Nuprl Lemma : involution-has-fixpoint
∀n:ℕ
∀[T:Type]. (T ~ ℕn ⇒ (∀f:T ⟶ T. ((∀x:T. ((f (f x)) = x ∈ T)) ⇒ ((n rem 2) = 1 ∈ ℤ) ⇒ (∃x:T. ((f x) = x ∈ T)))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
remainder: n rem m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
inject: Inj(A;B;f),
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
int_nzero: ℤ-o,
l_exists: (∃x∈L. P[x])
Lemmas referenced :
count-by-orbits,
equal-wf-T-base,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
all_wf,
equal_wf,
equipollent_wf,
int_seg_wf,
nat_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
orbit-of-involution,
decidable__l_exists,
list_wf,
length_wf,
decidable__equal_int,
not-l_exists,
l_member_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
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,
l_sum-sum,
sum_functionality,
length_wf_nat,
sum_constant,
sum_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
rem-exact,
nequal_wf,
singleton-orbit,
hd_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_functionElimination,
hypothesis,
independent_isectElimination,
intEquality,
remainderEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
voidElimination,
baseClosed,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
universeEquality,
imageElimination,
imageMemberEquality,
productElimination,
independent_pairFormation,
unionElimination,
setEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
hyp_replacement,
promote_hyp,
dependent_set_memberEquality,
functionExtensionality
Latex:
\mforall{}n:\mBbbN{}
\mforall{}[T:Type]
(T \msim{} \mBbbN{}n {}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T. ((\mforall{}x:T. ((f (f x)) = x)) {}\mRightarrow{} ((n rem 2) = 1) {}\mRightarrow{} (\mexists{}x:T. ((f x) = x)))))
Date html generated:
2019_06_20-PM-02_18_14
Last ObjectModification:
2018_09_22-PM-11_03_20
Theory : equipollence!!cardinality!
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