Nuprl Lemma : implies-equiv-props
∀L:ℙ List+. ((∀i:ℕ||L|| - 1. (L[i] ⇒ L[i + 1])) ⇒ (last(L) ⇒ hd(L)) ⇒ equiv-props(L))
Proof
Definitions occuring in Statement :
equiv-props: equiv-props(L),
last: last(L),
select: L[n],
listp: A List+,
length: ||as||,
hd: hd(l),
int_seg: {i..j-},
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
listp: A List+,
prop: ℙ,
uimplies: b supposing a,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
subtract: n - m,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
not: ¬A,
false: False,
cons: [a / b],
bfalse: ff,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
guard: {T},
last: last(L),
cand: A c∧ B,
equiv-props: equiv-props(L),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
last_wf,
listp_properties,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
istype-void,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
hd_wf,
int_seg_wf,
subtract_wf,
length_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
decidable__lt,
subtract-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
subtype_rel_self,
itermAdd_wf,
int_term_value_add_lemma,
listp_wf,
istype-less_than,
primrec-wf2,
less_than_wf,
nat_properties,
istype-nat,
select0,
istype-le,
add-associates,
add-swap,
add-commutes,
zero-add,
int_seg_subtype_nat,
istype-false,
reduce_hd_cons_lemma,
subtract-add-cancel,
decidable__equal_int,
intformeq_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_minus_lemma,
squash_wf,
le_wf,
list_wf,
istype-universe,
minus-add,
minus-minus,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
because_Cache,
imageElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
applyEquality,
addEquality,
Error :setIsType,
functionEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hyp_replacement,
productEquality,
imageMemberEquality,
multiplyEquality,
Error :inhabitedIsType
Latex:
\mforall{}L:\mBbbP{} List\msupplus{}. ((\mforall{}i:\mBbbN{}||L|| - 1. (L[i] {}\mRightarrow{} L[i + 1])) {}\mRightarrow{} (last(L) {}\mRightarrow{} hd(L)) {}\mRightarrow{} equiv-props(L))
Date html generated:
2019_06_20-PM-01_50_24
Last ObjectModification:
2019_03_26-AM-11_01_01
Theory : list_1
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