Nuprl Lemma : rel-is-immediate
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(∀x,y:T. (R x y ⇐⇒ R+! x y)) supposing
((∀a,b,c:T. (((R a b) ∧ (R a c)) ⇒ (b = c ∈ T))) and
(∀x,y:T. ((R+ x y) ⇒ (¬(R+ y x)))))
Proof
Definitions occuring in Statement :
rel-immediate: R!,
rel_plus: R+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rel-immediate: R!,
cand: A c∧ B,
infix_ap: x f y,
exists: ∃x:A. B[x],
or: P ∨ Q,
rel_plus: R+,
nat_plus: ℕ+,
decidable: Dec(P),
sq_type: SQType(T),
guard: {T},
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
le: A ≤ B
Lemmas referenced :
rel_plus_wf,
rel-immediate_wf,
all_wf,
equal_wf,
not_wf,
rel-rel-plus,
rel_plus_iff2,
rel_star_wf,
rel-star-iff-rel-plus-or,
nat_plus_properties,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
eq_int_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
assert_wf,
bnot_wf,
equal-wf-base,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
less_than_wf,
rel_exp_one,
rel_exp_wf,
nat_plus_subtype_nat,
subtract_wf,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
zero-add,
le-add-cancel,
condition-implies-le,
minus-add,
add-swap,
minus-zero,
le-add-cancel2,
minus-minus,
minus-one-mul,
minus-one-mul-top
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
applyEquality,
extract_by_obid,
isectElimination,
cumulativity,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
rename,
axiomEquality,
productEquality,
because_Cache,
lambdaFormation,
independent_pairFormation,
independent_functionElimination,
productElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
unionElimination,
setElimination,
natural_numberEquality,
instantiate,
intEquality,
independent_isectElimination,
equalityTransitivity,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
baseClosed,
impliesFunctionality,
dependent_set_memberEquality,
imageMemberEquality,
addEquality,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(\mforall{}x,y:T. (R x y \mLeftarrow{}{}\mRightarrow{} R\msupplus{}! x y)) supposing
((\mforall{}a,b,c:T. (((R a b) \mwedge{} (R a c)) {}\mRightarrow{} (b = c))) and
(\mforall{}x,y:T. ((R\msupplus{} x y) {}\mRightarrow{} (\mneg{}(R\msupplus{} y x)))))
Date html generated:
2016_10_25-AM-11_01_19
Last ObjectModification:
2016_07_12-AM-07_08_22
Theory : general
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