Nuprl Lemma : rel_star_iff
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y:T. (x (R^*) y ⇐⇒ (∃z:T. ((x (R^*) z) ∧ (z R y))) ∨ (x = y ∈ T))
Proof
Definitions occuring in Statement :
rel_star: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
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,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
rel_exp: R^n,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt
Lemmas referenced :
false_wf,
infix_ap_wf,
less_than_wf,
add-subtract-cancel,
decidable__lt,
int_term_value_add_lemma,
itermAdd_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
subtract_wf,
rel_exp_iff,
equal_wf,
and_wf,
or_wf,
rel_exp_wf,
nat_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
applyEquality,
hypothesisEquality,
unionElimination,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
inlFormation,
dependent_pairFormation,
dependent_set_memberEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
inrFormation,
addEquality,
because_Cache,
productEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}x,y:T. (x (R\^{}*) y \mLeftarrow{}{}\mRightarrow{} (\mexists{}z:T. ((x (R\^{}*) z) \mwedge{} (z R y))) \mvee{} (x = y))
Date html generated:
2016_05_14-PM-03_52_39
Last ObjectModification:
2016_01_14-PM-11_11_01
Theory : relations2
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