Nuprl Lemma : islist-iff-length-has-value
∀[T:Type]. ∀[t:colist(T)]. uiff((is-list(t))↓;(||t||)↓)
Proof
Definitions occuring in Statement :
is-list: is-list(t),
length: ||as||,
colist: colist(T),
has-value: (a)↓,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
has-value: (a)↓,
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
is-list-fun: is-list-fun(),
prop: ℙ,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
is-list: is-list(t),
is-list-approx: is-list-approx(j),
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
ext-eq: A ≡ B,
unit: Unit,
list_ind: list_ind,
length: ||as||,
nil: [],
it: ⋅,
cons: [a / b],
so_apply: x[s],
so_lambda: λ2x.t[x],
pi2: snd(t),
guard: {T},
subtype_rel: A ⊆r B,
bool: 𝔹,
btrue: tt,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
compose: f o g,
less_than': less_than'(a;b),
le: A ≤ B
Lemmas referenced :
istype-universe,
colist_wf,
subtract-1-ge-0,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
strictness-apply,
fun_exp0_lemma,
has-value_wf_base,
bottom_diverge,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
is-list-approx-step,
colist-ext,
subtype_rel_b-union-right,
unit_wf2,
b-union_wf,
subtype_rel_transitivity,
unit_subtype_colist,
co-list-cases,
is-exception_wf,
length_of_cons_lemma,
termination,
nat_wf,
int-value-type,
le_wf,
set-value-type,
colength_wf,
length-is-colength,
value-type-has-value,
is-list-approx_wf,
decidable__le,
has-value_wf-partial,
bool_wf,
union-value-type,
is-list_wf,
fun_exp_unroll_1,
istype-sqequal,
istype-le,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
int_formula_prop_eq_lemma,
intformeq_wf,
istype-assert,
not_wf,
bnot_wf,
assert_wf,
int_subtype_base,
equal-wf-base,
eq_int_wf,
fun_exp_unroll,
bottom_wf-partial,
add-wf-partial-nat,
istype-false,
nat-partial-nat
Rules used in proof :
universeEquality,
instantiate,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
universeIsType,
hypothesis,
axiomSqleEquality,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
independent_pairFormation,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation_alt,
intWeakElimination,
rename,
setElimination,
compactness,
lambdaEquality,
voidEquality,
isect_memberEquality,
baseClosed,
cumulativity,
because_Cache,
unionElimination,
dependent_set_memberEquality_alt,
productEquality,
equalityElimination,
productElimination,
hypothesis_subsumption,
sqleReflexivity,
divergentSqle,
intEquality,
closedConclusion,
applyEquality,
addEquality,
equalityIsType1,
promote_hyp,
equalityIstype,
functionIsType,
sqequalBase,
baseApply,
callbyvalueAdd
Latex:
\mforall{}[T:Type]. \mforall{}[t:colist(T)]. uiff((is-list(t))\mdownarrow{};(||t||)\mdownarrow{})
Date html generated:
2020_05_20-AM-09_08_02
Last ObjectModification:
2020_02_03-AM-11_34_30
Theory : eval!all
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