/**
* @covers PHPixme\__PRIVATE__::curryExactly3
*/
public function test_curryExactly3()
{
// "The flow of time itself is convoluted; with heroes centuries old phasing in and out." -- Solaire of Astora
$arity3 = internal::curryExactly3(function (...$args) {
return $args;
});
$expectedOutput = [1, 2, 3];
// Establishing some basic qualities of curryExactly3
self::assertInstanceOf(Closure, $arity3);
/*
* A note about the notation.
* => means imply.
* (x)->a is a function.
* X1 means that the first argument is set
* _1 means it is explicitly void in a call
* -> ... -> between argument states means any number of identity operations caused by no arguments
* ≅ means 'is essentially the same as' or more formally the categorically congruent to.
* ∴ means 'therefore'
* ∎ means end of a proof or Q.E.D.
* In variable naming __ means a step in the sequence
*/
/** Show that curryExactly3((x y z) -> a) ≅ (x y z) -> a **/
self::assertInstanceOf(Closure, $arity3());
self::assertSame($arity3, $arity3());
self::assertSame($arity3, $arity3(_()));
self::assertSame($arity3, $arity3(_(), _()));
self::assertSame($arity3, $arity3(_(), _(), _()));
// => curryExactly3((x y z)->a) -> ... -> a
self::assertEquals($expectedOutput, $arity3(1, 2, 3, 4, 5));
/** ∴ curryExactly3((x y z) -> a) ≅ (x y z) -> a ∎ **/
/** show that (X1)->(X2)->(X3) ≅ (X1 X2 X3) **/
$step1 = $arity3(1);
// Prove that step 1 is a thunk
self::assertInstanceOf(Closure, $step1);
self::assertSame($step1, $step1());
self::assertSame($step1, $step1(_()));
self::assertSame($step1, $step1(_(), _()));
$step2 = $step1(2);
// Prove that step2 is a thunk
self::assertInstanceOf(Closure, $step2);
self::assertSame($step2, $step2());
self::assertSame($step2, $step2(_()));
self::assertSame($step2, $step2(_(), _()));
self::assertEquals($expectedOutput, $step2(3, 4, 5));
// => (X1)->...-> (X2) -> ...-> (X3) -> a
unset($step1, $step2);
/** ∴ (X1)->(X2)->(X3) ≅ (X1 X2 X3) ∎ **/
/** Show that (X1 X2 X3)
* ≅ (X1) -> (X2 X3)
* ≅ (X1 _2)-> (X2 X3)
* ≅ (X1 _2 _3) -> (X2 X3)
**/
// The step above proves that (X1) is a thunk.
$step1X1_2 = $arity3(1, _());
$step1X1_2_3 = $arity3(1, _(), _());
// (X1) properties were proven previously as a thunk and a closure
self::assertInstanceOf(Closure, $step1X1_2);
self::assertInstanceOf(Closure, $step1X1_2_3);
// (X1 _2) is a thunk
self::assertSame($step1X1_2, $step1X1_2());
self::assertSame($step1X1_2, $step1X1_2(_()));
self::assertSame($step1X1_2, $step1X1_2(_(), _()));
// (X1 _2 _3) is a thunk
self::assertSame($step1X1_2_3, $step1X1_2_3());
self::assertSame($step1X1_2_3, $step1X1_2_3(_()));
self::assertSame($step1X1_2_3, $step1X1_2_3(_(), _()));
