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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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Cedille

Cedille declarations, each row carrying the statement and, where present, the proof.

Source

Schema

Column Type Description
statement string Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof string Verbatim proof/body, empty if the declaration has none
type string Declaration keyword
symbolic_name string Declaration identifier
library string Sub-library
filename string Repository-relative source path
imports list[string] File-level Require/Import modules
deps list[string] Intra-corpus identifiers referenced
docstring string Preceding documentation comment, empty if absent
source_url string Upstream repository
commit string Upstream commit extracted

Statistics

  • Entries: 1,094
  • With proof: 1,093 (99.9%)
  • With docstring: 0 (0.0%)
  • Libraries: 37

By type

Type Count
function 1,051
data 43

Example

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] .
  • type: function | symbolic_name: out' | encodings/mendler/C-extras.ced

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{cedille_dataset,
  title  = {Cedille},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/cedille/cedille, commit 78ca684859bb},
  url    = {https://huggingface.co/datasets/phanerozoic/Cedille}
}
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