### Nuprl Lemma : mrec-induction2

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

Proof

Definitions occuring in Statement :  mrecind: `mrecind(L;x.P[x])` mobj: `mobj(L)` mrec_spec: `MutualRectypeSpec` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` mrecind: `mrecind(L;x.P[x])` mkinds: `mKinds` 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` true: `True` ext-eq: `A ≡ B` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` mrec: `mrec(L;i)` outl: `outl(x)` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` sq_type: `SQType(T)` guard: `{T}` prec-arg-types: `prec-arg-types(lbl,p.a[lbl; p];i;lbl)` less_than': `less_than'(a;b)` cand: `A c∧ B` mrec-lt: `x < y` prec_sub+: `prec_sub+(P;lbl,p.a[lbl; p])` prec_sub: `prec_sub(P;lbl,p.a[lbl; p])` prec-sub: `prec-sub(P;lbl,p.a[lbl; p];j;x;i;y)` dest-prec: `dest-prec(x)` mk-prec: `mk-prec(lbl;x)` let: let bfalse: `ff` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` outr: `outr(x)` bnot: `¬bb` l_all: `(∀x∈L.P[x])`
Lemmas referenced :  istype-mrecind mobj_wf mrec_spec_wf mrec-induction select_wf mrec-spec_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 length_wf intformless_wf int_formula_prop_less_lemma int_seg_wf mrec-lt_wf mobj-ext mk-prec_wf subtype_rel_self mrec_wf tuple-type_wf prec-arg-types_wf istype-atom istype-less_than mkinds_wf subtype_base_sq union_subtype_base atom_subtype_base outl_wf equal_wf assert_wf btrue_wf bfalse_wf btrue_neq_bfalse select-map subtype_rel_list top_wf select-tuple_wf int_seg_subtype_nat istype-false length-map implies-rel_plus prec_wf prec_sub_wf int_subtype_base istype-true squash_wf true_wf istype-universe inl-one-one not-0-eq-1 inr-one-one iff_weakening_equal outr_wf bnot_wf list_wf l_member_wf select_member mobj-sq mobj-kind_wf mobj-label_wf mobj-tuple_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :isect_memberFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule Error :lambdaEquality_alt,  applyEquality Error :universeIsType,  hypothesis Error :functionIsType,  Error :TYPEIsType,  independent_functionElimination dependent_functionElimination setElimination rename instantiate closedConclusion unionEquality cumulativity atomEquality universeEquality because_Cache independent_isectElimination productElimination imageElimination natural_numberEquality unionElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :inhabitedIsType,  Error :equalityIstype,  equalityTransitivity equalitySymmetry Error :setIsType,  Error :dependent_pairEquality_alt,  Error :TYPEMemberIsType,  Error :dependent_set_memberEquality_alt,  Error :productIsType,  applyLambdaEquality promote_hyp hyp_replacement Error :inlEquality_alt,  baseApply baseClosed sqequalBase productEquality Error :inlFormation_alt,  Error :unionIsType,  intEquality imageMemberEquality Error :inrEquality_alt,  Error :inrFormation_alt

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

Date html generated: 2019_06_20-PM-02_16_16
Last ObjectModification: 2019_03_12-PM-11_30_30

Theory : tuples

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