Nuprl Lemma : decidable__all_length
∀[T:Type]. ∀[P:(T List) ⟶ ℙ].
((∀L:T List. Dec(P[L])) ⇒ (∃k:ℕ. T ~ ℕk) ⇒ (∀n:ℕ. Dec(∀L:T List. ((||L|| = n ∈ ℤ) ⇒ P[L]))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
false: False,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
istype: istype(T),
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
decidable_wf,
int_seg_wf,
equipollent_wf,
istype-nat,
istype-void,
subtype_rel_self,
subtype_rel_dep_function,
equipollent-list,
exp_wf4,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
length_wf_nat,
equal-wf-base,
list_wf,
decidable__all_finite
Rules used in proof :
equalityTransitivity,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :inrFormation_alt,
Error :functionIsType,
Error :inlFormation_alt,
unionElimination,
rename,
setElimination,
equalitySymmetry,
sqequalBase,
baseClosed,
closedConclusion,
baseApply,
Error :equalityIstype,
Error :setIsType,
Error :universeIsType,
universeEquality,
cumulativity,
instantiate,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
natural_numberEquality,
Error :lambdaEquality_alt,
sqequalRule,
applyEquality,
intEquality,
hypothesis,
hypothesisEquality,
setEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}L:T List. Dec(P[L])) {}\mRightarrow{} (\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. Dec(\mforall{}L:T List. ((||L|| = n) {}\mRightarrow{} P[L]))))
Date html generated:
2019_06_20-PM-02_19_35
Last ObjectModification:
2019_06_06-PM-00_17_31
Theory : equipollence!!cardinality!
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