fact stringlengths 17 3.77k | statement stringlengths 2 515 | proof stringlengths 0 3.49k | type stringclasses 4
values | symbolic_name stringlengths 1 23 | library stringclasses 3
values | filename stringclasses 56
values | imports listlengths 0 0 | deps listlengths 0 7 | docstring stringclasses 0
values | line_start int64 1 323 | line_end int64 1 337 | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
SigEquivToPath(#l:lvl) :
(->
[ty/a : (U #l kan)]
(* [ty/b : (U #l kan)] (Equiv ty/a ty/b))
(* [ty/b : (U #l kan)] (path [_] (U #l kan) ty/a ty/b)))
by {
lam ty/a {ty/b,equiv} =>
{ use ty/b
, abs x => `(V x ty/a ty/b equiv)
}
}. | SigEquivToPath(#l:lvl) :
(->
[ty/a : (U #l kan)]
(* [ty/b : (U #l kan)] (Equiv ty/a ty/b))
(* [ty/b : (U #l kan)] (path [_] (U #l kan) ty/a ty/b))) | by {
lam ty/a {ty/b,equiv} =>
{ use ty/b
, abs x => `(V x ty/a ty/b equiv)
}
}. | theorem | SigEquivToPath | example | example/univalence.prl | [] | [
"Equiv"
] | null | 277 | 287 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
SigPathToEquiv(#l:lvl) :
(->
[ty/a : (U #l kan)]
(* [ty/b : (U #l kan)] (path [_] (U #l kan) ty/a ty/b))
(* [ty/b : (U #l kan)] (Equiv ty/a ty/b)))
by {
lam ty/a {ty/b,p} =>
{ use ty/b
, use (PathToEquiv #l) [`ty/a, `ty/b, `p]
}
}. | SigPathToEquiv(#l:lvl) :
(->
[ty/a : (U #l kan)]
(* [ty/b : (U #l kan)] (path [_] (U #l kan) ty/a ty/b))
(* [ty/b : (U #l kan)] (Equiv ty/a ty/b))) | by {
lam ty/a {ty/b,p} =>
{ use ty/b
, use (PathToEquiv #l) [`ty/a, `ty/b, `p]
}
}. | theorem | SigPathToEquiv | example | example/univalence.prl | [] | [
"Equiv",
"PathToEquiv"
] | null | 289 | 299 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
UARetSig(#l:lvl) :
(->
[ty/a : (U #l kan)]
(Retract
(* [ty/b : (U #l kan)] (Equiv ty/a ty/b))
($ (SigEquivToPath #l) ty/a)
($ (SigPathToEquiv #l) ty/a)))
by {
lam ty/a {ty/b,equiv} =>
unfold SigPathToEquiv SigEquivToPath;
abs x =>
{ use ty/b
, use (UARet #l) [`ty/a, `ty/b, `equiv, `x]
... | UARetSig(#l:lvl) :
(->
[ty/a : (U #l kan)]
(Retract
(* [ty/b : (U #l kan)] (Equiv ty/a ty/b))
($ (SigEquivToPath #l) ty/a)
($ (SigPathToEquiv #l) ty/a))) | by {
lam ty/a {ty/b,equiv} =>
unfold SigPathToEquiv SigEquivToPath;
abs x =>
{ use ty/b
, use (UARet #l) [`ty/a, `ty/b, `equiv, `x]
}
}. | theorem | UARetSig | example | example/univalence.prl | [] | [
"Equiv",
"Retract",
"SigEquivToPath",
"SigPathToEquiv",
"UARet"
] | null | 301 | 315 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Univalence(#l:lvl) :
(->
[ty/a : (U #l kan)]
(IsContr (* [ty/b : (U #l kan)] (Equiv ty/a ty/b))))
by {
lam ty/a =>
use (RetIsContr (++ #l))
[ `(* [ty/b : (U #l kan)] (Equiv ty/a ty/b))
, `(* [ty/b : (U #l kan)] (path [_] (U #l kan) ty/a ty/b))
, use (SigEquivToPath #l) [`ty/a]
, use (SigPathToEq... | Univalence(#l:lvl) :
(->
[ty/a : (U #l kan)]
(IsContr (* [ty/b : (U #l kan)] (Equiv ty/a ty/b)))) | by {
lam ty/a =>
use (RetIsContr (++ #l))
[ `(* [ty/b : (U #l kan)] (Equiv ty/a ty/b))
, `(* [ty/b : (U #l kan)] (path [_] (U #l kan) ty/a ty/b))
, use (SigEquivToPath #l) [`ty/a]
, use (SigPathToEquiv #l) [`ty/a]
, use (UARetSig #l) [`ty/a]
, use (IsContrPath #l) [`ty/a]
]
}. | theorem | Univalence | example | example/univalence.prl | [] | [
"Equiv",
"IsContr",
"IsContrPath",
"RetIsContr",
"SigEquivToPath",
"SigPathToEquiv",
"UARetSig"
] | null | 323 | 337 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
MalformedTube :
wbool true
by {
`(hcom 0~>1 wbool tt)
}. | MalformedTube :
wbool true | by {
`(hcom 0~>1 wbool tt)
}. | theorem | MalformedTube | test/failure | test/failure/bad-hcom-empty.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Bool :
wbool true
by {
`(hcom 0~>1 wbool tt [0=1 [_] tt]);
auto
}. | Bool :
wbool true | by {
`(hcom 0~>1 wbool tt [0=1 [_] tt]);
auto
}. | theorem | Bool | test/failure | test/failure/bad-hcom-stuck.prl | [] | [] | null | 1 | 6 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Foo(#a : exp, #b : exp) : exp = (-> #a #b) . | Foo(#a : exp, #b : exp) : exp | = (-> #a #b) . | define | Foo | test/failure | test/failure/bad-op.prl | [] | [] | null | 1 | 1 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Foo-bool-type : (Foo bool) typeby {
auto
}. | Foo-bool-type : (Foo bool) typeby {
auto
}. | theorem | Foo-bool-type | test/failure | test/failure/bad-op.prl | [] | [
"Foo"
] | null | 3 | 5 | false | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c | |
Cmp(#f : exp, #g : exp) : exp =
(lam [x] ($ #f ($ #h x)))
. | Cmp(#f : exp, #g : exp) : exp | =
(lam [x] ($ #f ($ #h x)))
. | define | Cmp | test/failure | test/failure/freemeta.prl | [] | [] | null | 1 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
FreeVar : x true by {
auto
}. | FreeVar : x true | by {
auto
}. | theorem | FreeVar | test/failure | test/failure/freevar.prl | [] | [] | null | 1 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Foo : tt = tt in bool by {
auto
}. | Foo : tt = tt in bool | by {
auto
}. | theorem | Foo | test/failure | test/failure/incremental-parse.prl | [] | [] | null | 6 | 8 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Bar : tt = tt in bool ] by {
auto
}.
