Nuprl Lemma : rel-star-iff-rel-plus-or
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y:T. (x (R^*) y ⇐⇒ (x R+ y) ∨ (x = y ∈ T))
Proof
Definitions occuring in Statement :
rel_plus: R+,
rel_star: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
rel_plus: R+,
rel_star: R^*,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
or: P ∨ Q,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
nat_plus: ℕ+,
le: A ≤ B,
not: ¬A,
false: False,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
rel_exp: R^n,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
exists_wf,
nat_wf,
rel_exp_wf,
or_wf,
nat_plus_wf,
nat_plus_subtype_nat,
equal_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality,
applyEquality,
hypothesisEquality,
unionElimination,
functionEquality,
cumulativity,
universeEquality,
productElimination,
dependent_functionElimination,
setElimination,
rename,
natural_numberEquality,
instantiate,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
inrFormation,
inlFormation,
dependent_pairFormation,
dependent_set_memberEquality,
voidElimination,
addEquality,
isect_memberEquality,
voidEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:T. (x rel\_star(T; R) y \mLeftarrow{}{}\mRightarrow{} (x R\msupplus{} y) \mvee{} (x = y))
Date html generated:
2016_05_14-PM-03_52_46
Last ObjectModification:
2015_12_26-PM-06_57_02
Theory : relations2
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