### Nuprl Lemma : iseg_extend

`∀[T:Type]. ∀l1:T List. ∀v:T. ∀l2:T List.  (l1 ≤ l2 `` l1 @ [v] ≤ l2 supposing ||l1|| < ||l2|| c∧ (l2[||l1||] = v ∈ T))`

Proof

Definitions occuring in Statement :  iseg: `l1 ≤ l2` select: `L[n]` length: `||as||` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` cand: `A c∧ B` all: `∀x:A. B[x]` implies: `P `` Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  iseg: `l1 ≤ l2` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` cand: `A c∧ B` exists: `∃x:A. B[x]` prop: `ℙ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` so_lambda: `λ2x.t[x]` so_apply: `x[s]` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` int_seg: `{i..j-}` lelt: `i ≤ j < k` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` subtract: `n - m` sq_type: `SQType(T)` rev_implies: `P `` Q` nat: `ℕ` less_than': `less_than'(a;b)` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cons: `[a / b]`
Lemmas referenced :  member-less_than length_wf tl_wf equal_wf list_wf append_wf cons_wf nil_wf less_than_wf select_wf non_neg_length decidable__le satisfiable-full-omega-tt 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 exists_wf append_assoc length-append decidable__lt add-is-int-iff intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma false_wf length_wf_nat nat_wf squash_wf true_wf select_append_back lelt_wf iff_weakening_equal minus-one-mul add-mul-special zero-mul subtype_base_sq int_subtype_base and_wf list_induction length_of_nil_lemma list_ind_cons_lemma stuck-spread base_wf list_ind_nil_lemma reduce_tl_nil_lemma length_of_cons_lemma reduce_tl_cons_lemma
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut introduction sqequalHypSubstitution productElimination thin independent_pairEquality extract_by_obid isectElimination cumulativity hypothesisEquality hypothesis independent_isectElimination axiomEquality rename dependent_pairFormation productEquality because_Cache dependent_functionElimination natural_numberEquality unionElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll universeEquality hyp_replacement equalitySymmetry applyLambdaEquality imageElimination pointwiseFunctionality equalityTransitivity promote_hyp baseApply closedConclusion baseClosed dependent_set_memberEquality addEquality setElimination applyEquality imageMemberEquality independent_functionElimination instantiate functionEquality

Latex:
\mforall{}[T:Type]
\mforall{}l1:T  List.  \mforall{}v:T.  \mforall{}l2:T  List.
(l1  \mleq{}  l2  {}\mRightarrow{}  l1  @  [v]  \mleq{}  l2  supposing  ||l1||  <  ||l2||  c\mwedge{}  (l2[||l1||]  =  v))

Date html generated: 2017_04_17-AM-07_31_05
Last ObjectModification: 2017_02_27-PM-04_08_58

Theory : list_1

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