Thm Baz : [ tt = tt in bool by {
auto
}. | Bar : tt = tt in bool ] by {
auto
}.
Thm Baz : [ tt = tt in bool | by {
auto
}. | theorem | Bar | test/failure | test/failure/incremental-parse.prl | [] | [] | null | 10 | 16 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Path/Symm(#l:lvl) :
ty : (U #l)
>>
ty type with hcom
by {
auto
}. | Path/Symm(#l:lvl) :
ty : (U #l)
>>
ty type with hcom | by {
auto
}. | theorem | Path/Symm | test/failure | test/failure/kind-hcom.prl | [] | [] | null | 1 | 7 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
LexicalError : (-> bool bool) true by {
(lam x => `_tt); auto
}. | LexicalError : (-> bool bool) true | by {
(lam x => `_tt); auto
}. | theorem | LexicalError | test/failure | test/failure/lexical-error.prl | [] | [] | null | 1 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NegOne :
-1 in nat
by {
auto
}. | NegOne :
-1 in nat | by {
auto
}. | theorem | NegOne | test/failure | test/failure/num.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest :
tuple in (record [a : bool])
by {
auto
}. | RecordTest :
tuple in (record [a : bool]) | by {
auto
}. | theorem | RecordTest | test/failure | test/failure/record0.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest :
(! a tuple) in bool
by {
auto
}. | RecordTest :
(! a tuple) in bool | by {
auto
}. | theorem | RecordTest | test/failure | test/failure/record1.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest :
(! a (tuple [b tt])) in bool
by {
auto
}. | RecordTest :
(! a (tuple [b tt])) in bool | by {
auto
}. | theorem | RecordTest | test/failure | test/failure/record2.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
DuplicateLabel :
(tuple [a tt]) in (record [a a : bool])
by {
auto
}. | DuplicateLabel :
(tuple [a tt]) in (record [a a : bool]) | by {
auto
}. | theorem | DuplicateLabel | test/failure | test/failure/record3.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
DuplicateLabel :
(tuple [a tt] [a tt]) in (record [a : bool])
by {
auto
}. | DuplicateLabel :
(tuple [a tt] [a tt]) in (record [a : bool]) | by {
auto
}. | theorem | DuplicateLabel | test/failure | test/failure/record4.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Not : exp =
(lam [x] (if x ff tt))
. | Not : exp | =
(lam [x] (if x ff tt))
. | define | Not | test/failure | test/failure/undef-custom.prl | [] | [] | null | 1 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Foo : Bar = Bar in bool by {
auto
}. | Foo : Bar = Bar in bool | by {
auto
}. | theorem | Foo | test/failure | test/failure/undef-custom.prl | [] | [
"Bar"
] | null | 10 | 12 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Test : (* bool bool) = (* bool bool) type by {
auto
}. | Test : (* bool bool) = (* bool bool) type | by {
auto
}. | theorem | Test | test/success | test/success/bool-pair-test.prl | [] | [] | null | 1 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Ident-test/ : bool by {
`tt
}. | Ident-test/ : bool | by {
`tt
}. | theorem | Ident-test/ | test/success | test/success/dashes-n-slashes.prl | [] | [] | null | 2 | 4 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Decomposition :
(->
(record [rcd : (record [a : bool] [b : (* bool int)])] [circ : S1])
bool)
by {
lam x =>
let {rcd = {a = a, b = {welp}}, circ = circ} = x;
use welp
}. | Decomposition :
(->
(record [rcd : (record [a : bool] [b : (* bool int)])] [circ : S1])
bool) | by {
lam x =>
let {rcd = {a = a, b = {welp}}, circ = circ} = x;
use welp
}. | theorem | Decomposition | test/success | test/success/decomposition.prl | [] | [] | null | 1 | 9 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Apply :
(->
(->
bool
bool
(path [_]
(record [a : S1])
(tuple [a base])
(tuple [a base])))
S1)
by {
lam f =>
let {a = a} = f [`tt, `ff, `(dim 0)];
use a
}. | Apply :
(->
(->
bool
bool
(path [_]
(record [a : S1])
(tuple [a base])
(tuple [a base])))
S1) | by {
lam f =>
let {a = a} = f [`tt, `ff, `(dim 0)];
use a
}. | theorem | Apply | test/success | test/success/decomposition.prl | [] | [] | null | 13 | 27 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
UseHypTest :
(-> bool bool)
by {
lam x =>
claim p : (-> bool S1 bool) by {lam b c => use b};
use p [use x, `(loop 0)]
}. | UseHypTest :
(-> bool bool) | by {
lam x =>
claim p : (-> bool S1 bool) by {lam b c => use b};
use p [use x, `(loop 0)]
}. | theorem | UseHypTest | test/success | test/success/decomposition.prl | [] | [] | null | 33 | 39 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
UseLemmaTest :
(-> bool bool)
by {
lam x =>
use UseHypTest [use x]
}. | UseLemmaTest :
(-> bool bool) | by {
lam x =>
use UseHypTest [use x]
}. | theorem | UseLemmaTest | test/success | test/success/decomposition.prl | [] | [
"UseHypTest"
] | null | 43 | 48 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Discrete/reflection(#l:lvl) :
(->
[ty : (U #l discrete)]
[a b : ty]
[p : (path [_] ty a b)]
(= ty a b))
by {
lam ty a b p => `(coe 0~>1 [x] (= ty a (@ p x)) ax)
}. | Discrete/reflection(#l:lvl) :
(->
[ty : (U #l discrete)]
[a b : ty]
[p : (path [_] ty a b)]
(= ty a b)) | by {
lam ty a b p => `(coe 0~>1 [x] (= ty a (@ p x)) ax)
}. | theorem | Discrete/reflection | test/success | test/success/discrete-types.prl | [] | [] | null | 1 | 9 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
GetHole(#c : [exp].exp, #t : [exp].tac) = {
query gl <- concl;
match gl {
[hole | #jdg{(#c %hole)} => (#t %hole)]
}
}. | GetHole(#c : [exp].exp, #t : [exp].tac) | = {
query gl <- concl;
match gl {
[hole | #jdg{(#c %hole)} => (#t %hole)]
}
}. | tactic | GetHole | test/success | test/success/equality-elim.prl | [] | [] | null | 1 | 6 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Rewrite(#c : [exp].exp, #n, #a, #t : tac) = {
(GetHole [x] (#c x) [hole] #tac{
claim p : hole = #n in #a by {#t};
// Use the elimination rule for equality. We bind a new hypothesis which will represent the location
// in the goal #c which is being rewritten.
