Nuprl Lemma : basic-bar-induction
∀[T:Type]. ∀[R,A:(T List) ⟶ ℙ].
((∀s:T List. Dec(R[s]))
⇒ (∀s:T List. (R[s] ⇒ A[s]))
⇒ (∀s:T List. ((∀t:T. A[s @ [t]]) ⇒ A[s]))
⇒ (∀alpha:ℕ ⟶ T. (↓∃n:ℕ. R[map(alpha;upto(n))]))
⇒ A[[]])
Proof
Definitions occuring in Statement :
upto: upto(n),
map: map(f;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
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],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
all: ∀x:A. B[x],
guard: {T},
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3]
Lemmas referenced :
bar-induction,
all_wf,
nat_wf,
squash_wf,
exists_wf,
map_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
upto_wf,
list_wf,
append_wf,
cons_wf,
nil_wf,
decidable_wf,
list_ind_nil_lemma
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
functionEquality,
cumulativity,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
universeEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R,A:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}s:T List. Dec(R[s]))
{}\mRightarrow{} (\mforall{}s:T List. (R[s] {}\mRightarrow{} A[s]))
{}\mRightarrow{} (\mforall{}s:T List. ((\mforall{}t:T. A[s @ [t]]) {}\mRightarrow{} A[s]))
{}\mRightarrow{} (\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}n:\mBbbN{}. R[map(alpha;upto(n))]))
{}\mRightarrow{} A[[]])
Date html generated:
2016_05_14-PM-03_19_30
Last ObjectModification:
2015_12_26-PM-01_41_16
Theory : list_1
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