### Nuprl Lemma : pairwise-cons

`∀[T:Type]. ∀x:T. ∀L:T List.  ∀[P:T ⟶ T ⟶ ℙ']. ((∀x,y∈[x / L].  P[x;y]) `⇐⇒` (∀x,y∈L.  P[x;y]) ∧ (∀y∈L.P[x;y]))`

Proof

Definitions occuring in Statement :  pairwise: `(∀x,y∈L.  P[x; y])` l_all: `(∀x∈L.P[x])` cons: `[a / b]` list: `T List` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` pairwise: `(∀x,y∈L.  P[x; y])` member: `t ∈ T` int_seg: `{i..j-}` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` uimplies: `b supposing a` lelt: `i ≤ j < k` top: `Top` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` subtract: `n - m` ge: `i ≥ j ` le: `A ≤ B` less_than: `a < b` squash: `↓T` l_all: `(∀x∈L.P[x])` less_than': `less_than'(a;b)` select: `L[n]` cons: `[a / b]` sq_type: `SQType(T)`
Lemmas referenced :  int_formula_prop_eq_lemma intformeq_wf subtract-add-cancel int_subtype_base subtype_base_sq decidable__equal_int select-cons-tl false_wf select_cons_tl_sq add-subtract-cancel lelt_wf int_term_value_add_lemma itermAdd_wf decidable__lt non_neg_length int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermConstant_wf itermSubtract_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt subtract_wf decidable__le int_seg_properties length_of_cons_lemma add-member-int_seg2 list_wf l_member_wf l_all_wf and_wf cons_wf pairwise_wf2 length_wf int_seg_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation sqequalHypSubstitution cut lemma_by_obid isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis instantiate cumulativity sqequalRule lambdaEquality applyEquality productElimination setEquality functionEquality universeEquality dependent_functionElimination because_Cache independent_isectElimination dependent_set_memberEquality isect_memberEquality voidElimination voidEquality unionElimination dependent_pairFormation int_eqEquality intEquality computeAll addEquality imageElimination independent_functionElimination equalityTransitivity equalitySymmetry

Latex:
\mforall{}[T:Type]
\mforall{}x:T.  \mforall{}L:T  List.
\mforall{}[P:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}'].  ((\mforall{}x,y\mmember{}[x  /  L].    P[x;y])  \mLeftarrow{}{}\mRightarrow{}  (\mforall{}x,y\mmember{}L.    P[x;y])  \mwedge{}  (\mforall{}y\mmember{}L.P[x;y]))

Date html generated: 2016_05_14-PM-01_49_46
Last ObjectModification: 2016_01_15-AM-08_19_22

Theory : list_1

Home Index