Nuprl Lemma : W-type-induction
∀[A:Type]
((∀x,y:A. Dec(x = y ∈ A))
⇒ (∀[B:A ⟶ Type]. ∀[P:W-type(A; a.B[a]) ⟶ ℙ].
((∀a:A. ∀f:B[a] ⟶ W-type(A; a.B[a]). ((∀b:B[a]. P[f b]) ⇒ P[W-sup(a;f)])) ⇒ (∀w:W-type(A; a.B[a]). P[w]))))
Proof
Definitions occuring in Statement :
W-sup: W-sup(a;f),
W-type: W-type(A; a.B[a]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
Wselect: Wselect(w;s),
W-select: W-select(w;s),
ifthenelse: if b then t else f fi ,
null: null(as),
nil: [],
it: ⋅,
btrue: tt,
sq_stable: SqStable(P),
isr: isr(x),
assert: ↑b,
bfalse: ff,
false: False,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
guard: {T},
ext-eq: A ≡ B,
W-sup: W-sup(a;f),
deq: EqDecider(T),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
eqof: eqof(d),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
W-type: W-type(A; a.B[a]),
W-bars: W-bars(w;p)
Lemmas referenced :
bool-bar-induction,
unit_wf2,
Wselect_wf,
W-type_wf,
true_wf,
equal_wf,
list_wf,
isr_wf,
set_wf,
assert_wf,
all_wf,
append_wf,
cons_wf,
nil_wf,
not_wf,
nat_wf,
W-sup_wf,
decidable_wf,
sq_stable__assert,
false_wf,
deq-exists,
list_induction,
list_ind_nil_lemma,
W-type-ext,
subtype_rel_weakening,
list_ind_cons_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
subtype_rel-equal,
and_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
it_wf,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
bnot_wf,
eqof_wf,
member_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesisEquality,
unionEquality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
imageElimination,
imageMemberEquality,
baseClosed,
cumulativity,
universeEquality,
voidElimination,
natural_numberEquality,
productElimination,
promote_hyp,
isect_memberEquality,
voidEquality,
hypothesis_subsumption,
because_Cache,
productEquality,
equalityElimination,
inlEquality,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
dependent_pairFormation,
instantiate,
inrEquality,
functionExtensionality,
impliesFunctionality
Latex:
\mforall{}[A:Type]
((\mforall{}x,y:A. Dec(x = y))
{}\mRightarrow{} (\mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:W-type(A; a.B[a]) {}\mrightarrow{} \mBbbP{}].
((\mforall{}a:A. \mforall{}f:B[a] {}\mrightarrow{} W-type(A; a.B[a]). ((\mforall{}b:B[a]. P[f b]) {}\mRightarrow{} P[W-sup(a;f)]))
{}\mRightarrow{} (\mforall{}w:W-type(A; a.B[a]). P[w]))))
Date html generated:
2019_10_16-AM-11_38_01
Last ObjectModification:
2018_08_21-PM-02_00_11
Theory : bar!induction
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