Nuprl Lemma : longest-prefix-is-nil
∀[T:Type]. ∀[L:T List]. ∀[P:(T List) ⟶ 𝔹].
∀[L':T List]. (¬↑(P L')) supposing (L' < L and [] < L') supposing longest-prefix(P;L) = [] ∈ (T List)
Proof
Definitions occuring in Statement :
longest-prefix: longest-prefix(P;L),
proper-iseg: L1 < L2,
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
listp: A List+,
implies: P ⇒ Q,
not: ¬A,
false: False,
and: P ∧ Q,
or: P ∨ Q,
prop: ℙ,
guard: {T}
Lemmas referenced :
longest-prefix_property,
longest-prefix_wf,
subtype_rel_dep_function,
list_wf,
bool_wf,
listp_wf,
subtype_rel_self,
assert_wf,
proper-iseg_wf,
nil_wf,
equal-wf-T-base,
iseg_wf,
less_than_wf,
length_wf,
or_wf,
length_of_nil_lemma,
all_wf,
not_wf,
equal_wf,
and_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
cumulativity,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
lambdaFormation,
productElimination,
unionElimination,
functionExtensionality,
independent_functionElimination,
voidElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
productEquality,
isectEquality,
natural_numberEquality,
applyLambdaEquality,
functionEquality,
universeEquality,
hyp_replacement,
dependent_set_memberEquality,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbB{}].
\mforall{}[L':T List]. (\mneg{}\muparrow{}(P L')) supposing (L' < L and [] < L') supposing longest-prefix(P;L) = []
Date html generated:
2018_05_21-PM-06_42_04
Last ObjectModification:
2017_07_26-PM-04_54_04
Theory : general
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