Nuprl Lemma : rel_inverse_exp
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀n:ℕ. ∀x,y:T.  (x R^n^-1 y 
⇐⇒ x R^-1^n y)
Proof
Definitions occuring in Statement : 
rel_inverse: R^-1
, 
rel_exp: R^n
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
infix_ap: x f y
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
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
, 
so_apply: x[s]
, 
infix_ap: x f y
, 
rel_inverse: R^-1
, 
rel_exp: R^n
, 
eq_int: (i =z j)
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
guard: {T}
, 
exists: ∃x:A. B[x]
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
bfalse: ff
, 
sq_type: SQType(T)
Lemmas referenced : 
all_wf, 
iff_wf, 
infix_ap_wf, 
rel_inverse_wf, 
rel_exp_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, 
set_wf, 
less_than_wf, 
primrec-wf2, 
nat_wf, 
equal_wf, 
le_weakening2, 
eq_int_wf, 
bool_wf, 
equal-wf-base, 
int_subtype_base, 
assert_wf, 
less_than_transitivity1, 
le_weakening, 
less_than_irreflexivity, 
bnot_wf, 
not_wf, 
exists_wf, 
uiff_transitivity, 
eqtt_to_assert, 
assert_of_eq_int, 
iff_transitivity, 
iff_weakening_uiff, 
eqff_to_assert, 
assert_of_bnot, 
subtype_base_sq, 
rel_exp_add
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
thin, 
rename, 
setElimination, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
cumulativity, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
because_Cache, 
instantiate, 
universeEquality, 
dependent_set_memberEquality, 
natural_numberEquality, 
hypothesis, 
dependent_functionElimination, 
unionElimination, 
independent_pairFormation, 
voidElimination, 
productElimination, 
independent_functionElimination, 
independent_isectElimination, 
addEquality, 
applyEquality, 
isect_memberEquality, 
voidEquality, 
intEquality, 
minusEquality, 
functionExtensionality, 
functionEquality, 
equalitySymmetry, 
independent_pairEquality, 
axiomEquality, 
baseApply, 
closedConclusion, 
baseClosed, 
equalityTransitivity, 
productEquality, 
equalityElimination, 
impliesFunctionality, 
dependent_pairFormation
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}n:\mBbbN{}.  \mforall{}x,y:T.    (x  rel\_exp(T;  R;  n)\^{}-1  y  \mLeftarrow{}{}\mRightarrow{}  x  rel\_exp(T;  R\^{}-1;  n)  y)
Date html generated:
2017_04_14-AM-07_38_45
Last ObjectModification:
2017_02_27-PM-03_10_32
Theory : relations
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