Nuprl Lemma : rel_plus-iff-path
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,y:T. (x R+ y ⇐⇒ ∃L:T List. (1 < ||L|| ∧ rel-path-between(T;R;x;y;L)))
Proof
Definitions occuring in Statement :
rel-path-between: rel-path-between(T;R;x;y;L),
rel_plus: R+,
length: ||as||,
list: T List,
less_than: a < b,
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],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
rel_plus: R+,
infix_ap: x f y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
squash: ↓T
Lemmas referenced :
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
less_than_wf,
length_wf,
rel-path-between_wf,
exists_wf,
nat_plus_wf,
list_wf,
equal_wf,
subtract_wf,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-swap,
le-add-cancel,
decidable__equal_int,
itermSubtract_wf,
int_term_value_subtract_lemma,
rel_exp-iff-path,
nat_plus_subtype_nat,
rel_exp_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
addEquality,
dependent_set_memberEquality,
minusEquality,
applyEquality,
because_Cache,
imageElimination,
addLevel,
cumulativity,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}x,y:T. (x R\msupplus{} y \mLeftarrow{}{}\mRightarrow{} \mexists{}L:T List. (1 < ||L|| \mwedge{} rel-path-between(T;R;x;y;L)))
Date html generated:
2019_06_20-PM-02_02_13
Last ObjectModification:
2018_08_24-PM-11_35_59
Theory : relations2
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