### Nuprl Lemma : callbyvalueall-seq-fun1

`∀[L,K,G,F:Top]. ∀[n,m:ℕ]. ∀[k1,k2:ℕn + 1].`
`  (callbyvalueall-seq(λi.if i <z k1 then L[i] else K[i] fi ;G;F;n;m) ~ callbyvalueall-seq(λi.if i <z k2`
`                                                                                             then L[i]`
`                                                                                             else K[i]`
`                                                                                             fi ;G;F;n;m))`

Proof

Definitions occuring in Statement :  callbyvalueall-seq: `callbyvalueall-seq(L;G;F;n;m)` int_seg: `{i..j-}` nat: `ℕ` ifthenelse: `if b then t else f fi ` lt_int: `i <z j` uall: `∀[x:A]. B[x]` top: `Top` so_apply: `x[s]` lambda: `λx.A[x]` add: `n + m` natural_number: `\$n` sqequal: `s ~ t`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` guard: `{T}` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` and: `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`
Lemmas referenced :  decidable__le subtract_wf int_seg_properties nat_properties full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_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_wf le_wf decidable__equal_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma equal_wf subtype_base_sq int_subtype_base intformless_wf int_formula_prop_less_lemma ge_wf less_than_wf int_seg_wf nat_wf le_int_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot lt_int_wf assert_of_lt_int decidable__lt lelt_wf top_wf
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin setElimination rename because_Cache hypothesis hypothesisEquality unionElimination dependent_pairFormation dependent_set_memberEquality isectElimination natural_numberEquality addEquality productElimination independent_isectElimination approximateComputation independent_functionElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation instantiate cumulativity equalityTransitivity equalitySymmetry intWeakElimination lambdaFormation sqequalAxiom equalityElimination promote_hyp isect_memberFormation

Latex:
\mforall{}[L,K,G,F:Top].  \mforall{}[n,m:\mBbbN{}].  \mforall{}[k1,k2:\mBbbN{}n  +  1].
(callbyvalueall-seq(\mlambda{}i.if  i  <z  k1  then  L[i]  else  K[i]  fi  ;G;F;n;m)
\msim{}  callbyvalueall-seq(\mlambda{}i.if  i  <z  k2  then  L[i]  else  K[i]  fi  ;G;F;n;m))

Date html generated: 2018_05_21-PM-06_21_56
Last ObjectModification: 2018_05_19-PM-05_28_34

Theory : untyped!computation

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