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+# MIT License, Copyright (c) 2022 Marvin Borner
+# According to works of T.Æ. Mogensen
+
+:import std/Combinator .
+:import std/Pair .
+
+# checks whether balanced ternary number is zero
+zero? [0 T [F] [F] I]
+:test zero? +0 = T
+:test zero? -1 = F
+:test zero? +1 = F
+:test zero? +42 = F
+
+trit-neg [[[2]]]
+trit-pos [[[1]]]
+trit-zero [[[0]]]
+
+# shifts a negative trit into a balanced ternary number
+up-neg [[[[[2 (4 3 2 1 0)]]]]]
+:test up-neg +0 = -1
+:test up-neg -1 = -4
+:test up-neg +42 = +125
+
+# shifts a positive trit into a balanced ternary number
+up-pos [[[[[1 (4 3 2 1 0)]]]]]
+:test up-pos +0 = +1
+:test up-pos -1 = -2
+:test up-pos +42 = +127
+
+# shifts a zero trit into a balanced ternary number
+up-zero [[[[[0 (4 3 2 1 0)]]]]]
+:test up-zero +0 = [[[[0 3]]]]
+:test up-zero +1 = +3
+:test up-zero +42 = +126
+
+# shifts a specified trit into a balanced ternary number
+up [[[[[[5 2 1 0 (4 3 2 1 0)]]]]]]
+:test up trit-neg +42 = up-neg +42
+:test up trit-pos +42 = up-pos +42
+:test up trit-zero +42 = up-zero +42
+
+# negates a balanced ternary number
+negate [[[[[4 3 1 2 0]]]]]
+:test negate +0 = +0
+:test negate -1 = +1
+:test negate +42 = -42
+
+# extracts least significant trit from balanced ternary numbers
+lst [0 trit-zero [trit-neg] [trit-pos] [trit-zero]]
+:test lst +0 = trit-zero
+:test lst -1 = trit-neg
+:test lst +1 = trit-pos
+:test lst +42 = trit-zero
+
+# converts the normal balanced ternary representation into abstract
+# -> the abstract representation is used in add/sub/mul
+_abs-neg [[[[[2 4]]]]]
+_abs-pos [[[[[1 4]]]]]
+_abs-zero [[[[[0 4]]]]]
+abstractify [0 [[[[3]]]] _abs-neg _abs-pos _abs-zero]
+:test abstractify -3 = [[[[0 [[[[2 [[[[3]]]]]]]]]]]]
+:test abstractify +0 = [[[[3]]]]
+:test abstractify +3 = [[[[0 [[[[1 [[[[3]]]]]]]]]]]]
+
+# converts the abstracted balanced ternary representation back to normal
+# using Y/ω to solve recursion
+_normalize [[0 +0 [up-neg ([3 3 0] 0)] [up-pos ([3 3 0] 0)] [up-zero ([3 3 0] 0)]]]
+normalize ω _normalize
+:test normalize [[[[3]]]] = +0
+:test normalize (abstractify +42) = +42
+:test normalize (abstractify -42) = -42
+
+# checks whether two balanced ternary numbers are equal
+# -> removes leading 0s!
+_eq-z [zero? (normalize 0)]
+_eq-neg [[0 F [2 0] [F] [F]]]
+_eq-pos [[0 F [F] [2 0] [F]]]
+_eq-zero [[0 (1 0) [F] [F] [2 0]]]
+_eq-abs [0 _eq-z _eq-neg _eq-pos _eq-zero]
+eq? [[_eq-abs 1 (abstractify 0)]]
+:test eq? -42 -42 = T
+:test eq? -1 -1 = T
+:test eq? -1 +0 = F
+:test eq? +0 +0 = T
+:test eq? +1 +0 = F
+:test eq? +1 +1 = T
+:test eq? +42 +42 = T
+:test eq? [[[[(1 (0 (0 (0 (0 3)))))]]]] +1 = T
+
+# strips leading 0s from balanced ternary number
+_strip-z pair +0 T
+_strip-neg [0 [[pair (up-neg 1) F]]]
+_strip-pos [0 [[pair (up-pos 1) F]]]
+_strip-zero [0 [[pair (0 +0 (up-zero 1)) 0]]]
+strip [fst (0 _strip-z _strip-neg _strip-pos _strip-zero)]
+:test strip [[[[0 3]]]] = +0
+:test strip [[[[2 (0 (0 (0 (0 3))))]]]] = -1
+:test strip +42 = +42
+
+# I believe Mogensen's Paper has an error in its succ/pred definitions.
