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
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|---|---|---|---|---|---|---|---|---|---|---|
destruct-identity-type | {
@{ [H:identity-type(A) |- _] =>
unfold <identity-type>;
elim <H> <R,H'>;
elim <H'> <Refl, J>; thin <H'>
}
}. | tactic | destruct-identity-type | example | example/identity-types.jonprl | [] | [
"identity-type"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
destruct-hyp | {
@{ [H:A * B |- _] => elim <H>; thin <H>; reduce }
}. | tactic | destruct-hyp | example | example/identity-types.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
id-refl-wf : [{A:U{i}} {I:identity-type(A)} id-refl(I) ∈ is-reflexive(A; id-rel(I))] | {
auto; destruct-identity-type;
unfold <id-J id-refl id-rel fst snd>;
reduce; auto
}. | theorem | id-refl-wf | example | example/identity-types.jonprl | [] | [
"destruct-identity-type",
"fst",
"id-J",
"id-refl",
"id-rel",
"identity-type",
"is-reflexive",
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
id-J-wf : [{A:U{i}} {I:identity-type(A)} id-J(I) ∈ J-principle(A; id-rel(I); id-refl(I))] | {
auto; destruct-identity-type;
unfold <id-J id-refl id-rel fst snd>;
reduce; auto
}. | theorem | id-J-wf | example | example/identity-types.jonprl | [] | [
"J-principle",
"destruct-identity-type",
"fst",
"id-J",
"id-refl",
"id-rel",
"identity-type",
"snd"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
identity-type-symmetric : [{A:U{i}} (I:identity-type(A)) is-symmetric(A; id-rel(I))] | {
*{ intro; ?{ !{ auto } } };
destruct-identity-type;
unfold <id-rel fst>; reduce; auto;
unfold <J-principle>;
main { elim #5 [a] };
main { elim #9 [lam(y.lam(p.R y a))]};
main { elim #11 [Refl a]};
main { elim #13 [b] };
main { elim #15 [x] };
unfold <is-reflexive>; reduce; auto
}. | theorem | identity-type-symmetric | example | example/identity-types.jonprl | [] | [
"J-principle",
"destruct-identity-type",
"fst",
"id-rel",
"identity-type",
"is-reflexive",
"is-symmetric"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
identity-type-transitive : [{A:U{i}} (I:identity-type(A)) is-transitive(A; id-rel(I))] | {
*{ intro; ?{ !{ auto } } };
destruct-identity-type;
unfold <id-rel fst>; reduce; auto;
unfold <J-principle>;
main { elim #5 [b] };
main { elim #11 [lam(y.lam(p.R y c))] };
main { elim #13 [x'] };
main { elim #15 [a] };
reduce; auto;
*{ @{ [H : =(_;_;_) |- _] => thin <H> } };
assert [R b a] <... | theorem | identity-type-transitive | example | example/identity-types.jonprl | [] | [
"J-principle",
"assert",
"destruct-identity-type",
"fst",
"id-rel",
"identity-type",
"identity-type-symmetric",
"is-reflexive",
"is-symmetric",
"is-transitive",
"member",
"rel"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
7 "↓" := squash. | notation | squash | example | example/image.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
3 "∈" := member. | notation | member | example | example/image.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
4 "⇔" := iff. | notation | iff | example | example/image.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
squash_wf : [ {T:U{i}} ↓ T ∈ U{i} ] | {
unfold <squash>; auto
}. | theorem | squash_wf | example | example/image.jonprl | [] | [
"squash"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
implies_squash : [ {T:U{i}} T => ↓ T ] | {
*{intro}; aux {auto};
@{ [H: 'T |- _] => witness [lam(_.<>) H]};
unfold <squash>; auto
}. | theorem | implies_squash | example | example/image.jonprl | [] | [
"squash"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
approx_id_refl_wf : [
=(lam(x.x) ≼ lam(x.x) ; <> ≼ <> ; U{i})
] | {
unfold <member>; ext;
cut-lemma <implies_squash>;
elim #1 [lam(x.x) ≼ lam(x.x) ⇔ <> ≼ <>];
main {
assert [lam(x.x) ≼ lam(x.x) ⇔ <> ≼ <>];
main { elim #2 [H]; auto };
};
aux {unfold <iff and implies>; auto}
}. | theorem | approx_id_refl_wf | example | example/image.jonprl | [] | [
"assert",
"iff",
"implies",
"implies_squash",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
test_image_elim : [
