Nuprl Lemma : decidable__cmp-le
∀[T:Type]. ∀cmp:comparison(T). ∀x,y:cmp-type(T;cmp). Dec(cmp-le(cmp;x;y))
Proof
Definitions occuring in Statement :
cmp-le: cmp-le(cmp;x;y),
cmp-type: cmp-type(T;cmp),
comparison: comparison(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
cmp-type: cmp-type(T;cmp),
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
comparison: comparison(T),
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cmp-le: cmp-le(cmp;x;y),
decidable: Dec(P),
not: ¬A,
or: P ∨ Q,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
decidable_wf,
cmp-le_wf,
subtype_rel_self,
cmp-type_wf,
equal-wf-base,
equal-wf-T-base,
comparison_wf,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
minus_functionality_wrt_eq,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
subtype_rel-equal,
top_wf,
less_than_wf,
false_wf,
lt_int_wf,
assert_of_lt_int,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
less_than'_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
introduction,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
cut,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
hypothesis,
pertypeElimination,
productElimination,
cumulativity,
equalityTransitivity,
equalitySymmetry,
applyEquality,
dependent_functionElimination,
productEquality,
because_Cache,
intEquality,
setElimination,
rename,
baseClosed,
universeEquality,
instantiate,
independent_isectElimination,
independent_functionElimination,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
minusEquality,
unionElimination,
equalityElimination,
inlEquality,
independent_pairEquality,
lessCases,
independent_pairFormation,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
promote_hyp,
axiomEquality,
functionEquality,
inrEquality
Latex:
\mforall{}[T:Type]. \mforall{}cmp:comparison(T). \mforall{}x,y:cmp-type(T;cmp). Dec(cmp-le(cmp;x;y))
Date html generated:
2019_06_20-PM-01_41_56
Last ObjectModification:
2018_08_20-PM-09_32_17
Theory : list_1
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