Building Theories in Nuprl

Concl t THEN Lemma `fs unroll`THENM ReduceConcl ;;* THM fs add card>> 8A:D. 8s:FS(A). 8x:|A|. :(x fAg s) => |s +fAg x| = |s|+1 in Int* THM fs del exists>> 8A:D. 8s1:FS(A). 8x:|A|. 9s2:FS(A).(8y:|A|. y fAg s2 <=> y fAg s1 & :(y=x in |A|))& (x fAg s1 => |s2| = |s1|-1)* THM fs del>> A:D -> FS(A) -> |A| -> FS(A)Extraction:A s x. fs del exists(A,s,x).1* DEF fs del -fg ==fs del(,,)* THM fs del char>> 8A:D. 8x:|A|. 8s:FS(A). 8y:|A|. y fAg (s -fAg x) <=> :(x=y in |A|) & y fAg s* THM fs del card>> 8A:D. 8s:FS(A). 8x:|A|. x fAg s => |s -fAg x| = |s|-1 in Int* THM subset card>>8A:D. 8s1,s2:FS(A). s1 fAg s2 => |s1| |s2|* THM card fnl>> 8A:D. 8s1,s2:FS(A). s1 =fAg s2 => |s1| = |s2|* THM fs union exists>> 8A:D. 8r1,r2:FS(A). 9r:FS(A). 8x:|A|.x fAg r <=> x fAg r1 _ x fAg r2* THM fs union>> A:D -> FS(A) -> FS(A) -> FS(A)Extraction:A s1 s2. fs union exists(A,s1,s2).1* DEF fs union [fg ==fs union(,,)* THM fs union char>> 8A:D. 8s1,s2:FS(A). 8x:|A|.x fAg s1 [fAg s2 <=> x fAg s1 _ x fAg s2* THM eq union char>>8A:D. 8r1,r2,s:FS(A). r1 [fAg r2 =fAg s => 8x:|A|. x fAg r1 _ x fAg r2 => x fAg s* THM disjoint>> A:D -> FS(A) -> FS(A) -> U1Extraction:A s1 s2. :(9x:|A|. x fAg s1 & x fAg s2)26 * DEF disjointdisjfg(,)==disjoint(,,)* THM plus union commute>>8A:D. 8s1,s2:FS(A). 8x:|A|. s1 +fAg x [fAg s2 =fAg s1 [fAg s2 +fAg x* THM union card>> 8A:D. 8s1,s2:FS(A). disjfAg(s1,s2) => |s1 [fAg s2| = |s1| + |s2|* THM pick>> 8A:D. 8s:FS(A). 0<|s| => 9x:|A|. x fAg s* THM small card subset>>8A:D. 8n:N. 8s:FS(A). n |s| => 9r:FS(A). r fAg s & |r| = n* THM members>> A:D -> FS(A) -> U1Extraction:A s. f x:|A| | x fAg s g* DEF memberseltsfg()==members(,)* THM doubleton>>A:D->|A|->|A|->FS(A)Extraction:A x y. fg +fAg x +fAg y* DEF doubletonf,:g==doubleton(,,)* THM doubleton char>>8A:D. 8x,y,z:|A|. z fAg fx,y:Ag <=> z = x in |A| _ z = y in |A|* THM doubleton card>>8A:D. 8x,y:|A|. :(x = y in |A|) => |fx,y:Ag| = 2* DEF elt eq =fg ==(= in )* THM fs comp>>8A:D. 8s:FS(A). 8P:eltsfAg(s)->U1. 8x:eltsfAg(s). P(x) _ :(P(x)) => 9r1,r2:FS(A). 8x:eltsfAg(s).x fAg r1 <=> x fAg s & P(x) & x fAg r2 <=> x fAg s & :(P(x)) & disjfAg(r1,r2) & r1 [fAg r2=fAg s* THM Graph>> U2Extraction:A:D # V:FS(A) # E: (eltsfAg(V)->eltsfAg(V)->U1) #8x,y:eltsfAg(V). E(x,y) _ :(E(x,y)) &8x,y:eltsfAg(V). E(x,y) <=> E(y,x) & 8x:eltsfAg(V). :(E(x,x))* DEF GraphGraph==Graph* DEF Graph carrier27 ||==Graph carrier()* THM Graph carrier>>G:Graph->DExtraction:G. G.1* THM vertex>>G:Graph->FS(|G|)Extraction:G. G.2.1* DEF vertexV()==vertex(G)* THM edges>>G:Graph->eltsf|G|g(V(G))->eltsf|G|g(V(G))->U1Extraction:G. G.2.2.1* DEF edgesE()==edges()* THM ramfunc>>N+->f2..g->f2..g->U2Extraction:n l1 l2. 8G:Graph. n |V(G)| => 9s:FS(|G|). s f|G|g V(G) & |s| = l1 & 8x,y f|G|g s.:(x =f||G||g y) => E(G)(x,y) _ |s| = l2 & 8x,y f|G|g s. :(E(G)(x,y))* DEF ramfunc! (,)==ramfunc()()()* ML GraphElet GraphE newids i =GenProductE newids i THENUnfoldsInConcl ``Graph carrier vertex edges`` THEN ReduceConcl THEN Thin i ;;* THM fset prop>>8A:D. 8s:FS(A). 