{n:nat} {m:nat} {x : ↓ =(n; m; nat)} x ∈ ↓ =(n; m; nat) => =(n; m; nat)
] | {
unfold <squash member>;
*{intro}; aux {auto};
@{[H : image(_;_) |- _] => elim <H>};
auto
}. | theorem | test_image_elim | example | example/image.jonprl | [] | [
"member",
"squash"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
pi2 : (). | operator | pi2 | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[pi2] | =def= [lam(x. spread(x; u.v.v))]. | definition | pi2 | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
union : (0;1). | operator | union | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[union(A; x.B[x])] | =def= [image((x : A) * B[x]; pi2)]. | definition | union | example | example/list-from-union.jonprl | [] | [
"pi2"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
union-wf : [
{A : U{i}} {B : A -> U{i}} union(A; x.B x) ∈ U{i}
] | {
unfold <union pi2>; auto
}. | theorem | union-wf | example | example/list-from-union.jonprl | [] | [
"pi2",
"union"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
vec : (0;0). | operator | vec | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[vec(A; n)] | =def= [
natrec(n; void + unit; _.ih. (A * ih) + void)
]. | definition | vec | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
vec-wf : [{A:U{i}} {n:nat} vec(A; n) ∈ U{i}] | {
unfold <vec>; auto
}. | theorem | vec-wf | example | example/list-from-union.jonprl | [] | [
"vec"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list : (0). | operator | list | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[list(A)] | =def= [union(nat; i.vec(A; i))]. | definition | list | example | example/list-from-union.jonprl | [] | [
"union",
"vec"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-wf : [{A:U{i}} list(A) ∈ U{i}] | {
unfold <list union>; auto
}. | theorem | list-wf | example | example/list-from-union.jonprl | [] | [
"list",
"union"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nil : (). | operator | nil | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[nil] | =def= [inr(<>)]. | definition | nil | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
cons : (0;0). | operator | cons | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
-> 30 "#" := cons. | notation | cons | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[h # t] | =def= [inl(<h,t>)]. | definition | h | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nil-wf-vec : [{A:U{i}} nil ∈ vec(A; zero)] | {
unfold <nil vec>; reduce; auto
}. | theorem | nil-wf-vec | example | example/list-from-union.jonprl | [] | [
"nil",
"vec"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
cons-wf-vec : [{A:U{i}} {n:nat} {h:A} {t:vec(A; n)} h # t ∈ vec(A; succ(n))] | {
unfold <cons vec>; reduce; auto
}. | theorem | cons-wf-vec | example | example/list-from-union.jonprl | [] | [
"cons",
"vec"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nil-wf : [{A:U{i}} nil ∈ list(A)] | {
unfold <list union>; auto;
csubst [nil ~ pi2 <zero, nil>];
aux {unfold <pi2>; reduce; auto};
auto
}. | theorem | nil-wf | example | example/list-from-union.jonprl | [] | [
"list",
"nil",
"pi2",
"union"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
cons-wf : [{A:U{i}} {h:A} {t:list(A)} h # t ∈ list(A)] | {
auto; intro @i; auto;
unfold <list union>;
elim #3; elim #4;
main { csubst [pi2 <s, t'> ~ t'] };
main { csubst [(h # t') ~ pi2 <succ(s), h # t'>] };
aux { unfold <pi2>; reduce; auto };
auto
}. | theorem | cons-wf | example | example/list-from-union.jonprl | [] | [
"list",
"pi2",
"union"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-induction-principle : (). | operator | list-induction-principle | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[list-induction-principle] | =def= [
{A:U{i}} {C:list(A) -> U{i}} C nil -> ((h:A) (t:list(A)) C t -> C (h # t)) -> (l:list(A)) C l
]. | definition | list-induction-principle | example | example/list-from-union.jonprl | [] | [
"list",
"nil"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-induction-principle-wf : [list-induction-principle ∈ U{i'}] | {
unfold <list-induction-principle>; auto;
