### Nuprl Lemma : decidable__llex

`∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].`
`  ((∀a,b:A.  (Dec(<[a;b]) ∧ Dec(a = b ∈ A))) `` (∀L1,L2:A List.  Dec(L1 llex(A;a,b.<[a;b]) L2)))`

Proof

Definitions occuring in Statement :  llex: `llex(A;a,b.<[a; b])` list: `T List` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` llex: `llex(A;a,b.<[a; b])` infix_ap: `x f y` member: `t ∈ T` prop: `ℙ` and: `P ∧ Q` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` so_apply: `x[s]` nat: `ℕ` ge: `i ≥ j ` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` cand: `A c∧ B`
Lemmas referenced :  lelt_wf false_wf int_seg_subtype_nat not_wf decidable__exists_int_seg decidable__all_int_seg decidable__and decidable_wf and_wf list_wf nat_properties nat_wf exists_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 equal_wf int_seg_wf all_wf length_wf less_than_wf decidable__or
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalRule cut lemma_by_obid sqequalHypSubstitution isectElimination thin productEquality cumulativity hypothesisEquality hypothesis because_Cache natural_numberEquality lambdaEquality setElimination rename independent_isectElimination productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination introduction applyEquality universeEquality independent_functionElimination functionEquality instantiate inrFormation inlFormation dependent_set_memberEquality

Latex:
\mforall{}[A:Type].  \mforall{}[<:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}a,b:A.    (Dec(<[a;b])  \mwedge{}  Dec(a  =  b)))  {}\mRightarrow{}  (\mforall{}L1,L2:A  List.    Dec(L1  llex(A;a,b.<[a;b])  L2)))

Date html generated: 2016_05_15-PM-04_17_37
Last ObjectModification: 2016_01_16-AM-11_14_18

Theory : general

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