Nuprl Lemma : l_contains_pos_length
∀[T:Type]. ∀[A,B:T List]. (0 < ||B||) supposing (0 < ||A|| and A ⊆ B)
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
l_contains: A ⊆ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
implies: P ⇒ Q,
true: True,
false: False,
cons: [a / b],
top: Top,
bfalse: ff,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
guard: {T},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
l_member: (x ∈ l),
cand: A c∧ B,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
less_than_wf,
length_wf,
member-less_than,
l_all_wf,
l_member_wf,
istype-universe,
list_wf,
pos_length2,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
istype-void,
length_of_cons_lemma,
istype-false,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
le_wf,
stuck-spread,
istype-base,
length_of_nil_lemma,
nat_properties,
intformle_wf,
int_formula_prop_le_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
Error :isect_memberEquality_alt,
independent_isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :setIsType,
Error :inhabitedIsType,
universeEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :equalityIsType1,
Error :productIsType,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A,B:T List]. (0 < ||B||) supposing (0 < ||A|| and A \msubseteq{} B)
Date html generated:
2019_06_20-PM-01_26_34
Last ObjectModification:
2018_10_06-AM-11_23_28
Theory : list_1
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