ze"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat-code : (). | operator | nat-code | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[nat-code] | =def= [{a : atom | nat-decode(a) ⇓}]. | definition | nat-code | example | example/container.jonprl | [] | [
"nat-decode"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat-sig : (). | operator | nat-sig | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[nat-sig] | =def= [x : nat-code ◃ nat-decode(x)]. | definition | nat-sig | example | example/container.jonprl | [] | [
"nat-code",
"nat-decode"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat-code-wf : [nat-code ∈ U{i}] | {
unfold <nat-code has-value>; auto
}. | theorem | nat-code-wf | example | example/container.jonprl | [] | [
"has-value",
"nat-code"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat-decode-wf : [{a:nat-code} nat-decode(a) ∈ U{i}] | {
intro @i; auto;
unfold <nat-decode nat-code>; auto;
elim #1;
unfold <nat-decode>;
assert [bot ⇓];
aux { chyp-subst <- #4; auto };
bot-div #6
}. | theorem | nat-decode-wf | example | example/container.jonprl | [] | [
"assert",
"nat-code",
"nat-decode"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat-sig-wf : [nat-sig ∈ container{i}] | {
unfold <nat-sig nat-code nat-decode has-value>; auto;
elim #1;
assert [bot ⇓];
aux {chyp-subst <- #4 [h. h ⇓]; auto;};
bot-div #6
}. | theorem | nat-sig-wf | example | example/container.jonprl | [] | [
"assert",
"has-value",
"nat-code",
"nat-decode",
"nat-sig"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat' : (). | operator | nat' | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[nat'] | =def= [wtree(nat-sig)]. | definition | nat' | example | example/container.jonprl | [] | [
"nat-sig"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat'-wf : [nat' ∈ U{i}] | {
unfold <nat'>; auto
}. | theorem | nat'-wf | example | example/container.jonprl | [] | [
"nat'"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
ze-code-wf : ["ze" ∈ nat-code] | {
unfold <nat-code has-value nat-decode>;
auto; reduce; auto ; auto
}. | theorem | ze-code-wf | example | example/container.jonprl | [] | [
"has-value",
"nat-code",
"nat-decode",
"ze"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
su-code-wf : ["su" ∈ nat-code] | {
unfold <nat-code has-value nat-decode>;
auto; reduce; auto ; auto
}. | theorem | su-code-wf | example | example/container.jonprl | [] | [
"has-value",
"nat-code",
"nat-decode",
"su"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
ze : (). | operator | ze | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[ze] | =def= [sup("ze" ^ <>)]. | definition | ze | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
su : (0). | operator | su | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[su(N)] | =def= [sup("su" ^ N)]. | definition | su | example | example/container.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
ze-wf : [ze ∈ nat'] | {
unfold <ze nat' nat-sig>; auto; reduce; auto;
unfold <nat-decode>; reduce; auto
}. | theorem | ze-wf | example | example/container.jonprl | [] | [
"nat'",
"nat-decode",
"nat-sig",
"ze"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
su-wf : [{N:nat'} su(N) ∈ nat'] | {
intro @i; auto;
unfold <su nat'>;
auto; reduce; auto; auto;
unfold <nat-sig>; reduce; auto
}. | theorem | su-wf | example | example/container.jonprl | [] | [
"nat'",
"nat-sig",
"su"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nat'-elim-test : [nat' -> atom] | {
intro @i; auto;
unfold <nat'>;
elim #1; unfold <nat-sig>; reduce;
witness [a];
unfold <nat-code>; auto
}. | theorem | nat'-elim-test | example | example/container.jonprl | [] | [
"nat'",
"nat-code",
"nat-sig"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
50 "↓" := squash. | notation | squash | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
rel : (0). | operator | rel | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[rel(A)] | =def= [A -> A -> U{i}]. | definition | rel | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
rel-wf : [{A:U{i}} rel(A) ∈ U{i'}] | {
unfold <rel>; auto
}. | theorem | rel-wf | example | example/identity-types.jonprl | [] | [
