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# Untitled

a guest Jul 8th, 2017 184 Never
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1. Require Import ZArith.
2.
3. Module Type Zahl.
4.   Parameter Zahl : Type.
5.   Parameter plus : Zahl -> Zahl -> Zahl.
6.   Parameter zerop : Zahl -> bool.
7. End Zahl.
8.
9. Module ZDirect <: Zahl.
10.   Definition Zahl := Z.
11.   Definition plus := Zplus.
12.   Definition zerop := fun z => if Z_zerop z then true else false.
13. End ZDirect.
14.
15. Module ZQuotient <: Zahl.
16.   Definition Zahl := (nat * nat)%type.
17.   Definition plus x y :=
18.     match x, y with (a,b),(c,d) =>
19.       (a + c,b + d)
20.     end.
21.   Definition zerop x :=
22.     match x with (a,b) =>
23.       if eq_nat_dec a b
24.         then true
25.         else false
26.     end.
27. End ZQuotient.
28.
29. Module MyModule (Integer : Zahl).
30.   Import Integer.
31.   Definition funky (a : Zahl) :=
32.     if zerop (plus a a)
33.       then a
34.       else plus a (plus a a).
35. End MyModule.
36.
37. Module Z1 := MyModule ZDirect.
38. Module Z2 := MyModule ZQuotient.
39.
40. Check (Z1.funky).
41.
42.
43. Axiom R : (nat*nat)%type -> (nat*nat)%type -> Prop.
44. Axiom ZD_to_ZQ : Z -> (nat*nat)%type.
45. Axiom ZQ_to_ZD : (nat*nat)%type -> Z.
46. Axiom isomorphism1 : forall z,
47.   ZQ_to_ZD (ZD_to_ZQ z) = z.
48. Axiom isomorphism2 : forall z,
49.   R (ZD_to_ZQ (ZQ_to_ZD z)) z.
50. Axiom isomorphism_respectful : forall z z', R z z' -> ZQ_to_ZD z = ZQ_to_ZD z'.
51.
52. Theorem funky_respectful : forall a b : (nat*nat)%type, R a b -> R (Z2.funky a) (Z2.funky b).
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