Nuprl Lemma : callbyvalueall_seq-extend

∀[F,G,L,K:Top]. ∀[m:ℕ]. ∀[n:ℕm + 1].
  (callbyvalueall_seq(L;λf.mk_applies(f;K;n);λg.let x ⟵ F[g]
                                                in G[g;x];n;m) ~ callbyvalueall_seq(λn.if (n =z m)
                                                                                       then mk_lambdas_fun(λg.F[g];m)
                                                                                       else L n
                                                                                       fi ;λf.mk_applies(f;K;n)
                                                                                   ;λg.G[partial_ap(g;m
                                                                                       + 1;m);select_fun_ap(g;m + 1;m)]
                                                                                   ;n;m + 1))


Proof



Definitions occuring in Statement :  mk_applies: mk_applies(F;G;m),  select_fun_ap: select_fun_ap(g;n;m),  partial_ap: partial_ap(g;n;m),  mk_lambdas_fun: mk_lambdas_fun(F;m),  callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),  int_seg: {i..j-},  nat: ℕ,  callbyvalueall: callbyvalueall,  ifthenelse: if b then t else f fi ,  eq_int: (i =z j),  uall: ∀[x:A]. B[x],  top: Top,  so_apply: x[s1;s2],  so_apply: x[s],  apply: f a,  lambda: λx.A[x],  add: n + m,  natural_number: $n,  sqequal: s ~ t
Definitions unfolded in proof :  exists: ∃x:A. B[x],  member: t ∈ T,  nat: ℕ,  uall: ∀[x:A]. B[x],  int_seg: {i..j-},  guard: {T},  ge: i ≥ j ,  lelt: i ≤ j < k,  and: P ∧ Q,  all: ∀x:A. B[x],  decidable: Dec(P),  or: P ∨ Q,  uimplies: b supposing a,  not: ¬A,  implies: P ⇒ Q,  satisfiable_int_formula: satisfiable_int_formula(fmla),  false: False,  top: Top,  prop: ℙ,  sq_type: SQType(T),  callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  ifthenelse: if b then t else f fi ,  bfalse: ff,  bnot: ¬bb,  assert: ↑b,  nequal: a ≠ b ∈ T ,  partial_ap: partial_ap(g;n;m),  mk_lambdas: mk_lambdas(F;m),  mk_applies: mk_applies(F;G;m),  lt_int: i <z j,  subtract: n - m,  select_fun_ap: select_fun_ap(g;n;m),  nat_plus: ℕ+,  le: A ≤ B,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  subtype_rel: A ⊆r B,  less_than': less_than'(a;b),  true: True
Lemmas referenced :  subtract_wf,  int_seg_properties,  nat_properties,  decidable__le,  full-omega-unsat,  intformand_wf,  intformnot_wf,  intformle_wf,  itermConstant_wf,  itermSubtract_wf,  itermVar_wf,  intformless_wf,  itermAdd_wf,  int_formula_prop_and_lemma,  int_formula_prop_not_lemma,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_subtract_lemma,  int_term_value_var_lemma,  int_formula_prop_less_lemma,  int_term_value_add_lemma,  int_formula_prop_wf,  le_wf,  decidable__equal_int,  intformeq_wf,  int_formula_prop_eq_lemma,  equal_wf,  subtype_base_sq,  int_subtype_base,  ge_wf,  less_than_wf,  int_seg_wf,  nat_wf,  top_wf,  add-zero,  le_int_wf,  bool_wf,  eqtt_to_assert,  assert_of_le_int,  eqff_to_assert,  bool_cases_sqequal,  bool_subtype_base,  assert-bnot,  eq_int_wf,  assert_of_eq_int,  neg_assert_of_eq_int,  mk_applies_lambdas_fun0,  primrec1_lemma,  mk_applies_lambdas_fun1,  decidable__lt,  lelt_wf,  primrec-unroll,  primrec0_lemma,  mk_applies_fun,  mk_applies_lambdas1,  mk_applies_roll,  false_wf,  not-lt-2,  condition-implies-le,  minus-add,  minus-one-mul,  zero-add,  minus-one-mul-top,  add-commutes,  add_functionality_wrt_le,  add-associates,  le-add-cancel,  add-subtract-cancel,  assert_wf,  bnot_wf,  not_wf,  equal-wf-base,  mk_applies_unroll,  bool_cases,  iff_transitivity,  iff_weakening_uiff,  assert_of_bnot
Rules used in proof :  cut,  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  dependent_pairFormation,  dependent_set_memberEquality,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  setElimination,  rename,  because_Cache,  hypothesis,  hypothesisEquality,  natural_numberEquality,  addEquality,  productElimination,  dependent_functionElimination,  unionElimination,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  lambdaEquality,  int_eqEquality,  intEquality,  isect_memberEquality,  voidElimination,  voidEquality,  sqequalRule,  independent_pairFormation,  instantiate,  cumulativity,  equalityTransitivity,  equalitySymmetry,  intWeakElimination,  lambdaFormation,  sqequalAxiom,  isect_memberFormation,  equalityElimination,  promote_hyp,  applyEquality,  minusEquality,  baseApply,  closedConclusion,  baseClosed,  impliesFunctionality
Latex:
\mforall{}[F,G,L,K:Top].  \mforall{}[m:\mBbbN{}].  \mforall{}[n:\mBbbN{}m  +  1].
    (callbyvalueall\_seq(L;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.let  x  \mleftarrow{}{}  F[g]
                                                                                                in  G[g;x];n;m) 
    \msim{}  callbyvalueall\_seq(\mlambda{}n.if  (n  =\msubz{}  m)  then  mk\_lambdas\_fun(\mlambda{}g.F[g];m)  else  L  n  fi 
                                            ;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.G[partial\_ap(g;m  +  1;m);select\_fun\_ap(g;m  +  1;m)];n;m
                                            +  1))


Date html generated: 2018_05_21-PM-06_23_43
Last ObjectModification: 2018_05_19-PM-05_32_11

Theory : untyped!computation


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