statement stringlengths 3 176 | proof stringlengths 0 1.55k | type stringclasses 5
values | symbolic_name stringlengths 1 27 | library stringclasses 7
values | filename stringclasses 53
values | imports listlengths 0 0 | deps listlengths 0 13 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
UnitMonoidStruct : [MonoidSig] | {
unfold <MonoidSig>;
intro [unit]; auto
}. | theorem | UnitMonoidStruct | example | example/monoid.jonprl | [] | [
"MonoidSig"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
monoid-laws-unfold | {
unfold <MonoidLaws LeftUnit RightUnit Assoc>
}. | tactic | monoid-laws-unfold | example | example/monoid.jonprl | [] | [
"Assoc",
"LeftUnit",
"MonoidLaws",
"RightUnit"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
monoid-unfold | {
unfold <Monoid>; monoid-sig-unfold; monoid-laws-unfold.
}. | tactic | monoid-unfold | example | example/monoid.jonprl | [] | [
"Monoid",
"monoid-laws-unfold",
"monoid-sig-unfold"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
monoid-simplify | {
*{ monoid-unfold; reduce; auto }.
}. | tactic | monoid-simplify | example | example/monoid.jonprl | [] | [
"monoid-unfold"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
UnitMonoid : [Monoid] | {
unfold <Monoid>;
intro [UnitMonoidStruct] @i; unfold <UnitMonoidStruct>;
monoid-simplify; auto
}. | theorem | UnitMonoid | example | example/monoid.jonprl | [] | [
"Monoid",
"UnitMonoidStruct",
"monoid-simplify"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
UnitMonoid-LeftUnit : [LeftUnit(UnitMonoidStruct)] | {
unfold <UnitMonoidStruct>;
monoid-simplify; auto
}. | theorem | UnitMonoid-LeftUnit | example | example/monoid.jonprl | [] | [
"LeftUnit",
"UnitMonoidStruct",
"monoid-simplify"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
UnitMonoid-RightUnit : [RightUnit(UnitMonoidStruct)] | {
unfold <UnitMonoidStruct>;
monoid-simplify; auto
}. | theorem | UnitMonoid-RightUnit | example | example/monoid.jonprl | [] | [
"RightUnit",
"UnitMonoidStruct",
"monoid-simplify"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
10 "⇓" := has-value. | notation | has-value | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
natp-example : [nat] | {
witness [succ(zero)]; auto
}. | theorem | natp-example | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
minus-one : [nat -> nat] | {
intro <n>; aux {auto};
elim #1;
[ witness [zero]
, hypothesis #2
]; auto
}. | theorem | minus-one | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
minus-one-test : [=(minus-one succ(succ(zero)); succ(zero); nat)] | {
unfold <minus-one>; reduce; auto
}. | theorem | minus-one-test | example | example/nats.jonprl | [] | [
"minus-one"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
add : (0;0). | operator | add | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
-> 100 "N+" := add. | notation | add | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[M N+ N] | =def= [natrec(M; N; _.n.succ(n))]. | definition | M | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
add-id-left : [{n:nat} =(zero N+ n; n; nat)] | {
auto; unfold <add>; reduce; auto
}. | theorem | add-id-left | example | example/nats.jonprl | [] | [
"add"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
add-id-right : [{n:nat} =(n N+ zero; n; nat)] | {
auto; unfold <add>; elim #1; reduce; auto
}. | theorem | add-id-right | example | example/nats.jonprl | [] | [
"add"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
succ-right : [
{n:nat} {m:nat} =(n N+ succ(m); succ(n N+ m); nat)
] | {
auto; unfold <add>; elim #1; reduce; auto
}. | theorem | succ-right | example | example/nats.jonprl | [] | [
"add"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
add-commutes : [
{n:nat} {m:nat} =(n N+ m; m N+ n; nat)
] | {
||| Kick off the induction and do the boring computation thingies.
