Nuprl Lemma : rel_plus_implies
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y:T. ((x R+ y) ⇒ ((x R y) ∨ (∃z:T. ((x R+ z) ∧ (z R y)))))
Proof
Definitions occuring in Statement :
rel_plus: R+,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
rel_plus: R+,
infix_ap: x f y,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
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],
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
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,
or_wf,
exists_wf,
rel_plus_wf,
nat_plus_wf,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
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,
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,
less_than_wf,
decidable__lt,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal-wf-T-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
because_Cache,
universeEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
hypothesis,
functionExtensionality,
applyEquality,
rename,
setElimination,
dependent_functionElimination,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
productEquality,
independent_functionElimination,
inlFormation,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
inrFormation,
baseApply,
closedConclusion,
baseClosed,
equalityTransitivity,
equalityElimination,
impliesFunctionality,
imageMemberEquality,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:T. ((x R\msupplus{} y) {}\mRightarrow{} ((x R y) \mvee{} (\mexists{}z:T. ((x R\msupplus{} z) \mwedge{} (z R y)))))
Date html generated:
2017_04_17-AM-09_26_06
Last ObjectModification:
2017_02_27-PM-05_27_01
Theory : relations2
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