Nuprl Lemma : streamless-dec-equal
∀[T:Type]. (streamless(T) ⇒ (∀x,y:T. Dec(x = y ∈ T)))
Proof
Definitions occuring in Statement :
streamless: streamless(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
streamless: streamless(T),
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
nat: ℕ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
bfalse: ff,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
decidable: Dec(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
streamless_wf,
nat_wf,
exists_wf,
not_wf,
equal_wf,
all_wf,
ifthenelse_wf,
eq_int_wf,
assert_wf,
bnot_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
int_subtype_base,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesis,
universeEquality,
promote_hyp,
productElimination,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
functionEquality,
applyEquality,
functionExtensionality,
productEquality,
because_Cache,
rename,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setElimination,
intEquality,
unionElimination,
instantiate,
independent_isectElimination,
independent_pairFormation,
impliesFunctionality,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
computeAll,
inlFormation,
inrFormation,
equalityElimination
Latex:
\mforall{}[T:Type]. (streamless(T) {}\mRightarrow{} (\mforall{}x,y:T. Dec(x = y)))
Date html generated:
2018_05_21-PM-09_02_21
Last ObjectModification:
2017_07_26-PM-06_25_21
Theory : general
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