auto; unfold <add>; elim #1; reduce; auto;
||| The base case immediately follows from add-id-right
focus 0 #{
cut-lemma <add-id-right>; unfold <add>;
symmetry;
bhyp <add-id-right>; auto
};
||| In order to prove this we first ... | theorem | add-commutes | example | example/nats.jonprl | [] | [
"add",
"add-id-right",
"succ-right"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
pred : (0). | operator | pred | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[pred(n)] | =def= [natrec(n; bot; m._.m)]. | definition | pred | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
minus : (0;0). | operator | minus | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[minus(m; n)] | =def= [
natrec(n; m; n'.ih. pred(ih))
]. | definition | minus | example | example/nats.jonprl | [] | [
"pred"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
leq : (0;0). | operator | leq | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
10 "≤" := leq. | notation | leq | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[m ≤ n] | =def= [minus(n; m) ⇓]. | definition | m | example | example/nats.jonprl | [] | [
"minus"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
leq-wf : [{m:nat} {n:nat} (m ≤ n) ∈ U{i}] | {
unfold <leq>; auto
}. | theorem | leq-wf | example | example/nats.jonprl | [] | [
"leq"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
upto : (0). | operator | upto | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[upto(i)] | =def= [{j : nat | j ≤ i}]. | definition | upto | example | example/nats.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
upto-wf : [{n:nat} upto(n) ∈ U{i}] | {
unfold <upto>; auto
}. | theorem | upto-wf | example | example/nats.jonprl | [] | [
"upto"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
per-squash : (0). | operator | per-squash | example | example/per.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[per-squash(A)] | =def= [per(_._.A)]. | definition | per-squash | example | example/per.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
per-squash-wf : [{A:U{i}} member(per-squash(A); U{i})] | {
unfold <per-squash>; auto; reduce; auto
}. | theorem | per-squash-wf | example | example/per.jonprl | [] | [
"member",
"per-squash"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
iff_squash : [{A:U{i}} iff(per-squash(A);squash(A))] | {
auto;
[ unfold <per-squash>;auto;reduce;auto;
assert [=(x;x;per(_._.A))] <e>;[ auto; fail, id ];
elim <e>; reduce; [ id, auto; reduce; auto ];
witness [<>];auto;
unhide <y>;
csubst [ceq(<>;lam(_.<>)y)] [h.=(h;h;_)];[ reduce; auto, id ];
auto;fail
, id
, unfold <squash>;auto;fail
];
unfold <squ... | theorem | iff_squash | example | example/per.jonprl | [] | [
"assert",
"ceq",
"iff",
"member",
"per-squash",
"squash"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
quotient_wf: [
{T:U{i}}
{R:rel(T)}
is-symmetric(T;R)
-> is-transitive(T;R)
-> member(quotient(T;x.y.R x y);U{i})
] | {
auto;
unfold <is-symmetric is-transitive>;
auto;
unfold <quotient and>;eq-cd;
[ unfold <member>; eq-cd;
[ eq-base-tac;eq-cd;auto;reduce;auto;elim <x'''>;reduce;auto;fail
, eq-cd;
[ eq-base-tac;eq-cd;auto;reduce;auto;elim <x''''>;reduce;auto;fail
, auto;fail
]
]
, unfold <memb... | theorem | quotient_wf | example | example/per.jonprl | [] | [
"eq-base-tac",
"is-symmetric",
"is-transitive",
"member",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
polymorphic-id-unique : [
(F : {A : U{i}} A -> A) =(F; id; {A : U{i}} A -> A)
] | {
unfold <id>; auto;
elim <F> [A] <FA, eqFA>; auto;
ext; auto; reduce;
aux {
hyp-subst <- <eqFA> [h. =(h; h; _)];
auto
};
elim <F> [{y : A | =(y; x; A)}] <F', eqF'>; auto;
hyp-subst <- <eqF'> [H. =(H x; x; A)];
auto;
[ elim <F'> [x] <F'x, eqF'x>; auto;
hyp-subst <- <eqF'x> [J. =(J; x; ... | theorem | polymorphic-id-unique | example | example/polymorphic-id-unique.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
CTT : (0;0). | operator | CTT | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Paths : (0;0;0). | operator | Paths | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[CTT(A;B)] | =def= [A -> B]. | definition | CTT | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Paths(A;x;y)] | =def= [=(x;y;A)]. | definition | Paths | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Power : (0;0). | operator | Power | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Copower : (0;0). | operator | Copower | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Power(A;B)] | =def= [A -> B]. | definition | Power | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Copower(A;B)] | =def= [A * B]. | definition | Copower | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
End : (0;0). | operator | End | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Coend : (0;0). | operator | Coend | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[End(U;Hom)] | =def= [(A:U) Hom A A]. | definition | End | example | example/polynomials.jonprl | [] | [
"Hom"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Coend(U;Hom)] | =def= [(A:U) * Hom A A]. | definition | Coend | example | example/polynomials.jonprl | [] | [
"Hom"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
autoR | {
reduce; *{auto; reduce}
}. | tactic | autoR | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
End_wf : [
{U:U{i}} {Hom : U -> U -> U{i}} member(End(U;Hom); U{i})
] | {
unfold <End>; auto.
}. | theorem | End_wf | example | example/polynomials.jonprl | [] | [
"End",
"Hom",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Coend_wf : [
{U:U{i}} {Hom : U -> U -> U{i}} member(Coend(U;Hom); U{i})
] | {
unfold <Coend>; auto.