rewrite p;
[with x => `(#c x), id, auto... | Rewrite(#c : [exp].exp, #n, #a, #t : tac) | = {
(GetHole [x] (#c x) [hole] #tac{
claim p : hole = #n in #a by {#t};
// Use the elimination rule for equality. We bind a new hypothesis which will represent the location
// in the goal #c which is being rewritten.
rewrite p;
[with x => `(#c x), id, auto, auto]
})
}. | tactic | Rewrite | test/success | test/success/equality-elim.prl | [] | [
"GetHole"
] | null | 11 | 19 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
EqualityElimTest :
(-> [b : bool] (path [_] bool tt (if [_] bool tt tt ff)))
by {
// We're going to prove this in a silly way to illustrate equality elimination.
// We'll rewrite the goal by claiming (if tt tt ff) = tt in bool.
(Rewrite
[x] (-> bool (path [_] bool tt x))
tt bool #tac{auto});
// observ... | EqualityElimTest :
(-> [b : bool] (path [_] bool tt (if [_] bool tt tt ff))) | by {
// We're going to prove this in a silly way to illustrate equality elimination.
// We'll rewrite the goal by claiming (if tt tt ff) = tt in bool.
(Rewrite
[x] (-> bool (path [_] bool tt x))
tt bool #tac{auto});
// observe that the goal has now been rewritten!
?check-this-out;
lam b => abs _ => ... | theorem | EqualityElimTest | test/success | test/success/equality-elim.prl | [] | [
"Rewrite"
] | null | 21 | 32 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
EqualityKind0(#A) :
(->
[ty : (U 0 pre)]
[a b : ty]
(= (U 0 hcom) (= ty a b) (= ty a b)))
by {
lam ty a b => auto
}. | EqualityKind0(#A) :
(->
[ty : (U 0 pre)]
[a b : ty]
(= (U 0 hcom) (= ty a b) (= ty a b))) | by {
lam ty a b => auto
}. | theorem | EqualityKind0 | test/success | test/success/equality.prl | [] | [] | null | 1 | 8 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
EqualityKind1(#A) :
(->
[ty : (U 0 discrete)]
[a b : ty]
(= (U 0 kan) (= ty a b) (= ty a b)))
by {
lam ty a b => auto
}. | EqualityKind1(#A) :
(->
[ty : (U 0 discrete)]
[a b : ty]
(= (U 0 kan) (= ty a b) (= ty a b))) | by {
lam ty a b => auto
}. | theorem | EqualityKind1 | test/success | test/success/equality.prl | [] | [] | null | 10 | 17 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/bool :
(-> [i : dim]
(mem (U 0) (fcom 0~>1 bool [i=0 [j] bool] [i=1 [j] bool])))
by {
abs i => auto
}. | Fcom/bool :
(-> [i : dim]
(mem (U 0) (fcom 0~>1 bool [i=0 [j] bool] [i=1 [j] bool]))) | by {
abs i => auto
}. | theorem | Fcom/bool | test/success | test/success/fcom-types.prl | [] | [] | null | 1 | 6 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/Box :
(-> [i : dim]
(mem
(fcom 0~>1 bool [i=0 [j] bool] [i=1 [j] bool])
(box 0~>1 tt [i=0 tt] [i=1 tt])))
by {
abs i => auto
}. | Fcom/Box :
(-> [i : dim]
(mem
(fcom 0~>1 bool [i=0 [j] bool] [i=1 [j] bool])
(box 0~>1 tt [i=0 tt] [i=1 tt]))) | by {
abs i => auto
}. | theorem | Fcom/Box | test/success | test/success/fcom-types.prl | [] | [] | null | 10 | 17 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/Reduce :
(fcom 0~>1 bool [0=0 [j] bool]) = bool type
by {
auto
}. | Fcom/Reduce :
(fcom 0~>1 bool [0=0 [j] bool]) = bool type | by {
auto
}. | theorem | Fcom/Reduce | test/success | test/success/fcom-types.prl | [] | [] | null | 21 | 25 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/Cap1 :
tt in (fcom 0~>1 bool [0=0 [j] bool])
by {
auto
}. | Fcom/Cap1 :
tt in (fcom 0~>1 bool [0=0 [j] bool]) | by {
auto
}. | theorem | Fcom/Cap1 | test/success | test/success/fcom-types.prl | [] | [] | null | 27 | 31 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/Cap2 :
(cap 0<~1 (box 0~>1 tt [0=0 tt]) [0=0 [j] bool]) in bool
by {
auto
}. | Fcom/Cap2 :
(cap 0<~1 (box 0~>1 tt [0=0 tt]) [0=0 [j] bool]) in bool | by {
auto
}. | theorem | Fcom/Cap2 | test/success | test/success/fcom-types.prl | [] | [] | null | 33 | 37 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Hcom/Poly(#l:lvl) :
(->
[ty : (U #l hcom)]
[a b c d : ty]
(path [_] ty a b)
(path [_] ty a c)
(path [_] ty b d)
(path [_] ty c d))
by {
lam ty a b c d pab pac pbd =>
abs i =>
`(hcom 0~>1 ty (@ pab i)
[i=0 [j] (@ pac j)]
[i=1 [j] (@ pbd j)])
}. | Hcom/Poly(#l:lvl) :
(->
[ty : (U #l hcom)]
[a b c d : ty]
(path [_] ty a b)
(path [_] ty a c)
(path [_] ty b d)
(path [_] ty c d)) | by {
lam ty a b c d pab pac pbd =>
abs i =>
`(hcom 0~>1 ty (@ pab i)
[i=0 [j] (@ pac j)]
[i=1 [j] (@ pbd j)])
}. | theorem | Hcom/Poly | test/success | test/success/hcom.prl | [] | [] | null | 1 | 15 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Hcom/trans(#l:lvl) :
(->
[ty : (U #l hcom)]
[a b c : ty]
(path [_] ty a b)
(path [_] ty b c)
(path [_] ty a c))
by {
lam ty a b c pab pbc =>
abs i =>
`(hcom 0 ~> 1 ty (@ pab i)
[i=0 [_] a]
[i=1 [j] (@ pbc j)])
}. | Hcom/trans(#l:lvl) :
(->
[ty : (U #l hcom)]
[a b c : ty]
(path [_] ty a b)
