-include "Q/q/q.ma".
-include "list/list.ma".
-include "cprop_connectives.ma".
-
-
-notation "\rationals" non associative with precedence 99 for @{'q}.
-interpretation "Q" 'q = Q.
-
-record q_f : Type ≝ {
- start : ℚ;
- bars: list (ℚ × ℚ) (* base, height *)
-}.
-
-axiom qp : ℚ → ℚ → ℚ.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-
-axiom qm : ℚ → ℚ → ℚ.
-
-interpretation "Q minus" 'minus x y = (qm x y).
-
-axiom qlt : ℚ → ℚ → CProp.
-
-interpretation "Q less than" 'lt x y = (qlt x y).
-
-inductive q_comparison (a,b:ℚ) : CProp ≝
- | q_eq : a = b → q_comparison a b
- | q_lt : a < b → q_comparison a b
- | q_gt : b < a → q_comparison a b.
-
-axiom q_cmp:∀a,b:ℚ.q_comparison a b.
-
-definition qle ≝ λa,b:ℚ.a = b ∨ a < b.
-
-interpretation "Q less or equal than" 'le x y = (qle x y).
-
-notation "'nth'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp l \nbsp d \nbsp i" with precedence 71
-for @{'nth_appl $l $d $i}.
-interpretation "list nth" 'nth = (cic:/matita/list/list/nth.con _).
-interpretation "list nth" 'nth_appl l d i = (cic:/matita/list/list/nth.con _ l d i).
-
-notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-let rec make_list (A:Type) (def:nat→A) (n:nat) on n ≝
- match n with
- [ O ⇒ nil ?
- | S m ⇒ def m :: make_list A def m].
-
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-interpretation "'mk_list'" 'mk_list = (make_list _).
-notation < "'mk_list' \nbsp f \nbsp n"
-with precedence 71 for @{'mk_list_appl $f $n}.
-interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
-
-definition q0 : ℚ × ℚ ≝ 〈OQ,OQ〉.
-notation < "0 \sub \rationals" with precedence 90 for @{'q0}.
-interpretation "q0" 'q0 = q0.
-
-notation < "[ \rationals \sup 2]" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list (Prod Q Q)).
-notation < "[ \rationals \sup 2] \sup 2" with precedence 90 for @{'lq22}.
-interpretation "lq22" 'lq22 = (Prod (list (Prod Q Q)) (list (Prod Q Q))).
-
-
-notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = length.
-notation < "'len' \nbsp l" with precedence 70 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
-
-definition eject ≝
- λP.λp:∃x:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).P x.match p with [ex_introT p _ ⇒ p].
-coercion cic:/matita/dama/models/q_function/eject.con.
-definition inject ≝
- λP.λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).λh:P p. ex_introT ? P p h.
-(*coercion inject with 0 1 nocomposites.*)
-coercion cic:/matita/dama/models/q_function/inject.con 0 1 nocomposites.
-
-definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l q0 i),OQ〉) (length ? l)).