104 lines
3.3 KiB
Plaintext
104 lines
3.3 KiB
Plaintext
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%% div.elf - Divisibility on Natural Numbers
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%%
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%% We define:
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%% - divides D N: D divides N (D | N), meaning N = D * K for some K
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%% ============================================================
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%% DIVISIBILITY
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%% ============================================================
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%% D divides N if there exists K such that D * K = N
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divides : nat -> nat -> type.
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divides/intro : divides D N
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<- mult D K N.
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%% ============================================================
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%% BASIC DIVISIBILITY FACTS
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%% ============================================================
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%% 1 divides everything: 1 | N
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one-divides : {N:nat} divides (s z) N -> type.
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%mode one-divides +N -D.
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one-divides/base : one-divides N (divides/intro Dmult)
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<- mult-left-one N Dmult.
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%worlds () (one-divides _ _).
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%total {} (one-divides _ _).
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%% Everything divides 0: N | 0
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divides-zero : {N:nat} divides N z -> type.
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%mode divides-zero +N -D.
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%% For any N, mult N z z holds (mult/z pattern doesn't work here)
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%% We need to use mult-right-z
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divides-zero/base : divides-zero N (divides/intro D)
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<- mult-right-z N D.
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%worlds () (divides-zero _ _).
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%total {} (divides-zero _ _).
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%% N divides itself: N | N (reflexivity)
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divides-refl : {N:nat} divides N N -> type.
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%mode divides-refl +N -D.
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divides-refl/base : divides-refl N (divides/intro Dmult)
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<- mult-right-one N Dmult.
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%worlds () (divides-refl _ _).
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%total {} (divides-refl _ _).
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%% ============================================================
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%% DIVISION WITH REMAINDER (EUCLIDEAN DIVISION)
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%% ============================================================
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%% For D > 0 and any N, there exist unique Q, R with N = D*Q + R and R < D
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%% We represent this as: divmod N D Q R means N = D*Q + R and R < D
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divmod : nat -> nat -> nat -> nat -> type.
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%% Base case: if N < D, then Q = 0, R = N
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divmod/lt : divmod N D z N
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<- lt N D.
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%% Recursive case: if N >= D, we need to compute (N - D) div D
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%% This requires a subtraction relation, which we'll define
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%% Subtraction: sub A B C means A = B + C (so C = A - B when B <= A)
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sub : nat -> nat -> nat -> type.
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sub/z : sub N z N.
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sub/s : sub (s A) (s B) C
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<- sub A B C.
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%% Now we can define divmod for the recursive case
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divmod/ge : divmod N D (s Q) R
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<- le D N %% D <= N
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<- sub N D M %% M = N - D
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<- divmod M D Q R. %% M = D*Q + R, so N = D*(Q+1) + R
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%% ============================================================
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%% GCD (Greatest Common Divisor)
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%% ============================================================
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%% Euclidean algorithm: gcd(A, 0) = A, gcd(A, B) = gcd(B, A mod B)
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%% We define gcd using the Euclidean algorithm
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%% gcd A B G means G is the GCD of A and B
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gcd : nat -> nat -> nat -> type.
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gcd/z : gcd A z A. %% gcd(A, 0) = A
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gcd/s : gcd A (s B) G
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<- divmod A (s B) Q R %% A = (s B)*Q + R with R < s B
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<- gcd (s B) R G. %% G = gcd(s B, R)
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%% ============================================================
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%% COPRIMALITY
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%% ============================================================
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%% A and B are coprime if gcd(A, B) = 1
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coprime : nat -> nat -> type.
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coprime/intro : coprime A B
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<- gcd A B (s z).
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