(path [_] ty b c)
(path [_] ty a c)) | by {
lam ty a b c pab pbc =>
abs i =>
`(hcom 0 ~> 1 ty (@ pab i)
[i=0 [_] a]
[i=1 [j] (@ pbc j)])
}. | theorem | Hcom/trans | test/success | test/success/hcom.prl | [] | [] | null | 19 | 32 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Hcom/symm(#l:lvl) :
(->
[ty : (U #l hcom)]
[a b : ty]
(path [_] ty a b)
(path [_] ty b a))
by {
lam ty a b pab =>
abs i =>
`(hcom 0~>1 ty a
[i=0 [j] (@ pab j)]
[i=1 [_] a])
}. | Hcom/symm(#l:lvl) :
(->
[ty : (U #l hcom)]
[a b : ty]
(path [_] ty a b)
(path [_] ty b a)) | by {
lam ty a b pab =>
abs i =>
`(hcom 0~>1 ty a
[i=0 [j] (@ pab j)]
[i=1 [_] a])
}. | theorem | Hcom/symm | test/success | test/success/hcom.prl | [] | [] | null | 36 | 48 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Cap(#l:lvl) :
(->
[ty : (U #l hcom)]
[x : ty]
[i : dim]
(= ty
(hcom 0~>0 ty x [i=0 [_] x] [i=1 [_] x])
x))
by {
lam ty x => abs i => auto
}. | Cap(#l:lvl) :
(->
[ty : (U #l hcom)]
[x : ty]
[i : dim]
(= ty
(hcom 0~>0 ty x [i=0 [_] x] [i=1 [_] x])
x)) | by {
lam ty x => abs i => auto
}. | theorem | Cap | test/success | test/success/hcom.prl | [] | [] | null | 53 | 63 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Tube(#l:lvl) :
(->
[ty : (U #l hcom)]
[x : ty]
(= ty
(hcom 0~>1 ty x [1=1 [_] x] [0=0 [_] x])
x))
by {
lam ty x => auto
}. | Tube(#l:lvl) :
(->
[ty : (U #l hcom)]
[x : ty]
(= ty
(hcom 0~>1 ty x [1=1 [_] x] [0=0 [_] x])
x)) | by {
lam ty x => auto
}. | theorem | Tube | test/success | test/success/hcom.prl | [] | [] | null | 67 | 76 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
TrueByEvaluation :
(hcom 0~>0 bool tt) in bool
by {
auto
}. | TrueByEvaluation :
(hcom 0~>0 bool tt) in bool | by {
auto
}. | theorem | TrueByEvaluation | test/success | test/success/hcom.prl | [] | [] | null | 78 | 82 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
S1' : (U 0 kan)
{ base'
, loop' [x : dim] [x=0 (self base')] [x=1 (self base')]
}
by {
auto
}. | S1' : (U 0 kan)
{ base'
, loop' [x : dim] [x=0 (self base')] [x=1 (self base')]
} | by {
auto
}. | data | S1' | test/success | test/success/inductive-S1.prl | [] | [] | null | 1 | 7 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Loop' :
(path [_] (. S1' type) (. S1' base') (. S1' base'))
by {
abs u => `(. S1' loop' u)
}. | Loop' :
(path [_] (. S1' type) (. S1' base') (. S1' base')) | by {
abs u => `(. S1' loop' u)
}. | theorem | Loop' | test/success | test/success/inductive-S1.prl | [] | [
"S1'"
] | null | 9 | 13 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
S1' : (U 0 kan)
{ base'
, loop' [x : dim] [x=0 (self base')] [x=1 (self base')]
}
by {
auto
}. | S1' : (U 0 kan)
{ base'
, loop' [x : dim] [x=0 (self base')] [x=1 (self base')]
} | by {
auto
}. | data | S1' | test/success | test/success/inductive.prl | [] | [] | null | 1 | 7 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Pushout' (#l:lvl)
[a b c : (U #l coe)]
[f : (-> c a)]
[g : (-> c b)]
: (U #l kan)
{ left' a
, right' b
, glue' [x : c] [y : dim] [y=0 (self left' ($ f x))] [y=1 (self right' ($ g x))]
}
by {
auto
}. | Pushout' (#l:lvl)
[a b c : (U #l coe)]
[f : (-> c a)]
[g : (-> c b)]
: (U #l kan)
{ left' a
, right' b
, glue' [x : c] [y : dim] [y=0 (self left' ($ f x))] [y=1 (self right' ($ g x))]
} | by {
auto
}. | data | Pushout' | test/success | test/success/inductive.prl | [] | [] | null | 9 | 20 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
PropTrunc (#l:lvl) [a : (U 0 coe)] : (U 0 kan) {
pt a,
sq [x y : self] [z : dim] [z=0 x] [z=1 y]
} by { auto }. | PropTrunc (#l:lvl) [a : (U 0 coe)] : (U 0 kan) {
pt a,
sq [x y : self] [z : dim] [z=0 x] [z=1 y]
} | by { auto }. | data | PropTrunc | test/success | test/success/inductive.prl | [] | [] | null | 22 | 25 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Line/Test0 :
(->
[a : (U 0 kan)]
[l : (-> dim a)]
(= a (coe 0~>1 [_] a (@ l 0)) (@ (coe 0~>1 [_] (-> dim a) l) 0)))
by {
lam a l => `ax
}. | Line/Test0 :
(->
[a : (U 0 kan)]
[l : (-> dim a)]
(= a (coe 0~>1 [_] a (@ l 0)) (@ (coe 0~>1 [_] (-> dim a) l) 0))) | by {
lam a l => `ax
}. | theorem | Line/Test0 | test/success | test/success/lines.prl | [] | [] | null | 1 | 8 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Line/Test1 :
(->
[ty : (U 0 kan)]
[p : (line [_] ty)]
(path [_] ty (@ p 0) (@ p 1)))
by {
lam ty p => abs x => `(@ p x)
}. | Line/Test1 :
(->
[ty : (U 0 kan)]
[p : (line [_] ty)]
(path [_] ty (@ p 0) (@ p 1))) | by {
lam ty p => abs x => `(@ p x)
}. | theorem | Line/Test1 | test/success | test/success/lines.prl | [] | [] | null | 10 | 17 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Line/Trans :
(->
[ty : (U 0 kan)]
[p : (line [_] ty)]
[q : (line [_] ty)]
[eq : (= ty (@ p 1) (@ q 0))]
(path [_] ty (@ p 0) (@ q 1)))
