Nuprl Lemma : rel-comp-star
∀[T:Type]. ∀[R,S:T ⟶ T ⟶ ℙ]. (R o S)^* ⇐⇒ (R o ((S o R)^* o S)) ∨ (λx,y. (x = y ∈ T))
Proof
Definitions occuring in Statement :
rel-comp: (R1 o R2),
rel_equivalent: R1 ⇐⇒ R2,
rel_or: R1 ∨ R2,
rel_star: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rel_equivalent: R1 ⇐⇒ R2,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
rel_or: R1 ∨ R2,
infix_ap: x f y,
rel_star: R^*,
exists: ∃x:A. B[x],
nat: ℕ,
guard: {T},
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
false: False,
sq_type: SQType(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
rel-comp: (R1 o R2),
decidable: Dec(P),
sq_stable: SqStable(P),
squash: ↓T,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
int_upper: {i...}
Lemmas referenced :
rel-comp-exp,
rel_star_wf,
rel-comp_wf,
subtype_rel_self,
rel_or_wf,
equal_wf,
istype-universe,
eq_int_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
istype-assert,
istype-void,
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,
rel_exp_wf,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
not-equal-2,
sq_stable__le,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
minus-one-mul,
minus-one-mul-top,
minus-minus,
add-swap,
istype-le,
le_antisymmetry_iff,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
infix_ap_wf,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
add-subtract-cancel,
iff_weakening_equal,
rel_star_weakening
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :universeIsType,
applyEquality,
sqequalRule,
instantiate,
universeEquality,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
because_Cache,
Error :functionIsType,
productElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
Error :inrFormation_alt,
intEquality,
independent_isectElimination,
baseClosed,
Error :equalityIstype,
sqequalBase,
unionElimination,
cumulativity,
Error :inlFormation_alt,
Error :productIsType,
Error :dependent_set_memberEquality_alt,
voidElimination,
imageMemberEquality,
imageElimination,
addEquality,
Error :isect_memberEquality_alt,
minusEquality,
Error :dependent_pairFormation_alt,
promote_hyp,
equalityElimination,
closedConclusion,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]. \mforall{}[R,S:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(R o S)\^{}* \mLeftarrow{}{}\mRightarrow{} (R o ((S o R)\^{}* o S)) \mvee{} (\mlambda{}x,y. (x = y))
Date html generated:
2019_06_20-PM-00_31_28
Last ObjectModification:
2019_03_28-PM-03_48_44
Theory : relations
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