### Nuprl Lemma : sublist_append_front

`∀[T:Type]. ∀L,L1,L2:T List.  L ⊆ L1 @ L2 `` L ⊆ L1 supposing ¬(last(L) ∈ L2) supposing ¬↑null(L)`

Proof

Definitions occuring in Statement :  sublist: `L1 ⊆ L2` last: `last(L)` l_member: `(x ∈ l)` null: `null(as)` append: `as @ bs` list: `T List` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` and: `P ∧ Q` top: `Top` sublist: `L1 ⊆ L2` exists: `∃x:A. B[x]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` subtype_rel: `A ⊆r B` l_member: `(x ∈ l)` nat: `ℕ` cand: `A c∧ B` ge: `i ≥ j ` le: `A ≤ B` last: `last(L)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  l_member_wf last_wf not_wf assert_wf null_wf decidable__assert sublist_wf append_wf isect_wf list_wf assert_of_null nil-sublist non_nil_length not_functionality_wrt_uiff equal-wf-T-base decidable__lt int_seg_wf length_wf subtract_wf length-append decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf lelt_wf le_wf non_neg_length length_append subtype_rel_list top_wf length_wf_nat nat_properties int_seg_properties itermAdd_wf int_term_value_add_lemma less_than_wf equal_wf select_wf squash_wf true_wf select_append_back subtype_rel_self iff_weakening_equal add-is-int-iff false_wf increasing_implies_le nat_wf select_append_front increasing_wf all_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality lambdaEquality dependent_functionElimination voidElimination extract_by_obid independent_isectElimination hypothesis equalityTransitivity equalitySymmetry rename unionElimination universeEquality productElimination voidEquality hyp_replacement applyLambdaEquality because_Cache baseClosed independent_functionElimination applyEquality functionExtensionality natural_numberEquality cumulativity dependent_set_memberEquality independent_pairFormation imageElimination approximateComputation dependent_pairFormation int_eqEquality intEquality setElimination productEquality imageMemberEquality instantiate addEquality pointwiseFunctionality promote_hyp baseApply closedConclusion

Latex:
\mforall{}[T:Type].  \mforall{}L,L1,L2:T  List.    L  \msubseteq{}  L1  @  L2  {}\mRightarrow{}  L  \msubseteq{}  L1  supposing  \mneg{}(last(L)  \mmember{}  L2)  supposing  \mneg{}\muparrow{}null(L)

Date html generated: 2019_06_20-PM-01_22_51
Last ObjectModification: 2018_09_17-PM-06_04_14

Theory : list_1

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