// => (X1)-> ... -> (X2 X3) ≅ (X1 _2)-> ... -> (X2 X3) ≅ (X1 _2 _3)-> ... -> (X2 X3)
self::assertEquals($expectedOutput, $arity3(1)->__invoke(2, 3, 7));
self::assertEquals($expectedOutput, $step1X1_2(2, 3, 7));
self::assertEquals($expectedOutput, $step1X1_2_3(2, 3, 7));
unset($step1X1, $step1X1_2, $step1X1_2_3);
/** ∴ (X1)->(X2 X3) ≅ (X1 _2)-> (X2 X3) ≅ (X1 _2 _3) -> (X2 X3) ≅ (X1 X2 X3) ∎ **/
/** Show that (X1 X2 X3)
* ≅ (X1 X2) -> (X3)
* ≅ (X1 X2 _3)-> (X3)
**/
$step1X1X2 = $arity3(1, 2);
$step1X1X2_3 = $arity3(1, 2, _());
self::assertInstanceOf(Closure, $step1X1X2);
self::assertSame($step1X1X2, $step1X1X2());
self::assertSame($step1X1X2, $step1X1X2(_()));
// (X1 X2) is a thunk
self::assertInstanceOf(Closure, $step1X1X2_3);
self::assertSame($step1X1X2_3, $step1X1X2_3());
self::assertSame($step1X1X2_3, $step1X1X2_3(_()));
// (X1 X2 _3) is a thunk
// => (X1 X2 _3) -> ... -> (X3) ≅ (X1 X2) -> ... -> (X3)
self::assertEquals($expectedOutput, $step1X1X2(3, 4, 5));
self::assertEquals($expectedOutput, $step1X1X2_3(3, 2));
unset($step1X1X2, $step1X1X2_3);
/** ∴ (X1 X2 X3)
* ≅ (X1 X2) -> (X3)
* ≅ (X1 X2 _3) -> (X3) ∎
**/
/** Show that (X1 X2 X3)
* ≅ (_1 _2 X3) -> (X1 X2)
* ≅ (_1 _2 X3) -> (X1) -> (X2)
* ≅ (_1 _2 X3) -> (X1 _2) -> (X2)
**/
$step1_1_2X3 = $arity3(_(), _(), 3);
self::assertInstanceOf(Closure, $step1_1_2X3);
self::assertSame($step1_1_2X3, $step1_1_2X3());
self::assertSame($step1_1_2X3, $step1_1_2X3(_()));
self::assertSame($step1_1_2X3, $step1_1_2X3(_(), _()));
// (_1 _2 X3) is a thunk
$step2_1_2X3__X1 = $step1_1_2X3(1);
$step2_1_2X3__X1_3 = $step1_1_2X3(1, _());
self::assertInstanceOf(Closure, $step2_1_2X3__X1);
self::assertSame($step2_1_2X3__X1, $step2_1_2X3__X1());
self::assertSame($step2_1_2X3__X1, $step2_1_2X3__X1(_()));
// (_1 _2 X3) -> (X1) is a thunk
self::assertInstanceOf(Closure, $step2_1_2X3__X1_3);
self::assertSame($step2_1_2X3__X1_3, $step2_1_2X3__X1_3());
self::assertSame($step2_1_2X3__X1_3, $step2_1_2X3__X1_3(_()));
// (_1 _2 X3) -> (X1 _2) is a thunk
// => (_1 _2 X3) -> ... -> (X1 X2)
// ≅ (_1 _2 X3) -> ... -> (X1) -> ... -> (X2)
// ≅ (_1 _2 X3) -> ... -> (X1 _2) -> ... -> (X2)
self::assertEquals($expectedOutput, $step1_1_2X3(1, 2, 4));
self::assertEquals($expectedOutput, $step2_1_2X3__X1(2, 4));
self::assertEquals($expectedOutput, $step2_1_2X3__X1_3(2, 4));
unset($step1_1_2X3, $step2_1_2X3__X1, $step2_1_2X3__X1_3);
/** ∴ (X1 X2 X3)
* ≅ (_1 _2 X3) -> (X1 X2)
* ≅ (_1 _2 X3) -> (X1) -> (X2)
* ≅ (_1 _2 X3) -> (X1 _2) -> (X2) ∎
**/
/** Show (X1 X2 X3)
* ≅ (_1 X2) -> (X1 X3)
* ≅ (_1 X2) -> (X1) -> (X3)
* ≅ (_1 X2) -> (X1 _3) -> (X3)
* ≅ (_1 X2 _3) -> (X1 X3)
