Nuprl Lemma : decidable__rel_exp
∀k,n:ℕ. ∀[R:ℕn ⟶ ℕn ⟶ ℙ]. ((∀i,j:ℕn. Dec(i R j)) ⇒ (∀i,j:ℕn. Dec(i R^k j)))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
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],
natural_number: $n
Definitions unfolded in proof :
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,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
false: False,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
top: Top
Lemmas referenced :
decidable__equal_int_seg,
int_seg_wf,
all_wf,
decidable_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__exists_int_seg,
infix_ap_wf,
rel_exp_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
not-equal-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,
decidable__and2,
nat_wf,
uall_wf,
set_wf,
less_than_wf,
primrec-wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalRule,
isect_memberFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
lambdaEquality,
because_Cache,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
universeEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
productEquality,
dependent_set_memberEquality,
independent_pairFormation,
addEquality,
minusEquality,
isect_memberEquality,
voidEquality,
intEquality
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}[R:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n {}\mrightarrow{} \mBbbP{}]. ((\mforall{}i,j:\mBbbN{}n. Dec(i R j)) {}\mRightarrow{} (\mforall{}i,j:\mBbbN{}n. Dec(i rel\_exp(\mBbbN{}n; R; k) j)))
Date html generated:
2017_04_14-AM-07_38_07
Last ObjectModification:
2017_02_27-PM-03_10_13
Theory : relations
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