### Nuprl Lemma : pairwise-sublist

`∀[T:Type]. ∀L1,L2:T List.  ∀[P:T ⟶ T ⟶ ℙ']. (L1 ⊆ L2 `` (∀x,y∈L2.  P[x;y]) `` (∀x,y∈L1.  P[x;y]))`

Proof

Definitions occuring in Statement :  pairwise: `(∀x,y∈L.  P[x; y])` sublist: `L1 ⊆ L2` list: `T List` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` sublist: `L1 ⊆ L2` exists: `∃x:A. B[x]` pairwise: `(∀x,y∈L.  P[x; y])` and: `P ∧ Q` squash: `↓T` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` le: `A ≤ B` subtype_rel: `A ⊆r B` ge: `i ≥ j ` nat: `ℕ` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  pairwise_wf2 sublist_wf list_wf equal_wf squash_wf true_wf int_seg_properties length_wf decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_wf lelt_wf select_wf int_seg_wf non_neg_length decidable__le length_wf_nat nat_properties intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma iff_weakening_equal increasing_implies
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesisEquality sqequalRule lambdaEquality applyEquality functionExtensionality hypothesis functionEquality universeEquality productElimination imageElimination equalityTransitivity equalitySymmetry dependent_functionElimination setElimination rename dependent_set_memberEquality independent_pairFormation natural_numberEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll because_Cache applyLambdaEquality independent_functionElimination imageMemberEquality baseClosed

Latex:
\mforall{}[T:Type].  \mforall{}L1,L2:T  List.    \mforall{}[P:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}'].  (L1  \msubseteq{}  L2  {}\mRightarrow{}  (\mforall{}x,y\mmember{}L2.    P[x;y])  {}\mRightarrow{}  (\mforall{}x,y\mmember{}L1.    P[x;y]))

Date html generated: 2017_04_17-AM-07_44_32
Last ObjectModification: 2017_02_27-PM-04_16_58

Theory : list_1

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