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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