statement stringlengths 1 812 | proof stringlengths 0 1.13k | type stringclasses 2
values | symbolic_name stringlengths 1 31 | library stringclasses 37
values | filename stringclasses 126
values | imports listlengths 0 7 | deps listlengths 0 20 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
A ◂ ∀ X : ★. X ➔ (X ➔ X) ➔ (X ➔ X ➔ X) ➔ X | = Λ X. λ a. λ b. λ c. c (b (c a a)) (b (b a)). | function | A | cedille-tests/interactive.cedtest | cedille-tests/interactive.cedtest/code.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
test1 | = λ c : (Π x : A . B x) . a. | function | test1 | cedille-tests/printing.cedtest | cedille-tests/printing.cedtest/code.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
test2 | = λ x : (ι x : A . B x) . a . | function | test2 | cedille-tests/printing.cedtest | cedille-tests/printing.cedtest/code.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
a' | = a. | function | a' | cedille-tests/summary.cedtest | cedille-tests/summary.cedtest/code.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
b' | = b. | function | b' | cedille-tests/summary.cedtest | cedille-tests/summary.cedtest/code.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
A ◂ ∀ X : ★. X ➔ ∀ Y : ★. Y ➔ X | = Λ X. λ x. Λ Y. λ y. x. | function | A | cedille-tests/test1.cedtest | cedille-tests/test1.cedtest/code.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
out' ◂ ∀ F : ★ ➔ ★ . RecFunctor · F ➾ C · F ➔ F · (C · F) | =
Λ F . Λ fm . λ x .
(rUnfold · F -fm x).1 · (F · (C · F))
[ β{λ e . λ d . d} ,
Λ R . Λ reveal . λ e . λ d . cast · (F · R) · (F · (C · F))
-(fm · R · (C · F) reveal) d] . | function | out' | encodings/mendler | encodings/mendler/C-extras.ced | [
"C",
"is",
"top"
] | [
"RecFunctor",
"cast",
"rUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Alg ◂ (★ ➔ ★) ➔ ★ | =
λ X : ★ ➔ ★ .
∀ R : ★ .
(R ➔ X · R) ➔
F · R ➔
X · R. | function | Alg | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
rF ◂ ★ ➔ ★ | = λ C : ★ . ∀ X : ★ ➔ ★ . CastMap · X ➾ Alg · X ➔ X · C . | function | rF | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"Alg",
"CastMap"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
foldAlg : ∀ C: ★. ∀ X: ★ ➔ ★. CastMap ·X ➾ Alg ·X ➔ rF ·C ➔ X ·C | = Λ C. Λ X. Λ cmX. λ alg. λ x. x -cmX alg . | function | foldAlg | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"Alg",
"CastMap",
"rF"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
inAlg : Alg · rF | =
Λ R . λ eval . λ d . Λ X . Λ cmX . λ alg . alg (λ r. foldAlg -cmX alg (eval r)) d. | function | inAlg | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"Alg",
"eval",
"foldAlg",
"rF"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
C ◂ ★ | = Rec · rF. | function | C | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"Rec",
"rF"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
rFunctor ◂ CastMap · rF | =
Λ X . Λ Y . Λ c .
[λ x . Λ A . Λ fA . λ a .
cast -(fA -c)
(x · A -fA a),
β] . | function | rFunctor | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"CastMap",
"cast",
"rF"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
rFold ◂ rF · C ➔ C | =
recFold · rF -rFunctor . | function | rFold | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"rF",
"rFunctor",
"recFold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
rUnfold ◂ C ➔ rF · C | =
recUnfold · rF -rFunctor . | function | rUnfold | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"rF",
"rFunctor",
"recUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
fold ◂ ∀ X : ★ ➔ ★ . CastMap · X ➾ Alg · X ➔ C ➔ X · C | =
Λ X . Λ fX . λ alg . λ d . (rUnfold d) · X -fX alg . | function | fold | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"Alg",
"CastMap",
"rUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
in ◂ F · C ➔ C | =
λ d .