+# They use 3 abstractions in the _succ* functions, also we use switched +/0
+# in comparison to their implementation, yet the order of neg/pos/zero is
+# the same. Something's weird.
+
+# adds +1 to a balanced ternary number (can introduce leading 0s)
+_succ-z pair +0 +1
+_succ-neg [0 [[pair (up-neg 1) (up-zero 1)]]]
+_succ-zero [0 [[pair (up-zero 1) (up-pos 1)]]]
+_succ-pos [0 [[pair (up-pos 1) (up-neg 0)]]]
+succ [snd (0 _succ-z _succ-neg _succ-pos _succ-zero)]
+ssucc [strip (succ 0)]
+:test eq? (succ -42) -41 = T
+:test eq? (succ -1) +0 = T
+:test eq? (succ +0) +1 = T
+:test eq? (succ (succ (succ (succ (succ +0))))) +5 = T
+:test eq? (succ +42) +43 = T
+
+# subs +1 from a balanced ternary number (can introduce leading 0s)
+_pred-z pair +0 -1
+_pred-neg [0 [[pair (up-neg 1) (up-pos 0)]]]
+_pred-zero [0 [[pair (up-zero 1) (up-neg 1)]]]
+_pred-pos [0 [[pair (up-pos 1) (up-zero 1)]]]
+pred [snd (0 _pred-z _pred-neg _pred-pos _pred-zero)]
+spred [strip (pred 0)]
+:test pred -42 = -43
+:test pred +0 = -1
+:test spred (pred (pred (pred (pred +5)))) = +0
+:test spred +1 = +0
+:test pred +42 = +41
+
+# adds two balanced ternary numbers (can introduce leading 0s)
+_add-c [[1 0 trit-zero]]
+_add-b-neg2 [1 (up-zero (3 0 trit-neg)) (up-neg (3 0 trit-zero)) (up-pos (3 0 trit-neg))]
+_add-b-neg [1 (up-pos (3 0 trit-neg)) (up-zero (3 0 trit-zero)) (up-neg (3 0 trit-zero))]
+_add-b-zero [up 1 (3 0 trit-zero)]
+_add-b-pos [1 (up-zero (3 0 trit-zero)) (up-neg (3 0 trit-pos)) (up-pos (3 0 trit-zero))]
+_add-b-pos2 [1 (up-pos (3 0 trit-zero)) (up-zero (3 0 trit-pos)) (up-neg (3 0 trit-pos))]
+_add-a-neg [[[1 (_add-b-neg 1) _add-b-neg2 _add-b-zero _add-b-neg]]]
+_add-a-pos [[[1 (_add-b-pos 1) _add-b-zero _add-b-pos2 _add-b-pos]]]
+_add-a-zero [[[1 (_add-b-zero 1) _add-b-neg _add-b-pos _add-b-zero]]]
+_add-z [[0 (pred (normalize 1)) (succ (normalize 1)) (normalize 1)]]
+_add-abs [_add-c (0 _add-z _add-a-neg _add-a-pos _add-a-zero)]
+add [[_add-abs 1 (abstractify 0)]]
+sadd [[strip (add 1 0)]]
+:test eq? (add -42 -1) -43 = T
+:test eq? (add -5 +6) +1 = T
+:test eq? (add -1 +0) -1 = T
+:test eq? (add +0 +0) +0 = T
+:test eq? (add +1 +2) +3 = T
+:test eq? (add +42 +1) +43 = T
+
+# subs two balanced ternary numbers (can introduce leading 0s)
+sub [[add 1 (negate 0)]]
+ssub [[strip (sub 1 0)]]
+:test eq? (sub -42 -1) -41 = T
+:test eq? (sub -5 +6) -11 = T
+:test eq? (sub -1 +0) -1 = T
+:test eq? (sub +0 +0) +0 = T
+:test eq? (sub +1 +2) -1 = T
+:test eq? (sub +42 +1) +41 = T
+
+# muls two balanced ternary numbers (can introduce leading 0s)
+_mul-neg [sub (up-zero 0) 1]
+_mul-pos [add (up-zero 0) 1]
+_mul-zero [up-zero 0]
+mul [[1 +0 _mul-neg _mul-pos _mul-zero]]
+smul [[strip (mul 1 0)]]
+:test eq? (mul +42 +0) +0 = T
+:test eq? (mul -1 +42) -42 = T
+:test eq? (mul +3 +11) +33 = T
+:test eq? (mul +42 -4) -168 = T
+
+