Nuprl Lemma : primrec-as-fix

[n,b,c:Top].  (primrec(n;b;c) fix((λp,n. if (n) < (1)  then b  else (c (n 1) (p (n 1))))) n)


Proof




Definitions occuring in Statement :  primrec: primrec(n;b;c) uall: [x:A]. B[x] top: Top less: if (a) < (b)  then c  else d apply: a fix: fix(F) lambda: λx.A[x] subtract: m natural_number: $n sqequal: t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False and: P ∧ Q ge: i ≥  le: A ≤ B cand: c∧ B less_than: a < b squash: T guard: {T} uimplies: supposing a prop: top: Top bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  not: ¬A less_than': less_than'(a;b) true: True bfalse: ff exists: x:A. B[x] or: P ∨ Q sq_type: SQType(T) bnot: ¬bb assert: b iff: ⇐⇒ Q rev_implies:  Q decidable: Dec(P) primrec: primrec(n;b;c) primtailrec: primtailrec(n;i;b;f) subtract: m has-value: (a)↓
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf istype-less_than primrec0_lemma istype-void subtract-1-ge-0 primrec-unroll lt_int_wf eqtt_to_assert assert_of_lt_int less-iff-le add_functionality_wrt_le add-associates add-zero add-commutes le-add-cancel2 eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf less_than_wf iff_weakening_uiff assert_of_bnot istype-assert istype-nat istype-top decidable__le not-lt-2 zero-add le-add-cancel istype-false not-le-2 condition-implies-le minus-add minus-zero minus-one-mul minus-one-mul-top istype-le istype-int has-value_wf_base is-exception_wf exception-not-value value-type-has-value int-value-type
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination independent_pairFormation productElimination imageElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination Error :universeIsType,  sqequalRule Error :lambdaEquality_alt,  dependent_functionElimination Error :isect_memberEquality_alt,  axiomSqEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  because_Cache unionElimination equalityElimination equalityTransitivity equalitySymmetry addEquality Error :dependent_pairFormation_alt,  Error :equalityIstype,  promote_hyp instantiate cumulativity Error :functionIsType,  lessCases imageMemberEquality baseClosed callbyvalueReduce sqleReflexivity Error :dependent_set_memberEquality_alt,  minusEquality sqequalSqle divergentSqle callbyvalueLess baseApply closedConclusion lessExceptionCases axiomSqleEquality intEquality exceptionSqequal

Latex:
\mforall{}[n,b,c:Top].    (primrec(n;b;c)  \msim{}  fix((\mlambda{}p,n.  if  (n)  <  (1)    then  b    else  (c  (n  -  1)  (p  (n  -  1)))))  n)



Date html generated: 2019_06_20-AM-11_27_49
Last ObjectModification: 2019_01_28-PM-05_55_20

Theory : call!by!value_2


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