Nuprl Lemma : rel-comp-exp
∀[T:Type]. ∀[R,S:T ⟶ T ⟶ ℙ]. ∀n:ℕ. (R o S)^n ⇐⇒ if (n =z 0) then λx,y. (x = y ∈ T) else (R o ((S o R)^n - 1 o S)) fi\000C
Proof
Definitions occuring in Statement :
rel-comp: (R1 o R2),
rel_equivalent: R1 ⇐⇒ R2,
rel_exp: R^n,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
prop: ℙ,
so_lambda: λ2x.t[x],
ge: i ≥ j ,
int_upper: {i...},
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
so_apply: x[s],
rel_equivalent: R1 ⇐⇒ R2,
infix_ap: x f y,
rel-comp: (R1 o R2)
Lemmas referenced :
rel_equivalent_wf,
rel_exp_wf,
subtract_wf,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
istype-void,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
istype-le,
rel-comp_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
equal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-equal-2,
minus-zero,
le-add-cancel-alt,
istype-int,
istype-less_than,
primrec-wf2,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
sq_stable__le,
istype-nat,
istype-universe,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
subtype_rel_self,
le-add-cancel2,
infix_ap_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
sqequalRule,
rename,
setElimination,
Error :universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
hypothesis,
independent_pairFormation,
imageElimination,
productElimination,
dependent_functionElimination,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
Error :isect_memberEquality_alt,
minusEquality,
because_Cache,
closedConclusion,
Error :inhabitedIsType,
equalityElimination,
Error :lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
instantiate,
cumulativity,
Error :setIsType,
hypothesis_subsumption,
applyEquality,
imageMemberEquality,
baseClosed,
Error :functionIsType,
universeEquality,
independent_pairEquality,
axiomEquality,
Error :functionIsTypeImplies,
Error :productIsType,
hyp_replacement,
applyLambdaEquality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R,S:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}n:\mBbbN{}. (R o S)\^{}n \mLeftarrow{}{}\mRightarrow{} if (n =\msubz{} 0) then \mlambda{}x,y. (x = y) else (R o (rel\_exp(T; (S o R); n - 1) o S)) fi
Date html generated:
2019_06_20-PM-00_31_22
Last ObjectModification:
2019_03_27-PM-01_28_36
Theory : relations
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