Nuprl Lemma : decidable__equal_finite
Finite types have decidable equality.
We have to put a `guard` on the conclusion because otherwise the
tactic ProveDecidable will try to use this lemma to prove, for example,
that ⌜ℤ⌝ has decidable equality -- but ⌜ℤ⌝ is not finite, so the tactic fails.⋅
∀T:Type. (finite(T) ⇒ {∀x,y:T. Dec(x = y ∈ T)})
Proof
Definitions occuring in Statement :
finite: finite(T),
decidable: Dec(P),
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
finite: finite(T),
exists: ∃x:A. B[x],
equipollent: A ~ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
and: P ∧ Q,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
lelt: i ≤ j < k
Lemmas referenced :
decidable__int_equal,
int_seg_wf,
not_wf,
equal_wf,
finite_wf,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
int_seg_properties,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
hypothesis,
lambdaEquality,
setElimination,
rename,
isectElimination,
natural_numberEquality,
because_Cache,
unionElimination,
inlFormation,
inrFormation,
universeEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
Error :applyLambdaEquality
Latex:
\mforall{}T:Type. (finite(T) {}\mRightarrow{} \{\mforall{}x,y:T. Dec(x = y)\})
Date html generated:
2016_10_21-AM-11_00_10
Last ObjectModification:
2016_08_09-AM-10_54_55
Theory : equipollence!!cardinality!
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