1 include "sets/sets.ma".
3 nrecord Ax : Type[1] ≝ {
4 S:> setoid; (* Type[0]; *)
9 notation "𝐈 \sub( ❨a❩ )" non associative with precedence 70 for @{ 'I $a }.
10 notation "𝐂 \sub ( ❨a,\emsp i❩ )" non associative with precedence 70 for @{ 'C $a $i }.
12 notation > "𝐈 term 90 a" non associative with precedence 70 for @{ 'I $a }.
13 notation > "𝐂 term 90 a term 90 i" non associative with precedence 70 for @{ 'C $a $i }.
15 interpretation "I" 'I a = (I ? a).
16 interpretation "C" 'C a i = (C ? a i).
18 ndefinition cover_set ≝ λc:∀A:Ax.Ω^A → A → CProp[0].λA,C,U.
22 notation "hvbox(a break ◃ b)" non associative with precedence 45
23 for @{ 'covers $a $b }.
25 interpretation "covers set temp" 'covers C U = (cover_set ?? C U).
27 ninductive cover (A : Ax) (U : Ω^A) : A → CProp[0] ≝
28 | creflexivity : ∀a:A. a ∈ U → cover ? U a
29 | cinfinity : ∀a:A.∀i:𝐈 a. 𝐂 a i ◃ U → cover ? U a.
33 interpretation "covers" 'covers a U = (cover ? U a).
34 interpretation "covers set" 'covers a U = (cover_set cover ? a U).
36 ndefinition fish_set ≝ λf:∀A:Ax.Ω^A → A → CProp[0].
41 notation "hvbox(a break ⋉ b)" non associative with precedence 45
44 interpretation "fish set temp" 'fish A U = (fish_set ?? U A).
46 ncoinductive fish (A : Ax) (F : Ω^A) : A → CProp[0] ≝
47 | cfish : ∀a. a ∈ F → (∀i:𝐈 a .𝐂 a i ⋉ F) → fish A F a.
51 interpretation "fish set" 'fish A U = (fish_set fish ? U A).
52 interpretation "fish" 'fish a U = (fish ? U a).
54 nlet corec fish_rec (A:Ax) (U: Ω^A)
56 (H2: ∀a:A. a ∈ P → ∀j: 𝐈 a. 𝐂 a j ≬ P):
57 ∀a:A. ∀p: a ∈ P. a ⋉ U ≝ ?.
59 ##[ napply H1; napply p;
60 ##| #i; ncases (H2 a p i); #x; *; #xC; #xP; @; ##[napply x]
61 @; ##[ napply xC ] napply (fish_rec ? U P); nassumption;
65 notation "◃U" non associative with precedence 55
66 for @{ 'coverage $U }.
68 ndefinition coverage : ∀A:Ax.∀U:Ω^A.Ω^A ≝ λA,U.{ a | a ◃ U }.
70 interpretation "coverage cover" 'coverage U = (coverage ? U).
72 ndefinition cover_equation : ∀A:Ax.∀U,X:Ω^A.CProp[0] ≝ λA,U,X.
73 ∀a.a ∈ X ↔ (a ∈ U ∨ ∃i:𝐈 a.∀y.y ∈ 𝐂 a i → y ∈ X).
75 ntheorem coverage_cover_equation : ∀A,U. cover_equation A U (◃U).
77 ##[ nelim H; #b; (* manca clear *)
78 ##[ #bU; @1; nassumption;
79 ##| #i; #CaiU; #IH; @2; @ i; #c; #cCbi; ncases (IH ? cCbi);
81 ##| #_; napply CaiU; nassumption; ##] ##]
82 ##| ncases H; ##[ #E; @; nassumption]
83 *; #j; #Hj; @2 j; #w; #wC; napply Hj; nassumption;
87 ntheorem coverage_min_cover_equation :
88 ∀A,U,W. cover_equation A U W → ◃U ⊆ W.
