### Nuprl Lemma : length_firstn

`∀[A:Type]. ∀[as:A List]. ∀[n:{0...||as||}].  (||firstn(n;as)|| ~ n)`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` length: `||as||` list: `T List` int_iseg: `{i...j}` uall: `∀[x:A]. B[x]` natural_number: `\$n` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` top: `Top` so_apply: `x[s1;s2]` squash: `↓T` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` and: `P ∧ Q` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` nil: `[]` it: `⋅` less_than: `a < b` int_iseg: `{i...j}` cand: `A c∧ B` firstn: `firstn(n;as)` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` bfalse: `ff`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf int_iseg_wf length_wf equal-wf-T-base nat_wf colength_wf_list list-cases length_of_nil_lemma subtype_base_sq set_subtype_base le_wf int_subtype_base product_subtype_list spread_cons_lemma sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-commutes equal_wf subtract_wf not-ge-2 less-iff-le minus-minus add-swap length_of_cons_lemma list_wf lt_int_wf bool_wf assert_wf le_int_wf bnot_wf decidable__equal_int not-equal-2 minus-zero le_reflexive uiff_transitivity eqtt_to_assert assert_of_lt_int list_ind_nil_lemma eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int list_ind_cons_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality sqequalAxiom cumulativity applyEquality because_Cache unionElimination instantiate intEquality equalityTransitivity equalitySymmetry promote_hyp hypothesis_subsumption productElimination voidEquality applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality universeEquality productEquality equalityElimination

Latex:
\mforall{}[A:Type].  \mforall{}[as:A  List].  \mforall{}[n:\{0...||as||\}].    (||firstn(n;as)||  \msim{}  n)

Date html generated: 2017_04_14-AM-08_47_34
Last ObjectModification: 2017_02_27-PM-03_35_03

Theory : list_0

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