Nuprl Lemma : fset-size-proper-subset
∀[T:Type]. ∀eq:EqDecider(T). ∀x,ys:fset(T). (ys ⊆≠ x ⇒ ||ys|| < ||x||)
Proof
Definitions occuring in Statement :
fset-size: ||s||,
f-proper-subset: xs ⊆≠ ys,
fset: fset(T),
deq: EqDecider(T),
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
squash: ↓T,
fset: fset(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
f-proper-subset: xs ⊆≠ ys,
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
fset-size: ||s||,
so_lambda: λ2x.t[x],
so_apply: x[s],
l_subset: l_subset(T;as;bs),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
cand: A c∧ B,
false: False,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
f-proper-subset_wf,
fset_wf,
deq_wf,
member-less_than,
fset-size_wf,
nat_wf,
less_than_wf,
quotient-member-eq,
set-equal_wf,
set-equal-equiv,
list_wf,
list_subtype_fset,
equal-wf-base,
equal_wf,
set-equal-reflex,
member_wf,
squash_wf,
true_wf,
assert-deq-member,
l_member_wf,
length_wf,
remove-repeats_wf,
all_wf,
isect_wf,
l_subset-l_contains,
decidable__lt,
set-equal-l_contains,
remove-repeats-l_contains,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
remove-repeats-l_contains-iff,
no_repeats-same-length-l_contains,
remove-repeats_property,
l_contains_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
universeEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
productElimination,
productEquality,
instantiate,
addLevel,
allFunctionality,
levelHypothesis,
unionElimination,
independent_pairFormation,
voidElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
impliesFunctionality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}x,ys:fset(T). (ys \msubseteq{}\mneq{} x {}\mRightarrow{} ||ys|| < ||x||)
Date html generated:
2017_04_17-AM-09_22_26
Last ObjectModification:
2017_02_27-PM-05_25_25
Theory : finite!sets
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