89 #A; #U; #W; #H; #a; #aU; nelim aU; #b;
90 ##[ #bU; ncases (H b); #_; #H1; napply H1; @1; nassumption;
91 ##| #i; #CbiU; #IH; ncases (H b); #_; #H1; napply H1; @2; @i; napply IH;
95 notation "⋉F" non associative with precedence 55
98 ndefinition fished : ∀A:Ax.∀F:Ω^A.Ω^A ≝ λA,F.{ a | a ⋉ F }.
100 interpretation "fished fish" 'fished F = (fished ? F).
102 ndefinition fish_equation : ∀A:Ax.∀F,X:Ω^A.CProp[0] ≝ λA,F,X.
103 ∀a. a ∈ X ↔ a ∈ F ∧ ∀i:𝐈 a.∃y.y ∈ 𝐂 a i ∧ y ∈ X.
105 ntheorem fised_fish_equation : ∀A,F. fish_equation A F (⋉F).
106 #A; #F; #a; @; (* bug, fare case sotto diverso da farlo sopra *) #H; ncases H;
107 ##[ #b; #bF; #H2; @ bF; #i; ncases (H2 i); #c; *; #cC; #cF; @c; @ cC;
109 ##| #aF; #H1; @ aF; napply H1;
113 ntheorem fised_max_fish_equation : ∀A,F,G. fish_equation A F G → G ⊆ ⋉F.
114 #A; #F; #G; #H; #a; #aG; napply (fish_rec … aG);
115 #b; ncases (H b); #H1; #_; #bG; ncases (H1 bG); #E1; #E2; nassumption;
118 nrecord nAx : Type[2] ≝ {
119 nS:> setoid; (*Type[0];*)
121 nD: ∀a:nS. nI a → Type[0];
122 nd: ∀a:nS. ∀i:nI a. nD a i → nS
126 TYPE f A → B, g : B → A, f ∘ g = id, g ∘ g = id.
131 notation "𝐃 \sub ( ❨a,\emsp i❩ )" non associative with precedence 70 for @{ 'D $a $i }.
132 notation "𝐝 \sub ( ❨a,\emsp i,\emsp j❩ )" non associative with precedence 70 for @{ 'd $a $i $j}.
134 notation > "𝐃 term 90 a term 90 i" non associative with precedence 70 for @{ 'D $a $i }.
135 notation > "𝐝 term 90 a term 90 i term 90 j" non associative with precedence 70 for @{ 'd $a $i $j}.
137 interpretation "D" 'D a i = (nD ? a i).
138 interpretation "d" 'd a i j = (nd ? a i j).
139 interpretation "new I" 'I a = (nI ? a).
141 ndefinition image ≝ λA:nAx.λa:A.λi. { x | ∃j:𝐃 a i. x = 𝐝 a i j }.
143 notation > "𝐈𝐦 [𝐝 term 90 a term 90 i]" non associative with precedence 70 for @{ 'Im $a $i }.
144 notation "𝐈𝐦 [𝐝 \sub ( ❨a,\emsp i❩ )]" non associative with precedence 70 for @{ 'Im $a $i }.
146 interpretation "image" 'Im a i = (image ? a i).
148 ndefinition Ax_of_nAx : nAx → Ax.
149 #A; @ A (nI ?); #a; #i; napply (𝐈𝐦 [𝐝 a i]);
152 ninductive sigma (A : Type[0]) (P : A → CProp[0]) : Type[0] ≝
153 sig_intro : ∀x:A.P x → sigma A P.
155 interpretation "sigma" 'sigma \eta.p = (sigma ? p).
157 ndefinition nAx_of_Ax : Ax → nAx.
159 ##[ #a; #i; napply (Σx:A.x ∈ 𝐂 a i);
160 ##| #a; #i; *; #x; #_; napply x;
164 ninductive Ord (A : nAx) : Type[0] ≝
167 | oL : ∀a:A.∀i.∀f:𝐃 a i → Ord A. Ord A.