8P:eltsfAg(s)->U1. 8x fAg s. P(x) _ :(P(x)) => 8x fAg s. P(x) _ 9x fAgs. :(P(x))* THM fset2 prop>>8A:D. 8s:FS(A). 8P:eltsfAg(s)->eltsfAg(s)->U1. 8x,y fAg s. P(x)(y) _ :(P(x)(y)) => 8x,yfAg s. P(x)(y) _ 9x,y fAg s. :(P(x)(y))* ML DecideCaseslet MakeDecideCases v1 v2 n p =( Decide (make equal term (get type p v1) [v1;v2]) THENInstHyp [v1;v2] n THENM OnLastHyp E) p ;;let SolveDecideCases = %Decide Trivial Cases %Complete (ILeft THENM ILeft...) ORELSEComplete (ILeft THENM IRight ...) ORELSEComplete (IRight THENM ILeft...) ORELSEComplete (IRight THEN IRight...) ORELSE 28 % Double Negation Case %Complete (IRight THENM I THENM OnLastHyp E THENMBackchain...) ;;let DecideCases v1 v2 n =MakeDecideCases v1 v2 n THENM Try SolveDecideCases ;;* THM ramsey>>8l1,l2:f2..g. 9n:N+. n! (l1,l2)* EVAL demo% list procedures %let hd = l. [ nil ! "?"; h.t,v ! h; @ l] ;;let tl = l. [ nil ! nil; h.t,v ! t; @ l] ;;let nthtl = n l.x. [ 0 ! x ; i, v ! tl(v) ; @ n](l);;let member = x l tt ff. % i x in l return tt else ff %[ nil ! ff; h.t,v ! if x=h then tt else v; @ l] ;;let lookup = x y G tt ff. % lookup in adjacency list %l. member(y)(l)(tt)(ff) (hd (nthtl(x-1,G))) ;;let make list = n. [ 0 ! (nil) ; i, v ! (i.v) ; @ n] ;;let length = l. [ nil ! 0; h.t,v ! 1+v; @ l] ;;% Graph Making Procedures. Make an integer graph froman adjacency list. Recall a graph is a 4-tuple:Discrete Type # Vertex Set # Edge Relation #Edge Decision Procdure & Other Edge Properties %let make graph = l.<,< x y. lookup(x)(y)(l)(int)(void),< x y. lookup(x)(y)(l)(inl(0))(inr( z.axiom)),< x y. < z.0, z.0>, x . z.0>>>>> ;;let ram = term of(ramsey) ;;let ram n = l1 l2. ram(l1)(l2).1 ;;let ram clique = l1 l2 g. ram(l1)(l2).2(g)(axiom).1 ;;let ram decide = l1 l2 g.decide(ram(l1)(l2).2(g)(axiom).2.2; u."Clique";u."Independent Set") ;;29 Appendix 2 | Ramsey's Theorem>> 8l1,l2:f2..g. 9n:N+. n! (l1,l2)| BY (OnVar `l1` (NonNegInd `j1`)...*)| 1. j1:int| 2. j1 = 2|>> 8l2:f2..g. 9n:N+. n! (j1,l2)| | BY (I THENM ITerm 'l2'...)| | 3. l2:f2..g| | 4. G:Graph| | 5. l2 |V(G)|| |>> 9s:FS(|G|). s f|G|g V(G) & |s| = j1 & 8x,y f|G|g s. ((:(x=y in ||G||))->E(G)(x,y))_ |s| = l2 & 8x,y f|G|g s. (:(E(G)(x,y)))| | | BY (BringHyp 5 THEN GraphE ``A V E`` 4...)| | | 4. A:D| | | 5. V:FS(A)| | | 6. E:eltsfAg(V)->eltsfAg(V)->U1| | | 7. x5:8x,y:eltsfAg(V). E(x,y) _ :(E(x,y))| | | 8. x7:8x,y:eltsfAg(V). E(x,y) <=> E(y,x)| | | 9. x8:8x:eltsfAg(V). :(E(x,x))| | | 10. l2 |V|| | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s| = l2& 8x,y fAg s. (:(E(x,y)))| | | | BY (Cases ['8x,y fAg V. z w. z = w in |A| _ :(E(z,w))(x)(y)';'9x,y fAg V. :( zw. z = w in |A| _ :(E(z,w))(x)(y))']...)| | | |>> 8x,y fAg V. z w. z=w in |A| _ :(E(z,w))(x,y) _ 9x,y fAg V. :( z w. z=w in |A|_ :(E(z,w))(x,y))| | | | | BY (Lemma `fset2 prop` THEN ReduceConcl...) THEN (DecideCases 'x' 'y' 7...)| | | | 11. 