Nuprl Lemma : hdf-state-base3-3

[F1,F2,F3,f1,f2,f3,s:Top]. ∀[hdr1,hdr2,hdr3:Name].
  (hdf-state((λx,s. f1[x;s]) hdf-base(a.if name_eq(fst(a);hdr1) then [F1[a]] else [] fi )
             || x,s. f2[x;s]) hdf-base(a.if name_eq(fst(a);hdr2) then [F2[a]] else [] fi || x,s. f3[x;s])
                 hdf-base(a.if name_eq(fst(a);hdr3) then [F3[a]] else [] fi );[s]) 
     fix((λmk-hdf,s. (inl a.cbva_seq(λn.if name_eq(fst(a);hdr1) then f1[F1[a];s]
                                            if name_eq(fst(a);hdr2) then f2[F2[a];s]
                                            if name_eq(fst(a);hdr3) then f3[F3[a];s]
                                            else s
                                            fi ; λg.<mk-hdf (g x.x)), x.[x])>1))))) 
       s) supposing 
     ((¬(hdr1 hdr2 ∈ Name)) and 
     (hdr1 hdr3 ∈ Name)) and 
     (hdr2 hdr3 ∈ Name)))


Definitions occuring in Statement :  hdf-parallel: || Y hdf-state: hdf-state(X;bs) hdf-compose1: X hdf-base: hdf-base(m.F[m]) name_eq: name_eq(x;y) name: Name cons: [a b] nil: [] ifthenelse: if then else fi  uimplies: supposing a uall: [x:A]. B[x] top: Top so_apply: x[s1;s2] so_apply: x[s] pi1: fst(t) not: ¬A apply: a fix: fix(F) lambda: λx.A[x] pair: <a, b> inl: inl x natural_number: $n sqequal: t equal: t ∈ T cbva_seq: cbva_seq(L; F; m)
Definitions unfolded in proof :  uall: [x:A]. B[x] so_lambda: λ2x.t[x] member: t ∈ T top: Top so_apply: x[s] nat: le: A ≤ B and: P ∧ Q less_than': less_than'(a;b) false: False not: ¬A implies:  Q prop: cbva_seq: cbva_seq(L; F; m) select_fun_last: select_fun_last(g;m) select_fun_ap: select_fun_ap(g;n;m) mk_lambdas_fun: mk_lambdas_fun(F;m) bag-map: bag-map(f;bs) mk_lambdas: mk_lambdas(F;m) all: x:A. B[x] subtract: m callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m) le_int: i ≤j lt_int: i <j bnot: ¬bb ifthenelse: if then else fi  btrue: tt bfalse: ff eq_int: (i =z j) mk_lambdas-fun: mk_lambdas-fun(F;G;n;m) ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] decidable: Dec(P) or: P ∨ Q nat_plus: + callbyvalueall: callbyvalueall evalall: evalall(t) nil: [] it: bag-append: as bs append: as bs list_ind: list_ind cons: [a b] so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] empty-bag: {} subtype_rel: A ⊆B name: Name has-valueall: has-valueall(a) bag-null: bag-null(bs) has-value: (a)↓ bag-combine: x∈bs.f[x] bag-union: bag-union(bbs) concat: concat(ll) reduce: reduce(f;k;as) map: map(f;as) null: null(as) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2]

\mforall{}[F1,F2,F3,f1,f2,f3,s:Top].  \mforall{}[hdr1,hdr2,hdr3:Name].
    (hdf-state((\mlambda{}x,s.  f1[x;s])  o  hdf-base(a.if  name\_eq(fst(a);hdr1)  then  [F1[a]]  else  []  fi  )
                          ||  (\mlambda{}x,s.  f2[x;s])  o  hdf-base(a.if  name\_eq(fst(a);hdr2)  then  [F2[a]]  else  []  fi  )
                                ||  (\mlambda{}x,s.  f3[x;s])  o  hdf-base(a.if  name\_eq(fst(a);hdr3)  then  [F3[a]]  else  []  fi  );[s\000C]) 
          \msim{}  fix((\mlambda{}mk-hdf,s.  (inl  (\mlambda{}a.cbva\_seq(\mlambda{}n.if  name\_eq(fst(a);hdr1)  then  f1[F1[a];s]
                                                                                        if  name\_eq(fst(a);hdr2)  then  f2[F2[a];s]
                                                                                        if  name\_eq(fst(a);hdr3)  then  f3[F3[a];s]
                                                                                        else  s
                                                                                        fi  ;  \mlambda{}g.<mk-hdf  (g  (\mlambda{}x.x)),  g  (\mlambda{}x.[x])>  1))))) 
              s)  supposing 
          ((\mneg{}(hdr1  =  hdr2))  and 
          (\mneg{}(hdr1  =  hdr3))  and 
          (\mneg{}(hdr2  =  hdr3)))

Date html generated: 2016_05_16-AM-10_49_08
Last ObjectModification: 2016_01_17-AM-11_08_44

Theory : halting!dataflow

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