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 ⇒ (¬↑(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,
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],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
listp: A List+,
istype: istype(T),
or: P ∨ Q,
implies: P ⇒ Q,
longest-prefix: longest-prefix(P;L),
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
proper-iseg: L1 < L2,
iseg: l1 ≤ l2,
ge: i ≥ j ,
rev_implies: P ⇐ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bfalse: ff,
let: let,
cons: [a / b],
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
assert: ↑b,
true: True,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
lt_int: i <z j,
nil: [],
list_ind: list_ind,
length: ||as||,
decidable: Dec(P)
Lemmas referenced :
list_induction,
list_wf,
bool_wf,
iseg_wf,
longest-prefix_wf,
subtype_rel_dep_function,
listp_wf,
less_than_wf,
length_wf,
proper-iseg_wf,
equal-wf-T-base,
length_of_nil_lemma,
not_wf,
assert_wf,
null_nil_lemma,
reduce_tl_nil_lemma,
iseg_weakening,
nil_wf,
member-less_than,
istype-less_than,
proper-iseg-length,
non_neg_length,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-assert,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
length_of_cons_lemma,
cons_wf,
list-cases,
product_subtype_list,
istype-universe,
nil_iseg,
equal-wf-base-T,
btrue_neq_bfalse,
top_wf,
subtype_rel_list,
null_wf3,
equal_wf,
and_wf,
bfalse_wf,
btrue_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_term_value_add_lemma,
itermAdd_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
tl_wf,
true_wf,
iff_imp_equal_bool,
int_subtype_base,
assert_of_lt_int,
cons-proper-iseg,
all_wf,
equal-wf-base,
false_wf,
or_wf,
cons_iseg,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
iseg_nil
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesis,
productEquality,
applyEquality,
because_Cache,
inhabitedIsType,
universeIsType,
independent_isectElimination,
setElimination,
rename,
isectEquality,
natural_numberEquality,
unionEquality,
baseClosed,
applyLambdaEquality,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
imageElimination,
productElimination,
voidElimination,
inlFormation_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
productIsType,
functionIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
isectIsType,
unionIsType,
instantiate,
universeEquality,
sqequalBase,
addEquality,
lambdaEquality,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
lambdaFormation,
equalityElimination,
inrFormation_alt,
cumulativity,
intEquality,
dependent_pairFormation,
inrFormation,
hyp_replacement,
inlFormation
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:(T List) {}\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' < L {}\mRightarrow{} (\mneg{}\muparrow{}(P L')))))
\mvee{} ((\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:
2020_05_20-AM-08_06_31
Last ObjectModification:
2019_11_27-PM-02_18_14
Theory : general
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