by {
(lam ty p q eq => abs x =>
`(hcom 0~>1 ty (@ p x)
[x=0 [_] (@ p 0)]
[x=1 [y] (@ q y)]));
repeat {assumption || auto-step}
}. | Line/Trans :
(->
[ty : (U 0 kan)]
[p : (line [_] ty)]
[q : (line [_] ty)]
[eq : (= ty (@ p 1) (@ q 0))]
(path [_] ty (@ p 0) (@ q 1))) | by {
(lam ty p q eq => abs x =>
`(hcom 0~>1 ty (@ p x)
[x=0 [_] (@ p 0)]
[x=1 [y] (@ q y)]));
repeat {assumption || auto-step}
}. | theorem | Line/Trans | test/success | test/success/lines.prl | [] | [] | null | 20 | 34 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Line/Symm :
(->
[ty : (U 0 kan)]
[p : (line [_] ty)]
(path [_] ty (@ p 1) (@ p 0)))
by {
lam ty p => abs x =>
`(hcom 0~>1 ty (@ p 0)
[x=0 [y] (@ p y)]
[x=1 [_] (@ p 0)])
}. | Line/Symm :
(->
[ty : (U 0 kan)]
[p : (line [_] ty)]
(path [_] ty (@ p 1) (@ p 0))) | by {
lam ty p => abs x =>
`(hcom 0~>1 ty (@ p 0)
[x=0 [y] (@ p y)]
[x=1 [_] (@ p 0)])
}. | theorem | Line/Symm | test/success | test/success/lines.prl | [] | [] | null | 36 | 46 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Thm1 :
(->
[p q : (U 0)]
p q p)
by {
lam p q a b => use a
}. | Thm1 :
(->
[p q : (U 0)]
p q p) | by {
lam p q a b => use a
}. | theorem | Thm1 | test/success | test/success/logical-investigations.prl | [] | [] | null | 3 | 9 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Thm2 :
(->
[p q r : (U 0)]
(-> p q)
(-> p q r)
p r)
by {
lam p q r f g a =>
use g [use a, use f [use a]]
}. | Thm2 :
(->
[p q r : (U 0)]
(-> p q)
(-> p q r)
p r) | by {
lam p q r f g a =>
use g [use a, use f [use a]]
}. | theorem | Thm2 | test/success | test/success/logical-investigations.prl | [] | [] | null | 11 | 20 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Thm3/low-level :
(->
[p q r : (U 0)]
(-> p q)
(-> q r)
(-> p r))
by {
// fresh p q r pq qr x ->
repeat {refine fun/intro || id};
auto; with x qr pq r q p =>
elim qr; elim pq;
[ use x
, with _ y => use y
, use x
, with _ _ _ z => use z
]
}. | Thm3/low-level :
(->
[p q r : (U 0)]
(-> p q)
(-> q r)
(-> p r)) | by {
// fresh p q r pq qr x ->
repeat {refine fun/intro || id};
auto; with x qr pq r q p =>
elim qr; elim pq;
[ use x
, with _ y => use y
, use x
, with _ _ _ z => use z
]
}. | theorem | Thm3/low-level | test/success | test/success/logical-investigations.prl | [] | [] | null | 26 | 42 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Thm3/high-level :
(->
[p q r : (U 0)]
(-> p q)
(-> q r)
(-> p r))
by {
lam p q r f g x =>
use g [use f [use x]]
}. | Thm3/high-level :
(->
[p q r : (U 0)]
(-> p q)
(-> q r)
(-> p r)) | by {
lam p q r f g x =>
use g [use f [use x]]
}. | theorem | Thm3/high-level | test/success | test/success/logical-investigations.prl | [] | [] | null | 49 | 58 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Not(#A) = (-> #A void) . | Not(#A) | = (-> #A void) . | define | Not | test/success | test/success/logical-investigations.prl | [] | [] | null | 62 | 62 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Thm4 :
(-> [p q : (U 0)] (Not p) p q)
by {
lam p q r a =>
unfold Not;
let boom = r [use a];
elim boom
}. | Thm4 :
(-> [p q : (U 0)] (Not p) p q) | by {
lam p q r a =>
unfold Not;
let boom = r [use a];
elim boom
}. | theorem | Thm4 | test/success | test/success/logical-investigations.prl | [] | [
"Not"
] | null | 64 | 71 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Thm5 :
(-> [p : (U 0)] p (Not (Not p)))
by {
lam p a => unfold Not; lam r =>
use r [use a]
}. | Thm5 :
(-> [p : (U 0)] p (Not (Not p))) | by {
lam p a => unfold Not; lam r =>
use r [use a]
}. | theorem | Thm5 | test/success | test/success/logical-investigations.prl | [] | [
"Not"
] | null | 73 | 78 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Thm6(#A,#B) :
(-> [p q : (U 0)] (-> p q) (Not q) (Not p))
by {
lam p q f g => unfold Not; lam a =>
use g [use f [use a]]
}. | Thm6(#A,#B) :
(-> [p q : (U 0)] (-> p q) (Not q) (Not p)) | by {
lam p q f g => unfold Not; lam a =>
use g [use f [use a]]
}. | theorem | Thm6 | test/success | test/success/logical-investigations.prl | [] | [
"Not"
] | null | 84 | 89 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
QueryGoalType(#t : [exp].tac) = {
query gl <- concl;
match gl {
[a | #jdg{%a true} => (#t %a)]
}
}. | QueryGoalType(#t : [exp].tac) | = {
query gl <- concl;
match gl {
[a | #jdg{%a true} => (#t %a)]
}
}. | tactic | QueryGoalType | test/success | test/success/match.prl | [] | [] | null | 1 | 6 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
MatchGoal : (-> bool bool bool bool bool bool) by {
repeat {
(QueryGoalType [ty] #tac{
match ty {
[a b | (-> [x:%a] (%b x)) => refine fun/intro; [id, auto]]
}
})
};
with _ _ y => use y
}. | MatchGoal : (-> bool bool bool bool bool bool) | by {
repeat {
(QueryGoalType [ty] #tac{
