Nuprl Lemma : iseg_append_single

`∀[T:Type]. ∀l1,l2:T List. ∀x:T.  (l1 ≤ l2 @ [x] `⇐⇒` l1 ≤ l2 ∨ (l1 = (l2 @ [x]) ∈ (T List)))`

Proof

Definitions occuring in Statement :  iseg: `l1 ≤ l2` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` or: `P ∨ Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` or: `P ∨ Q` member: `t ∈ T` prop: `ℙ` guard: `{T}` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` top: `Top` so_apply: `x[s]` rev_implies: `P `` Q` squash: `↓T` true: `True` less_than: `a < b` less_than': `less_than'(a;b)` false: `False` cons: `[a / b]` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff` cand: `A c∧ B`
Lemmas referenced :  equal_wf list_wf append_wf cons_wf nil_wf iseg_wf or_wf exists_wf less_than_wf length_wf length-append iseg_append_iff iff_wf list-cases length_of_nil_lemma product_subtype_list length_of_cons_lemma cons_iseg iseg_nil null_nil_lemma null_cons_lemma length-singleton iseg_weakening
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut independent_pairFormation sqequalHypSubstitution unionElimination thin inlFormation hypothesis introduction extract_by_obid isectElimination cumulativity hypothesisEquality sqequalRule inrFormation productElimination lambdaEquality productEquality natural_numberEquality applyLambdaEquality isect_memberEquality voidElimination voidEquality addLevel impliesFunctionality because_Cache dependent_functionElimination independent_functionElimination universeEquality applyEquality imageElimination imageMemberEquality baseClosed hyp_replacement equalitySymmetry promote_hyp hypothesis_subsumption rename dependent_pairFormation

Latex:
\mforall{}[T:Type].  \mforall{}l1,l2:T  List.  \mforall{}x:T.    (l1  \mleq{}  l2  @  [x]  \mLeftarrow{}{}\mRightarrow{}  l1  \mleq{}  l2  \mvee{}  (l1  =  (l2  @  [x])))

Date html generated: 2017_04_17-AM-08_46_08
Last ObjectModification: 2017_02_27-PM-05_04_42

Theory : list_1

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