169 notation "Λ term 90 f" non associative with precedence 50 for @{ 'oL $f }.
170 notation "x+1" non associative with precedence 50 for @{'oS $x }.
172 interpretation "ordinals Lambda" 'oL f = (oL ? ? ? f).
173 interpretation "ordinals Succ" 'oS x = (oS ? x).
175 nlet rec famU (A : nAx) (U : Ω^A) (x : Ord A) on x : Ω^A ≝
178 | oS y ⇒ let Un ≝ famU A U y in Un ∪ { x | ∃i.𝐈𝐦[𝐝 x i] ⊆ Un}
179 | oL a i f ⇒ { x | ∃j.x ∈ famU A U (f j) } ].
181 notation < "term 90 U \sub (term 90 x)" non associative with precedence 50 for @{ 'famU $U $x }.
182 notation > "U ⎽ term 90 x" non associative with precedence 50 for @{ 'famU $U $x }.
184 interpretation "famU" 'famU U x = (famU ? U x).
186 ndefinition ord_coverage : ∀A:nAx.∀U:Ω^A.Ω^A ≝ λA,U.{ y | ∃x:Ord A. y ∈ famU ? U x }.
188 ndefinition ord_cover_set ≝ λc:∀A:nAx.Ω^A → Ω^A.λA,C,U.
189 ∀y.y ∈ C → y ∈ c A U.
191 interpretation "coverage new cover" 'coverage U = (ord_coverage ? U).
192 interpretation "new covers set" 'covers a U = (ord_cover_set ord_coverage ? a U).
193 interpretation "new covers" 'covers a U = (mem ? (ord_coverage ? U) a).
195 ntheorem new_coverage_reflexive:
196 ∀A:nAx.∀U:Ω^A.∀a. a ∈ U → a ◃ U.
197 #A; #U; #a; #H; @ (oO A); napply H;
201 ∀A:nAx.∀a:A.∀i,f,U.∀j:𝐃 a i.U⎽(f j) ⊆ U⎽(Λ f).
202 #A; #a; #i; #f; #U; #j; #b; #bUf; @ j; nassumption;
205 naxiom AC : ∀A,a,i,U.(∀j:𝐃 a i.∃x:Ord A.𝐝 a i j ∈ U⎽x) → (Σf.∀j:𝐃 a i.𝐝 a i j ∈ U⎽(f j)).
207 naxiom setoidification :
208 ∀A:nAx.∀a,b:A.∀U.a=b → b ∈ U → a ∈ U.
210 alias symbol "covers" = "new covers".
211 alias symbol "covers" = "new covers set".
212 alias symbol "covers" = "new covers".
213 ntheorem new_coverage_infinity:
214 ∀A:nAx.∀U:Ω^A.∀a:A. (∃i:𝐈 a. 𝐈𝐦[𝐝 a i] ◃ U) → a ◃ U.
215 #A; #U; #a; *; #i; #H; nnormalize in H;
216 ncut (∀y:𝐃 a i.∃x:Ord A.𝐝 a i y ∈ U⎽x); ##[
217 #y; napply H; @ y; napply #; ##] #H';
218 ncases (AC … H'); #f; #Hf;
219 ncut (∀j.𝐝 a i j ∈ U⎽(Λ f));
220 ##[ #j; napply (ord_subset … f … (Hf j));##] #Hf';
221 @ ((Λ f)+1); @2; nwhd; @i; #x; *; #d; #Hd;
222 napply (setoidification … Hd); napply Hf';
226 nlemma subseteq_union: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
227 #A; #U; #V; #W; #H; #H1; #x; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
230 nlemma new_coverage_min :
231 ∀A:nAx.∀U:qpowerclass A.∀V.U ⊆ V → (∀a:A.∀i.𝐈𝐦[𝐝 a i] ⊆ V → a ∈ V) → ◃(pc ? U) ⊆ V.