8x,y fAg V. z w. z=w in |A| _ :(E(z,w))(x,y)| | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s| =l2 & 8x,y fAg s. (:(E(x,y)))| | | | | BY (InstLemma `small card subset` ['A';'l2';'V']...) THEN OnLastHyp EThin| | | | | 12. a14:FS(A)| | | | | 13. a14 fAg V & |a14| = l2| | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s|= l2 & 8x,y fAg s. (:(E(x,y)))| | | | | | BY (ITerm 'a14'...) THEN (IRight...)| | | | | | 13. a14 fAg V| | | | | | 14. |a14| = l2| | | | | | 15. a14 fAg V| | | | | | 16. |a14| = l2| | | | | | 17. x:|A|| | | | | | 18. y:|A|| | | | | | 19. x fAg a14| | | | | | 20. y fAg a14| | | | | | 21. E(x,y)| | | | | |>> void| | | | | | | BY (InstHyp ['x';'y'] 11 THENS OnLastHyp E THEN Try Backchain...)| | | | | | | 22. x=y in |A| _ :(E(x,y))| | | | | | | 23. x=y in |A|| | | | | | |>> void| | | | | | | | BY (SubstForInHyp 'y=x in eltsfAg(V)' 21!)| | | | | | | | THEN OnLastHyp i. BringHyp i THEN (Backchain...)| | | | 11. 9x,y fAg V. :( z w. z=w in |A| _ :(E(z,w))(x,y))30 | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s| =l2 & 8x,y fAg s. (:(E(x,y)))| | | | | BY (RepeatProductE ``x y`` 11 THENM ITerm 'fx,y:Ag'...) THEN| | | | | ((OnLastHyp i. HypChar `doubleton char` i THENS OnLastHyp E !!)| | | | |ORELSE (ILeft...))| | | | | 11. x:|A|| | | | | 12. x fAg V| | | | | 13. y:|A|| | | | | 14. y fAg V| | | | | 15. :( z w. z=w in |A| _ :(E(z,w))(x,y))| | | | | 16. fx,y:Ag fAg V| | | | |>> |fx,y:Ag| = j1| | | | | | BY (SubstFor (h 2 p)...) THEN (Lemma `doubleton card`...)| | | | | | THEN (E 15 THENM ReduceConcl...) THEN (ILeft...)| | | | | 11. x:|A|| | | | | 12. x fAg V| | | | | 13. y:|A|| | | | | 14. y fAg V| | | | | 15. :( z w. z=w in |A| _ :(E(z,w))(x,y))| | | | | 16. fx,y:Ag fAg V| | | | | 17. |fx,y:Ag| = j1| | | | | 18. x@0:|A|| | | | | 19. y@0:|A|| | | | | 20. x@0 fAg fx,y:Ag| | | | | 21. y@0 fAg fx,y:Ag| | | | | 22. :(x@0=y@0 in |A|)| | | | |>> E(x@0,y@0)| | | | | | BY (Assert 'E(x,y)' THENM| | | | | |f. f 21 THEN f 20 ( i. HypChar `doubleton char` i)...)| | | | | | THEN Try (Complete (OnLastHyp E THEN UnfoldInConcl `doubleton` !!))| | | | | | 20. :(x@0=y@0 in |A|)| | | | | | 21. y@0=x in |A| _ y@0=y in |A|| | | | | | 22. x@0=x in |A| _ x@0=y in |A|| | | | | |>> E(x,y)| | | | | | | BY (Decide 'x=y in |A|' THEN InstHyp ['x';'y'] 7 THENM OnLastHyp E...)| | | | | | | THEN (E 15 THEN ReduceConcl...) THEN (IRight...)| | | | | | 20. :(x@0=y@0 in |A|)| | | | | | 21. E(x,y)| | | | | | 22. y@0=x in |A| _ y@0=y in |A|| | | | | | 23. x@0=x in |A| _ x@0=y in |A|| | | | | |>> E(x@0,y@0)| | | | | | | BY (E 22 THEN E 23...) THEN Try (Complete (E 20...))| | | | | | | 24. y@0=x in |A|| | | | | | | 25. x@0=y in |A|| | | | | | |>> E(x@0,y@0)| | | | | | | | BY (SubstForInHyp 'x=y@0 in eltsfAg(V)' 21 !) THEN| | | | | | | | (OnLastHyp (SubstForInHyp 'y=x@0 in eltsfAg(V)') !!)| | | | | | | | THEN (BackThruHyp 8!)| | | | | | | 24. y@0=y in |A|| | | | | | | 25. x@0=x in |A|| | | | | | |>> E(x@0,y@0)| | | | | | | | BY (SubstForInHyp 'x=x@0 in eltsfAg(V)' 21 !) THEN| | | | | | | | (OnLastHyp (SubstForInHyp 'y=y@0 in eltsfAg(V)') !!)| 1. j1:int| 2. 231 | 3. 