### Nuprl Lemma : istype-mrecind

`∀[L:MutualRectypeSpec]. ∀[P:mobj(L) ⟶ TYPE].  istype(mrecind(L;x.P[x]))`

Proof

Definitions occuring in Statement :  mrecind: `mrecind(L;x.P[x])` mobj: `mobj(L)` mrec_spec: `MutualRectypeSpec` istype: `istype(T)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` mrecind: `mrecind(L;x.P[x])` all: `∀x:A. B[x]` member: `t ∈ T` mkinds: `mKinds` prec-arg-types: `prec-arg-types(lbl,p.a[lbl; p];i;lbl)` implies: `P `` Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` int_seg: `{i..j-}` uimplies: `b supposing a` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than: `a < b` squash: `↓T` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` mrec: `mrec(L;i)` uiff: `uiff(P;Q)` outl: `outl(x)` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` prec: `prec(lbl,p.a[lbl; p];i)` so_apply: `x[s]` ext-eq: `A ≡ B` cand: `A c∧ B` outr: `outr(x)` bnot: `¬bb` bfalse: `ff` list: `T List` l_all: `(∀x∈L.P[x])`
Lemmas referenced :  mkinds_wf istype-less_than length_wf mrec-spec_wf tuple-type_wf map_wf prec_wf list_wf istype-universe mobj_wf mrec_spec_wf int_seg_wf select_wf int_seg_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 decidable__lt intformless_wf int_formula_prop_less_lemma select-tuple_wf int_seg_subtype_nat istype-false select-map subtype_rel_list top_wf equal_wf squash_wf true_wf inl-one-one outl_wf assert_wf btrue_wf bfalse_wf mrec_wf btrue_neq_bfalse not-0-eq-1 inr-one-one subtype_rel_self iff_weakening_equal mobj-ext map-length outr_wf bnot_wf istype-true mk-prec_wf prec-arg-types_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  sqequalRule Error :functionIsType,  Error :universeIsType,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename Error :setIsType,  Error :inhabitedIsType,  natural_numberEquality instantiate unionEquality cumulativity atomEquality universeEquality Error :lambdaFormation_alt,  Error :lambdaEquality_alt,  equalityTransitivity equalitySymmetry unionElimination Error :equalityIstype,  dependent_functionElimination independent_functionElimination Error :unionIsType,  applyLambdaEquality because_Cache Error :TYPEIsType,  closedConclusion independent_isectElimination productElimination imageElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation applyEquality Error :dependent_set_memberEquality_alt,  Error :productIsType,  promote_hyp hyp_replacement Error :inlEquality_alt,  imageMemberEquality baseClosed Error :TYPEMemberIsType,  Error :dependent_pairEquality_alt,  Error :inrEquality_alt

Latex:
\mforall{}[L:MutualRectypeSpec].  \mforall{}[P:mobj(L)  {}\mrightarrow{}  TYPE].    istype(mrecind(L;x.P[x]))

Date html generated: 2019_06_20-PM-02_16_02
Last ObjectModification: 2019_03_12-PM-11_10_11

Theory : tuples

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