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-reflexive : (0;0). | operator | is-reflexive | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[is-reflexive(A;R)] | =def= [(a:A) R a a]. | definition | is-reflexive | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-symmetric : (0;0). | operator | is-symmetric | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[is-symmetric(A;R)] | =def= [(a:A) (b:A) R a b -> R b a]. | definition | is-symmetric | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-transitive : (0;0). | operator | is-transitive | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[is-transitive(A;R)] | =def= [(a:A) (b:A) (c:A) R a b -> R b c -> R a c]. | definition | is-transitive | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-reflexive-wf : [{A:U{i}} {R:rel(A)} is-reflexive(A;R) ∈ U{i}] | {
unfold <is-reflexive rel>;
auto
}. | theorem | is-reflexive-wf | example | example/identity-types.jonprl | [] | [
"is-reflexive",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-symmetric-wf : [{A:U{i}} {R:rel(A)} is-symmetric(A;R) ∈ U{i}] | {
unfold <is-symmetric rel>; auto
}. | theorem | is-symmetric-wf | example | example/identity-types.jonprl | [] | [
"is-symmetric",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-transitive-wf : [{A:U{i}} {R:rel(A)} is-transitive(A;R) ∈ U{i}] | {
unfold <is-transitive rel>; auto
}. | theorem | is-transitive-wf | example | example/identity-types.jonprl | [] | [
"is-transitive",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
J-principle : (0;0;0). | operator | J-principle | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[J-principle(A; R; Refl)] | =def= [
(M:A) (C : (y:A) R M y -> U{i})
C M (Refl M) -> (N:A) (P : R M N) C N P
]. | definition | J-principle | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
J-principle-wf : [{A:U{i}} {R:rel(A)} {Refl : is-reflexive(A; R)} J-principle(A; R; Refl) ∈ U{i'}] | {
auto; unfold <J-principle>;
intro @i; auto;
unfold <is-reflexive>; auto
}. | theorem | J-principle-wf | example | example/identity-types.jonprl | [] | [
"J-principle",
"is-reflexive",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
identity-type : (0). | operator | identity-type | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[identity-type(A)] | =def= [
(R : rel(A))
* (Refl : is-reflexive(A; R))
* J-principle(A; R; Refl)
]. | definition | identity-type | example | example/identity-types.jonprl | [] | [
"J-principle",
"is-reflexive",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
identity-type-wf : [{A:U{i}} identity-type(A) ∈ U{i'}] | {
unfold <identity-type>; auto;
cum @i; auto
}. | theorem | identity-type-wf | example | example/identity-types.jonprl | [] | [
"identity-type"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
id-rel : (0). | operator | id-rel | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[id-rel(I)] | =def= [fst(I)]. | definition | id-rel | example | example/identity-types.jonprl | [] | [
"fst"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
id-refl : (0). | operator | id-refl | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[id-refl(I)] | =def= [fst(snd(I))]. | definition | id-refl | example | example/identity-types.jonprl | [] | [
"fst",
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
id-J : (0). | operator | id-J | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[id-J(I)] | =def= [snd(snd(I))]. | definition | id-J | example | example/identity-types.jonprl | [] | [
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
id-rel-wf : [{A:U{i}} {I:identity-type(A)} id-rel(I) ∈ rel(A)] | {
auto; unfold <id-rel identity-type fst>; auto
}. | theorem | id-rel-wf | example | example/identity-types.jonprl | [] | [
"fst",
"id-rel",
"identity-type",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 |
End of preview. Expand in Data Studio
JonPRL
Declarations from JonPRL, a Nuprl-family proof refinement logic for computational type theory.
Source
- Repository: https://github.com/jonsterling/JonPRL
- Commit:
f937a5461955fff8cb22331ecb6fdeae8df636d5 - Files: 53
- 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: 771
- With proof: 516 (66.9%)
- With docstring: 0 (0.0%)
- Libraries: 7
By type
| Type | Count |
|---|---|
| theorem | 243 |
| definition | 227 |
| operator | 215 |
| tactic | 46 |
| notation | 40 |
Example
bottom-diverges : [ ¬ bot ⇓ ]
{
unfold <not implies>;
intro; aux {auto; unfold <has-value>; auto};
bot-div #1
}.
- type: theorem | symbolic_name:
bottom-diverges| example/approx.jonprl
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{jonprl_dataset,
title = {JonPRL},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/jonsterling/JonPRL, commit f937a5461955},
url = {https://huggingface.co/datasets/phanerozoic/JonPRL}
}
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