Nuprl Lemma : rel_plus_closure
∀[T:Type]. ∀[R,R2:T ⟶ T ⟶ ℙ].
(Trans(T)(R2[_1;_2]) ⇒ (∀x,y:T. ((x R y) ⇒ (x R2 y))) ⇒ (∀x,y:T. ((x R+ y) ⇒ (x R2 y))))
Proof
Definitions occuring in Statement :
rel_plus: R+,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
rel_plus: R+,
infix_ap: x f y,
exists: ∃x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rel_exp: R^n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
rev_implies: P ⇐ Q,
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
rel_exp_one,
infix_ap_wf,
rel_exp_wf,
false_wf,
le_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
all_wf,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
nat_plus_wf,
rel_plus_wf,
trans_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
bnot_wf,
not_wf,
exists_wf,
subtract_wf,
add-subtract-cancel,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
promote_hyp,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
introduction,
extract_by_obid,
isectElimination,
instantiate,
cumulativity,
because_Cache,
universeEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
functionExtensionality,
applyEquality,
rename,
setElimination,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
baseApply,
closedConclusion,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productEquality,
equalityElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T)(R2[$_{1}$;$_{2}$]) {}\mRightarrow{} (\mforall{}x,y:T. ((x R y) {}\mRightarrow{} (x R\000C2 y))) {}\mRightarrow{} (\mforall{}x,y:T. ((x R\msupplus{} y) {}\mRightarrow{} (x R2 y))))
Date html generated:
2017_04_17-AM-09_26_53
Last ObjectModification:
2017_02_27-PM-05_27_50
Theory : relations2
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