Nuprl Lemma : rel_star_of_equiv
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. ∀x,y:T. (EquivRel(T)(_1 E _2) ⇒ (x (E^*) y) ⇒ (x E y))
Proof
Definitions occuring in Statement :
rel_star: R^*,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rel_star: R^*,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
rel_exp: R^n,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
trans: Trans(T;x,y.E[x; y]),
nequal: a ≠ b ∈ T
Lemmas referenced :
exists_wf,
nat_wf,
rel_exp_wf,
equiv_rel_wf,
all_wf,
infix_ap_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
subtype_rel_self,
set_wf,
less_than_wf,
primrec-wf2,
equal_wf,
and_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-equal-2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
applyEquality,
hypothesisEquality,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
universeEquality,
rename,
setElimination,
cumulativity,
because_Cache,
functionEquality,
instantiate,
dependent_set_memberEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
functionExtensionality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityElimination,
equalityTransitivity,
dependent_pairFormation,
promote_hyp,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:T. (EquivRel(T)($_{1}$ E $_{2\mbackslash{}ff\000C7d$) {}\mRightarrow{} (x rel\_star(T; E) y) {}\mRightarrow{} (x E y))
Date html generated:
2019_06_20-PM-00_30_48
Last ObjectModification:
2018_09_26-PM-00_46_12
Theory : relations
Home
Index