#a; #b; #c; //; nqed.
naxiom Qle_refl: ∀q1. q1≤q1.
naxiom Qle_trans: ∀x,y,z. x≤y → y≤z → x≤z.
+naxiom Qlt_trans: ∀x,y,z. x < y → y < z → x < z.
+naxiom Qle_lt_trans1: ∀x,y,z. x ≤ y → y < z → x < z.
+naxiom Qle_lt_trans2: ∀x,y,z. x < y → y ≤ z → x < z.
naxiom Qle_plus_compat: ∀x,y,z,t. x≤y → z≤t → x+z ≤ y+t.
naxiom Qmult_zero: ∀q:Q. 0 * q = 0.
nrecord pre_order (A: Type[0]) : Type[1] ≝
{ pre_r :2> A → A → CProp[0];
- pre_sym: reflexive … pre_r;
+ pre_refl: reflexive … pre_r;
pre_trans: transitive … pre_r
}.
#A; #F; #a; #H; ncases H; /2/.
nqed.
+alias symbol "I" (instance 7) = "I".
+alias symbol "I" (instance 18) = "I".
+alias symbol "I" (instance 18) = "I".
+alias symbol "I" (instance 18) = "I".
+nlet corec ftfish_coind
+ (A: Ax_pro) (F: Ω^A) (P: A → CProp[0])
+ (Hcorefl: ∀a. P a → a ∈ F)
+ (Hcoleqleft: ∀a. P a → ∀b. a ≤ b → P b)
+ (Hcoleqinfinity: ∀a. P a → ∀b. a ≤ b → ∀i:𝐈 b. ∃x. x ∈ 𝐂 b i ↓ (singleton … a) ∧ P x)
+: ∀a:A. P a → a ⋉ F ≝ ?.
+ #a; #H; @
+ [ /2/
+ | #b; #H; napply (ftfish_coind … Hcorefl Hcoleqleft Hcoleqinfinity); /2/
+ | #b; #H1; #i; ncases (Hcoleqinfinity a H ? H1 i); #x; *; #H2; #H3;
+ @ x; @; //; napply (ftfish_coind … Hcorefl Hcoleqleft Hcoleqinfinity); //]
+nqed.
+
(*CSC: non serve manco questo (vedi sotto) *)
nlemma auto_hint3: ∀A. S__o__AAx A = S (AAx A).
#A; //.
{ pt_set: Ω^A;
pt_inhabited: ∃a. a ∈ pt_set;
pt_filtering: ∀a,b. a ∈ pt_set → b ∈ pt_set → ∃c. c ∈ (singleton … a) ↓ (singleton … b) → c ∈ pt_set;
- pt_closed: {b | b ⋉ pt_set} ⊆ pt_set
+ pt_closed: pt_set ⊆ {b | b ⋉ pt_set}
}.
ndefinition Rd ≝ Pt Rax.
[ napply { c | fst … c < q ∧ q < snd … c }
| @ [ @ (Qminus q 1) (Qplus q 1) | ncases daemon ]
##| #c; #d; #Hc; #Hd; @ [ @ (Qmin (fst … c) (fst … d)) (Qmax (snd … c) (snd … d)) | ncases daemon]
-##| #a; #H; nlapply (ftcoreflexivity … H); /2/ ]
+##| #a; #H; napply (ftfish_coind Rax ? (λa. fst … a < q ∧ q < snd … a)); /2/
+ [ /5/ | #b; *; #H1; #H2; #c; *; #H3; #H4; #i; ncases i
+ [ #w; nnormalize;
+ ##| nnormalize;
+ ]
nqed.