Nuprl Lemma : no_repeats-same-length-l_contains
∀[T:Type]. ∀as,bs:T List. (no_repeats(T;as) ⇒ (||as|| = ||bs|| ∈ ℤ) ⇒ as ⊆ bs ⇒ bs ⊆ as)
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
l_contains: A ⊆ B,
l_member: (x ∈ l),
l_all: (∀x∈L.P[x]),
exists: ∃x:A. B[x],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
cand: A c∧ B,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
pi1: fst(t),
inject: Inj(A;B;f),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
surject: Surj(A;B;f),
less_than: a < b,
sq_type: SQType(T)
Lemmas referenced :
l_contains_wf,
equal_wf,
length_wf,
no_repeats_wf,
list_wf,
lelt_wf,
select_wf,
int_seg_properties,
nat_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_wf,
exists_wf,
all_wf,
non_neg_length,
length_wf_nat,
and_wf,
less_than_wf,
injection-is-surjection,
squash_wf,
true_wf,
iff_weakening_equal,
le_wf,
decidable__equal_int_seg,
int_seg_subtype_nat,
false_wf,
nat_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
l_member_wf,
select_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
intEquality,
universeEquality,
sqequalRule,
dependent_functionElimination,
productElimination,
dependent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination,
natural_numberEquality,
unionElimination,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
applyEquality,
functionExtensionality,
applyLambdaEquality,
independent_functionElimination,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
productEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}as,bs:T List. (no\_repeats(T;as) {}\mRightarrow{} (||as|| = ||bs||) {}\mRightarrow{} as \msubseteq{} bs {}\mRightarrow{} bs \msubseteq{} as)
Date html generated:
2017_04_17-AM-07_29_53
Last ObjectModification:
2017_02_27-PM-04_07_56
Theory : list_1
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