rFold
(Λ X . Λ fX . λ alg . alg · C (fold · X -fX alg) d). | function | in | encodings/mendler | encodings/mendler/C-simple.ced | [
"stdlib"
] | [
"fold",
"rFold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
foldU ◂ Top ➔ Top | = λ alg . β{|λ d . d alg|} . | function | foldU | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Top"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
inU ◂ Top | = β{|λ d . λ alg . alg (foldU alg) d|}. | function | inU | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Top",
"foldU"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
inAlgU ◂ Top | = β{|λ _ . λ d . inU d|}. | function | inAlgU | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Top",
"inU"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
outU ◂ Top | = β{|λ x. x (λ _. λ xs. xs)|}. | function | outU | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Top",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Alg ◂ ★ ➔ (★ ➔ ★) ➔ ★ | =
λ C : ★ . λ X : ★ ➔ ★ .
ι alg : Top . ∀ R : ★ .
Cast · R · C ➾
View · (R ➔ F · R) outU ➾
View · (R ➔ X · R) (foldU alg) ➔
F · R ➔
X · R. | function | Alg | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Cast",
"Top",
"View",
"foldU",
"outU"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
rF' ◂ ★ ➔ ★ | = λ C : ★ . ∀ X : ★ ➔ ★ . CastMap · X ➾ Alg · C · X ➔ X · C . | function | rF' | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Alg",
"CastMap"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
rF ◂ ★ ➔ ★ | = λ C : ★ . ι x : rF' · C . { x inAlgU ≃ x } . | function | rF | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"inAlgU",
"rF'"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
rFunctor ◂ CastMap · rF | =
Λ X . Λ Y . Λ c .
[λ x . [Λ A . Λ fA . λ a .
cast -(fA -c)
(x.1 · A -fA
[β{|a|} ,
Λ R . Λ reveal . Λ o. λ e . λ d .
a.2 · R -(castTrans -reveal -c) -o e d ]),
x.2],
β] . | function | rFunctor | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"CastMap",
"cast",
"castTrans",
"rF"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
fold ◂ ∀ X : ★ ➔ ★ . CastMap · X ➾ Alg · C · X ➔ C ➔ X · C | =
Λ X . Λ fX . λ alg . λ d . (rUnfold d).1 · X -fX alg . | function | fold | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Alg",
"CastMap",
"rUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
cRec ◂ C ➔ rF' · C | =
λ t . (rUnfold t).1 . | function | cRec | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"rF'",
"rUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
out ◂ C ➔ F · C | =
λ x .
(rUnfold x).1 · F -fm
[β{|λ _ . λ d . d|} , Λ R . Λ reveal . Λ o. λ eval . λ d . d] . | function | out | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"eval",
"rUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
in ◂ F · C ➔ C | =
λ d .
rFold
[Λ X . Λ fX . λ alg . alg.2 · C -(castRefl · C ) -[ out , β{|out|} ]
[fold · X -fX alg , β{|fold alg|} ] d,
β{|λ alg . (alg (fold alg) d)|}]. | function | in | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"castRefl",
"fold",
"out",
"rFold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
inAlg ◂ Alg · C · (λ _ : ★ . C) | =
[ β{|λ eval . λ d . in d|} ,
Λ R . Λ reveal . Λ o . λ eval . λ d . in (cast -(fm -reveal) d)]. | function | inAlg | encodings/mendler | encodings/mendler/C.ced | [
"stdlib"
] | [
"Alg",
"cast",
"eval",
"in"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
PrfAlg : ★ ➔ (C ➔ ★ ➔ ★) ➔ ★ | =
λ D: ★. λ P : C ➔ ★ ➔ ★ .
ι alg : Top .
∀ R : ★ .
∀ reveal : Cast · R · C. ∀ revealh: Cast · R · D.
View · (R ➔ F · R) outU ➾
View · (Π r : R . P (cast -reveal r) · R) (foldU alg) ➔
Π p : F · R .
P (in (cast · (F · R) · (F · C) -(fm -reveal) p)) · R . | function | PrfAlg | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"Cast",
"Top",
"View",
"cast",
"foldU",
"in",
"outU"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Inductive : ★ ➔ C ➔ ★ | =
λ D : ★ . λ d : C.
