Nuprl Lemma : dl-sem_functionality1

∀x:dl-Obj()
  ∀[K:Type]. ∀[R:ℕ ⟶ K ⟶ K ⟶ ℙ]. ∀[P,P':ℕ ⟶ K ⟶ ℙ].
    ((∀n∈dl-prop-atoms() x.∀k:K. (P[n] k ⇐⇒ P'[n] k))
    ⇒ dl-same-sem(x;K;dl-sem(K;n.R[n];m.P[m]) x;dl-sem(K;n.R[n];m.P'[m]) x))


Proof



Definitions occuring in Statement :  dl-prop-atoms: dl-prop-atoms(),  dl-same-sem: dl-same-sem(x;K;r;s),  dl-sem: dl-sem(K;n.R[n];m.P[m]),  dl-Obj: dl-Obj(),  l_all: (∀x∈L.P[x]),  nat: ℕ,  uall: ∀[x:A]. B[x],  prop: ℙ,  so_apply: x[s],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  implies: P ⇒ Q,  apply: f a,  function: x:A ⟶ B[x],  universe: Type
Definitions unfolded in proof :  dl-induction,  uall: ∀[x:A]. B[x],  so_lambda: λ2x.t[x],  member: t ∈ T,  prop: ℙ,  implies: P ⇒ Q,  all: ∀x:A. B[x],  so_apply: x[s],  nat: ℕ,  ge: i ≥ j ,  decidable: Dec(P),  or: P ∨ Q,  uimplies: b supposing a,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  false: False,  top: Top,  and: P ∧ Q,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  subtype_rel: A ⊆r B,  dl-same-sem: dl-same-sem(x;K;r;s),  dl-sem: dl-sem(K;n.R[n];m.P[m]),  so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),  so_apply: x[s1;s2;s3;s4],  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  dl-kind: dl-kind(d),  mobj-kind: mobj-kind(x),  pi1: fst(t),  dl-prog-obj: prog(x),  uiff: uiff(P;Q),  eq_atom: x =a y,  assert: ↑b,  ifthenelse: if b then t else f fi ,  bfalse: ff,  dl-prop-atoms: dl-prop-atoms(),  dl-ind: dl-ind,  mrec_ind: mrec_ind(L;h;x),  genrec-ap: genrec-ap,  decidable__atom_equal,  dl-aprog: atm(x),  mk-prec: mk-prec(lbl;x),  btrue: tt,  nil: [],  it: ⋅,  dl-prog-sem: [|alpha|],  dl-prop-sem: [|phi|],  dl-prop-obj: prop(x),  guard: {T},  rel_implies: R1 => R2,  infix_ap: x f y,  l_member: (x ∈ l),  select: L[n],  cons: [a / b],  cand: A c∧ B,  less_than: a < b,  squash: ↓T,  less_than': less_than'(a;b),  true: True
Lemmas referenced :  dl-induction,  uall_wf,  nat_wf,  l_all_wf,  dl-prop-atoms_wf,  iff_wf,  nat_properties,  decidable__le,  full-omega-unsat,  intformand_wf,  intformnot_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  istype-int,  int_formula_prop_and_lemma,  istype-void,  int_formula_prop_not_lemma,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_wf,  istype-le,  istype-nat,  l_member_wf,  dl-same-sem_wf,  dl-sem_wf,  istype-universe,  subtype-TYPE,  dl-Obj_wf,  dl-ind-dl-aprog,  subtype_rel_self,  istype-atom,  set_subtype_base,  le_wf,  int_subtype_base,  assert_of_eq_atom,  nil_wf,  dl-ind-dl-comp,  l_all_append,  dl-prog-obj_wf,  dl-prog-sem_wf,  dl-kind_wf,  dl-comp_wf,  cons_wf,  atom_subtype_base,  dl-prog_wf,  dl-ind-dl-choose,  dl-choose_wf,  dl-ind-dl-iterate,  dl-iterate_wf,  dl-ind-dl-test,  dl-prop-sem_wf,  dl-test_wf,  dl-prop-obj_wf,  dl-prop_wf,  dl-ind-dl-aprop,  dl-aprop_wf,  dl-ind-dl-false,  dl-false_wf,  dl-implies_wf,  dl-ind-dl-implies,  dl-and_wf,  dl-ind-dl-and,  dl-or_wf,  dl-ind-dl-or,  dl-box_wf,  dl-ind-dl-box,  dl-diamond_wf,  dl-ind-dl-diamond,  rel_star_functionality_wrt_rel_implies,  rel_star_wf,  l_all_iff,  length_of_cons_lemma,  length_of_nil_lemma,  istype-less_than,  length_wf,  list_subtype_base,  decidable__atom_equal
Rules used in proof :  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isectElimination,  thin,  sqequalRule,  lambdaEquality_alt,  instantiate,  universeEquality,  isectEquality,  functionEquality,  cumulativity,  hypothesis,  hypothesisEquality,  applyEquality,  dependent_set_memberEquality_alt,  setElimination,  rename,  dependent_functionElimination,  natural_numberEquality,  unionElimination,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  dependent_pairFormation_alt,  int_eqEquality,  isect_memberEquality_alt,  voidElimination,  independent_pairFormation,  universeIsType,  because_Cache,  setIsType,  lambdaFormation_alt,  isect_memberFormation_alt,  inhabitedIsType,  equalityIstype,  baseApply,  closedConclusion,  baseClosed,  intEquality,  sqequalBase,  equalitySymmetry,  equalityTransitivity,  productElimination,  functionIsType,  productIsType,  promote_hyp,  atomEquality,  tokenEquality,  isectIsType,  independent_pairEquality,  functionIsTypeImplies,  unionIsType,  inlFormation_alt,  inrFormation_alt,  imageMemberEquality
Latex:
\mforall{}x:dl-Obj()
    \mforall{}[K:Type].  \mforall{}[R:\mBbbN{}  {}\mrightarrow{}  K  {}\mrightarrow{}  K  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[P,P':\mBbbN{}  {}\mrightarrow{}  K  {}\mrightarrow{}  \mBbbP{}].
        ((\mforall{}n\mmember{}dl-prop-atoms()  x.\mforall{}k:K.  (P[n]  k  \mLeftarrow{}{}\mRightarrow{}  P'[n]  k))
        {}\mRightarrow{}  dl-same-sem(x;K;dl-sem(K;n.R[n];m.P[m])  x;dl-sem(K;n.R[n];m.P'[m])  x))


Date html generated: 2019_10_15-AM-11_43_59
Last ObjectModification: 2019_03_26-AM-11_51_57

Theory : dynamic!logic


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