Nuprl Lemma : preserved_by_star
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[R:T ⟶ T ⟶ ℙ]. (R preserves P ⇒ R^* preserves P)
Proof
Definitions occuring in Statement :
rel_star: R^*,
preserved_by: R preserves P,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
preserved_by: R preserves P,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
rel_star: R^*,
infix_ap: x f y,
exists: ∃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,
prop: ℙ,
and: P ∧ Q,
member: t ∈ T,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
uimplies: b supposing a,
false: False,
guard: {T},
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s]
Lemmas referenced :
and_wf,
equal_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
all_wf,
infix_ap_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,
rel_star_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
addLevel,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyLambdaEquality,
setElimination,
rename,
applyEquality,
levelHypothesis,
cumulativity,
because_Cache,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
functionExtensionality,
dependent_pairFormation,
promote_hyp,
instantiate,
lambdaEquality,
functionEquality,
universeEquality,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (R preserves P {}\mRightarrow{} rel\_star(T; R) preserves P)
Date html generated:
2017_04_14-AM-07_38_30
Last ObjectModification:
2017_02_27-PM-03_10_19
Theory : relations
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