∀ P : C ➔ ★ ➔ ★ .
(Π c : C . CastMap · (P c)) ➾
PrfAlg · D · P ➔
P d · D . | function | Inductive | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"CastMap",
"PrfAlg"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
DF : ★ ➔ ★ | = λ D : ★ . ι x : C . Inductive · D x. | function | DF | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"Inductive"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
D : ★ | = Rec · DF . | function | D | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"DF",
"Rec"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
indFunctor : ∀ x : C . CastMap · (λ X : ★ . Inductive · X x) | =
Λ x . Λ A . Λ B . Λ c .
[ λ d . Λ P . Λ fP . λ palg .
[palg' : PrfAlg ·A ·P =
[ β{palg}
, Λ R. Λ cc. Λ cd. Λ o. λ ih. λ xs.
palg.2 -cc -([ λ r. elimCast -c (elimCast -cd r) , β ]) -o ih xs]] -
cast -(fP x -c) (d · P -fP palg') , β ] . | function | indFunctor | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"CastMap",
"Inductive",
"PrfAlg",
"cast",
"elimCast",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
dFunctor : CastMap · DF | =
Λ X . Λ Y . Λ c .
[ λ d . [ d.1 , cast -(indFunctor -d.1 · X · Y -c) d.2] , β]. | function | dFunctor | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"CastMap",
"DF",
"cast",
"indFunctor"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
dFold ◂ DF · D ➔ D | =
recFold · DF -dFunctor . | function | dFold | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"DF",
"dFunctor",
"recFold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
dUnfold ◂ D ➔ DF · D | =
recUnfold · DF -dFunctor . | function | dUnfold | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"DF",
"dFunctor",
"recUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
out' : D ➔ F · D | = λ d.
(dUnfold d).2 ·(λ _: C. λ _: ★. F · D) -(λ x. Λ X. Λ Y. Λ c. castRefl ·(F ·D))
[ β{λ ih. λ xs. xs} , Λ R. Λ c. Λ d. Λ o. λ ih. λ xs. castMap -fm -d xs ]. | function | out' | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"castMap",
"castRefl",
"dUnfold",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
in' : F · D ➔ D | =
λ d .
dFold
[ in
(cast · (F · D) · (F · C)
-(fm · D · C -[λ x . (dUnfold x).1 , β]) d) ,
Λ P . Λ fP . λ palg . palg.2 · D
-[λ x . (dUnfold x).1 , β] -[ λ x. x , β ] -[out' , β{out'}]
[ λ r . (dUnfold r).2 · P -fP palg , β{λ r . r palg}]... | function | in' | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"cast",
"dFold",
"dUnfold",
"in",
"out'"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
inAlg' : Alg · C · (λ _ : ★ . D) | =
[ β{λ eval . λ p . (in p)} ,
Λ R . Λ reveal . Λ o. λ eval . λ p .
in' (cast -(fm
-(intrCast -eval.1
-(λ r . ρ eval.2 - ρ (rUnfold (cast -reveal r)).2 - β))) p) ] . | function | inAlg' | encodings/mendler | encodings/mendler/D.ced | [
"public",
"stdlib"
] | [
"Alg",
"cast",
"eval",
"in",
"in'",
"intrCast",
"rUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
IsHom : Π X1 : ★ . (Alg · C · X1) ➔
Π X2 : ★ . (Alg · C · X2) ➔
Π h : X1 ➔ X2 . ★ | =
λ X1 : ★ . λ alg1 : Alg · C · X1 .
λ X2 : ★ . λ alg2 : Alg · C · X2 .
λ h : X1 ➔ X2 .
∀ R : ★ . Cast · R · C ➾
Π fr : F · R . { h (alg1 (foldU alg1) fr) ≃ alg2 (foldU alg2) fr }. | function | IsHom | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"Cast",
"foldU"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
IsHom2 : Π X1 : ★ . (Alg · C · X1) ➔
Π X2 : ★ . (Alg · C · X2) ➔
Π h : X1 ➔ X2 . ★ | =
λ X1 : ★ . λ alg1 : Alg · C · X1 .
