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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
RecordTest4 :
(! a (tuple [a tt] [b ff])) = tt in bool
by {
auto
}. | RecordTest4 :
(! a (tuple [a tt] [b ff])) = tt in bool | by {
auto
}. | theorem | RecordTest4 | test/success | test/success/record.prl | [] | [] | null | 33 | 37 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest5(#p) :
(-> [p : record] (= record p tuple))
by {
lam _ => auto
}. | RecordTest5(#p) :
(-> [p : record] (= record p tuple)) | by {
lam _ => auto
}. | theorem | RecordTest5 | test/success | test/success/record.prl | [] | [] | null | 39 | 43 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest6 :
(->
[p : (record [a : bool] [b c : record])]
bool)
by {
lam {a = a} => use a
}. | RecordTest6 :
(->
[p : (record [a : bool] [b c : record])]
bool) | by {
lam {a = a} => use a
}. | theorem | RecordTest6 | test/success | test/success/record.prl | [] | [] | null | 45 | 51 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordTest7 :
(record
[a : S1]
[b : (path [_] S1 a a)])
by {
{a = `base, b = abs i => `(loop i)}
}. | RecordTest7 :
(record
[a : S1]
[b : (path [_] S1 a a)]) | by {
{a = `base, b = abs i => `(loop i)}
}. | theorem | RecordTest7 | test/success | test/success/record.prl | [] | [] | null | 53 | 59 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
RecordElimTest :
(->
(record
[b : bool]
[c : S1]
[p : (path [_] bool b b)])
(* [b : bool] (path [_] bool b b)))
by {
lam {b = welp, p = hello} =>
{use welp, use hello}
}. | RecordElimTest :
(->
(record
[b : bool]
[c : S1]
[p : (path [_] bool b b)])
(* [b : bool] (path [_] bool b b))) | by {
lam {b = welp, p = hello} =>
{use welp, use hello}
}. | theorem | RecordElimTest | test/success | test/success/record.prl | [] | [] | null | 61 | 71 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/trans3 :
(-> [a b c d : S1]
(path [_] S1 a b)
(path [_] S1 a c)
(path [_] S1 b d)
(path [_] S1 c d))
by {
lam a b c d pab pac pbd =>
abs i => `(fcom 0~>1 (@ pab i) [i=0 [j] (@ pac j)] [i=1 [j] (@ pbd j)])
}. | Fcom/trans3 :
(-> [a b c d : S1]
(path [_] S1 a b)
(path [_] S1 a c)
(path [_] S1 b d)
(path [_] S1 c d)) | by {
lam a b c d pab pac pbd =>
abs i => `(fcom 0~>1 (@ pab i) [i=0 [j] (@ pac j)] [i=1 [j] (@ pbd j)])
}. | theorem | Fcom/trans3 | test/success | test/success/S1-fcom.prl | [] | [] | null | 1 | 10 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/trans2 :
(-> [a b c : S1]
(path [_] S1 a b)
(path [_] S1 b c)
(path [_] S1 a c))
by {
lam a b c pab pbc =>
abs i => `(fcom 0~>1 (@ pab i) [i=0 [_] a] [i=1 [j] (@ pbc j)])
}. | Fcom/trans2 :
(-> [a b c : S1]
(path [_] S1 a b)
(path [_] S1 b c)
(path [_] S1 a c)) | by {
lam a b c pab pbc =>
abs i => `(fcom 0~>1 (@ pab i) [i=0 [_] a] [i=1 [j] (@ pbc j)])
}. | theorem | Fcom/trans2 | test/success | test/success/S1-fcom.prl | [] | [] | null | 14 | 22 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fcom/symm :
(-> [a b : S1]
(path [_] S1 a b)
(path [_] S1 b a))
by {
lam a b pab =>
abs i => `(fcom 0~>1 a [i=0 [j] (@ pab j)] [i=1 [_] a])
}. | Fcom/symm :
(-> [a b : S1]
(path [_] S1 a b)
(path [_] S1 b a)) | by {
lam a b pab =>
abs i => `(fcom 0~>1 a [i=0 [j] (@ pab j)] [i=1 [_] a])
}. | theorem | Fcom/symm | test/success | test/success/S1-fcom.prl | [] | [] | null | 24 | 31 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Tube :
(->
[x : S1]
(= S1 (fcom 0~>1 x [1=1 [_] x] [0=0 [_] x]) x))
by {
lam x => auto
}. | Tube :
