Nuprl Lemma : transitive-loop2
∀[T:Type]
∀L:T List
∀[R:{x:T| (x ∈ L)} ⟶ {x:T| (x ∈ L)} ⟶ ℙ]
(Trans({x:T| (x ∈ L)} ;x,y.R[x;y])
⇒ (∀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),
l_member: (x ∈ 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,
set: {x:A| B[x]} ,
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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
true: True,
false: False,
cons: [a / b],
bfalse: ff,
guard: {T},
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
listp: A List+,
so_apply: x[s],
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]
Lemmas referenced :
transitive-loop,
l_member_wf,
list-subtype,
isect_wf,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
last_wf,
hd_wf,
listp_properties,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
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,
all_wf,
int_seg_wf,
subtract_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
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,
trans_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
functionExtensionality,
unionElimination,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
setElimination,
rename,
addEquality,
independent_pairFormation,
intEquality,
minusEquality,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
pointwiseFunctionality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[R:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]
(Trans(\{x:T| (x \mmember{} L)\} ;x,y.R[x;y])
{}\mRightarrow{} (\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_13
Last ObjectModification:
2017_07_26-PM-05_15_13
Theory : general
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