8l2:f2..g. 9n:N+. n! (j1-1,l2)|>> 8l2:f2..g. 9n:N+. n! (j1,l2)| | BY (OnVar `l2` (NonNegInd `j2`)...)| | 4. j2:int| | 5. j2 = 2| |>> 9n:N+. n! (j1,j2)| | | BY (ITerm 'j1'...)| | | 6. G:Graph| | | 7. j1 |V(G)|| | |>> 9s:FS(|G|). s f|G|g V(G) & |s| = j1 & 8x,y f|G|g s. ((:(x=y in ||G||))->E(G)(x,y))_ |s| = j2 & 8x,y f|G|g s. (:(E(G)(x,y)))| | | | BY (BringHyp 7 THEN GraphE ``A V E`` 6...)| | | | 6. A:D| | | | 7. V:FS(A)| | | | 8. E:eltsfAg(V)->eltsfAg(V)->U1| | | | 9. x6:8x,y:eltsfAg(V). E(x,y) _ :(E(x,y))| | | | 10. x8:8x,y:eltsfAg(V). E(x,y) <=> E(y,x)| | | | 11. x9:8x:eltsfAg(V). :(E(x,x))| | | | 12. j1 |V|| | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s| =j2 & 8x,y fAg s. (:(E(x,y)))| | | | | BY (Cases ['8x,y fAg V. z w. z = w in |A| _ E(z,w)(x)(y)';'9x,y fAg V. :( z w.z = w in |A| _ E(z,w)(x)(y))']...)| | | | |>> 8x,y fAg V. z w. z=w in |A| _ E(z,w)(x,y) _ 9x,y fAg V. :( z w. z=w in |A|_ E(z,w)(x,y))| | | | | | BY (Lemma `fset2 prop` THEN ReduceConcl...) THEN (DecideCases 'x' 'y' 9...)| | | | | 13. 8x,y fAg V. z w. z=w in |A| _ E(z,w)(x,y)| | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s|= j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | BY (InstLemma `small card subset` ['A';'j1';'V']...) THEN (OnLastHyp EThin...)| | | | | | 14. a13:FS(A)| | | | | | 15. a13 fAg V| | | | | | 16. |a13| = j1| | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s|= j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | BY (ITerm 'a13' ...) THEN (ILeft...)| | | | | | | 17. a13 fAg V| | | | | | | 18. |a13| = j1| | | | | | | 19. x:|A|| | | | | | | 20. y:|A|| | | | | | | 21. x fAg a13| | | | | | | 22. y fAg a13| | | | | | | 23. :(x=y in |A|)| | | | | | |>> E(x,y)| | | | | | | | BY (InstHyp ['x';'y'] 13 THENS OnLastHyp E THEN Try Backchain...)| | | | | | | | 24. x=y in |A| _ E(x,y)| | | | | | | | 25. x=y in |A|| | | | | | | |>> E(x,y)| | | | | | | | | BY Contradiction| | | | | 13. 9x,y fAg V. :( z w. z=w in |A| _ E(z,w)(x,y))| | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s|= j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | BY (RepeatProductE ``x y`` 13 THENM ITerm 'fx,y:Ag'...)| | | | | | THEN ((OnLastHyp (HypChar `doubleton char`) THENS OnLastHyp E!)| | | | | |ORELSE (IRight...))| | | | | | 13. x:|A|32 | | | | | | 14. x fAg V| | | | | | 15. y:|A|| | | | | | 16. y fAg V| | | | | | 17. :( z w. z=w in |A| _ E(z,w)(x,y))| | | | | | 18. fx,y:Ag fAg V| | | | | |>> |fx,y:Ag| = j2| | | | | | | BY (SubstFor (h 5 p) ...) THEN (Lemma `doubleton card`...)| | | | | | | THEN (E 17 THENM ReduceConcl...) THEN (ILeft...)| | | | | | 13. x:|A|| | | | | | 14. x fAg V| | | | | | 15. y:|A|| | | | | | 16. y fAg V| | | | | | 17. :( z w. z=w in |A| _ E(z,w)(x,y))| | | | | | 18. fx,y:Ag fAg V| | | | | | 19. |fx,y:Ag| = j2| | | | | | 20. x@0:|A|| | | | | | 21. y@0:|A|| | | | | | 22. x@0 fAg fx,y:Ag| | | | | | 23. y@0 fAg fx,y:Ag| | | | | | 24. E(x@0,y@0)| | | | | |>> void| | | | | | | BY ( f. f 23 THEN f 22( i. HypChar `doubleton char` i)...)