λ X2 : ★ . λ alg2 : Alg · C · X2 .
λ h : X1 ➔ X2 .
Π r : C .
{ h (fold alg1 r) ≃ fold alg2 r }. | function | IsHom2 | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"fold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
HomHom2 : ∀ X1 : ★ . ∀ alg1 : Alg · C · X1 .
∀ X2 : ★ . ∀ alg2 : Alg · C · X2 .
∀ h : X1 ➔ X2 .
IsHom · X1 alg1 · X2 alg2 h ➔
IsHom2 · X1 alg1 · X2 alg2 h | =
Λ X1 . Λ alg1 . Λ X2 . Λ alg2 . Λ h . λ H .
λ r .
θ<r> (Ind r)
[ β{λ ih . λ fr . H fr} ,
Λ R . Λ reveal . λ ih . λ fr .
(H · R -reveal fr)]. | function | HomHom2 | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"Ind",
"IsHom",
"IsHom2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Hom2Hom : ∀ X1 : ★ . ∀ alg1 : Alg · C · X1 .
∀ X2 : ★ . ∀ alg2 : Alg · C · X2 .
∀ h : X1 ➔ X2 .
IsHom2 · X1 alg1 · X2 alg2 h ➔
IsHom · X1 alg1 · X2 alg2 h | =
Λ X1 . Λ alg1 . Λ X2 . Λ alg2 . Λ h . λ H .
Λ R . Λ c . λ r .
ρ (H (in -fm (cast -(fm c) r))) - β. | function | Hom2Hom | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"IsHom",
"IsHom2",
"cast",
"in"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
IdHom : ∀ X : ★ . ∀ alg : Alg · C · X .
IsHom · X alg · X alg (λ x . x) | =
Λ X . Λ alg . Λ R . Λ _ . λ fr . β. | function | IdHom | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"IsHom"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
ComposeHom : ∀ X1 : ★ . ∀ alg1 : Alg · C · X1 .
∀ X2 : ★ . ∀ alg2 : Alg · C · X2 .
∀ X3 : ★ . ∀ alg3 : Alg · C · X3 .
∀ h1 : X1 ➔ X2 .
∀ h2 : X2 ➔ X3 .
IsHom · X1 alg1 · X2 alg2 h1 ➔
IsHom · X2 alg2 · X3 alg3 h2 ➔
I... | =
Λ X1 . Λ alg1 .
Λ X2 . Λ alg2 .
Λ X3 . Λ alg3 .
Λ h1 . Λ h2 . λ H1 . λ H2 .
Λ R . Λ c . λ fr . ρ+ (H1 · R -c fr) - ρ (H2 · R -c fr) - β. | function | ComposeHom | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"IsHom"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
WeakInitiality : ∀ X : ★ . ∀ alg : Alg · C · X .
IsHom · C (inAlg -fm) · X alg (fold -fm · X alg) | =
Λ X . Λ alg .
Λ R . Λ c . λ fr . β. | function | WeakInitiality | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"IsHom",
"fold",
"inAlg"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Initiality : ∀ X : ★ . ∀ alg : Alg · C · X .
∀ h : C ➔ X .
IsHom · C (inAlg -fm) · X alg h ➔
Π c : C . { h c ≃ fold alg c } | =
Λ X . Λ alg . Λ h . λ hh . λ c .