(->
[x : S1]
(= S1 (fcom 0~>1 x [1=1 [_] x] [0=0 [_] x]) x)) | by {
lam x => auto
}. | theorem | Tube | test/success | test/success/S1-fcom.prl | [] | [] | null | 33 | 39 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
TrueByEvaluation :
(fcom 0~>0 base) in S1
by {
auto
}. | TrueByEvaluation :
(fcom 0~>0 base) in S1 | by {
auto
}. | theorem | TrueByEvaluation | test/success | test/success/S1-fcom.prl | [] | [] | null | 41 | 45 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Loop :
(path [_] S1 base base)
by {
abs u => `(loop u)
}. | Loop :
(path [_] S1 base base) | by {
abs u => `(loop u)
}. | theorem | Loop | test/success | test/success/S1.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
LoopBetaEasiest(#i:lvl) :
(-> [u : dim] [a : (U #i)] [x : a]
(= a (S1-rec [_] a (loop u) x [_] x) x))
by {
abs u => lam a x =>
refine s1/beta/loop;
auto
}. | LoopBetaEasiest(#i:lvl) :
(-> [u : dim] [a : (U #i)] [x : a]
(= a (S1-rec [_] a (loop u) x [_] x) x)) | by {
abs u => lam a x =>
refine s1/beta/loop;
auto
}. | theorem | LoopBetaEasiest | test/success | test/success/S1.prl | [] | [] | null | 7 | 14 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
LoopBetaEasier(#i:lvl) :
(-> [u : dim] [a : (-> S1 (U #i))]
[b : ($ a base)] [l : (path [v] ($ a (loop v)) b b)]
(= ($ a (loop u)) (S1-rec [x] a (loop u) b [v] (@ l v)) (@ l u)))
by {
abs u => lam a b l =>
refine s1/beta/loop;
auto
}. | LoopBetaEasier(#i:lvl) :
(-> [u : dim] [a : (-> S1 (U #i))]
[b : ($ a base)] [l : (path [v] ($ a (loop v)) b b)]
(= ($ a (loop u)) (S1-rec [x] a (loop u) b [v] (@ l v)) (@ l u))) | by {
abs u => lam a b l =>
refine s1/beta/loop;
auto
}. | theorem | LoopBetaEasier | test/success | test/success/S1.prl | [] | [] | null | 16 | 24 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Cmp(#m, #n) =
(lam [x] ($ #m ($ #n x)))
. | Cmp(#m, #n) | =
(lam [x] ($ #m ($ #n x)))
. | define | Cmp | test/success | test/success/strict-bool.prl | [] | [] | null | 1 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Bool/Not =
(lam [x] (if [_] bool x ff tt))
. | Bool/Not | =
(lam [x] (if [_] bool x ff tt))
. | define | Bool/Not | test/success | test/success/strict-bool.prl | [] | [] | null | 5 | 7 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Bool/Not-Not-Id :
(Cmp Bool/Not Bool/Not) = (lam [x] x) in (-> bool bool)
by {
auto
}. | Bool/Not-Not-Id :
(Cmp Bool/Not Bool/Not) = (lam [x] x) in (-> bool bool) | by {
auto
}. | theorem | Bool/Not-Not-Id | test/success | test/success/strict-bool.prl | [] | [
"Bool/Not",
"Cmp"
] | null | 9 | 13 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
SBool/Not-Not-Id-Path :
(path
[_] (-> bool bool)
(Cmp Bool/Not Bool/Not)
(lam [x] x))
by {
abs i => lam x => use x
}. | SBool/Not-Not-Id-Path :
(path
[_] (-> bool bool)
(Cmp Bool/Not Bool/Not)
(lam [x] x)) | by {
abs i => lam x => use x
}. | theorem | SBool/Not-Not-Id-Path | test/success | test/success/strict-bool.prl | [] | [
"Bool/Not",
"Cmp"
] | null | 15 | 22 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Times(#A, #B) =
(* #A #B)
. | Times(#A, #B) | =
(* #A #B)
. | define | Times | test/success | test/success/unfold.prl | [] | [] | null | 1 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Proj1(#z) = {
let {x} = #z;
use x
}. | Proj1(#z) | = {
let {x} = #z;
use x
}. | tactic | Proj1 | test/success | test/success/unfold.prl | [] | [] | null | 5 | 8 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Proj2(#z) = {
let {welp, x} = #z;