* ≅ (_1 X2 _3) -> (X1) -> (X3)
* ≅ (_1 X2 _3) -> (X1 _3) -> (X3)
**/
$step1_1X2 = $arity3(_(), 2);
self::assertInstanceOf(Closure, $step1_1X2);
self::assertSame($step1_1X2, $step1_1X2());
self::assertSame($step1_1X2, $step1_1X2(_()));
self::assertSame($step1_1X2, $step1_1X2(_(), _()));
// (_1 X2) is a thunk
$step1_1X2_3 = $arity3(_(), 2, _());
self::assertInstanceOf(Closure, $step1_1X2_3);
self::assertSame($step1_1X2_3, $step1_1X2_3());
self::assertSame($step1_1X2_3, $step1_1X2_3(_()));
self::assertSame($step1_1X2_3, $step1_1X2_3(_(), _()));
// (_1 X2 _3) is a thunk
$step2_1X2__X1 = $step1_1X2(1);
self::assertInstanceOf(Closure, $step2_1X2__X1);
self::assertSame($step2_1X2__X1, $step2_1X2__X1());
self::assertSame($step2_1X2__X1, $step2_1X2__X1(_()));
// (_1 X2) -> (X1) is a thunk
$step2_1X2_3__X1 = $step1_1X2_3(1);
self::assertInstanceOf(Closure, $step2_1X2_3__X1);
self::assertSame($step2_1X2_3__X1, $step2_1X2_3__X1());
self::assertSame($step2_1X2_3__X1, $step2_1X2_3__X1(_()));
// (_1 X2 _3) -> (X1) is a thunk
$step2_1X2__X1_3 = $step1_1X2(1, _());
self::assertInstanceOf(Closure, $step2_1X2__X1_3);
self::assertSame($step2_1X2__X1_3, $step2_1X2__X1_3());
self::assertSame($step2_1X2__X1_3, $step2_1X2__X1_3(_()));
// (_1 X2) -> (X1 _3) is a thunk
$step2_1X2_3__X1_3 = $step1_1X2_3(1, _());
self::assertInstanceOf(Closure, $step2_1X2_3__X1_3);
self::assertSame($step2_1X2_3__X1_3, $step2_1X2_3__X1_3());
self::assertSame($step2_1X2_3__X1_3, $step2_1X2_3__X1_3(_()));
// (_1 X2 _3) -> (X1 _3) is a thunk
// => (_1 X2) -> ... -> (X1 X3)
// ≅ (_1 X2 _3) -> ... -> (X1 X3)
// ≅ (_1 X2) -> ... -> (X1) -> ... -> (X3)
// ≅ (_1 X2 _3) -> ... ->(X1) -> ... -> (X3)
// ≅ (_1 X2) -> ... -> (X1 _3) -> ... -> (X3)
// ≅ (_1 X2 _3) -> ... -> (X1 _3) -> ... -> (X3)
self::assertEquals($expectedOutput, $step1_1X2(1, 3, 7));
self::assertEquals($expectedOutput, $step1_1X2_3(1, 3, 7));
self::assertEquals($expectedOutput, $step2_1X2__X1(3, 7));
self::assertEquals($expectedOutput, $step2_1X2_3__X1(3, 7));
self::assertEquals($expectedOutput, $step2_1X2__X1_3(3, 7));
self::assertEquals($expectedOutput, $step2_1X2_3__X1_3(3, 7));
unset($step1_1X2, $step1_1X2_3, $step2_1X2__X1, $step2_1X2_3__X1, $step2_1X2__X1_3, $step2_1X2_3__X1_3);
/** ∴ (X1 X2 X3)
* ≅ (_1 X2) -> (X1 X3)
* ≅ (_1 X2) -> (X1) -> (X3)
* ≅ (_1 X2) -> (X1 _3) -> (X3)
* ≅ (_1 X2 _3) -> (X1 X3)
* ≅ (_1 X2 _3) -> (X1) -> (X3)
* ≅ (_1 X2 _3) -> (X1 _3) -> (X3) ∎
**/
/** Show that (X1 X2 X3)
* ≅ (X1) -> (_2 X3) -> (X2)
* ≅ (X1 _2) -> (_2 X3) -> (X2)
* ≅ (X1 _2 _3) -> (_2 X3) -> (X2)
* ≅ (X1 _2 X3) -> (X2)
*/
// Previously proven: (X1) === (X1 _2) === (X1 _2 _3) Skipping step.
