Nuprl Lemma : null-segment
∀[T:Type]. ∀[as:T List]. ∀[i:{0...||as||}]. ∀[j:{i...||as||}]. null(as[i..j-]) = (i =z j)
Proof
Definitions occuring in Statement :
segment: as[m..n-],
length: ||as||,
null: null(as),
list: T List,
int_iseg: {i...j},
eq_int: (i =z j),
bool: 𝔹,
uall: ∀[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,
squash: ↓T,
prop: ℙ,
int_iseg: {i...j},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
uiff: uiff(P;Q)
Lemmas referenced :
length_segment,
equal_wf,
squash_wf,
true_wf,
bool_wf,
null-length-zero,
segment_wf,
subtype_rel_list,
top_wf,
eq_int_wf,
iff_weakening_equal,
int_iseg_wf,
length_wf,
list_wf,
iff_imp_equal_bool,
subtract_wf,
int_iseg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
equal-wf-T-base,
assert_of_eq_int,
assert_wf,
iff_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
cumulativity,
setElimination,
rename,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
intEquality,
independent_pairFormation,
lambdaFormation,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
addLevel,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[as:T List]. \mforall{}[i:\{0...||as||\}]. \mforall{}[j:\{i...||as||\}]. null(as[i..j\msupminus{}]) = (i =\msubz{} j)
Date html generated:
2017_04_17-AM-07_36_24
Last ObjectModification:
2017_02_27-PM-04_10_28
Theory : list_1
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