use x
}. | Proj2(#z) | = {
let {welp, x} = #z;
use x
}. | tactic | Proj2 | test/success | test/success/unfold.prl | [] | [] | null | 10 | 13 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Times/Proj :
(-> [ty : (U 0)] (Times bool ty) ty)
by {
lam ty x => (Proj2 x)
}. | Times/Proj :
(-> [ty : (U 0)] (Times bool ty) ty) | by {
lam ty x => (Proj2 x)
}. | theorem | Times/Proj | test/success | test/success/unfold.prl | [] | [
"Proj2",
"Times"
] | null | 15 | 19 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Univ0(#i:lvl, #j:lvl) :
(U #i) in (U (++ (lmax #i #j)))
by {
auto
}. | Univ0(#i:lvl, #j:lvl) :
(U #i) in (U (++ (lmax #i #j))) | by {
auto
}. | theorem | Univ0 | test/success | test/success/universes.prl | [] | [] | null | 1 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Univ1(#i:lvl) :
nat in (U #i discrete)
by {
auto
}. | Univ1(#i:lvl) :
nat in (U #i discrete) | by {
auto
}. | theorem | Univ1 | test/success | test/success/universes.prl | [] | [] | null | 7 | 11 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Univ2 :
(->
[a : (U 0 discrete)]
(= (U 1 kan) a a))
by {
lam a => auto
}. | Univ2 :
(->
[a : (U 0 discrete)]
(= (U 1 kan) a a)) | by {
lam a => auto
}. | theorem | Univ2 | test/success | test/success/universes.prl | [] | [] | null | 13 | 19 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Monoid(#i:lvl) : (U (++ #i)) by {
`(record
[ob : (U #i)]
[one : ob]
[mul : (-> ob ob ob)]
[idn/l : (-> [m : ob] (= ob ($ mul one m) m))]
[idn/r : (-> [m : ob] (= ob ($ mul m one) m))]
[assoc :
(->
[l m n : ob]
(= ob
($ mul l ($ mul m n))
($ mul ($ mul m n) l... | Monoid(#i:lvl) : (U (++ #i)) | by {
`(record
[ob : (U #i)]
[one : ob]
[mul : (-> ob ob ob)]
[idn/l : (-> [m : ob] (= ob ($ mul one m) m))]
[idn/r : (-> [m : ob] (= ob ($ mul m one) m))]
[assoc :
(->
[l m n : ob]
(= ob
($ mul l ($ mul m n))
($ mul ($ mul m n) l)))])
}. | theorem | Monoid | test/success | test/success/universes.prl | [] | [] | null | 21 | 34 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
HasAllPathsTo (#C,#c) = (-> [c' : #C] (path [_] #C c' #c)). | HasAllPathsTo (#C,#c) | = (-> [c' : #C] (path [_] #C c' #c)). | define | HasAllPathsTo | test/success | test/success/V-types.prl | [] | [] | null | 1 | 1 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IsContr (#C) = (* [c : #C] (HasAllPathsTo #C c)). | IsContr (#C) | = (* [c : #C] (HasAllPathsTo #C c)). | define | IsContr | test/success | test/success/V-types.prl | [] | [
"HasAllPathsTo"
] | null | 3 | 3 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Fiber (#A,#B,#f,#b) = (* [a : #A] (path [_] #B ($ #f a) #b)). | Fiber (#A,#B,#f,#b) | = (* [a : #A] (path [_] #B ($ #f a) #b)). | define | Fiber | test/success | test/success/V-types.prl | [] | [] | null | 5 | 5 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IsEquiv (#A,#B,#f) = (-> [b : #B] (IsContr (Fiber #A #B #f b))). | IsEquiv (#A,#B,#f) | = (-> [b : #B] (IsContr (Fiber #A #B #f b))). | define | IsEquiv | test/success | test/success/V-types.prl | [] | [
"Fiber",
"IsContr"
] | null | 7 | 7 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Equiv (#A,#B) = (* [f : (-> #A #B)] (IsEquiv #A #B f)). | Equiv (#A,#B) | = (* [f : (-> #A #B)] (IsEquiv #A #B f)). | define | Equiv | test/success | test/success/V-types.prl | [] | [
"IsEquiv"
] | null | 9 | 9 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Id = (lam [a] a). | Id | = (lam [a] a). | define | Id | test/success | test/success/V-types.prl | [] | [] | null | 11 | 11 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IdIsEquiv(#l:lvl) :