θ<c> (Ind c)
[ β{λ rec . λ d . β} ,
Λ R . Λ reveal . λ rec . λ d . ρ+ (hh · R -reveal d) - β ]. | function | Initiality | encodings/mendler | encodings/mendler/Hom.ced | [
"Ind",
"stdlib"
] | [
"Alg",
"Ind",
"IsHom",
"fold",
"inAlg",
"rec"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
toD : C ➔ D | =
λ x . (rUnfold x).1 · (λ _ : ★ . D) -(constCastMap · D) inAlg' . | function | toD | encodings/mendler | encodings/mendler/Ind.ced | [
"public",
"stdlib"
] | [
"constCastMap",
"inAlg'",
"rUnfold"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
reflection : Π x : C . { toD x ≃ x } | =
λ x . (rUnfold x).2. | function | reflection | encodings/mendler | encodings/mendler/Ind.ced | [
"public",
"stdlib"
] | [
"rUnfold",
"toD"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
castD : Cast · C · D | =
intrCast · C · D -toD -reflection. | function | castD | encodings/mendler | encodings/mendler/Ind.ced | [
"public",
"stdlib"
] | [
"Cast",
"intrCast",
"reflection",
"toD"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Ind : Π c : C . Inductive · C c | =
λ c . cast -(indFunctor -c · D · C -[λ d . (dUnfold d).1 , β]) (dUnfold (cast -castD c)).2. | function | Ind | encodings/mendler | encodings/mendler/Ind.ced | [
"public",
"stdlib"
] | [
"Inductive",
"cast",
"castD",
"dUnfold",
"indFunctor"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
lambek1 : Π x : F · C . { out (in x) ≃ x} | = λ x . β . | function | lambek1 | encodings/mendler | encodings/mendler/Ind.ced | [
"public",
"stdlib"
] | [
"in",
"out"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
lambek2 : Π x : C . { in (out x) ≃ x } | =
λ x .
Ind x · (λ x : C . λ _ : ★ . { in (out x) ≃ x }) -(λ c . constCastMap · { in (out c) ≃ c })
[β{λ _ . λ _ . λ x .x} , Λ R . Λ cc . Λ cd . Λ o. λ ih . λ p . β] . | function | lambek2 | encodings/mendler | encodings/mendler/Ind.ced | [
"public",
"stdlib"
] | [
"Ind",
"constCastMap",
"in",
"out"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Id : ★ ➔ ★ ➔ ★ | = Cast. | function | Id | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Cast"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
intrId | = intrCast. | function | intrId | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"intrCast"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
elimId | = elimCast. | function | elimId | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"elimCast"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
IdMapping : (★ ➔ ★) ➔ ★ | = CastMap. | function | IdMapping | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"CastMap"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
imap | = castMap. | function | imap | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"castMap"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Fix : Π F: ★ ➔ ★. IdMapping ·F ➔ ★ | = λ F: ★ ➔ ★. λ pos: IdMapping ·F. Rec ·Ind.rF. | function | Fix | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"IdMapping",
"Ind",
"Rec",
"rF"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
D : ★ | = Fix ·F pos. | function | D | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Fix"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
in : F ·D ➔ D | = Ind.in. | function | in | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Ind"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
out : D ➔ F ·D | = Ind.out. | function | out | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Ind"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
PrfAlg : (D ➔ ★ ➔ ★) ➔ ★ | = λ P: D ➔ ★ ➔ ★.
∀ R: ★. ∀ c: Id ·R ·D. Π o: R ➔ F ·R. ∀ oeq: { o ≃ out }.
(Π x: R. P (elimCast -c x) ·R) ➔
Π xs: F ·R. P (in (castMap -pos -c xs)) ·R. | function | PrfAlg | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Id",
"castMap",
"elimCast",
"in",
"out",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
induction: ∀ P: D ➔ ★ ➔ ★. (Π x: D. IdMapping ·(P x)) ➾ PrfAlg ·P ➔ Π d: D. P d ·D | = Λ P. Λ posP. λ alg. λ d.
Ind.Ind d ·P -posP
[ β{λ v. λ xs. alg out v xs}
, Λ R. Λ cc. Λ cd. Λ o. λ ih. λ xs. alg -cc (elimView β{out} -o) -β ih.1 xs ]. | function | induction | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"IdMapping",
"Ind",
"PrfAlg",
"elimView",
"out",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
inductionComp : ∀ P: D ➔ ★ ➔ ★. Π alg: PrfAlg ·P. Π xs: F ·D. {induction alg (in xs) ≃ alg out (induction alg) xs} | = Λ P. λ alg. λ xs. β. | function | inductionComp | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"PrfAlg",
"in",
"induction",
"out",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
lambek1 : Π xs: F ·D. {xs ≃ out (in xs)} | = λ xs. ς (Ind.lambek1 xs). | function | lambek1 | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Ind",
"in",
"out",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
lambek2 : Π x: D. {x ≃ in (out x)} | = λ x.