| | | | | | | 22. E(x@0,y@0)| | | | | | | 23. y@0=x in |A| _ y@0=y in |A|| | | | | | | 24. x@0=x in |A| _ x@0=y in |A|| | | | | | |>> void| | | | | | | | BY (EThin 24 THEN EThin 23 ...)| | | | | | | | 23. x@0=x in |A|| | | | | | | | 24. y@0=x in |A|| | | | | | | |>> void| | | | | | | | | BY (SubstForInHyp 'x@0=x in eltsfAg(V)' 22!) THEN| | | | | | | | | (OnLastHyp (SubstForInHyp 'y@0=x in eltsfAg(V)')!)| | | | | | | | | THEN (EOn 'x' 11!) THEN (Contradiction...)| | | | | | | | 23. x@0=x in |A|| | | | | | | | 24. y@0=y in |A|| | | | | | | |>> void| | | | | | | | | BY (E 17 ...) THEN (IRight...) THEN| | | | | | | | | (SubstFor 'x=x@0 in eltsfAg(V)'!) THEN| | | | | | | | | (SubstFor 'y=y@0 in eltsfAg(V)'!)| | | | | | | | 23. x@0=y in |A|| | | | | | | | 24. y@0=x in |A|| | | | | | | |>> void| | | | | | | | | BY (E 17 ...) THEN (IRight...) THEN| | | | | | | | | (SubstFor 'x=y@0 in eltsfAg(V)'!) THEN| | | | | | | | | (SubstFor 'y=x@0 in eltsfAg(V)'!) THEN| | | | | | | | | (BackThruHyp 10!)| | | | | | | | 23. x@0=y in |A|| | | | | | | | 24. y@0=y in |A|| | | | | | | |>> void| | | | | | | | | BY (SubstForInHyp 'x@0=y in eltsfAg(V)' 22!) THEN| | | | | | | | | (OnLastHyp (SubstForInHyp 'y@0=y in eltsfAg(V)')!)| | | | | | | | | THEN (EOn 'y' 11!) THEN (Contradiction...)| | 4. j2:int| | 5. 2| | 6. 9n:N+. n! (j1,j2-1)33 | |>> 9n:N+. n! (j1,j2)| | | BY (EOnThin 'j2' 3 THEN EThin 5 THEN EThin 5...)| | | 3. j2:int| | | 4. 2| | | 5. a4:N+| | | 6. a4! (j1,j2-1)| | | 7. a5:N+| | | 8. a5! (j1-1,j2)| | |>> 9n:N+. n! (j1,j2)| | | | BY (ITerm 'a4 + a5'...) THEN Try (E 7 THEN E 5 THEN MonoFromHyps `+`...*)| | | | 9. G:Graph| | | | 10. a4+a5 |V(G)|| | | |>> 9s:FS(|G|). s f|G|g V(G) & |s| = j1 & 8x,y f|G|g s. ((:(x=y in ||G||))->E(G)(x,y))_ |s| = j2 & 8x,y f|G|g s. (:(E(G)(x,y)))| | | | | BY (BringHyp 10 THEN GraphE ``A V E`` 9...)| | | | | 9. A:D| | | | | 10. V:FS(A)| | | | | 11. E:eltsfAg(V)->eltsfAg(V)->U1| | | | | 12. x9:8x,y:eltsfAg(V). E(x,y) _ :(E(x,y))| | | | | 13. x11:8x,y:eltsfAg(V). E(x,y) <=> E(y,x)| | | | | 14. x12:8x:eltsfAg(V). :(E(x,x))| | | | | 15. a4+a5 |V|| | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s|= j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | BY InstLemma `pick` ['A';'V'] THEN Try (E 7 THEN E 5 THENS MonoFromHyps `+`...*)| | | | | | THEN OnLastHyp EThin| | | | | | 16. v0:|A|| | | | | | 17. v0 fAg V| | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _ |s|= j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | BY (InstLemma `fs comp` ['A';'V -fAg v0';'E(v0)'] THENM Try Backchain...)| | | | | | | THEN Try (Ext...) THEN (Id!)| | | | | | | 18. 9r1,r2:FS(A). 8x:eltsfAg(V -fAg v0). x fAg r1 <=> x fAg V -fAg v0 & E(v0,x)& x fAg r2 <=> x fAg V -fAg v0 & :(E(v0,x)) & disjfAg(r1,r2) & r1 [fAg r2 =fAg V -fAg v0| | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y)) _|s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | BY (RepeatProductE ``r1 r2`` 18 THENM Assert 'a4+a5-1 |r1|+|r2|'...)| | | | | | | | 18. r1:FS(A)| | | | | | | | 19. r2:FS(A)| | | | | | | | 20. 