}. | theorem | Coend_wf | example | example/polynomials.jonprl | [] | [
"Coend",
"Hom",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Ran : (0;0;0;0;0). | operator | Ran | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Lan : (0;0;0;0;0). | operator | Lan | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Ran(E;B;f;φ;b)] | =def= [End(E; lam(X. lam(Y. Power(Paths(B; b; f X); φ Y))))]. | definition | Ran | example | example/polynomials.jonprl | [] | [
"End",
"Paths",
"Power"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Lan(E;B;f;φ;b)] | =def= [Coend(E; lam(X. lam(Y. Copower(Paths(B; f X; b); φ Y))))]. | definition | Lan | example | example/polynomials.jonprl | [] | [
"Coend",
"Copower",
"Paths"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Ran_wf : [
{E:U{i}} {B:U{i}} {f : E -> B} {φ : E -> U{i}} {b:B}
member(Ran(E;B;f;φ;b); U{i})
] | {
unfold <Ran Power End Paths>; autoR.
}. | theorem | Ran_wf | example | example/polynomials.jonprl | [] | [
"End",
"Paths",
"Power",
"Ran",
"autoR",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Lan_wf : [
{E:U{i}} {B:U{i}} {f : E -> B} {φ : E -> U{i}} {b:B}
member(Lan(E;B;f;φ;b); U{i})
] | {
unfold <Lan Copower Coend Paths>; autoR.
}. | theorem | Lan_wf | example | example/polynomials.jonprl | [] | [
"Coend",
"Copower",
"Lan",
"Paths",
"autoR",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Fam : (0). | operator | Fam | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
dom : (0). | operator | dom | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
map : (0;0). | operator | map | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Fam(B)] | =def= [(Dom:U{i}) * Dom -> B]. | definition | Fam | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[dom(f)] | =def= [fst(f)]. | definition | dom | example | example/polynomials.jonprl | [] | [
"fst"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[map(f;b)] | =def= [snd(f) b]. | definition | map | example | example/polynomials.jonprl | [] | [
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Fam_wf : [{B:U{i}} member(Fam(B); U{i'})] | {
unfold <Fam>; auto
}. | theorem | Fam_wf | example | example/polynomials.jonprl | [] | [
"Fam",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
dom_wf : [{B:U{i}} {f:Fam(B)} member(dom(f); U{i})] | {
unfold <Fam map dom fst>; auto;
unfold <fst>;
unfold <fst>
}. | theorem | dom_wf | example | example/polynomials.jonprl | [] | [
"Fam",
"dom",
"fst",
"map",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
map_wf : [
{B:U{i}} {f:Fam(B)} {x:dom(f)} member(map(f;x); B)
] | {
unfold <Fam dom map fst snd>; autoR.
}. | theorem | map_wf | example | example/polynomials.jonprl | [] | [
"Fam",
"autoR",
"dom",
"fst",
"map",
"member",
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Fiber : (0;0;0). | operator | Fiber | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Fiber(B; f; b)] | =def= [(x : dom(f)) * =(b; map(f; x); B)]. | definition | Fiber | example | example/polynomials.jonprl | [] | [
"dom",
"map"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Fiber_wf : [
{B:U{i}} {f:Fam(B)} {b:B} member(Fiber(B;f;b); U{i})
] | {
unfold <Fam Fiber dom map fst snd>; autoR.
}. | theorem | Fiber_wf | example | example/polynomials.jonprl | [] | [
"Fam",
"Fiber",
"autoR",
"dom",
"fst",
"map",
"member",
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Exists : (0;0;0). | operator | Exists | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Invert : (0). | operator | Invert | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Forall : (0;0;0). | operator | Forall | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Exists(X;Y;f)] | =def= [lam(φ. lam(y. Lan(X;Y;f;φ;y)))]. | definition | Exists | example | example/polynomials.jonprl | [] | [
"Lan"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Invert(f)] | =def= [lam(φ. lam(x. φ (f x)))]. | definition | Invert | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Forall(X;Y;f)] | =def= [lam(φ. lam(y. Ran(X;Y;f;φ;y)))]. | definition | Forall | example | example/polynomials.jonprl | [] | [
"Ran"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Exists_wf : [
{X:U{i}} {Y:U{i}} {f : X -> Y} {φ : X -> U{i}}
member(Exists(X;Y;f) φ; Y -> U{i})
] | {
unfold <Exists Lan Coend Copower Paths>; autoR.
}. | theorem | Exists_wf | example | example/polynomials.jonprl | [] | [
"Coend",
"Copower",
"Exists",
"Lan",
"Paths",
"autoR",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Invert_wf : [
{X:U{i}} {Y:U{i}} {f : X -> Y} {φ : Y -> U{i}}
member(Invert(f) φ; X -> U{i})
] | {
unfold <Invert>; autoR.