(-> [ty : (U #l hcom)] (IsEquiv ty ty Id))
by {
lam ty a =>
{ {use a, abs _ => use a}
, lam {_,c'} => abs i =>
{ `(hcom 1~>0 ty a
[i=0 [j] (@ c' j)]
[i=1 [j] a])
, abs j =>
`(hcom 1~>j ty a
[i=0 [j] (@ c' j)]
[i=1 [j] a])
... | IdIsEquiv(#l:lvl) :
(-> [ty : (U #l hcom)] (IsEquiv ty ty Id)) | by {
lam ty a =>
{ {use a, abs _ => use a}
, lam {_,c'} => abs i =>
{ `(hcom 1~>0 ty a
[i=0 [j] (@ c' j)]
[i=1 [j] a])
, abs j =>
`(hcom 1~>j ty a
[i=0 [j] (@ c' j)]
[i=1 [j] a])
}
}
}. | theorem | IdIsEquiv | test/success | test/success/V-types.prl | [] | [
"Id",
"IsEquiv"
] | null | 13 | 28 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IdEquiv(#l:lvl) :
(-> [ty : (U #l hcom)] (Equiv ty ty))
by {
lam ty =>
{`Id, use (IdIsEquiv #l) [use ty]}
}. | IdEquiv(#l:lvl) :
(-> [ty : (U #l hcom)] (Equiv ty ty)) | by {
lam ty =>
{`Id, use (IdIsEquiv #l) [use ty]}
}. | theorem | IdEquiv | test/success | test/success/V-types.prl | [] | [
"Equiv",
"Id",
"IdIsEquiv"
] | null | 30 | 35 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IdV(#i:dim, #l:lvl, #ty) =
(V #i #ty #ty ($ (IdEquiv #l) #ty))
. | IdV(#i:dim, #l:lvl, #ty) | =
(V #i #ty #ty ($ (IdEquiv #l) #ty))
. | define | IdV | test/success | test/success/V-types.prl | [] | [
"IdEquiv"
] | null | 39 | 41 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IdV/Wf(#l:lvl) :
(->
[i : dim]
[ty : (U #l hcom)]
(mem (U #l) (IdV i #l ty)))
by {
abs i => lam ty => auto
}. | IdV/Wf(#l:lvl) :
(->
[i : dim]
[ty : (U #l hcom)]
(mem (U #l) (IdV i #l ty))) | by {
abs i => lam ty => auto
}. | theorem | IdV/Wf | test/success | test/success/V-types.prl | [] | [
"IdV"
] | null | 43 | 50 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IdV/Test0(#l:lvl) :
(->
[i : dim]
[ty : (U #l hcom)]
[a : ty]
(mem (IdV i #l ty) (Vin i a a)))
by {
abs i => lam ty a => auto
}. | IdV/Test0(#l:lvl) :
(->
[i : dim]
[ty : (U #l hcom)]
[a : ty]
(mem (IdV i #l ty) (Vin i a a))) | by {
abs i => lam ty a => auto
}. | theorem | IdV/Test0 | test/success | test/success/V-types.prl | [] | [
"IdV"
] | null | 52 | 60 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IdV/Test1(#l:lvl) :
(->
[ty : (U #l hcom)]
[a : ty]
(= ty (Vproj (dim 0) (Vin (dim 0) a a) Id) a))
by {
lam ty a => auto
}. | IdV/Test1(#l:lvl) :
(->
[ty : (U #l hcom)]
[a : ty]
(= ty (Vproj (dim 0) (Vin (dim 0) a a) Id) a)) | by {
lam ty a => auto
}. | theorem | IdV/Test1 | test/success | test/success/V-types.prl | [] | [
"Id"
] | null | 62 | 69 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
IdV/Test2(#l:lvl) :
(->
[ty : (U #l kan)]
[a : ty]
(= ty (coe 0~>1 [x] (IdV x #l ty) a)
(coe 0~>1 [_] ty a)))
by {
lam ty a => auto
}. | IdV/Test2(#l:lvl) :
(->
[ty : (U #l kan)]
[a : ty]
(= ty (coe 0~>1 [x] (IdV x #l ty) a)
(coe 0~>1 [_] ty a))) | by {
lam ty a => auto
}. | theorem | IdV/Test2 | test/success | test/success/V-types.prl | [] | [
"IdV"
] | null | 71 | 79 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Not = (lam [b] (if [_] bool b ff tt)). | Not | = (lam [b] (if [_] bool b ff tt)). | define | Not | test/success | test/success/V-types.prl | [] | [] | null | 83 | 83 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Bool/reflect :
(->
[a b : bool]
[p : (path [_] bool a b)]
(= bool a b))
by {
lam a b p => `(coe 0~>1 [x] (= bool a (@ p x)) ax)
}. | Bool/reflect :
(->
[a b : bool]
[p : (path [_] bool a b)]
(= bool a b)) | by {
