pyrites/twelf-experiments/peano/theorems.elf

%% theorems.elf - Verified Computations and Theorem Summary
%%
%% This file demonstrates computational verification using %solve
%% and summarizes the proof status of all theorems.

%% ============================================================
%% COMPUTATIONAL VERIFICATIONS
%% ============================================================

%% Basic arithmetic
%solve add-2-3 : add two three N.
%% Result: N = five

%solve add-3-2 : add three two N.
%% Result: N = five (commutativity)

%solve mult-2-3 : mult two three N.
%% Result: N = s (s (s (s (s (s z))))) = 6

%solve mult-3-2 : mult three two N.
%% Result: N = 6 (commutativity)

%% Exponentiation
%solve exp-2-3 : exp two three N.
%% Result: N = 8

%solve exp-3-2 : exp three two N.
%% Result: N = 9

%% Factorial (defined in prime.elf)
%solve fact-3 : factorial three N.
%% Result: N = 6

%solve fact-4 : factorial four N.
%% Result: N = 24

%% Order relations
%solve two-lt-five : lt two five.
%% Result: lt/s (lt/s lt/z)

%solve three-le-three : le three three.
%% Result: le/s (le/s (le/s le/z))

%% Divisibility
%solve two-div-four : divides two four.
%% Result: divides/intro (mult two two four)

%% ============================================================
%% VERIFIED THEOREMS (Twelf %total checked)
%% ============================================================

%% From nat.elf:
%%   nat-case-total: every nat is z or s of something

%% From add.elf:
%%   add-total:       addition always produces a result
%%   add-right-z:     N + 0 = N
%%   add-right-s:     M + (S N) = S(M + N)
%%   add-comm:        M + N = N + M
%%   add-assoc:       (A + B) + C = A + (B + C)
%%   add-cancel-left: A + B = A + C implies B = C
%%   add-func:        addition is deterministic

%% From mult.elf:
%%   mult-total:      multiplication always produces a result
%%   mult-right-z:    N * 0 = 0
%%   mult-left-one:   1 * N = N
%%   mult-right-one:  N * 1 = N

%% From exp.elf:
%%   exp-total:       exponentiation always produces a result
%%   exp-one:         N^1 = N
%%   exp-base-one:    1^N = 1
%%   exp-base-zero:   0^(S N) = 0

%% From order.elf:
%%   le-refl:         N <= N
%%   le-trans:        A <= B and B <= C implies A <= C
%%   le-antisym:      A <= B and B <= A implies A = B
%%   lt-to-le:        A < B implies A <= B
%%   lt-s-to-le:      A < S(B) implies A <= B
%%   le-to-lt-s:      A <= B implies A < S(B)
%%   trichotomy-total: for all M, N: M < N or M = N or M > N
%%   lt-trans:        A < B and B < C implies A < C

%% From div.elf:
%%   one-divides:     1 | N for all N
%%   divides-zero:    N | 0 for all N
%%   divides-refl:    N | N for all N

%% From prime.elf:
%%   factorial-total: factorial always produces a result

%% ============================================================
%% THEOREMS STATED BUT NOT FULLY PROVEN
%% ============================================================

%% Multiplication:
%%   mult-comm:       M * N = N * M
%%                    (Requires mult-right-s lemma with careful
%%                    associativity/commutativity reasoning)
%%
%%   mult-assoc:      (A * B) * C = A * (B * C)
%%                    (Requires distributivity)
%%
%%   mult-distrib-l:  A * (B + C) = A*B + A*C
%%                    (Provable with induction on A)

%% Number theory:
%%   has-prime-divisor-total: every N > 1 has a prime divisor
%%                    (Requires strong/well-founded induction)
%%
%%   infinitude-of-primes: for any prime P, there exists prime Q > P
%%                    (Classic Euclidean proof using factorial)
%%
%%   euclid-lemma:    if P prime and P | AB then P | A or P | B
%%                    (Requires Bezout's identity or equivalent)
%%
%%   fundamental-theorem-existence: every N > 1 has prime factorization
%%                    (Follows from has-prime-divisor with induction)
%%
%%   fundamental-theorem-uniqueness: prime factorization is unique
%%                    (Requires Euclid's lemma)