8x:eltsfAg(V -fAg v0). x fAg r1 <=> x fAg V -fAg v0 & E(v0,x) & x fAg r2<=> x fAg V -fAg v0 & :(E(v0,x))| | | | | | | | 21. disjfAg(r1,r2)| | | | | | | | 22. r1 [fAg r2 =fAg V -fAg v0| | | | | | | | 23. |r1|+|r2|<(a4+a5)-1| | | | | | | |>> void| | | | | | | | | BY (FLemma `union card` [21] THENS FLemma `card fnl` [22]| | | | | | | | | THENS InstLemma `fs del card` ['A';'V';'v0']...)| | | | | | | | 18. r1:FS(A)| | | | | | | | 19. r2:FS(A)| | | | | | | | 20. 8x:eltsfAg(V -fAg v0). x fAg r1 <=> x fAg V -fAg v0 & E(v0,x) & x fAg r2<=> x fAg V -fAg v0 & :(E(v0,x))| | | | | | | | 21. disjfAg(r1,r2)| | | | | | | | 22. r1 [fAg r2 =fAg V -fAg v0| | | | | | | | 23. (a4+a5)-1 |r1|+|r2|| | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))34 | | | | | | | | | BY (Assert 'a5 |r1| _ a4 |r2|' THENS OnLastHyp EThin...)| | | | | | | | | THENO ((Decide '|r1|| | | | | | | | | 23. a5 |r1|| | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | BY (EOnThin '>>>>' 8...)| | | | | | | | | | THEN OnLastHyp (UnfoldsInHyp ``Graph carrier vertex edges``)| | | | | | | | | | THEN Try (Complete (RepeatExt!))| | | | | | | | | | 8. A:D| | | | | | | | | | 9. V:FS(A)| | | | | | | | | | 10. E:eltsfAg(V)->eltsfAg(V)->U1| | | | | | | | | | 11. x9:8x,y:eltsfAg(V). E(x,y) _ :(E(x,y))| | | | | | | | | | 12. x11:8x,y:eltsfAg(V). E(x,y) <=> E(y,x)| | | | | | | | | | 13. x12:8x:eltsfAg(V). :(E(x,x))| | | | | | | | | | 14. a4+a5 |V|| | | | | | | | | | 15. v0:|A|| | | | | | | | | | 16. v0 fAg V| | | | | | | | | | 17. r1:FS(A)| | | | | | | | | | 18. r2:FS(A)| | | | | | | | | | 19. 8x:eltsfAg(V -fAg v0). x fAg r1 <=> x fAg V -fAg v0 & E(v0,x) & x fAgr2 <=> x fAg V -fAg v0 & :(E(v0,x))| | | | | | | | | | 20. disjfAg(r1,r2)| | | | | | | | | | 21. r1 [fAg r2 =fAg V -fAg v0| | | | | | | | | | 22. a5 |r1|| | | | | | | | | | 23. (a5 |r1|)->9s:FS(A). s fAg r1 & |s| = j1-1 & 8x,y fAg s. ((:(x=yin |A|))->E(x,y)) _ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | | BY (EThin 23 ...) THEN Repeat (OnLastHyp EThin)| | | | | | | | | | | 23. a16:FS(A)| | | | | | | | | | | 24. a16 fAg r1| | | | | | | | | | | 25. |a16| = j1-1| | | | | | | | | | | 26. 8x,y fAg a16.((:(x=y in |A|))->E(x,y))| | | | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | | | BY ((ITerm 'a16 +fAg v0'...)!) THEN (ILeft...)| | | | | | | | | | | | 27. a16 +fAg v0 fAg V| | | | | | | | | | | |>> |a16 +fAg v0| = j1| | | | | | | | | | | | | BY (InstLemma `fs add card` ['A';'a16';'v0']...) THEN| | | | | | | | | | | | | (EOnThin 'v0' 24 THENS OnLastHyp EThin...) THEN (EThin 21...)| | | | | | | | | | | | | 21. a5 |r1|| | | | | | | | | | | | | 22. a16:FS(A)| | | | | | | | | | | | | 23. |a16| = j1-1| | | | | | | | | | | | | 24. 8x,y fAg a16. ((:(x=y in |A|))->E(x,y))| | | | | | | | | | | | | 25. a16 +fAg v0 fAg V| | | | | | | | | | | | | 26. 8A:D. 8s:FS(A). 8x:|A|. (:(x fAg s))->(|s +fAg x| = |s|+1)| | | | | | | | | | | | | 27. v0 fAg a16| | | | | | | | | | | | | 28. v0 fAg r1| | | | | | | | | | | | | 29. r1 [fAg r2 fAg V -fAg v0| | | | | | | | | | | | | 30. V -fAg v0 fAg r1 [fAg r2| | | | | | | | | | | | |>> void| | | | | | | | | | | | | | BY (EOnThin 'v0' 29...) THEN (OnLastHyp EThin!)