Nuprl Lemma : longest-prefix_property'
∀[T:Type]
∀L:T List. ∀P:T List+ ⟶ 𝔹.
(longest-prefix(P;L) ≤ L
∧ longest-prefix(P;L) < L supposing 0 < ||L||
∧ (((longest-prefix(P;L) = [] ∈ (T List)) ∧ (∀L':T List. ([] < L' ⇒ L' < L ⇒ (¬↑(P L')))))
∨ (0 < ||longest-prefix(P;L)||
∧ (↑(P longest-prefix(P;L)))
∧ (∀L':T List. (longest-prefix(P;L) < L' ⇒ L' < L ⇒ (¬↑(P L')))))))
Proof
Definitions occuring in Statement :
longest-prefix: longest-prefix(P;L),
proper-iseg: L1 < L2,
iseg: l1 ≤ l2,
listp: A List+,
length: ||as||,
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
uimplies: b supposing a,
listp: A List+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
longest-prefix: longest-prefix(P;L),
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
less_than': less_than'(a;b),
proper-iseg: L1 < L2,
iseg: l1 ≤ l2,
ge: i ≥ j ,
le: A ≤ B,
bfalse: ff,
let: let,
cons: [a / b],
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True,
lt_int: i <z j
Lemmas referenced :
list_induction,
list_wf,
all_wf,
listp_wf,
bool_wf,
iseg_wf,
longest-prefix_wf,
less_than_wf,
length_wf,
proper-iseg_wf,
or_wf,
equal-wf-T-base,
not_wf,
assert_wf,
proper-iseg-length,
length_of_nil_lemma,
nil_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
null_nil_lemma,
reduce_tl_nil_lemma,
iseg_weakening,
member-less_than,
non_neg_length,
intformle_wf,
int_formula_prop_le_lemma,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
length_of_cons_lemma,
cons_wf_listp,
list-cases,
product_subtype_list,
equal_wf,
nil_iseg,
cons_wf,
equal-wf-base,
btrue_wf,
bfalse_wf,
and_wf,
null_wf3,
subtype_rel_list,
top_wf,
btrue_neq_bfalse,
equal-wf-base-T,
itermAdd_wf,
int_term_value_add_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_ind_cons_lemma,
list_ind_nil_lemma,
iff_imp_equal_bool,
true_wf,
tl_wf,
int_subtype_base,
assert_of_lt_int,
subtype_rel-equal,
cons-proper-iseg,
cons_iseg,
iseg_nil,
add-is-int-iff,
false_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
independent_functionElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
cumulativity,
universeEquality,
lambdaEquality,
functionEquality,
productEquality,
functionExtensionality,
applyEquality,
isectEquality,
natural_numberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
productElimination,
unionElimination,
imageElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
inlFormation,
setElimination,
comment,
promote_hyp,
hypothesis_subsumption,
addEquality,
applyLambdaEquality,
equalityElimination,
instantiate,
inrFormation,
imageMemberEquality,
hyp_replacement,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:T List\msupplus{} {}\mrightarrow{} \mBbbB{}.
(longest-prefix(P;L) \mleq{} L
\mwedge{} longest-prefix(P;L) < L supposing 0 < ||L||
\mwedge{} (((longest-prefix(P;L) = []) \mwedge{} (\mforall{}L':T List. ([] < L' {}\mRightarrow{} L' < L {}\mRightarrow{} (\mneg{}\muparrow{}(P L')))))
\mvee{} (0 < ||longest-prefix(P;L)||
\mwedge{} (\muparrow{}(P longest-prefix(P;L)))
\mwedge{} (\mforall{}L':T List. (longest-prefix(P;L) < L' {}\mRightarrow{} L' < L {}\mRightarrow{} (\mneg{}\muparrow{}(P L')))))))
Date html generated:
2018_05_21-PM-06_41_58
Last ObjectModification:
2017_07_26-PM-04_53_55
Theory : general
Home
Index