Nuprl Lemma : member_sublist
∀[T:Type]. ∀L1,L2:T List. (L1 ⊆ L2 ⇒ {∀x:T. ((x ∈ L1) ⇒ (x ∈ L2))})
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
sublist: L1 ⊆ L2,
guard: {T},
exists: ∃x:A. B[x],
and: P ∧ Q,
l_member: (x ∈ l),
cand: A c∧ B,
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
prop: ℙ,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_wf,
lelt_wf,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
length_wf_nat,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
less_than_wf,
select_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
exists_wf,
nat_wf,
sublist_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairFormation,
cut,
applyEquality,
functionExtensionality,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
cumulativity,
dependent_functionElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
voidElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. (L1 \msubseteq{} L2 {}\mRightarrow{} \{\mforall{}x:T. ((x \mmember{} L1) {}\mRightarrow{} (x \mmember{} L2))\})
Date html generated:
2017_04_14-AM-09_29_13
Last ObjectModification:
2017_02_27-PM-04_01_41
Theory : list_1
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