induction ·(λ x: D. λ _: ★. {x ≃ in (out x)})
-(λ x. Λ X. Λ Y. Λ c. castRefl ·{ x ≃ in (out x) }) (Λ R. Λ c. λ o. Λ oeq. λ ih. λ xs. β) x. | function | lambek2 | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"castRefl",
"in",
"induction",
"out",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
IsD : ★ ➔ ★ | = λ R: ★. Pair ·(Id ·R ·D) ·(View ·(R ➔ F ·R) β{out}). | function | IsD | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Id",
"Pair",
"View",
"out"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
isD : IsD ·D | = pair (intrId -(λ x. x) -(λ x. β)) (intrView β{out} -out -β). | function | isD | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"IsD",
"intrId",
"intrView",
"out",
"pair"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
toD : ∀ R: ★. ∀ _: IsD ·R. R ➔ D | = Λ R. Λ is. λ r. elimId -(fst is) r. | function | toD | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"IsD",
"elimId",
"fst",
"is"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
toFD : ∀ R: ★. ∀ _: IsD ·R. F ·R ➔ F ·D | = Λ R. Λ is. λ xs. imap -pos -(fst is) xs. | function | toFD | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"IsD",
"fst",
"imap",
"is",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
ByCases: (D ➔ ★) ➔ Π R: ★. IsD ·R ➔ ★ | = λ P: D ➔ ★. λ R: ★. λ is: IsD ·R. Π xs: F ·R. P (in (toFD -is xs)). | function | ByCases | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"IsD",
"in",
"is",
"toFD",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
mu': ∀ R: ★. ∀ is: IsD ·R. Π r: R. ∀ P: D ➔ ★. ByCases ·P ·R is ➔ P (toD -is r) | = Λ R. Λ is. λ r. Λ P. λ case. ρ (lambek2 (toD -is r)) - case (elimView β{out} -(snd is) r). | function | mu' | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"ByCases",
"IsD",
"case",
"elimView",
"is",
"lambek2",
"out",
"snd",
"toD"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
ByInd: (D ➔ Π R: ★. IsD ·R ➔ ★) ➔ ★ | = λ P: D ➔ Π R: ★. IsD ·R ➔ ★. ∀ R: ★. ∀ is: IsD ·R. (Π r: R. P (toD -is r) ·R is)
➔ ByCases ·(λ x: D. P x ·R is) ·R is. | function | ByInd | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"ByCases",
"IsD",
"is",
"toD"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
IdMappingO : (Π R: ★. IsD ·R ➔ ★) ➔ ★ | = λ P: Π R: ★. IsD ·R ➔ ★. ∀ X: ★. ∀ Y: ★.
Cast ·X ·Y ➔ Cast ·(∀ isX: IsD ·X. P ·X isX) ·(∀ isY: IsD ·Y. P ·Y isY). | function | IdMappingO | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"Cast",
"IsD"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
mu: Π d: D. ∀ P: D ➔ Π R: ★. IsD ·R ➔ ★. (Π x: D. IdMappingO ·(P x)) ➾ ByInd ·P ➔ P d ·D isD | = λ d. Λ P. Λ imP. λ ind.
induction ·(λ x: D. λ R: ★. ∀ is: IsD ·R. P x ·R is)
-(λ x. Λ X. Λ Y. Λ c. [ λ f. Λ is. (elimCast -(imP x c) f) -is , β ])
(Λ R. Λ c. λ o. Λ oeq. λ ih. λ xs. Λ is.
ind -is (λ x. ih x -is) xs)
d -isD. | function | mu | encodings/mendler | encodings/mendler/interface.ced | [
"Ind",
"data/sigma",
"stdcore"
] | [
"ByInd",
"IdMappingO",
"IsD",
"elimCast",
"induction",
"is",
"isD",
"xs"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
cast | = elimCast. | function | cast | encodings/mendler | encodings/mendler/libmods.ced | [
"stdlib"
] | [
"elimCast"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
constCastMap : ∀ X : ★ . CastMap · (λ _ : ★ . X) | =
Λ X .
