Nuprl Lemma : decidable-last-rel
∀[T:Type]. ∀[P:(T List) ⟶ T ⟶ ℙ].
((∀L:T List. ∀x:T. Dec(P[L;x])) ⇒ (∀L:T List. Dec(∃L':T List. ∃x:T. ((L = (L' @ [x]) ∈ (T List)) ∧ P[L';x]))))
Proof
Definitions occuring in Statement :
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃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],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
guard: {T},
not: ¬A,
false: False,
exists: ∃x:A. B[x],
and: P ∧ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
cons: [a / b],
bfalse: ff,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
sq_type: SQType(T),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
list_wf,
all_wf,
decidable_wf,
list-cases,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
append_is_nil,
cons_wf,
nil_wf,
and_wf,
equal_wf,
btrue_neq_bfalse,
product_subtype_list,
exists_wf,
append_wf,
length_wf,
length-append,
last_lemma,
last_wf,
not_wf,
last_singleton_append,
assert_elim,
not_assert_elim,
assert_wf,
firstn_append,
non_neg_length,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
lelt_wf,
firstn_length,
firstn_wf,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
length_append,
iff_weakening_equal,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__equal_int,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
sqequalRule,
unionElimination,
cumulativity,
functionExtensionality,
functionEquality,
universeEquality,
inrFormation,
productElimination,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
applyLambdaEquality,
setElimination,
rename,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
productEquality,
inlFormation,
dependent_pairFormation,
addLevel,
hyp_replacement,
levelHypothesis,
natural_numberEquality,
int_eqEquality,
intEquality,
computeAll,
addEquality,
instantiate,
imageElimination,
imageMemberEquality,
baseClosed,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}L:T List. \mforall{}x:T. Dec(P[L;x]))
{}\mRightarrow{} (\mforall{}L:T List. Dec(\mexists{}L':T List. \mexists{}x:T. ((L = (L' @ [x])) \mwedge{} P[L';x]))))
Date html generated:
2018_05_21-PM-07_21_22
Last ObjectModification:
2017_07_26-PM-05_05_16
Theory : general
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