### Nuprl Lemma : sq_stable__pairwise

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

Proof

Definitions occuring in Statement :  pairwise: `(∀x,y∈L.  P[x; y])` list: `T List` sq_stable: `SqStable(P)` 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 :  pairwise: `(∀x,y∈L.  P[x; y])` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` so_apply: `x[s1;s2]` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T` so_apply: `x[s]`
Lemmas referenced :  list_wf sq_stable_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le int_seg_properties select_wf all_wf length_wf int_seg_wf sq_stable__all
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut thin instantiate lemma_by_obid sqequalHypSubstitution isectElimination cumulativity natural_numberEquality hypothesisEquality hypothesis lambdaEquality setElimination rename applyEquality universeEquality independent_isectElimination productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll because_Cache imageElimination independent_functionElimination introduction functionEquality

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

Date html generated: 2016_05_14-PM-01_49_10
Last ObjectModification: 2016_01_15-AM-08_17_46

Theory : list_1

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