### Nuprl Lemma : length-nat-if-has-value

`∀l:Base. ((||l||)↓ `` (||l|| ∈ ℕ))`

Proof

Definitions occuring in Statement :  length: `||as||` nat: `ℕ` has-value: `(a)↓` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` base: `Base`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` length: `||as||` list_ind: list_ind uall: `∀[x:A]. B[x]` nat: `ℕ` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` top: `Top` not: `¬A` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` nat_plus: `ℕ+` has-value: `(a)↓` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf has-value_wf_base int_subtype_base base_wf fun_exp0_lemma strictness-apply bottom_diverge decidable__le subtract_wf false_wf not-ge-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel fun_exp_unroll_1 has-value-implies-dec-ispair-2 top_wf value-type-has-value int-value-type nat_wf add_nat_wf le_wf sq_stable__le equal_wf has-value-implies-dec-isaxiom-2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalHypSubstitution sqequalRule hypothesis compactness thin introduction extract_by_obid isectElimination hypothesisEquality setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination lambdaEquality dependent_functionElimination axiomEquality equalityTransitivity equalitySymmetry baseApply closedConclusion baseClosed applyEquality isect_memberEquality voidEquality unionElimination independent_pairFormation productElimination addEquality intEquality minusEquality because_Cache dependent_set_memberEquality callbyvalueCallbyvalue callbyvalueReduce callbyvalueAdd imageMemberEquality imageElimination

Latex:
\mforall{}l:Base.  ((||l||)\mdownarrow{}  {}\mRightarrow{}  (||l||  \mmember{}  \mBbbN{}))

Date html generated: 2017_04_14-AM-08_35_32
Last ObjectModification: 2017_02_27-PM-03_29_02

Theory : list_0

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