match ty {
[a b | (-> [x:%a] (%b x)) => refine fun/intro; [id, auto]]
}
})
};
with _ _ y => use y
}. | theorem | MatchGoal | test/success | test/success/match.prl | [] | [
"QueryGoalType"
] | null | 8 | 18 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
One :
(int 1) in int
by {
auto
}. | One :
(int 1) in int | by {
auto
}. | theorem | One | test/success | test/success/num.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NegOne :
(int -1) in int
by {
auto
}. | NegOne :
(int -1) in int | by {
auto
}. | theorem | NegOne | test/success | test/success/num.prl | [] | [] | null | 7 | 11 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NatOne :
(nat 1) in nat
by {
auto
}. | NatOne :
(nat 1) in nat | by {
auto
}. | theorem | NatOne | test/success | test/success/num.prl | [] | [] | null | 13 | 17 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NatIsInt :
(-> [x : nat] (mem int (pos x)))
by {
lam x => auto
}. | NatIsInt :
(-> [x : nat] (mem int (pos x))) | by {
lam x => auto
}. | theorem | NatIsInt | test/success | test/success/num.prl | [] | [] | null | 19 | 23 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Pred :
(-> nat nat)
by {
lam a =>
elim a;
[ `zero ];
[ with a' ind => `a' ]
}. | Pred :
(-> nat nat) | by {
lam a =>
elim a;
[ `zero ];
[ with a' ind => `a' ]
}. | theorem | Pred | test/success | test/success/num.prl | [] | [] | null | 25 | 32 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus :
(-> nat nat nat)
by {
lam a =>
elim a;
[ lam x => use x
, with ind a' =>
lam x =>
let ih/x = ind [use x]; `(succ ih/x)
]
}. | Plus :
(-> nat nat nat) | by {
lam a =>
elim a;
[ lam x => use x
, with ind a' =>
lam x =>
let ih/x = ind [use x]; `(succ ih/x)
]
}. | theorem | Plus | test/success | test/success/num.prl | [] | [] | null | 34 | 44 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus/wf :
Plus in (-> nat nat nat)
by {
auto
}. | Plus/wf :
Plus in (-> nat nat nat) | by {
auto
}. | theorem | Plus/wf | test/success | test/success/num.prl | [] | [
"Plus"
] | null | 46 | 50 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus/zeroL :
(-> [n : nat] (= nat ($ Plus (nat 0) n) n))
by {
lam n => auto
}. | Plus/zeroL :
(-> [n : nat] (= nat ($ Plus (nat 0) n) n)) | by {
lam n => auto
}. | theorem | Plus/zeroL | test/success | test/success/num.prl | [] | [
"Plus"
] | null | 52 | 56 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus/zero/R :
(-> [n : nat] (= nat ($ Plus n (nat 0)) n))
by {
lam n =>
elim n;
[ `ax
, with ind n' =>
rewrite ind at left;
[ with x => `(succ x) ];
auto
]
}. | Plus/zero/R :
(-> [n : nat] (= nat ($ Plus n (nat 0)) n)) | by {
lam n =>
elim n;
[ `ax
, with ind n' =>
rewrite ind at left;
[ with x => `(succ x) ];
auto
]
}. | theorem | Plus/zero/R | test/success | test/success/num.prl | [] | [
"Plus"
] | null | 58 | 69 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus/succ/L :
(-> [n m : nat] (= nat ($ Plus (succ n) m) (succ ($ Plus n m))))
by {
lam n m => auto
}. | Plus/succ/L :
(-> [n m : nat] (= nat ($ Plus (succ n) m) (succ ($ Plus n m)))) | by {
lam n m => auto
}. | theorem | Plus/succ/L | test/success | test/success/num.prl | [] | [
"Plus"
] | null | 71 | 75 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus/succ/R :
(-> [n m : nat] (= nat ($ Plus n (succ m)) (succ ($ Plus n m))))
by {
lam n m => elim n;
[ auto
, with n'/ih n' => rewrite ($ Plus/succ/L n' (succ m)) at left;
[ with x => `x
, rewrite ($ Plus/succ/L n' m) at right;
[ with x => `(succ x)
, rewrite n'/ih at left;
... | Plus/succ/R :
(-> [n m : nat] (= nat ($ Plus n (succ m)) (succ ($ Plus n m)))) | by {
lam n m => elim n;
[ auto
, with n'/ih n' => rewrite ($ Plus/succ/L n' (succ m)) at left;
[ with x => `x
, rewrite ($ Plus/succ/L n' m) at right;
[ with x => `(succ x)
, rewrite n'/ih at left;
[ with x => `(succ x) ]
]
]
];
auto
}. | theorem | Plus/succ/R | test/success | test/success/num.prl | [] | [
"Plus",
"Plus/succ/L"
] | null | 77 | 93 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus/test0 :
(-> [n m : nat] [eq : (= nat ($ Plus n zero) m)] (= nat n m))
by {
lam n m eq =>
rewrite ($ Plus/zero/R n) in eq at left;
[ with x => `x ]; auto; use eq
}. | Plus/test0 :
(-> [n m : nat] [eq : (= nat ($ Plus n zero) m)] (= nat n m)) | by {
lam n m eq =>
rewrite ($ Plus/zero/R n) in eq at left;
[ with x => `x ]; auto; use eq
}. | theorem | Plus/test0 | test/success | test/success/num.prl | [] | [
"Plus",
"Plus/zero/R"
] | null | 95 | 101 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Eq/sym :
(-> [ty : (U 0)] [a b : ty] (= ty a b) (= ty b a))
by {
lam ty a b eq => symmetry; use eq
}. | Eq/sym :
(-> [ty : (U 0)] [a b : ty] (= ty a b) (= ty b a)) | by {
lam ty a b eq => symmetry; use eq
