Nuprl Lemma : count-by-equiv
∀[A:Type]. ∀[E:A ⟶ A ⟶ ℙ].
(EquivRel(A;x,y.E[x;y])
⇒ (∀L:A List. (∀a:A. (∃b∈L. E[a;b])) ⇒ A ~ i:ℕ||L|| × {a:A| E[a;L[i]]} supposing (∀a,b∈L. ¬E[a;b])))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
pairwise: (∀x,y∈L. P[x; y]),
l_exists: (∃x∈L. P[x]),
select: L[n],
length: ||as||,
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
pairwise: (∀x,y∈L. P[x; y]),
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
cand: A c∧ B,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
pi1: fst(t),
equipollent: A ~ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
pi2: snd(t),
le: A ≤ B,
respects-equality: respects-equality(S;T),
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y])
Lemmas referenced :
l_exists_iff,
l_member_wf,
l_exists_wf,
pairwise_wf2,
not_wf,
list_wf,
equiv_rel_wf,
istype-universe,
subtype_rel_self,
istype-less_than,
istype-nat,
length_wf,
select_wf,
non_neg_length,
decidable__le,
length_wf_nat,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
istype-le,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
respects-equality-product,
nat_wf,
subtype-base-respects-equality,
set_subtype_base,
le_wf,
int_subtype_base,
respects-equality-set-trivial,
equal_functionality_wrt_subtype_rel2,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
biject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
hypothesis,
extract_by_obid,
isectElimination,
because_Cache,
setElimination,
applyEquality,
Error :setIsType,
Error :universeIsType,
productElimination,
independent_functionElimination,
promote_hyp,
Error :functionIsType,
instantiate,
cumulativity,
universeEquality,
functionExtensionality,
Error :productIsType,
Error :equalityIstype,
independent_isectElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
natural_numberEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
independent_pairFormation,
Error :equalityIsType1,
hyp_replacement,
Error :dependent_pairEquality_alt,
Error :dependent_set_memberEquality_alt,
imageElimination,
independent_pairEquality,
setEquality,
intEquality,
closedConclusion,
sqequalBase,
productEquality
Latex:
\mforall{}[A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(EquivRel(A;x,y.E[x;y])
{}\mRightarrow{} (\mforall{}L:A List
(\mforall{}a:A. (\mexists{}b\mmember{}L. E[a;b])) {}\mRightarrow{} A \msim{} i:\mBbbN{}||L|| \mtimes{} \{a:A| E[a;L[i]]\} supposing (\mforall{}a,b\mmember{}L. \mneg{}E[a;b])))
Date html generated:
2019_06_20-PM-02_18_08
Last ObjectModification:
2018_11_23-PM-02_08_10
Theory : equipollence!!cardinality!
Home
Index