Nuprl Lemma : rel_exp_one
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y:T. (x R^1 y ⇐⇒ x R y)
Proof
Definitions occuring in Statement :
rel_exp: R^n,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtract: n - m,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
true: True,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
infix_ap: x f y,
rel_exp: R^n,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
false_wf,
or_wf,
exists_wf,
less_than_wf,
infix_ap_wf,
rel_exp_wf,
le_wf,
equal-wf-base,
equal_wf,
rel_exp_iff,
iff_wf,
subtract_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
sqequalRule,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
productElimination,
hypothesis,
addLevel,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
voidElimination,
levelHypothesis,
promote_hyp,
hypothesisEquality,
lambdaEquality,
productEquality,
because_Cache,
universeEquality,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
functionEquality,
baseClosed,
impliesFunctionality,
hyp_replacement,
inlFormation,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:T. (x rel\_exp(T; R; 1) y \mLeftarrow{}{}\mRightarrow{} x R y)
Date html generated:
2017_04_17-AM-09_26_49
Last ObjectModification:
2017_02_27-PM-05_27_48
Theory : relations2
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