$step2X1___2X3 = $arity3(1)->__invoke(_(), 3);
self::assertInstanceOf(Closure, $step2X1___2X3);
self::assertSame($step2X1___2X3, $step2X1___2X3());
self::assertSame($step2X1___2X3, $step2X1___2X3(_()));
// (X1) -> (_2 X3) is a thunk
$step1X1_2X3 = $arity3(1, _(), 3);
self::assertInstanceOf(Closure, $step1X1_2X3);
self::assertSame($step1X1_2X3, $step1X1_2X3());
self::assertSame($step1X1_2X3, $step1X1_2X3(_()));
// (X1 _2 X3) is a thunk
// => (X1)-> ... -> (_2 X3) -> ... -> (X2)
// ≅ (X1 _2 X3) -> ... -> (X2)
self::assertEquals($expectedOutput, $step2X1___2X3(2, 7));
self::assertEquals($expectedOutput, $step1X1_2X3(2, 7));
unset($step2X1___2X3, $step1X1_2X3);
/** ∴ (X1 X2 X3)
* ≅ (X1) -> (_2 X3) -> (X2)
* ≅ (X1 _2) -> (_2 X3) -> (X2)
* ≅ (X1 _2 _3) -> (_2 X3) -> (X2)
* ≅ (X1 _2 X3) -> (X2) ∎
*/
/** Show that (X1 X2 X3)
* ≅ (_1 X2 X3) -> (X1)
* ≅ (_1 X2)->(_1 X3) -> (X1)
* ≅ (_1 X2 _3) -> (_1 X3) -> (X1)
**/
$step1_1X2X3 = $arity3(_(), 2, 3);
self::assertInstanceOf(Closure, $step1_1X2X3);
self::assertSame($step1_1X2X3, $step1_1X2X3());
self::assertSame($step1_1X2X3, $step1_1X2X3(_()));
// (_1 X2 X3) is a thunk
// Previously we showed that (_1 X2) === (_1 X2 _3) and showed they were a thunk.
// Skipping proofs for both of these steps
$step2_1X2___1X3 = $arity3(_(), 2)->__invoke(_(), 3);
self::assertInstanceOf(Closure, $step2_1X2___1X3);
self::assertSame($step2_1X2___1X3, $step2_1X2___1X3());
self::assertSame($step2_1X2___1X3, $step2_1X2___1X3(_()));
// (_1 X2) -> (_1 X3) is a thunk
$step2_1X2_3___1X3 = $arity3(_(), 2)->__invoke(_(), 3);
self::assertInstanceOf(Closure, $step2_1X2_3___1X3);
self::assertSame($step2_1X2_3___1X3, $step2_1X2_3___1X3());
self::assertSame($step2_1X2_3___1X3, $step2_1X2_3___1X3(_()));
// (_1 X2 _3) -> (_1 X3) is a thunk
// => (_1 X2 X3) -> ... -> (X1)
// ≅ (_1 X2)-> ... -> (_1 X3) -> ... -> (X1)
// ≅ (_1 X2 _3) -> ... -> (_1 X3) -> ... ->(X1)
self::assertEquals($expectedOutput, $step1_1X2X3(1, 7));
self::assertEquals($expectedOutput, $step2_1X2___1X3(1, 7));
self::assertEquals($expectedOutput, $step2_1X2_3___1X3(1, 7));
unset($step1_1X2X3, $step2_1X2___1X3, $step2_1X2_3___1X3);
/** ∴ (X1 X2 X3)
* ≅ (_1 X2 X3) -> (X1)
* ≅ (_1 X2)->(_1 X3) -> (X1)
* ≅ (_1 X2 _3) -> (_1 X3) -> (X1) ∎
**/
/** Show that (X1 X2 X3) ≅ (_1 _2 X3) -> (_1 X2) -> (X1) **/
// Previously we proved that (_1 _2 X3) is a thunk.
// Skipping proof for this step
$step2_1_2X3___1X2 = $arity3(_(), _(), 3)->__invoke(_(), 2);
self::assertInstanceOf(Closure, $step2_1_2X3___1X2);
self::assertSame($step2_1_2X3___1X2, $step2_1_2X3___1X2());
self::assertSame($step2_1_2X3___1X2, $step2_1_2X3___1X2(_()));
// (_1 _2 X3) -> (_1 X2) is a thunk
// => (_1 _2 X3) -> (_1 X2) -> (X1)
self::assertEquals($expectedOutput, $step2_1_2X3___1X2(1, 7));
unset($step2_1_2X3___1X2);
/** ∴ (X1 X2 X3) ≅ (_1 _2 X3) -> (_1 X2) -> (X1) ∎ **/
// Praise the sun! \o/
// Let us now go forth and engage in jolly cooperation!
}