| | | | | | | | | | | | | | THEN (OnLastHyp (HypChar `fs del char`) THENS Contradiction...)| | | | | | | | | | | | 27. a16 +fAg v0 fAg V| | | | | | | | | | | | 28. |a16 +fAg v0| = j135 | | | | | | | | | | | | 29. x:|A|| | | | | | | | | | | | 30. y:|A|| | | | | | | | | | | | 31. x fAg a16 +fAg v0| | | | | | | | | | | | 32. y fAg a16 +fAg v0| | | | | | | | | | | | 33. :(x=y in |A|)| | | | | | | | | | | |>> E(x,y)| | | | | | | | | | | | | BY (Decide 'x=v0 in |A|' THEN Decide 'y=v0 in |A|'...)| | | | | | | | | | | | | 34. x=v0 in |A|| | | | | | | | | | | | | 35. y=v0 in |A|| | | | | | | | | | | | |>> E(x,y)| | | | | | | | | | | | | | BY (E 33...)| | | | | | | | | | | | | 34. x=v0 in |A|| | | | | | | | | | | | | 35. :(y=v0 in |A|)| | | | | | | | | | | | |>> E(x,y)| | | | | | | | | | | | | | BY (SubstFor 'x=v0 in eltsfAg(V)'!) THEN (BackThruHyp 19!)| | | | | | | | | | | | | 34. :(x=v0 in |A|)| | | | | | | | | | | | | 35. y=v0 in |A|| | | | | | | | | | | | |>> E(x,y)| | | | | | | | | | | | | | BY (BackThruHyp 12...) THEN| | | | | | | | | | | | | | (SubstFor 'y=v0 in eltsfAg(V)'!) THEN (BackThruHyp 19!)| | | | | | | | | | | | | 34. :(x=v0 in |A|)| | | | | | | | | | | | | 35. :(y=v0 in |A|)| | | | | | | | | | | | |>> E(x,y)| | | | | | | | | | | | | | BY (BackThruHyp 26!)| | | | | | | | | | | 23. a16:FS(A)| | | | | | | | | | | 24. a16 fAg r1| | | | | | | | | | | 25. |a16| = j2| | | | | | | | | | | 26. 8x,y fAg a16. (:(E(x,y)))| | | | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | | | BY ((ITerm 'a16'...)!) THEN (IRight...)| | | | | | | | | 23. a4 |r2|| | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | BY (EOnThin '>>>>' 6...)| | | | | | | | | | THEN OnLastHyp (UnfoldsInHyp ``Graph carrier vertex edges``)| | | | | | | | | | THEN Try (Complete (RepeatExt!))| | | | | | | | | | 6. a5:N+| | | | | | | | | | 7. a5! (j1-1,j2)| | | | | | | | | | 8. A:D| | | | | | | | | | 9. V:FS(A)| | | | | | | | | | 10. E:eltsfAg(V)->eltsfAg(V)->U1| | | | | | | | | | 11. x9:8x,y:eltsfAg(V). E(x,y) _ :(E(x,y))| | | | | | | | | | 12. x11:8x,y:eltsfAg(V). E(x,y) <=> E(y,x)| | | | | | | | | | 13. x12:8x:eltsfAg(V). :(E(x,x))| | | | | | | | | | 14. a4+a5 |V|| | | | | | | | | | 15. v0:|A|| | | | | | | | | | 16. v0 fAg V| | | | | | | | | | 17. r1:FS(A)| | | | | | | | | | 18. r2:FS(A)| | | | | | | | | | 19. 8x:eltsfAg(V -fAg v0). x fAg r1 <=> x fAg V -fAg v0 & E(v0,x) & x fAgr2 <=> x fAg V -fAg v0 & :(E(v0,x))| | | | | | | | | | 20. disjfAg(r1,r2)| | | | | | | | | | 21. r1 [fAg r2 =fAg V -fAg v0| | | | | | | | | | 22. a4 |r2|36 | | | | | | | | | | 23. (a4 |r2|)->9s:FS(A). s fAg r2 & |s| = j1 & 8x,y fAg s. ((:(x=yin |A|))->E(x,y)) _ |s| = j2-1 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | | BY (EThin 23...) THEN Repeat (OnLastHyp EThin)| | | | | | | | | | | 23. a16:FS(A)| | | | | | | | | | | 24. a16 fAg r2| | | | | | | | | | | 25. |a16| = j1| | | | | | | | | | | 26. 