}. | theorem | Invert_wf | example | example/polynomials.jonprl | [] | [
"Invert",
"autoR",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Forall_wf : [
{X:U{i}} {Y:U{i}} {f : X -> Y} {φ : X -> U{i}}
member(Forall(X;Y;f) φ; Y -> U{i})
] | {
unfold <Forall Ran End Power Paths>; autoR
}. | theorem | Forall_wf | example | example/polynomials.jonprl | [] | [
"End",
"Forall",
"Paths",
"Power",
"Ran",
"autoR",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
δ : (). | operator | δ | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Θ : (0;0). | operator | Θ | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[δ] | =def= [lam(x.<x, x>)]. | definition | δ | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Θ(X;p)] | =def= [Exists(X;X*X; δ) lam(_. unit) p]. | definition | Θ | example | example/polynomials.jonprl | [] | [
"Exists"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
eq_unit_unit : [Θ(unit; <<>,<>>)] | {
unfold <Θ δ Exists Lan Coend Copower Paths unit>; autoR;
intro [<>]; auto.
}. | theorem | eq_unit_unit | example | example/polynomials.jonprl | [] | [
"Coend",
"Copower",
"Exists",
"Lan",
"Paths",
"autoR"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Polynomial : (). | operator | Polynomial | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
base-space : (0). | operator | base-space | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
fam : (0). | operator | fam | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Polynomial] | =def= [(B:U{i}) * Fam(B)]. | definition | Polynomial | example | example/polynomials.jonprl | [] | [
"Fam"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[base-space(E)] | =def= [fst(E)]. | definition | base-space | example | example/polynomials.jonprl | [] | [
"fst"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[fam(E)] | =def= [snd(E)]. | definition | fam | example | example/polynomials.jonprl | [] | [
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Polynomial_wf : [member(Polynomial; U{i'})] | {
unfold <Polynomial Fam>; auto
}. | theorem | Polynomial_wf | example | example/polynomials.jonprl | [] | [
"Fam",
"Polynomial",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Embed : (0). | operator | Embed | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Embed(B)] | =def= [lam(A. CTT(A;B))]. | definition | Embed | example | example/polynomials.jonprl | [] | [
"CTT"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Embed_wf : [
{B:U{i}} {X:U{i}} member(Embed(B) X; U{i})
] | {
unfold <Embed CTT>; autoR
}. | theorem | Embed_wf | example | example/polynomials.jonprl | [] | [
"CTT",
"Embed",
"autoR",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Ext : (0;0). | operator | Ext | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Ext(p;X)] | =def= [base-space(p) * unit]. | definition | Ext | example | example/polynomials.jonprl | [] | [
"base-space"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Ext_wf : [
{p:Polynomial} {X:U{i}} member(Ext(p;X); U{i'})
] | {
unfold <Polynomial Ext Fam base-space Embed CTT Fiber fam dom map fst snd unit>; auto
}. | theorem | Ext_wf | example | example/polynomials.jonprl | [] | [
"CTT",
"Embed",
"Ext",
"Fam",
"Fiber",
"Polynomial",
"base-space",
"dom",
"fam",
"fst",
"map",
"member",
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Pullback : (0;0;0). | operator | Pullback | example | example/polynomials.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[Pullback(B;f;g)] | =def= [(x : dom(f)) * Fiber(B; g; map(f;x))]. | definition | Pullback | example | example/polynomials.jonprl | [] | [
"Fiber",
"dom",
"map"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Pullback_wf : [
{B:U{i}} {f:Fam(B)} {g:Fam(B)} member(Pullback(B;f;g); U{i})
] | {
unfold <Pullback Fam Fiber map dom fst snd>; autoR.
}. | theorem | Pullback_wf | example | example/polynomials.jonprl | [] | [
"Fam",
"Fiber",
"Pullback",
"autoR",
"dom",
"fst",
"map",
"member",
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
proj-equal : [
{A:U{i}} {B:U{i}} {a:A} {a':A} {b:B} {b':B}
=(<a,b>; <a',b'>; A * B) => =(a;a';A)
] | {
auto;
assert [=(spread(<a,b>; x.y.x);
spread(<a',b'>; x.y.x);
A)];
aux {hyp-subst → #7 [h. =(spread(h; x.y.x); spread(<a',b'>; x.y.x); A)]};
reduce;
auto
}. | theorem | proj-equal | example | example/proj-equal.jonprl | [] | [
"assert"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
easy-with-fiat : [void] | {
auto
}. | theorem | easy-with-fiat | example | example/resource.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 |
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