Nuprl Lemma : rel_exp_add_iff
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀m,n:ℕ. ∀x,z:T. (x R^m + n z ⇐⇒ ∃y:T. ((x R^m y) ∧ (y R^n z)))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
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,
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
rel_exp: R^n,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
infix_ap: x f y,
exists: ∃x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
cand: A c∧ B,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
nat_plus: ℕ+,
less_than: a < b,
nequal: a ≠ b ∈ T
Lemmas referenced :
all_wf,
nat_wf,
iff_wf,
infix_ap_wf,
rel_exp_wf,
subtract_wf,
add_nat_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,
sq_stable__le,
equal_wf,
exists_wf,
set_wf,
less_than_wf,
primrec-wf2,
and_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_weakening2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
minus-zero,
add-mul-special,
zero-mul,
int_subtype_base,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
mul-associates,
omega-shadow,
nat_properties,
general_arith_equation1,
not-equal-2,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
hypothesisEquality,
because_Cache,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
cumulativity,
instantiate,
universeEquality,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
productEquality,
functionEquality,
dependent_pairFormation,
addLevel,
hyp_replacement,
applyLambdaEquality,
levelHypothesis,
equalityElimination,
promote_hyp,
multiplyEquality,
impliesFunctionality,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}m,n:\mBbbN{}. \mforall{}x,z:T. (x R\^{}m + n z \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((x R\^{}m y) \mwedge{} (y rel\_exp(T; R; n) z)))
Date html generated:
2017_04_14-AM-07_38_17
Last ObjectModification:
2017_02_27-PM-03_10_39
Theory : relations
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