Nuprl Lemma : longest-prefix_property2
∀[T:Type]
∀L:T List. ∀P:(T List) ⟶ 𝔹. ∀L2:T List.
0 < ||L2|| supposing 0 < ||L||
∧ (∀L':T List. ([] < L' ⇒ L' < L2 ⇒ (¬↑(P (longest-prefix(P;L) @ L')))))
∧ ((↑(P longest-prefix(P;L))) ∨ (↑null(longest-prefix(P;L))))
supposing L = (longest-prefix(P;L) @ L2) ∈ (T List)
Proof
Definitions occuring in Statement :
longest-prefix: longest-prefix(P;L),
proper-iseg: L1 < L2,
length: ||as||,
null: null(as),
append: as @ bs,
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],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
listp: A List+,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
false: False,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
rev_implies: P ⇐ Q
Lemmas referenced :
longest-prefix_property,
equal_wf,
list_wf,
append_wf,
longest-prefix_wf,
subtype_rel_dep_function,
bool_wf,
listp_wf,
subtype_rel_self,
less_than_wf,
length_wf,
proper-iseg-length,
length_wf_nat,
nat_wf,
length-append,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_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_term_value_add_lemma,
int_formula_prop_wf,
false_wf,
assert_wf,
proper-iseg_wf,
nil_wf,
and_wf,
null_wf3,
subtype_rel_list,
top_wf,
null_nil_lemma,
btrue_wf,
subtype_base_sq,
bool_subtype_base,
list_ind_nil_lemma,
squash_wf,
true_wf,
proper-iseg-append,
append-nil,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
axiomEquality,
rename,
productElimination,
independent_pairFormation,
cumulativity,
because_Cache,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setElimination,
functionEquality,
universeEquality,
natural_numberEquality,
independent_functionElimination,
dependent_set_memberEquality,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
unionElimination,
imageElimination,
pointwiseFunctionality,
equalityTransitivity,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
functionExtensionality,
setEquality,
instantiate,
inrFormation,
inlFormation,
imageMemberEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:(T List) {}\mrightarrow{} \mBbbB{}. \mforall{}L2:T List.
0 < ||L2|| supposing 0 < ||L||
\mwedge{} (\mforall{}L':T List. ([] < L' {}\mRightarrow{} L' < L2 {}\mRightarrow{} (\mneg{}\muparrow{}(P (longest-prefix(P;L) @ L')))))
\mwedge{} ((\muparrow{}(P longest-prefix(P;L))) \mvee{} (\muparrow{}null(longest-prefix(P;L))))
supposing L = (longest-prefix(P;L) @ L2)
Date html generated:
2016_10_25-AM-10_47_17
Last ObjectModification:
2016_07_12-AM-06_56_06
Theory : general
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