Nuprl Lemma : callbyvalueall_seq-combine2

[F,L1,L2,K:Top]. ∀[m1,m2:ℕ]. ∀[n:ℕm1 1].
  (callbyvalueall_seq(L1;λf.mk_applies(f;K;n);λg.callbyvalueall_seq(L2[g];λx.x;F;0;m2);n;m1) 
  callbyvalueall_seq(λi.if i <m1 then L1 else mk_lambdas_fun(λg.(L2[g] (i m1));m1) fi f.mk_applies(f;K;n)
                      g.(F partial_ap_gen(g;m1 m2;m1;m2));n;m1 m2))


Proof




Definitions occuring in Statement :  mk_applies: mk_applies(F;G;m) partial_ap_gen: partial_ap_gen(g;n;s;m) mk_lambdas_fun: mk_lambdas_fun(F;m) callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m) int_seg: {i..j-} nat: ifthenelse: if then else fi  lt_int: i <j uall: [x:A]. B[x] top: Top so_apply: x[s] apply: a lambda: λx.A[x] subtract: m add: m natural_number: $n sqequal: t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat: int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q exists: x:A. B[x] ge: i ≥  all: x:A. B[x] decidable: Dec(P) or: P ∨ Q uimplies: supposing a not: ¬A implies:  Q satisfiable_int_formula: satisfiable_int_formula(fmla) false: False top: Top prop: sq_type: SQType(T) guard: {T} callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m) bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  le: A ≤ B less_than': less_than'(a;b) so_lambda: λ2x.t[x] so_apply: x[s] iff: ⇐⇒ Q rev_implies:  Q assert: b bfalse: ff partial_ap_gen: partial_ap_gen(g;n;s;m) mk_lambdas: mk_lambdas(F;m) partial_ap: partial_ap(g;n;m) bnot: ¬bb
Lemmas referenced :  int_seg_properties subtract_wf 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 zero-add callbyvalueall_seq-shift false_wf callbyvalueall_seq-shift-init0 mk_applies_ite callbyvalueall_seq-fun1 lt_int_wf iff_imp_equal_bool bfalse_wf assert_of_lt_int assert_wf iff_wf mk_applies_lambdas_fun0 callbyvalueall_seq-eta add-subtract-cancel mk_applies_lambdas decidable__lt lelt_wf primrec0_lemma callbyvalueall_seq-partial-ap-all0 eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot mk_applies_roll
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin natural_numberEquality addEquality setElimination rename because_Cache hypothesis hypothesisEquality productElimination dependent_pairFormation dependent_set_memberEquality 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 addLevel impliesFunctionality promote_hyp

Latex:
\mforall{}[F,L1,L2,K:Top].  \mforall{}[m1,m2:\mBbbN{}].  \mforall{}[n:\mBbbN{}m1  +  1].
    (callbyvalueall\_seq(L1;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.callbyvalueall\_seq(L2[g];\mlambda{}x.x;F;0;m2);n;m1) 
    \msim{}  callbyvalueall\_seq(\mlambda{}i.if  i  <z  m1  then  L1  i  else  mk\_lambdas\_fun(\mlambda{}g.(L2[g]  (i  -  m1));m1)  fi 
                                            ;\mlambda{}f.mk\_applies(f;K;n);\mlambda{}g.(F  partial\_ap\_gen(g;m1  +  m2;m1;m2));n;m1  +  m2))



Date html generated: 2018_05_21-PM-06_23_27
Last ObjectModification: 2018_05_19-PM-05_31_27

Theory : untyped!computation


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