Nuprl Lemma : null_remove_leading
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. uiff(↑null(remove_leading(x.P[x];L));(∀x∈L.↑P[x]))
Proof
Definitions occuring in Statement :
remove_leading: remove_leading(a.P[a];L),
l_all: (∀x∈L.P[x]),
null: null(as),
list: T List,
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
prop: ℙ,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
true: True,
false: False,
cons: [a / b],
bfalse: ff,
guard: {T},
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
sq_type: SQType(T),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
squash: ↓T
Lemmas referenced :
remove_leading_property,
remove_leading_wf,
set_wf,
list_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
hd_wf,
listp_properties,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
less_than_wf,
length_wf,
append-nil,
l_all_iff,
l_member_wf,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
l_all_wf2,
assert_witness,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
true_wf,
append_wf,
nil_wf,
l_all_append,
cons_wf,
l_all_cons,
reduce_hd_cons_lemma,
subtype_rel_set
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
productElimination,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
unionElimination,
independent_functionElimination,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
lambdaFormation,
setElimination,
rename,
addEquality,
independent_pairFormation,
intEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
setEquality,
addLevel,
levelHypothesis,
instantiate,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageElimination,
axiomEquality,
independent_pairEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. uiff(\muparrow{}null(remove\_leading(x.P[x];L));(\mforall{}x\mmember{}L.\muparrow{}P[x]))
Date html generated:
2018_05_21-PM-06_44_10
Last ObjectModification:
2017_07_26-PM-04_54_52
Theory : general
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