Nuprl Lemma : remove_leading_wf
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. (remove_leading(x.P[x];L) ∈ {L:T List| (¬↑null(L)) ⇒ (¬↑P[hd(L)])} )
Proof
Definitions occuring in Statement :
remove_leading: remove_leading(a.P[a];L),
hd: hd(l),
null: null(as),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
remove_leading: remove_leading(a.P[a];L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
bool: 𝔹,
unit: Unit,
bnot: ¬bb
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
bool_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
list_wf,
list_ind_nil_lemma,
nil_wf,
subtype_rel_list,
null_nil_lemma,
null_cons_lemma,
assert_wf,
hd_wf,
length_of_nil_lemma,
length_of_cons_lemma,
length_wf_nat,
sq_stable__le,
not_wf,
null_wf3,
top_wf,
false_wf,
not-ge-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
list_ind_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
reduce_hd_cons_lemma,
length_cons_ge_one,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
universeEquality,
functionExtensionality,
imageMemberEquality,
minusEquality,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
(remove\_leading(x.P[x];L) \mmember{} \{L:T List| (\mneg{}\muparrow{}null(L)) {}\mRightarrow{} (\mneg{}\muparrow{}P[hd(L)])\} )
Date html generated:
2018_05_21-PM-06_43_01
Last ObjectModification:
2017_07_26-PM-04_54_36
Theory : general
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