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1.4.0 casep.txt clarified, smoothed, tied to oSigma.txt
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orcmid committed Jun 8, 2024
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126 changes: 60 additions & 66 deletions docs/oFrugal/casep.txt
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@@ -1,4 +1,4 @@
// casep.txt 1.3.0 UTF-8 2024-05-12
// casep.txt 1.4.0 UTF-8 2024-06-08
//--|----1----|----2----|----3----|----4----|----5----|----6----|----7----|--*
//
// OBAP ABSTRACTION FUNCTIONS
Expand Down Expand Up @@ -51,7 +51,7 @@
// In Frugalese pseudo-notation, this can be reformed into a form in
// which the lambda-abstractions are explicitly applied.
//
// casep = λ.v ρ.casepv λ.L (a)
// casep = λ.v ρ.casepv λ.L // (1)
// ( if is-singleton(L)
// then L
// else if (.b .a L) v
Expand Down Expand Up @@ -99,35 +99,36 @@
// so the first condition-form in casep (1a) translates to the oFrugal
// form
//
// .ev :: (.d :: L :: .b :: L) (a)
// .ev :: (.d :: L :: .b :: L) // (4)
// :: `( L
// :: not-satisfied-part )
//
//
// 5. THE CURRENT CASE-CONDITION CHECKING PART
//
// The not-satisfied-part of (a) is another conditional form in (1a),
// The not-satisfied-part of (4) is another conditional form of (1),
// thus
//
// .ev :: ((.b :: .a :: L) :: v) (a)
// .ev :: ((.b :: .a :: L) :: ` v) // (5)
// :: `( (.a :: L)
// :: casepv :: .b :: L)
//
// We are slowly boiling the Frugalese pseudo-code down to a canonical
// oFrugal form in which the only remaining lindies are those which are
// to be abstracted away.
//
// One way to keep track of everything is by introduction of comments
// and also making of some intermediate oFrugal definitions to avoid
// recopying of portions already figured out.
// Note that v is enclosed, ensuring that at the level where (5) is
// evaluated, the previously-substituted operand value for v is simply
// used at that point, not evaluated further.
//
//
//
// 6. PUTTING IT ALL TOGETHER
//
// casep = λ.v ρ.casepv λ.L (a)
// casep = λ.v ρ.casepv λ.L // (6a)
// (.ev :: (.d :: L :: .b :: L)
// :: `( L
// :: .ev :: ((.b :: .a :: L) :: v)
// :: .ev :: ((.b :: .a :: L) :: ` v)
// :: `( (.a :: L)
// :: casepv :: .b :: L)
// )
Expand All @@ -142,27 +143,27 @@
// oFrugal. With availability of an operating oFrugal implementation,
// a dependable confirmation can also be established.

!def ob ^casep
!def ob ^casep // (6b)

= ^λ.v ^ρ.casepv ^λ.L

( // if is-singleton(L)
( // if is-singleton(L)
.ev :: (.d :: L :: .b :: L)

:: `( // then L
:: `( // then L
L
::
// else if (.b .a L) v
.ev :: ((.b :: .a :: L) :: ` v)

// else if (.b .a L) v
:: .ev :: ((.b :: .a :: L) :: v)

:: `( // then .a L
(.a :: L)

// else casepv .b L
:: casepv :: .b :: L
)
)
);
:: `( // then .a L
(.a :: L)
::
// else casepv .b L
casepv :: .b :: L
)
)
);

// assuming already-defined ^λ and ^ρ.
//
Expand All @@ -178,31 +179,31 @@
//
// When a script-form is not yet for an applicative function taking any
// operands, the λ.L heuristic will simply rewrite that script so that
// every occurrence of L is replaced by the primitive .ARG.
// every occurrence of L is replaced by the primitive .arg.
//
// 7.1 Abstracting Operand L

!def ob ^casep
!def ob ^casep // (7.1)