232 #A; #U; #V; #HUV; #Im; #b; *; #o; ngeneralize in match b; nchange with ((pc ? U)⎽o ⊆ V);
234 ##[ #b; #bU0; napply HUV; napply bU0;
235 ##| #p; #IH; napply subseteq_union; ##[ nassumption; ##]
236 #x; *; #i; #H; napply (Im ? i); napply (subseteq_trans … IH); napply H;
237 ##| #a; #i; #f; #IH; #x; *; #d; napply IH; ##]
240 nlet rec famF (A: nAx) (F : Ω^A) (x : Ord A) on x : Ω^A ≝
243 | oS o ⇒ let Fo ≝ famF A F o in Fo ∩ { x | ∀i:𝐈 x.∃j:𝐃 x i.𝐝 x i j ∈ Fo }
244 | oL a i f ⇒ { x | ∀j:𝐃 a i.x ∈ famF A F (f j) }
247 interpretation "famF" 'famU U x = (famF ? U x).
249 ndefinition ord_fished : ∀A:nAx.∀F:Ω^A.Ω^A ≝ λA,F.{ y | ∀x:Ord A. y ∈ F⎽x }.
251 interpretation "fished new fish" 'fished U = (ord_fished ? U).
252 interpretation "new fish" 'fish a U = (mem ? (ord_fished ? U) a).
254 ntheorem new_fish_antirefl:
255 ∀A:nAx.∀F:Ω^A.∀a. a ⋉ F → a ∈ F.
256 #A; #F; #a; #H; nlapply (H (oO ?)); #aFo; napply aFo;
259 nlemma co_ord_subset:
260 ∀A:nAx.∀F:Ω^A.∀a,i.∀f:𝐃 a i → Ord A.∀j. F⎽(Λ f) ⊆ F⎽(f j).
261 #A; #F; #a; #i; #f; #j; #x; #H; napply H;
265 ∀A:nAx.∀a:A.∀i,F. (∀f:𝐃 a i → Ord A.∃x:𝐃 a i.𝐝 a i x ∈ F⎽(f x)) → ∃j:𝐃 a i.∀x:Ord A.𝐝 a i j ∈ F⎽x.
268 ntheorem new_fish_compatible:
269 ∀A:nAx.∀F:Ω^A.∀a. a ⋉ F → ∀i:𝐈 a.∃j:𝐃 a i.𝐝 a i j ⋉ F.
270 #A; #F; #a; #aF; #i; nnormalize;
272 nlapply (aF (Λf+1)); #aLf;
273 nchange in aLf with (a ∈ F⎽(Λ f) ∧ ∀i:𝐈 a.∃j:𝐃 a i.𝐝 a i j ∈ F⎽(Λ f));
274 ncut (∃j:𝐃 a i.𝐝 a i j ∈ F⎽(f j));##[
275 ncases aLf; #_; #H; nlapply (H i); *; #j; #Hj; @j; napply Hj;##] #aLf';
280 nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
281 #A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
284 nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
285 #A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
288 ntheorem max_new_fished:
289 ∀A:nAx.∀G,F:Ω^A.G ⊆ F → (∀a.a ∈ G → ∀i.𝐈𝐦[𝐝 a i] ≬ G) → G ⊆ ⋉F.
290 #A; #G; #F; #GF; #H; #b; #HbG; #o; ngeneralize in match HbG; ngeneralize in match b;
291 nchange with (G ⊆ F⎽o);
294 ##| #p; #IH; napply (subseteq_intersection_r … IH);
295 #x; #Hx; #i; ncases (H … Hx i); #c; *; *; #d; #Ed; #cG;
296 @d; napply IH; napply (setoidification … Ed^-1); napply cG;
297 ##| #a; #i; #f; #Hf; nchange with (G ⊆ { y | ∀x. y ∈ F⎽(f x) });
298 #b; #Hb; #d; napply (Hf d); napply Hb;