Λ A . Λ B . Λ c . castRefl · X . | function | constCastMap | encodings/mendler | encodings/mendler/libmods.ced | [
"stdlib"
] | [
"CastMap",
"castRefl"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
append : ∀ A : ★ . List · A ➔ List · A ➔ List · A | =
Λ A . λ l1 . λ l2 .
listRec l1 · (List · A)
[ β{λ e . λ d . d (λ u . l2) (λ p . p (λ h . λ t . cons h (e t)))} ,
Λ R . Λ c . λ e . λ d .
d.1 · (List · A)
(λ u . l2)
(λ p . p.1 · (List · A) (λ h . λ t . cons · A h (e.1 t)))]. | function | append | encodings/mendler | encodings/mendler/list-examples.ced | [
"cnat",
"list"
] | [
"List",
"cons",
"listRec"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
appendAssoc : ∀ A : ★ . Π l1 : List · A . ∀ l2 : List · A . ∀ l3 : List · A .
{ append l1 (append l2 l3) ≃ append (append l1 l2) l3 } | =
Λ A . λ l1 . Λ l2 . Λ l3 .
θ<l1> (listInd l1)
[ β{λ ih . λ d . ind-Sum d (λ u . λ x . x) (λ p . ind-Product p (λ h . λ t . λ x . x))} ,
Λ R . Λ _ . λ ih . λ d .
θ<d> (ind-Sum d) (λ u . β) (λ p . θ<p> (ind-Product p)
(λ h . λ t . ρ+ (ih.1 t) - β)) ]. | function | appendAssoc | encodings/mendler | encodings/mendler/list-examples.ced | [
"cnat",
"list"
] | [
"List",
"append",
"ind-Product",
"ind-Sum",
"listInd"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
testList : List · cNat | = cons (cS (cS cZ)) (cons (cS cZ) (cons cZ (nil · cNat))) . | function | testList | encodings/mendler | encodings/mendler/list-examples.ced | [
"cnat",
"list"
] | [
"List",
"cNat",
"cS",
"cZ",
"cons",
"nil"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
ListFunctor ◂ RecFunctor · ListF | =
Λ X . Λ Y . λ c .
caste (λ d : ListF · X. d.1 · (ListF · Y)
(λ _ . in1 · Unit · (Product · A · Y) unit)
(λ p . p.1 · (ListF · Y)
(λ h . λ t . in2 · Unit · (Product · A · Y) (pair h (cast -c t)))))
-(λ a . θ<a> (ind-Sum a) (λ x . ρ (eta-Unit x) - β)
... | function | ListFunctor | encodings/mendler | encodings/mendler/list.ced | [
"Ind",
"cnat",
"public"
] | [
"ListF",
"Product",
"RecFunctor",
"Unit",
"cast",
"caste",
"eta-Unit",
"in1",
"in2",
"ind-Product",
"ind-Sum",
"pair",
"unit"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
List ◂ ★ | = C. | function | List | encodings/mendler | encodings/mendler/list.ced | [
"Ind",
"cnat",
"public"
] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
listRec : C ➔ rF' · C | = cRec -ListFunctor. | function | listRec | encodings/mendler | encodings/mendler/list.ced | [
"Ind",
"cnat",
"public"
] | [
"ListFunctor",
"cRec",
"rF'"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
listInd : Π c : List . Inductive ListFunctor c | = Ind -ListFunctor. | function | listInd | encodings/mendler | encodings/mendler/list.ced | [
"Ind",
"cnat",
"public"
] | [
"Ind",
"Inductive",
"List",
"ListFunctor"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
nil ◂ List | = in -ListFunctor (in1 unit) . | function | nil | encodings/mendler | encodings/mendler/list.ced | [
"Ind",
"cnat",
"public"
] | [
"List",
"ListFunctor",
"in",
"in1",
"unit"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
cons ◂ A ➔ List ➔ List | =
λ h . λ t . in -ListFunctor (in2 (pair h t)) . | function | cons | encodings/mendler | encodings/mendler/list.ced | [
"Ind",
"cnat",
"public"
] | [
"List",
"ListFunctor",
"in",
"in2",
"pair"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
ListF ◂ ★ ➔ ★ | = λ X : ★ . Sum · Unit · (Product · A · X) . | function | ListF | encodings/mendler | encodings/mendler/listf.ced | [
"public"
] | [
"Product",
"Sum",
"Unit"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
𝒌123 | = ★. | function | 𝒌123 | issues/issue-114 | issues/issue-114/kind-defs-broken.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
𝒌 | = ★ ➔ ★ . | function | 𝒌 | issues/issue-12 | issues/issue-12/cast2.ced | [] | [] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Cast2 ◂ 𝒌 ➔ 𝒌 ➔ ★ | = λ A : 𝒌 . λ B : 𝒌 .