cum @i; auto
}. | theorem | list-induction-principle-wf | example | example/list-from-union.jonprl | [] | [
"list-induction-principle"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-recursion : (). | operator | list-recursion | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[list-recursion] | =def= [
lam(n. lam(c. lam(l. decide(l; ht.c ht; _.n))))
]. | definition | list-recursion | example | example/list-from-union.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-recursion-wf : [list-recursion ∈ list-induction-principle] | {
unfold <list-recursion list-induction-principle>; auto;
eq-cd @i'; aux { auto; cum @i; auto };
eq-cd <nil-case>; auto;
eq-cd <cons-case> @i; auto;
eq-cd <l> @i; auto;
unfold <list union>;
elim #5; elim #6;
unfold <pi2 vec>; reduce;
elim #7; reduce; auto;
elim #8; reduce; auto;
unfold <nil unit>;... | theorem | list-recursion-wf | example | example/list-from-union.jonprl | [] | [
"list",
"list-induction-principle",
"list-recursion",
"nil",
"pi2",
"union",
"vec"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-induction : [list-induction-principle] | {
witness [list-recursion]; auto
}. | theorem | list-induction | example | example/list-from-union.jonprl | [] | [
"list-induction-principle",
"list-recursion"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
2 "∈" := member. | notation | member | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
option : (0). | operator | option | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
10 "?" := option. | notation | option | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[A ?] | =def= [A + unit]. | definition | A | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
option-wf : [{A:U{i}} A? ∈ U{i}] | {
unfold <option>; auto
}. | theorem | option-wf | example | example/list.jonprl | [] | [
"option"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
some : (0). | operator | some | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[some(M)] | =def= [inl(M)]. | definition | some | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
some-wf : [{A:U{i}} {M:A} some(M) ∈ A?] | {
unfold <some option>; auto
}. | theorem | some-wf | example | example/list.jonprl | [] | [
"option",
"some"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
none : (). | operator | none | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[none] | =def= [inr(<>)]. | definition | none | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
none-wf : [{A:U{i}} none ∈ A?] | {
unfold <none option>; auto
}. | theorem | none-wf | example | example/list.jonprl | [] | [
"none",
"option"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-some : (0). | operator | is-some | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[is-some(M)] | =def= [decide(M; _.unit; _.void)]. | definition | is-some | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
is-some-wf : [{A:U{i}} {M:A?} is-some(M) ∈ U{i}] | {
auto; intro @i; auto;
unfold <is-some option>; auto;
elim #2; reduce; auto
}. | theorem | is-some-wf | example | example/list.jonprl | [] | [
"is-some",
"option"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-sig : (0). | operator | list-sig | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[list-sig(A)] | =def= [a:A? <: is-some(a)]. | definition | list-sig | example | example/list.jonprl | [] | [
"is-some"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-sig-wf : [{A:U{i}} list-sig(A) ∈ container{i}] | {
auto; unfold <list-sig>; auto;
cut-lemma <is-some-wf>;
elim #3 [A]; auto; unfold <member>;
bhyp #4; auto
}. | theorem | list-sig-wf | example | example/list.jonprl | [] | [
"is-some-wf",
"list-sig",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
5 "list" := list. | notation | list | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[A list] | =def= [wtree(list-sig(A))]. | definition | A | example | example/list.jonprl | [] | [
"list",
"list-sig"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
list-wf : [{A:U{i}} A list ∈ U{i}] | {
auto; unfold <list>; auto;
elim #2; reduce;
