### Nuprl Lemma : length_of_not_nil

`∀[A:Type]. ∀[as:A List].  uiff(¬(as = [] ∈ (A List));||as|| ≥ 1 )`

Proof

Definitions occuring in Statement :  length: `||as||` nil: `[]` list: `T List` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` ge: `i ≥ j ` not: `¬A` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` or: `P ∨ Q` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` false: `False` ge: `i ≥ j ` le: `A ≤ B` prop: `ℙ` less_than': `less_than'(a;b)` true: `True` cons: `[a / b]` top: `Top` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtract: `n - m` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` decidable: `Dec(P)`
Lemmas referenced :  list-cases length_of_nil_lemma nil_wf less_than'_wf not_wf equal-wf-base list_wf ge_wf product_subtype_list length_of_cons_lemma length_wf equal-wf-T-base cons_wf cons_neq_nil non_neg_length length_wf_nat nat_wf set_subtype_base le_wf int_subtype_base equal_wf add-commutes add_functionality_wrt_le subtract_wf le_reflexive minus-one-mul zero-add one-mul add-mul-special add-associates two-mul mul-distributes-right zero-mul not-ge-2 false_wf add-swap omega-shadow less_than_wf nat_properties decidable__le
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity hypothesisEquality introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis dependent_functionElimination unionElimination sqequalRule independent_pairFormation isect_memberFormation independent_functionElimination cumulativity voidElimination productElimination independent_pairEquality lambdaEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry baseClosed because_Cache lambdaFormation promote_hyp hypothesis_subsumption isect_memberEquality voidEquality addEquality universeEquality dependent_pairFormation sqequalIntensionalEquality applyEquality intEquality independent_isectElimination multiplyEquality minusEquality dependent_set_memberEquality imageMemberEquality setElimination rename

Latex:
\mforall{}[A:Type].  \mforall{}[as:A  List].    uiff(\mneg{}(as  =  []);||as||  \mgeq{}  1  )

Date html generated: 2017_04_14-AM-08_36_19
Last ObjectModification: 2017_02_27-PM-03_28_31

Theory : list_0

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