### Nuprl Lemma : tuple-decomp

`∀[n:ℕ]. ∀[F:Top].  (tuple(n;i.F[i]) ~ if (n =z 0) then ⋅ if (n =z 1) then F[0] else <F[0], tuple(n - 1;i.F[i + 1])> fi )`

Proof

Definitions occuring in Statement :  tuple: `tuple(n;i.F[i])` nat: `ℕ` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` it: `⋅` uall: `∀[x:A]. B[x]` top: `Top` so_apply: `x[s]` pair: `<a, b>` subtract: `n - m` add: `n + m` natural_number: `\$n` sqequal: `s ~ t`
Definitions unfolded in proof :  tuple: `tuple(n;i.F[i])` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` upto: `upto(n)` from-upto: `[n, m)` ifthenelse: `if b then t else f fi ` lt_int: `i <z j` bfalse: `ff` eq_int: `(i =z j)` subtract: `n - m` btrue: `tt` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` append: `as @ bs` bool: `𝔹` unit: `Unit` it: `⋅` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` compose: `f o g` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf top_wf map_nil_lemma list_ind_nil_lemma decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma decidable__equal_int subtype_base_sq int_subtype_base upto_decomp1 map_cons_lemma list_ind_cons_lemma null_nil_lemma upto_decomp2 nat_wf le_wf intformeq_wf int_formula_prop_eq_lemma eq_int_wf bool_wf uiff_transitivity equal-wf-base assert_wf eqtt_to_assert assert_of_eq_int iff_transitivity bnot_wf not_wf iff_weakening_uiff eqff_to_assert assert_of_bnot equal_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int null-map null-upto map-map
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination sqequalAxiom unionElimination instantiate cumulativity because_Cache dependent_set_memberEquality imageMemberEquality baseClosed equalityElimination baseApply closedConclusion applyEquality equalityTransitivity equalitySymmetry productElimination impliesFunctionality promote_hyp

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[F:Top].
(tuple(n;i.F[i])  \msim{}  if  (n  =\msubz{}  0)  then  \mcdot{}
if  (n  =\msubz{}  1)  then  F[0]
else  <F[0],  tuple(n  -  1;i.F[i  +  1])>
fi  )

Date html generated: 2017_04_17-AM-09_29_16
Last ObjectModification: 2017_02_27-PM-05_29_27

Theory : tuples

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