}. | theorem | Eq/sym | test/success | test/success/num.prl | [] | [] | null | 103 | 107 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Plus/comm :
(-> [n m : nat] (= nat ($ Plus n m) ($ Plus m n)))
by {
lam n m => elim n;
[ symmetry; `($ Plus/zero/R m)
, with n'/ih n' => rewrite ($ Plus/succ/L n' m) at left;
[ with x => `x
, rewrite n'/ih at left;
[ with x => `(succ x)
, symmetry; `($ Plus/succ/R m n')
... | Plus/comm :
(-> [n m : nat] (= nat ($ Plus n m) ($ Plus m n))) | by {
lam n m => elim n;
[ symmetry; `($ Plus/zero/R m)
, with n'/ih n' => rewrite ($ Plus/succ/L n' m) at left;
[ with x => `x
, rewrite n'/ih at left;
[ with x => `(succ x)
, symmetry; `($ Plus/succ/R m n')
]
]
];
auto
}. | theorem | Plus/comm | test/success | test/success/num.prl | [] | [
"Plus",
"Plus/succ/L",
"Plus/succ/R",
"Plus/zero/R"
] | null | 109 | 124 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NatSymm :
(->
[a b : nat]
(path [_] nat a b)
(path [_] nat b a))
by {
lam a b pab =>
abs i =>
`(hcom 0~>1 nat a
[i=0 [j] (@ pab j)]
[i=1 [_] a])
}. | NatSymm :
(->
[a b : nat]
(path [_] nat a b)
(path [_] nat b a)) | by {
lam a b pab =>
abs i =>
`(hcom 0~>1 nat a
[i=0 [j] (@ pab j)]
[i=1 [_] a])
}. | theorem | NatSymm | test/success | test/success/num.prl | [] | [] | null | 126 | 137 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IntPred :
(-> int int)
by {
lam a => elim a;
[ with n => elim n;
[ `(int -1)
, with _ n' => `(pos n')
]
, with n => `(negsucc (succ n))
];
}. | IntPred :
(-> int int) | by {
lam a => elim a;
[ with n => elim n;
[ `(int -1)
, with _ n' => `(pos n')
]
, with n => `(negsucc (succ n))
];
}. | theorem | IntPred | test/success | test/success/num.prl | [] | [] | null | 139 | 149 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IntSucc :
(-> int int)
by {
lam a => elim a;
[ with n => `(pos (succ n))
, with n => elim n;
[ `(int 0)
, with _ n' => `(negsucc n')
]
]
}. | IntSucc :
(-> int int) | by {
lam a => elim a;
[ with n => `(pos (succ n))
, with n => elim n;
[ `(int 0)
, with _ n' => `(negsucc n')
]
]
}. | theorem | IntSucc | test/success | test/success/num.prl | [] | [] | null | 151 | 161 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IntPlus :
(-> int int int)
by {
lam a => elim a;
[ with n => elim n;
[ lam b => use b
, with ind a' => lam b => `($ IntSucc ($ ind b))
]
, with n => elim n;
[ lam b => `($ IntPred b)
, with ind a' => lam b => `($ IntPred ($ ind b))
]
]
}. | IntPlus :
(-> int int int) | by {
lam a => elim a;
[ with n => elim n;
[ lam b => use b
, with ind a' => lam b => `($ IntSucc ($ ind b))
]
, with n => elim n;
[ lam b => `($ IntPred b)
, with ind a' => lam b => `($ IntPred ($ ind b))
]
]
}. | theorem | IntPlus | test/success | test/success/num.prl | [] | [
"IntPred",
"IntSucc"
] | null | 163 | 176 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Int4Plus3 :
($ IntPlus (int 4) (int 3)) = (int 7) in int
by { auto }. | Int4Plus3 :
($ IntPlus (int 4) (int 3)) = (int 7) in int | by { auto }. | theorem | Int4Plus3 | test/success | test/success/num.prl | [] | [
"IntPlus"
] | null | 178 | 180 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Int-6Plus10 :
($ IntPlus (int -6) (int 10)) = (int 4) in int
by { auto }. | Int-6Plus10 :
($ IntPlus (int -6) (int 10)) = (int 4) in int | by { auto }. | theorem | Int-6Plus10 | test/success | test/success/num.prl | [] | [
"IntPlus"
] | null | 182 | 184 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Int-1Plus-9 :
($ IntPlus (int -1) (int -9)) = (int -10) in int
by { auto }. | Int-1Plus-9 :
($ IntPlus (int -1) (int -9)) = (int -10) in int | by { auto }. | theorem | Int-1Plus-9 | test/success | test/success/num.prl | [] | [
"IntPlus"
] | null | 186 | 188 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
PathApConst :
(-> (path [_] bool tt tt) bool)
by {
lam p => use p [`(dim 0)]
}. | PathApConst :
(-> (path [_] bool tt tt) bool) | by {
lam p => use p [`(dim 0)]
}. | theorem | PathApConst | test/success | test/success/path-ap-const.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
PrimitiveSequencingTest : (-> bool bool bool bool) by {
repeat {refine fun/intro || id}; auto;
with z y x => use y
}. | PrimitiveSequencingTest : (-> bool bool bool bool) | by {
repeat {refine fun/intro || id}; auto;
with z y x => use y
}. | theorem | PrimitiveSequencingTest | test/success | test/success/primitive-sequencing.prl | [] | [] | null | 1 | 4 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Pushout/Test0 :
(pushout record record bool [_] tuple [_] tuple)
by {
`(left tuple)
}. | Pushout/Test0 :
(pushout record record bool [_] tuple [_] tuple) | by {
`(left tuple)
}. | theorem | Pushout/Test0 | test/success | test/success/pushout.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Pushout/Test1 :
