Nuprl Lemma : ml-length-sq
∀[T:Type]. ∀[l:T List]. (ml-length(l) ~ ||l||) supposing valueall-type(T)
Proof
Definitions occuring in Statement :
ml-length: ml-length(l),
length: ||as||,
list: T List,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
ml-length: ml-length(l),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
or: P ∨ Q,
ml_apply: f(x),
spreadcons: spreadcons,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
nil: [],
it: ⋅,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
top: Top,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
subtract: n - m,
subtype_rel: A ⊆r B,
has-value: (a)↓,
has-valueall: has-valueall(a),
bfalse: ff
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
istype-void,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
add-associates,
istype-int,
add-commutes,
add-swap,
zero-add,
length_of_cons_lemma,
valueall-type-has-valueall,
list_wf,
list-valueall-type,
cons_wf,
evalall-reduce,
null_cons_lemma,
length_wf,
le_weakening2,
valueall-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomSqEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
callbyvalueReduce,
sqleReflexivity,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :isect_memberEquality_alt,
Error :equalityIsType1,
because_Cache,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
applyEquality,
minusEquality,
Error :equalityIsType4,
baseApply,
closedConclusion,
addEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[l:T List]. (ml-length(l) \msim{} ||l||) supposing valueall-type(T)
Date html generated:
2019_06_20-PM-00_44_27
Last ObjectModification:
2018_10_07-AM-00_35_09
Theory : ML
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