%% prime.elf - Prime Numbers and Related Theorems
%%
%% A prime number P > 1 is a natural number whose only divisors are 1 and P.

%% ============================================================
%% DEFINITION OF PRIME
%% ============================================================

%% A number is "greater than 1"
gt-one : nat -> type.
gt-one/intro : gt-one (s (s N)).          %% N >= 2 means N > 1

%% A number is composite if it has a non-trivial divisor
%% composite N means: exists D such that 1 < D < N and D | N
composite : nat -> type.
composite/intro : composite N
                   <- gt-one D           %% D > 1
                   <- lt D N             %% D < N
                   <- divides D N.       %% D | N

%% A number is prime if P > 1 and not composite
%% We'll define small primes directly and provide a general characterization

prime : nat -> type.
prime/two   : prime (s (s z)).                    %% 2 is prime
prime/three : prime (s (s (s z))).                %% 3 is prime
prime/five  : prime (s (s (s (s (s z))))).        %% 5 is prime
prime/seven : prime (s (s (s (s (s (s (s z))))))). %% 7 is prime

%% ============================================================
%% FACTORIAL
%% ============================================================

factorial : nat -> nat -> type.
%mode factorial +N -R.

factorial/z : factorial z (s z).         %% 0! = 1
factorial/s : factorial (s N) R
               <- factorial N F          %% F = N!
               <- mult (s N) F R.        %% R = (N+1) * N! = (N+1)!

factorial-total : {N:nat} factorial N F -> type.
%mode factorial-total +N -D.

factorial-total/z : factorial-total z factorial/z.
factorial-total/s : factorial-total (s N) (factorial/s Dmult Dfact)
                     <- factorial-total N Dfact
                     <- mult-total (s N) _ Dmult.

%worlds () (factorial-total _ _).
%total N (factorial-total N _).

%% ============================================================
%% PRIME FACTORIZATION (Representation)
%% ============================================================

%% A list of natural numbers
nat-list : type.
nil  : nat-list.
cons : nat -> nat-list -> nat-list.

%% Product of a list
list-product : nat-list -> nat -> type.
%mode list-product +L -P.

list-product/nil  : list-product nil (s z).
list-product/cons : list-product (cons N L) P
                     <- list-product L Q
                     <- mult N Q P.

%% All elements of a list are prime
all-prime : nat-list -> type.
all-prime/nil  : all-prime nil.
all-prime/cons : all-prime (cons P L)
                  <- prime P
                  <- all-prime L.

%% A prime factorization of N is a list L where:
%% 1. All elements of L are prime
%% 2. The product of L equals N
prime-factorization : nat -> nat-list -> type.
prime-factorization/intro : prime-factorization N L
                             <- all-prime L
                             <- list-product L N.

%% ============================================================
%% THEOREMS (Statements)
%% ============================================================

%% Theorem: Every N > 1 has a prime divisor
%% has-prime-divisor N P means P is prime and P | N
has-prime-divisor : nat -> nat -> type.
has-prime-divisor/intro : has-prime-divisor N P
                           <- prime P
                           <- divides P N.

%% Theorem (Euclid): There are infinitely many primes
%% For any N, there exists a prime P > N
%% This is expressed as: for any list of primes, there's a prime not in the list

%% Theorem (Fundamental Theorem of Arithmetic):
%% 1. Existence: Every N > 1 has a prime factorization
%% 2. Uniqueness: The factorization is unique up to ordering

%% Euclid's Lemma: If P is prime and P | AB, then P | A or P | B
%% This is key to proving uniqueness of factorization

%% ============================================================
%% COMPUTATIONAL EXAMPLES
%% ============================================================

%% These can be verified using %solve:
%% 2! = 2
%% 3! = 6
%% 4! = 24
%% 5! = 120

%% Example factorizations:
%% 6 = 2 * 3 = cons 2 (cons 3 nil)
%% 12 = 2 * 2 * 3 = cons 2 (cons 2 (cons 3 nil))