(pushout bool bool bool [x] x [x] x)
by {
`(right tt)
}. | Pushout/Test1 :
(pushout bool bool bool [x] x [x] x) | by {
`(right tt)
}. | theorem | Pushout/Test1 | test/success | test/success/pushout.prl | [] | [] | null | 7 | 11 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Pushout/Test2 :
(-> dim (pushout bool bool bool [x] x [x] x))
by {
abs u => `(glue u tt tt tt)
}. | Pushout/Test2 :
(-> dim (pushout bool bool bool [x] x [x] x)) | by {
abs u => `(glue u tt tt tt)
}. | theorem | Pushout/Test2 | test/success | test/success/pushout.prl | [] | [] | null | 13 | 17 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
S1' = (pushout record record bool [_] tuple [_] tuple). | S1' | = (pushout record record bool [_] tuple [_] tuple). | define | S1' | test/success | test/success/pushout.prl | [] | [] | null | 19 | 19 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
PushoutToS1 :
(-> S1' S1)
by {
lam p => elim p;
[ `base
, `base
, with c u:dim =>
elim c;
[ `(loop u)
, `base
]
];
auto
}. | PushoutToS1 :
(-> S1' S1) | by {
lam p => elim p;
[ `base
, `base
, with c u:dim =>
elim c;
[ `(loop u)
, `base
]
];
auto
}. | theorem | PushoutToS1 | test/success | test/success/pushout.prl | [] | [
"S1'"
] | null | 22 | 35 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
PushoutToS1/Test0 :
(= (-> S1' S1) PushoutToS1 PushoutToS1)
by {
unfold PushoutToS1; // otherwise too easy
refine fun/eq/lam;
[ refine pushout/eq/pushout-rec; auto
, auto
]
}. | PushoutToS1/Test0 :
(= (-> S1' S1) PushoutToS1 PushoutToS1) | by {
unfold PushoutToS1; // otherwise too easy
refine fun/eq/lam;
[ refine pushout/eq/pushout-rec; auto
, auto
]
}. | theorem | PushoutToS1/Test0 | test/success | test/success/pushout.prl | [] | [
"PushoutToS1",
"S1'"
] | null | 38 | 46 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
PushoutBetaEasiest(#i:lvl) :
(-> [a b c d : (U #i)] [w : d]
[u : dim] [x : a] [y : b] [z : c]
(= d (pushout-rec [_] d (glue u z x y) [_] w [_] w [_ _] w) w))
by {
lam a b c d w => abs u => lam x y z =>
refine pushout/beta/glue;
auto
}. | PushoutBetaEasiest(#i:lvl) :
(-> [a b c d : (U #i)] [w : d]
[u : dim] [x : a] [y : b] [z : c]
(= d (pushout-rec [_] d (glue u z x y) [_] w [_] w [_ _] w) w)) | by {
lam a b c d w => abs u => lam x y z =>
refine pushout/beta/glue;
auto
}. | theorem | PushoutBetaEasiest | test/success | test/success/pushout.prl | [] | [] | null | 48 | 56 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
PushoutBetaEasier(#i:lvl) :
(-> [a b c : (U #i)] [f : (-> c a)] [g : (-> c b)]
[d : (-> (pushout a b c [z] ($ f z) [z] ($ g z)) (U #i))]
[wl : (-> [x : a] ($ d (left x)))] [wr : (-> [y : b] ($ d (right y)))]
[wg : (-> [z : c] (path [v] ($ d (glue v z ($ f z) ($ g z))) ($ wl ($ f z)) ($ wr ($ g z))))... | PushoutBetaEasier(#i:lvl) :
(-> [a b c : (U #i)] [f : (-> c a)] [g : (-> c b)]
[d : (-> (pushout a b c [z] ($ f z) [z] ($ g z)) (U #i))]
[wl : (-> [x : a] ($ d (left x)))] [wr : (-> [y : b] ($ d (right y)))]
[wg : (-> [z : c] (path [v] ($ d (glue v z ($ f z) ($ g z))) ($ wl ($ f z)) ($ wr ($ g z))))... | by {
lam a b c f g d wl wr wg => abs u => lam m =>
refine pushout/beta/glue;
auto
}. | theorem | PushoutBetaEasier | test/success | test/success/pushout.prl | [] | [] | null | 58 | 71 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTypeTest :
(record [a : bool] [b : (path [_] bool a a)] [c : bool] [d : S1]) type
by {
auto
}. | RecordTypeTest :
(record [a : bool] [b : (path [_] bool a a)] [c : bool] [d : S1]) type | by {
auto
}. | theorem | RecordTypeTest | test/success | test/success/record.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest0 :
tuple in record
by {
auto
}. | RecordTest0 :
tuple in record | by {
auto
}. | theorem | RecordTest0 | test/success | test/success/record.prl | [] | [] | null | 9 | 13 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest1 :
(tuple [a tt]) in (record [a : bool])
by {
auto
}. | RecordTest1 :
(tuple [a tt]) in (record [a : bool]) | by {
auto
}. | theorem | RecordTest1 | test/success | test/success/record.prl | [] | [] | null | 15 | 19 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest2 :
(tuple [a tt] [b tuple]) in (record [b : record] [a : bool])
by {
auto
}. | RecordTest2 :
(tuple [a tt] [b tuple]) in (record [b : record] [a : bool]) | by {
auto
}. | theorem | RecordTest2 | test/success | test/success/record.prl | [] | [] | null | 21 | 25 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest3 :
(tuple [a tt] [b ff]) in (record [b a : bool])
by {
auto
}. | RecordTest3 :
(tuple [a tt] [b ff]) in (record [b a : bool]) | by {
auto
}. | theorem | RecordTest3 | test/success | test/success/record.prl | [] | [] | null | 27 | 31 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.