Nuprl Lemma : rel_exp_iff
∀n:ℕ
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀x,y:T. (x R^n y ⇐⇒ (∃z:T. (0 < n c∧ ((x R^n - 1 z) ∧ (z R y)))) ∨ ((n = 0 ∈ ℤ) ∧ (x = y ∈ T)))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
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],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_apply: x[s],
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
infix_ap: x f y,
rel_exp: R^n,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
subtract: n - m
Lemmas referenced :
uall_wf,
all_wf,
iff_wf,
infix_ap_wf,
rel_exp_wf,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
or_wf,
exists_wf,
less_than_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
set_wf,
primrec-wf2,
nat_properties,
equal-wf-T-base,
nat_wf,
false_wf,
eq_int_wf,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
intformeq_wf,
int_formula_prop_eq_lemma,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
and_wf,
decidable__equal_int,
subtype_base_sq,
bool_cases,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
setElimination,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesisEquality,
because_Cache,
dependent_set_memberEquality,
dependent_functionElimination,
natural_numberEquality,
hypothesis,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
applyEquality,
productEquality,
baseApply,
closedConclusion,
baseClosed,
isect_memberFormation,
inrFormation,
imageElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
independent_functionElimination,
impliesFunctionality,
inlFormation,
addLevel,
hyp_replacement,
applyLambdaEquality,
levelHypothesis
Latex:
\mforall{}n:\mBbbN{}
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}x,y:T.
(x R\^{}n y \mLeftarrow{}{}\mRightarrow{} (\mexists{}z:T. (0 < n c\mwedge{} ((x R\^{}n - 1 z) \mwedge{} (z R y)))) \mvee{} ((n = 0) \mwedge{} (x = y)))
Date html generated:
2017_04_17-AM-09_26_24
Last ObjectModification:
2017_02_27-PM-05_28_16
Theory : relations2
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