auto
}. | theorem | list-wf | example | example/list.jonprl | [] | [
"list"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[nil] | =def= [sup(none ^ <>)]. | definition | nil | example | example/list.jonprl | [] | [
"none"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[cons(x; xs)] | =def= [sup(some(x) ^ xs)]. | definition | cons | example | example/list.jonprl | [] | [
"some"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
-> 5 "#" := cons. | notation | cons | example | example/list.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
list-wf-tac | {
unfold <list-sig>; reduce; auto
}. | tactic | list-wf-tac | example | example/list.jonprl | [] | [
"list-sig"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
nil-wf : [{A:U{i}} nil ∈ A list] | {
unfold <nil list>; auto;
list-wf-tac;
unfold <is-some none>; reduce; auto
}. | theorem | nil-wf | example | example/list.jonprl | [] | [
"is-some",
"list",
"list-wf-tac",
"nil",
"none"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
cons-wf : [{A:U{i}} {x:A} {xs:A list} cons(x; xs) ∈ A list] | {
auto; intro @i; auto;
unfold <cons list>; auto;
list-wf-tac
}. | theorem | cons-wf | example | example/list.jonprl | [] | [
"cons",
"list",
"list-wf-tac"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
test : [("1" # "2" # "4" # nil) ∈ atom list] | {
auto
}. | theorem | test | example | example/list.jonprl | [] | [
"list",
"nil"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
match_example : [{A : U{i}} {B : U{i}} A -> B -> A] | {
intro; [intro, auto]; [intro, auto]; [intro, auto]; [id, auto];
@{[H : A |- A] => witness [H]; auto}
}. | theorem | match_example | example | example/match.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
match_example2 : [(A : U{i})(x : A) A] | {
intro; [intro, auto]; [assert [member(x; A)], auto]; [auto, id];
(* The 's here at the start of A assert that we want to bind it
* to something in the context vs generate a lexically scoped name.
* It's like the opposite of Coq's ? mechanism.
*)
@{[H : member(X; 'A) |- 'A] => witness [X]; assumption}
... | theorem | match_example2 | example | example/match.jonprl | [] | [
"assert",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
modus_ponens : [{A : U{i}}{B : U{i}}((B -> A) * B) -> A] | {
auto; *{
@{ [H' : A, H : A -> B |- B] => elim <H> [H']; auto
| [H : X * B |- C] => elim <H>
}
}
}. | theorem | modus_ponens | example | example/match.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
squash-intro | {
@{ [|- squash(A)] =>
assert [A]; main {
@{[H : A |- _] => unfold <squash member>; witness [lam(_.<>) H]; auto}
}
}
}. | tactic | squash-intro | example | example/match.jonprl | [] | [
"assert",
"member",
"squash"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
test-binding : [(A : U{i}) -> squash(unit)] | {
(* Note that squash-intro correctly isn't confused by the A
* already in the context and correctly binds a fresh variable
*)
intro; main {squash-intro}; auto
}. | theorem | test-binding | example | example/match.jonprl | [] | [
"squash",
"squash-intro"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
MonoidSig : (). | operator | MonoidSig | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
car : (0). | operator | car | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
ze : (0). | operator | ze | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
op : (0;0;0). | operator | op | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
MonoidLaws : (0). | operator | MonoidLaws | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
LeftUnit : (0). | operator | LeftUnit | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
RightUnit : (0). | operator | RightUnit | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Assoc : (0). | operator | Assoc | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
Monoid : (). | operator | Monoid | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | ||
