104 lines
4.5 KiB
Text
104 lines
4.5 KiB
Text
-- --------------------------------------------------------------------------------------------------------------
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-- | BASIC LOGIC |
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-- --------------------------------------------------------------------------------------------------------------
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@include logic.pg
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-- --------------------------------------------------------------------------------------------------------------
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-- | BASIC DEFINITIONS |
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-- --------------------------------------------------------------------------------------------------------------
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-- note we leave off the type ascriptions for most of these, as the type isn't
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-- very interesting
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-- I'd always strongly recommend including the type ascriptions for theorems
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-- a unary operation on a set `A` is a function `A -> A`
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unop (A : *) := A -> A;
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-- a binary operation on a set `A` is a function `A -> A -> A`
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binop (A : *) := A -> A -> A;
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-- a binary operation is associative if ...
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assoc (A : *) (op : binop A) :=
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forall (a b c : A), eq A (op (op a b) c) (op a (op b c));
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-- an element `e : A` is a left identity with respect to binop `op` if `∀ a, e * a = a`
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id_l (A : *) (op : binop A) (e : A) :=
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forall (a : A), eq A (op e a) a;
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-- likewise for right identity
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id_r (A : *) (op : binop A) (e : A) :=
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forall (a : A), eq A (op a e) a;
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-- an element is an identity element if it is both a left and right identity
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id (A : *) (op : binop A) (e : A) := and (id_l A op e) (id_r A op e);
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-- b is a left inverse for a if `b * a = e`
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-- NOTE: we don't require `e` to be an identity in this definition.
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-- this definition is purely for convenience's sake
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inv_l (A : *) (op : binop A) (e : A) (a b : A) := eq A (op b a) e;
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-- likewise for right inverse
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inv_r (A : *) (op : binop A) (e : A) (a b : A) := eq A (op a b) e;
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-- and full-on inverse
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inv (A : *) (op : binop A) (e : A) (a b : A) := and (inv_l A op e a b) (inv_r A op e a b);
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-- --------------------------------------------------------------------------------------------------------------
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-- | ALGEBRAIC STRUCTURES |
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-- --------------------------------------------------------------------------------------------------------------
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-- a set `S` with binary operation `op` is a semigroup if its operation is associative
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semigroup (S : *) (op : binop S) : * := assoc S op;
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-- a set `M` with binary operation `op` and element `e` is a monoid
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monoid (M : *) (op : binop M) (e : M) : * :=
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and (semigroup M op) (id M op e);
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-- some "getters" for `monoid` so we don't have to do a bunch of very verbose
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-- and-eliminations every time we want to use something
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id_lm (M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e) : id_l M op e :=
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and_elim_l (id_l M op e) (id_r M op e)
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(and_elim_r (semigroup M op) (id M op e) Hmonoid);
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id_rm (M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e) : id_r M op e :=
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and_elim_r (id_l M op e) (id_r M op e)
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(and_elim_r (semigroup M op) (id M op e) Hmonoid);
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-- now we can prove that, for any monoid, if `a` is a left identity, then it
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-- must be "the" identity
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monoid_id_l_implies_identity
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(M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e)
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(a : M) (H : id_l M op a) : eq M a e :=
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-- WTS a = a * e = e
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-- we can use `eq_trans` to glue proofs of `a = a * e` and `a * e = e` together
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eq_trans M a (op a e) e
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-- first, `a = a * e`, but we'll use `eq_sym` to flip it around
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(eq_sym M (op a e) a
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-- now the goal is to show `a * e = a`, which follows immediately from `id_r`
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(id_rm M op e Hmonoid a))
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-- now we need to show that `a * e = e`, but this immediately follows from `H`
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(H e);
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-- the analogous result for right identities
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monoid_id_r_implies_identity
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(M : *) (op : binop M) (e : M) (Hmonoid : monoid M op e)
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(a : M) (H : id_r M op a) : eq M a e :=
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-- this time, we'll show `a = e * a = e`
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eq_trans M a (op e a) e
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-- first, `a = e * a`
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(eq_sym M (op e a) a
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-- this time, it immediately follows from `id_l`
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(id_lm M op e Hmonoid a))
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-- and `e * a = e`
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(H e);
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-- groups are just monoids with inverses
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group (G : *) (op : binop G) (e : G) (i : unop G) : * :=
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and (monoid G op e)
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(forall (a : G), inv G op e a (i a));
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