Nuprl Lemma : finite-decidable-inhabited
∀[T:Type]. (finite(T) ⇒ (T ∨ (¬T)))
Proof
Definitions occuring in Statement :
finite: finite(T),
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
universe: Type
Definitions unfolded in proof :
surject: Surj(A;B;f),
biject: Bij(A;B;f),
squash: ↓T,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
and: P ∧ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
int_seg: {i..j-},
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
false: False,
not: ¬A,
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
member: t ∈ T,
all: ∀x:A. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
finite: finite(T),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-less_than,
decidable__lt,
int_formula_prop_less_lemma,
intformless_wf,
int_seg_wf,
subtype_rel_self,
istype-le,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
le_wf,
set_subtype_base,
nat_wf,
istype-universe,
finite_wf,
istype-void,
int_subtype_base,
subtype_base_sq,
decidable__equal_int
Rules used in proof :
Error :productIsType,
imageElimination,
baseClosed,
imageMemberEquality,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
applyEquality,
universeEquality,
Error :functionIsType,
sqequalRule,
Error :inlFormation_alt,
Error :universeIsType,
Error :inrFormation_alt,
independent_functionElimination,
because_Cache,
independent_isectElimination,
intEquality,
cumulativity,
isectElimination,
instantiate,
unionElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. (finite(T) {}\mRightarrow{} (T \mvee{} (\mneg{}T)))
Date html generated:
2019_06_20-PM-02_18_49
Last ObjectModification:
2019_06_12-PM-03_05_33
Theory : equipollence!!cardinality!
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