lam a b p => `(coe 0~>1 [x] (= bool a (@ p x)) ax)
}. | theorem | Bool/reflect | test/success | test/success/V-types.prl | [] | [] | null | 85 | 92 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
Bool/contra/inverse (#p:exp) = {
query gl <- concl;
match gl {
[a b | #jdg{%a = %b in bool} =>
claim eq : (= bool %b %a) by {use Bool/reflect [`%b, `%a, `#p]; auto};
symmetry; auto
]
[a | %[a:jdg] => id]
}
}. | Bool/contra/inverse (#p:exp) | = {
query gl <- concl;
match gl {
[a b | #jdg{%a = %b in bool} =>
claim eq : (= bool %b %a) by {use Bool/reflect [`%b, `%a, `#p]; auto};
symmetry; auto
]
[a | %[a:jdg] => id]
}
}. | tactic | Bool/contra/inverse | test/success | test/success/V-types.prl | [] | [
"Bool/reflect"
] | null | 94 | 103 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NotIsEquiv :
(IsEquiv bool bool Not)
by {
lam b =>
{ {`($ Not b), abs _ => use b}
, lam {_,p'} =>
(abs i =>
{ `($ Not (hcom 1~>0 bool b
[i=0 [j] (@ p' j)]
[i=1 [j] b]))
, abs j =>
`(hcom 1~>j bool b
[i=0 [j] (@ p' j)]
... | NotIsEquiv :
(IsEquiv bool bool Not) | by {
lam b =>
{ {`($ Not b), abs _ => use b}
, lam {_,p'} =>
(abs i =>
{ `($ Not (hcom 1~>0 bool b
[i=0 [j] (@ p' j)]
[i=1 [j] b]))
, abs j =>
`(hcom 1~>j bool b
[i=0 [j] (@ p' j)]
[i=1 [j] b])
}
); ... | theorem | NotIsEquiv | test/success | test/success/V-types.prl | [] | [
"Bool/contra/inverse",
"IsEquiv",
"Not"
] | null | 105 | 122 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NotEquiv :
(Equiv bool bool)
by {
{`Not, `NotIsEquiv}
}. | NotEquiv :
(Equiv bool bool) | by {
{`Not, `NotIsEquiv}
}. | theorem | NotEquiv | test/success | test/success/V-types.prl | [] | [
"Equiv",
"Not",
"NotIsEquiv"
] | null | 124 | 128 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NotV(#i:dim) = (V #i bool bool NotEquiv). | NotV(#i:dim) | = (V #i bool bool NotEquiv). | define | NotV | test/success | test/success/V-types.prl | [] | [
"NotEquiv"
] | null | 130 | 130 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NotV/Wf :
(-> [i : dim] (mem (U 0 kan) (NotV i)))
by {
abs i => auto
}. | NotV/Wf :
(-> [i : dim] (mem (U 0 kan) (NotV i))) | by {
abs i => auto
}. | theorem | NotV/Wf | test/success | test/success/V-types.prl | [] | [
"NotV"
] | null | 132 | 136 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NotV/Test0 :
(->
[i : dim]
[a : bool]
(mem (NotV i) (Vin i ($ Not a) a)))
by {
abs i => lam a => auto
}. | NotV/Test0 :
(->
[i : dim]
[a : bool]
(mem (NotV i) (Vin i ($ Not a) a))) | by {
abs i => lam a => auto
}. | theorem | NotV/Test0 | test/success | test/success/V-types.prl | [] | [
"Not",
"NotV"
] | null | 138 | 145 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NotV/Test1 :
(->
[a : bool]
(= bool (coe 0~>1 [x] (NotV x) a) ($ Not a)))
by {
lam a => auto
}. | NotV/Test1 :
(->
[a : bool]
(= bool (coe 0~>1 [x] (NotV x) a) ($ Not a))) | by {
lam a => auto
}. | theorem | NotV/Test1 | test/success | test/success/V-types.prl | [] | [
"Not",
"NotV"
] | null | 147 | 153 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
NotV/Test2 :
(->
[a : bool]
(= bool (coe 1~>0 [x] (NotV x) a) ($ Not a)))
by {
lam a => auto
}. | NotV/Test2 :
(->
[a : bool]
(= bool (coe 1~>0 [x] (NotV x) a) ($ Not a))) | by {
lam a => auto
}. | theorem | NotV/Test2 | test/success | test/success/V-types.prl | [] | [
"Not",
"NotV"
] | null | 155 | 161 | true | https://github.com/RedPRL/sml-redprl | c72190de76f7ed1cfbe1d2046c96e99ac5022b0c |
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