Nuprl Lemma : remove_leading_eq_nil
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. uiff(remove_leading(x.P[x];L) = [] ∈ (T List);(∀x∈L.↑P[x]))
Proof
Definitions occuring in Statement :
remove_leading: remove_leading(a.P[a];L),
l_all: (∀x∈L.P[x]),
nil: [],
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,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
top: Top,
implies: P ⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
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,
less_than: a < b,
squash: ↓T,
l_all: (∀x∈L.P[x]),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
equal-wf-T-base,
list_wf,
remove_leading_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
subtype_rel_set,
not_wf,
hd_wf,
uiff_wf,
l_all_wf2,
l_member_wf,
select_wf,
int_seg_properties,
length_wf,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
bool_wf,
assert_witness,
iff_weakening_uiff,
assert_of_null,
null_remove_leading
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
hypothesis,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
functionEquality,
universeEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
independent_pairEquality,
independent_functionElimination,
axiomEquality,
addLevel
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. uiff(remove\_leading(x.P[x];L) = [];(\mforall{}x\mmember{}L.\muparrow{}P[x]))
Date html generated:
2018_05_21-PM-06_44_16
Last ObjectModification:
2017_07_26-PM-04_54_56
Theory : general
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