ι cast2 : ∀ X : ★ . A · X ➔ B · X . {cast2 ≃ λ x . x}. | function | Cast2 | issues/issue-12 | issues/issue-12/cast2.ced | [] | [
"cast2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
cast2 ◂ ∀ A : 𝒌 . ∀ B : 𝒌 . Cast2 · A · B ➾ ∀ X : ★ . A · X ➔ B · X | =
Λ A . Λ B . Λ c . Λ X . λ a . φ (ρ c.2 - β) - (c.1 · X a) {a} . | function | cast2 | issues/issue-12 | issues/issue-12/cast2.ced | [] | [
"Cast2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
caste2 ◂ ∀ A : 𝒌 . ∀ B : 𝒌 . Π f : ∀ X : ★ . A · X ➔ B · X . (∀ X : ★ . Π a : A · X . {f a ≃ a}) ➾ Cast2 · A · B | =
Λ A . Λ B . λ f . Λ e . [ Λ X . λ a . φ (e · X a) - (f · X a) {a} , β ] . | function | caste2 | issues/issue-12 | issues/issue-12/cast2.ced | [] | [
"Cast2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
castCompose2 ◂ ∀ A : 𝒌 . ∀ B : 𝒌 . ∀ C : 𝒌 . Cast2 · A · B ➔ Cast2 · B · C ➔ Cast2 · A · C | =
Λ A . Λ B . Λ C . λ c1 . λ ca . [ Λ X . λ a . cast2 · B · C -ca · X (cast2 · A · B -c1 · X a) , β] . | function | castCompose2 | issues/issue-12 | issues/issue-12/cast2.ced | [] | [
"Cast2",
"cast2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
castId2 ◂ ∀ A : 𝒌 . Cast2 · A · A | =
Λ A . [ Λ X . λ a . a , β]. | function | castId2 | issues/issue-12 | issues/issue-12/cast2.ced | [] | [
"Cast2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
Trmc
◂ (★ ➔ ★) ➔ ★ ➔ ★ | = λ Trmc : ★ ➔ ★ . λ X : ★ .
(ι l : Top .
Π f : X ➔ X .
∀ g : ∀ X : ★ . X ➔ Trmc · X .
(∀ x : X . { f x ≃ g x l }) ➾ X)
➔ X . | function | Trmc | issues/issue-12 | issues/issue-12/issue-12.ced | [
"RecType2",
"cast2",
"top"
] | [
"Top"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
TrmcF
◂ RecFunctor2 ·Trmc | = Λ X . Λ Y . λ c .
[ Λ Z . λ a . λ l .
a [ ●
, λ f . Λ g . Λ e . l.2 f -(Λ Q . λ q . cast2 -c (g · Q q)) -●]
, ● ]. | function | TrmcF | issues/issue-12 | issues/issue-12/issue-12.ced | [
"RecType2",
"cast2",
"top"
] | [
"RecFunctor2",
"Trmc",
"cast2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 | |
RecFunctor2 ◂ ★ | =
∀ X : 𝒌 . ∀ Y : 𝒌 . Cast2 · X · Y ➔ Cast2 · (F · X) · (F · Y) . | function | RecFunctor2 | issues/issue-12 | issues/issue-12/RecType2.ced | [
"cast2",
"top"
] | [
"Cast2"
] | https://github.com/cedille/cedille | 78ca684859bb2c5e166d65279b0a352267138c89 |
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