Nuprl Lemma : rel_exp-iff-path
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀n:ℕ. ∀x,y:T. (x R^n y ⇐⇒ ∃L:T List. ((||L|| = (n + 1) ∈ ℤ) ∧ rel-path-between(T;R;x;y;L)))
Proof
Definitions occuring in Statement :
rel-path-between: rel-path-between(T;R;x;y;L),
length: ||as||,
list: T List,
rel_exp: R^n,
nat: ℕ,
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,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
infix_ap: x f y,
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
cons: [a / b],
select: L[n],
last: last(L),
rel-path-between: rel-path-between(T;R;x;y;L),
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
rel-path: rel-path(R;L),
le: A ≤ B,
ge: i ≥ j ,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
assert: ↑b
Lemmas referenced :
infix_ap_wf,
rel_exp_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
list_wf,
length_wf,
int_subtype_base,
rel-path-between_wf,
less_than_wf,
primrec-wf2,
all_wf,
iff_wf,
exists_wf,
equal_wf,
nat_wf,
equal-wf-T-base,
length-singleton,
nil_wf,
cons_wf,
reduce_hd_cons_lemma,
length_of_nil_lemma,
length_of_cons_lemma,
int_seg_wf,
satisfiable-full-omega-tt,
int_seg_properties,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
non_neg_length,
product_subtype_list,
false_wf,
subtype_base_sq,
list-cases,
eq_int_wf,
equal-wf-base,
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,
subtype_rel_self,
subtract-add-cancel,
decidable__equal_int,
null_wf,
iff_weakening_equal,
squash_wf,
true_wf,
rel-path-between-cons,
null_nil_lemma,
null_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
sqequalRule,
rename,
setElimination,
Error :functionIsType,
Error :universeIsType,
hypothesisEquality,
Error :inhabitedIsType,
Error :productIsType,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
because_Cache,
universeEquality,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
functionExtensionality,
applyEquality,
Error :equalityIsType3,
baseApply,
closedConclusion,
baseClosed,
Error :setIsType,
productEquality,
intEquality,
addEquality,
lambdaEquality,
productElimination,
lambdaFormation,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
equalitySymmetry,
imageMemberEquality,
computeAll,
hypothesis_subsumption,
promote_hyp,
levelHypothesis,
equalityTransitivity,
addLevel,
Error :equalityIsType4,
equalityElimination,
Error :equalityIsType1,
hyp_replacement,
applyLambdaEquality,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}n:\mBbbN{}. \mforall{}x,y:T. (x R\^{}n y \mLeftarrow{}{}\mRightarrow{} \mexists{}L:T List. ((||L|| = (n + 1)) \mwedge{} rel-path-between(T;R;x;y;L)))
Date html generated:
2019_06_20-PM-02_02_08
Last ObjectModification:
2018_09_30-PM-02_47_37
Theory : relations2
Home
Index