[MonoidSig] | =def= [(A:U{i}) * A * (A -> A -> A)]. | definition | MonoidSig | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[car(M)] | =def= [spread(M; x.y.x)]. | definition | car | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[ze(M)] | =def= [spread(spread(M; x.y.y); x.y.x)]. | definition | ze | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[op(M;X;Y)] | =def= [spread(spread(M; x.y.y); x.y.y) X Y]. | definition | op | example | example/monoid.jonprl | [] | [] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[LeftUnit(M)] | =def= [{m:car(M)} =(m; op(M; ze(M); m); car(M))]. | definition | LeftUnit | example | example/monoid.jonprl | [] | [
"car",
"op",
"ze"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[RightUnit(M)] | =def= [{m:car(M)} =(op(M; ze(M); m); m; car(M))]. | definition | RightUnit | example | example/monoid.jonprl | [] | [
"car",
"op",
"ze"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Assoc(M)] | =def= [{x:car(M)} {y:car(M)} {z:car(M)} =(op(M;op(M;x;y);z); op(M;x;op(M;y;z));car(M))]. | definition | Assoc | example | example/monoid.jonprl | [] | [
"car",
"op"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[MonoidLaws(M)] | =def= [LeftUnit(M) * RightUnit(M) * Assoc(M)]. | definition | MonoidLaws | example | example/monoid.jonprl | [] | [
"Assoc",
"LeftUnit",
"RightUnit"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
[Monoid] | =def= [{M : MonoidSig | MonoidLaws(M)}]. | definition | Monoid | example | example/monoid.jonprl | [] | [
"MonoidLaws",
"MonoidSig"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
monoid-sig-unfold | {
unfold <MonoidSig car op ze>
}. | tactic | monoid-sig-unfold | example | example/monoid.jonprl | [] | [
"MonoidSig",
"car",
"op",
"ze"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
MonoidSig-wf : [member(MonoidSig; U{i'})] | {
monoid-sig-unfold; auto.
}. | theorem | MonoidSig-wf | example | example/monoid.jonprl | [] | [
"MonoidSig",
"member",
"monoid-sig-unfold"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
car-wf : [{M:MonoidSig} member(car(M); U{i})] | {
intro @i'; auto;
unfold <MonoidSig car>; auto
}. | theorem | car-wf | example | example/monoid.jonprl | [] | [
"MonoidSig",
"car",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
ze-wf : [{M:MonoidSig} member(ze(M); car(M))] | {
intro @i'; auto;
unfold <MonoidSig ze car>;
auto; reduce; auto
}. | theorem | ze-wf | example | example/monoid.jonprl | [] | [
"MonoidSig",
"car",
"member",
"ze"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
op-wf : [{M:MonoidSig} {X:car(M)} {Y:car(M)} member(op(M; X; Y); car(M))] | {
intro @i'; auto; *{ intro @i; auto };
unfold <MonoidSig car op>;
auto; reduce; auto
}. | theorem | op-wf | example | example/monoid.jonprl | [] | [
"MonoidSig",
"car",
"member",
"op"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
LeftUnit-wf : [{M:MonoidSig} member(LeftUnit(M); U{i})] | {
intro @i'; auto;
unfold <LeftUnit>;
auto
}. | theorem | LeftUnit-wf | example | example/monoid.jonprl | [] | [
"LeftUnit",
"MonoidSig",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
RightUnit-wf : [{M:MonoidSig} member(RightUnit(M);U{i})] | {
intro @i'; auto;
unfold <RightUnit>; auto
}. | theorem | RightUnit-wf | example | example/monoid.jonprl | [] | [
"MonoidSig",
"RightUnit",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Assoc-wf : [{M:MonoidSig} member(Assoc(M); U{i})] | {
intro @i'; auto;
unfold <Assoc>; auto
}. | theorem | Assoc-wf | example | example/monoid.jonprl | [] | [
"Assoc",
"MonoidSig",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
MonoidLaws-wf : [{M:MonoidSig} member(MonoidLaws(M); U{i})] | {
intro @i'; auto;
unfold <MonoidLaws>; auto
}. | theorem | MonoidLaws-wf | example | example/monoid.jonprl | [] | [
"MonoidLaws",
"MonoidSig",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 | |
Monoid-wf : [member(Monoid; U{i'})] | {
unfold <Monoid>; auto; cum @i; auto
}. | theorem | Monoid-wf | example | example/monoid.jonprl | [] | [
"Monoid",
"member"
] | https://github.com/jonsterling/JonPRL | f937a5461955fff8cb22331ecb6fdeae8df636d5 |
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