Nuprl Lemma : transitive-loop
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Trans(T;x,y.R[x;y])
⇒ (∀L:T List. ((∀i:ℕ||L|| - 1. R[L[i];L[i + 1]]) ⇒ R[last(L);hd(L)] supposing ¬↑null(L) ⇒ (∀a∈L.(∀b∈L.R[a;b])))))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
last: last(L),
select: L[n],
hd: hd(l),
length: ||as||,
null: null(as),
list: T List,
trans: Trans(T;x,y.E[x; y]),
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uimplies: b supposing a,
not: ¬A,
false: False,
cons: [a / b],
top: Top,
bfalse: ff,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
sq_type: SQType(T),
ge: i ≥ j ,
trans: Trans(T;x,y.E[x; y]),
l_member: (x ∈ l),
cand: A c∧ B,
last: last(L),
select: L[n],
nil: [],
it: ⋅
Lemmas referenced :
list-cases,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
product_subtype_list,
null_cons_lemma,
false_wf,
l_member_wf,
l_all_iff,
l_all_wf2,
all_wf,
isect_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
last_wf,
hd_wf,
listp_properties,
length_of_nil_lemma,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
less_than_wf,
length_wf,
int_seg_wf,
subtract_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
subtract-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
list_wf,
trans_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
subtype_rel_self,
set_wf,
primrec-wf2,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
add-member-int_seg2,
le_wf,
squash_wf,
true_wf,
iff_weakening_equal,
add-swap,
add-mul-special,
zero-mul,
stuck-spread,
base_wf,
reduce_hd_cons_lemma,
select0,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
addLevel,
allFunctionality,
impliesFunctionality,
because_Cache,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
levelHypothesis,
allLevelFunctionality,
impliesLevelFunctionality,
functionEquality,
functionExtensionality,
cumulativity,
natural_numberEquality,
addEquality,
independent_pairFormation,
intEquality,
minusEquality,
dependent_set_memberEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
pointwiseFunctionality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
universeEquality,
instantiate,
hyp_replacement,
imageMemberEquality,
productEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}L:T List
((\mforall{}i:\mBbbN{}||L|| - 1. R[L[i];L[i + 1]])
{}\mRightarrow{} R[last(L);hd(L)] supposing \mneg{}\muparrow{}null(L)
{}\mRightarrow{} (\mforall{}a\mmember{}L.(\mforall{}b\mmember{}L.R[a;b])))))
Date html generated:
2018_05_21-PM-07_41_04
Last ObjectModification:
2018_05_19-PM-04_48_32
Theory : general
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