8x,y fAg a16. ((:(x=y in |A|))->E(x,y))| | | | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | | | BY ((ITerm 'a16'...)!) THEN (ILeft...)| | | | | | | | | | | 23. a16:FS(A)| | | | | | | | | | | 24. a16 fAg r2| | | | | | | | | | | 25. |a16| = j2-1| | | | | | | | | | | 26. 8x,y fAg a16. (:(E(x,y)))| | | | | | | | | | |>> 9s:FS(A). s fAg V & |s| = j1 & 8x,y fAg s. ((:(x=y in |A|))->E(x,y))_ |s| = j2 & 8x,y fAg s. (:(E(x,y)))| | | | | | | | | | | | BY ((ITerm 'a16 +fAg v0'...)!) THEN (IRight...)| | | | | | | | | | | | 27. a16 +fAg v0 fAg V| | | | | | | | | | | |>> |a16 +fAg v0| = j2| | | | | | | | | | | | | BY (InstLemma `fs add card` ['A';'a16';'v0']...) THEN| | | | | | | | | | | | | (EOnThin 'v0' 24 THENS OnLastHyp EThin...) THEN| | | | | | | | | | | | | (EThin 21...)| | | | | | | | | | | | | 21. a4 |r2|| | | | | | | | | | | | | 22. a16:FS(A)| | | | | | | | | | | | | 23. |a16| = j2-1| | | | | | | | | | | | | 24. 8x,y fAg a16. (:(E(x,y)))| | | | | | | | | | | | | 25. a16 +fAg v0 fAg V| | | | | | | | | | | | | 26. 8A:D. 8s:FS(A). 8x:|A|. (:(x fAg s))->(|s +fAg x| = |s|+1)| | | | | | | | | | | | | 27. v0 fAg a16| | | | | | | | | | | | | 28. v0 fAg r2| | | | | | | | | | | | | 29. r1 [fAg r2 fAg V -fAg v0| | | | | | | | | | | | | 30. V -fAg v0 fAg r1 [fAg r2| | | | | | | | | | | | |>> void| | | | | | | | | | | | | | BY (EOnThin 'v0' 29 ...) THEN (OnLastHyp EThin!) THEN| | | | | | | | | | | | | | (OnLastHyp (HypChar `fs del char`) THENS Contradiction...)| | | | | | | | | | | | 27. a16 +fAg v0 fAg V| | | | | | | | | | | | 28. |a16 +fAg v0| = j2| | | | | | | | | | | | 29. x:|A|| | | | | | | | | | | | 30. y:|A|| | | | | | | | | | | | 31. x fAg a16 +fAg v0| | | | | | | | | | | | 32. y fAg a16 +fAg v0| | | | | | | | | | | | 33. E(x,y)| | | | | | | | | | | |>> void| | | | | | | | | | | | | BY (Decide 'x=v0 in |A|' THEN Decide 'y=v0 in |A|'...)| | | | | | | | | | | | | THEN BringHyp 33| | | | | | | | | | | | | 34. x=v0 in |A|| | | | | | | | | | | | | 35. y=v0 in |A|| | | | | | | | | | | | |>> :(E(x,y))| | | | | | | | | | | | | | BY SubstFor 'x=v0 in eltsfAg(V)' THENM SubstFor 'y=v0 in eltsfAg(V)'| | | | | | | | | | | | | | THEN (Try Backchain!)| | | | | | | | | | | | | 34. x=v0 in |A|| | | | | | | | | | | | | 35. :(y=v0 in |A|)| | | | | | | | | | | | |>> :(E(x,y)) 37 | | | | | | | | | | | | | | BY SubstFor 'x=v0 in eltsfAg(V)' THEN (Try (BackThruHyp 19)!)| | | | | | | | | | | | | 34. :(x=v0 in |A|)| | | | | | | | | | | | | 35. y=v0 in |A|| | | | | | | | | | | | |>> :(E(x,y))| | | | | | | | | | | | | | BY (Assert 'E(v0,x)' THENO BackThruHyp 12 THENO SubstFor 'v0=y ineltsfAg(V)'!)| | | | | | | | | | | | | | 36. E(v0,x)| | | | | | | | | | | | | | 37. E(x,y)| | | | | | | | | | | | | |>> void| | | | | | | | | | | | | | | BY BringHyp 36 THEN (BackThruHyp 19!)| | | | | | | | | | | | | 34. :(x=v0 in |A|)| | | | | | | | | | | | | 35. :(y=v0 in |A|)| | | | | | | | | | | | |>> :(E(x,y))| | | | | | | | | | | | | | BY (BackThruHyp 26!)38