Nuprl Lemma : rel_plus-restriction-equiv
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ]. ((∀x,y:T. ((P[y] ∧ (R x y)) ⇒ P[x])) ⇒ (∀x,y:T. (R|P+ x y ⇐⇒ R+|P x y)))
Proof
Definitions occuring in Statement :
rel_plus: R+,
rel-restriction: R|P,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rel_implies: R1 => R2,
infix_ap: x f y,
nat_plus: ℕ+,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
rel-restriction: R|P,
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
squash: ↓T,
true: True,
trans: Trans(T;x,y.E[x; y]),
rel_plus: R+
Lemmas referenced :
rel_plus_wf,
rel-restriction_wf,
all_wf,
rel_plus-of-restriction,
nat_plus_properties,
rel_exp_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
nat_plus_wf,
false_wf,
rel-rel-plus,
itermAdd_wf,
int_term_value_add_lemma,
eq_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
add-subtract-cancel,
exists_wf,
equal_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
true_wf,
nat_wf,
and_wf,
less_than_wf,
rel_plus_trans
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
applyEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
hypothesis,
lambdaEquality,
sqequalRule,
functionEquality,
universeEquality,
productEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
rename,
setElimination,
dependent_set_memberEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
addEquality,
equalityTransitivity,
baseApply,
closedConclusion,
baseClosed,
instantiate,
impliesFunctionality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. ((P[y] \mwedge{} (R x y)) {}\mRightarrow{} P[x])) {}\mRightarrow{} (\mforall{}x,y:T. (R|P\msupplus{} x y \mLeftarrow{}{}\mRightarrow{} R\msupplus{}|P x y)))
Date html generated:
2017_04_17-AM-09_28_04
Last ObjectModification:
2017_02_27-PM-05_28_30
Theory : relations2
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