Nuprl Lemma : firstn_is_iseg

`∀[T:Type]. ∀L1,L2:T List.  (L1 ≤ L2 `⇐⇒` ∃n:ℕ||L2|| + 1. (L1 = firstn(n;L2) ∈ (T List)))`

Proof

Definitions occuring in Statement :  iseg: `l1 ≤ l2` firstn: `firstn(n;as)` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  iseg: `l1 ≤ l2` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` false: `False` le: `A ≤ B` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` guard: `{T}` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` uiff: `uiff(P;Q)` int_iseg: `{i...j}` cand: `A c∧ B` firstn: `firstn(n;as)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` less_than: `a < b` squash: `↓T` sq_type: `SQType(T)` true: `True`
Lemmas referenced :  exists_wf list_wf equal_wf append_wf int_seg_wf length_wf firstn_wf non_neg_length decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma 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 intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma lelt_wf int_seg_subtype false_wf length_append subtype_rel_list top_wf length-append int_seg_properties list_induction length_of_nil_lemma list_ind_nil_lemma length_of_cons_lemma list_ind_cons_lemma length_zero length_firstn_eq le_wf lt_int_wf bool_wf equal-wf-T-base assert_wf less_than_wf le_int_wf bnot_wf uiff_transitivity eqtt_to_assert assert_of_lt_int eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int subtype_base_sq int_subtype_base decidable__equal_int add-is-int-iff intformeq_wf itermSubtract_wf int_formula_prop_eq_lemma int_term_value_subtract_lemma cons_wf squash_wf true_wf nth_tl_wf append_firstn_lastn subtype_rel_sets
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination hypothesisEquality hypothesis lambdaEquality natural_numberEquality addEquality setElimination rename Error :universeIsType,  universeEquality dependent_pairFormation dependent_set_memberEquality dependent_functionElimination because_Cache unionElimination independent_isectElimination approximateComputation independent_functionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality applyEquality equalityTransitivity equalitySymmetry applyLambdaEquality cumulativity hyp_replacement productEquality baseClosed equalityElimination instantiate pointwiseFunctionality promote_hyp imageElimination baseApply closedConclusion imageMemberEquality setEquality

Latex:
\mforall{}[T:Type].  \mforall{}L1,L2:T  List.    (L1  \mleq{}  L2  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbN{}||L2||  +  1.  (L1  =  firstn(n;L2)))

Date html generated: 2019_06_20-PM-01_28_19
Last ObjectModification: 2018_09_26-PM-05_39_29

Theory : list_1

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