= ^λ.v ^ρ.casepv // ^λ.L

( // if is-singleton(L)
.ev :: (.d :: .arg :: .b :: .arg)
( // if is-singleton(L)
.ev :: (.d :: L :: .b :: .arg)

:: `( // then L
:: `( // then L
.arg
::
// else if (.b .a L) v
.ev :: ((.b :: .a :: .arg) :: ` v)

// else if (.b .a L) v
:: .ev :: ((.b :: .a :: .arg) :: v)

:: `( // then .a L
(.a :: .arg)

// else casepv .b L
:: casepv :: .b :: .arg
)
)
);
:: `( // then .a L
(.a :: .arg)
::
// else casepv .b L
casepv :: .b :: .arg
)
)
);
//
// 7.2 Abstracting the Recursion
//
Expand All @@ -215,25 +216,26 @@
//
!def ob ^casep

= ^λ.v // ^ρ.casepv ^λ.L
= ^λ.v // ^ρ.casepv ^λ.L // (7.2)

( // if is-singleton(L)
.ev :: (.d :: .arg :: .b :: .arg)
( // if is-singleton(L)
.ev :: (.d :: L :: .b :: .arg)

:: `( // then L
.arg
::
// else if (.b .a L) v
.ev :: ((.b :: .a :: .arg) :: ` v)

// else if (.b .a L) v
:: .ev :: ((.b :: .a :: .arg) :: v)

:: `( // then .a L
:: `( // then .a L
(.a :: .arg)

// else casepv .b L
:: .self :: .b :: .arg
)
::
// else casepv .b L
.self :: .b :: .arg
)
)
);
);

//
// 7.3 Abstracting Operand v
//
Expand All @@ -242,18 +244,11 @@
// all occurrences of v by .ARG is incorrect.
//
// The required behavior in this case is to create a script that when
// applied to an operand, V, yields the above script but with `V in
// place of v. The quotation is required because we don't want to
// evaluate V, we want simply to use it in the form that is supplied
// as an operand.
//
// Forms of the abstraction function, σ.v, are useful in this case.
// Here, we provide that function manually. We will, at this stage,
// have a loss of similarity with the Frugalese pseudo-language form
// that we started with.
// applied to an operand, V, yields the above script but with V in
// place of v.
//
// This version is obtained by manually applying the algorithm in
// sigma.txt <https://orcmid.github.io/miser/oFrugal/sigma.txt>.
// oSigma.txt <https://orcmid.github.io/miser/lambda/oSigma.txt>.

!def ob ^casep

Expand All @@ -280,8 +275,7 @@
);

// demonstrating why it is important to have the abstraction functions
// up-and-running, with easier confirmation and, if necessary, saving
// definitions to a text file.
// up-and-running, with easier confirmation and simpler expresssion.
//
//
//--|----1----|----2----|----3----|----4----|----5----|----6----|----7----|--*
Expand All @@ -304,7 +298,7 @@
// ATTRIBUTION
//
// Hamilton, Dennis E. Hand-Compiled casep(v, L). Miser Theory
// Conception text file casep.txt version 1.3.0 dated 2024-05-12,
// Conception text file casep.txt version 1.4.0 dated 2024-06-08,
// available on the Internet as a version of
// <https://orcmid.github.io/miser/oFrugal/casep.txt>
//
Expand All @@ -314,10 +308,10 @@
//
// * Test this with oFrugal when implementation is available.
//
// * Continue making this more-readable following section 4.
//
//--|----1----|----2----|----3----|----4----|----5----|----6----|----7----|--*
//
// 1.4.0 2024-06-98T16:06Z More clarifications, better layout, tying in oSigma
// 1.3.0 2024-05-12T20:41Z Cleanit it all up
// 1.2.4 2024-05-11T21:36Z Completed polishing ^casep for now.
// 1.2.3 2024-05-11T16:46Z Use intermediates to make conditionals more
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