Nuprl Lemma : fset-induction
∀[T:Type]
∀eq:EqDecider(T)
∀[P:fset(T) ⟶ ℙ]
((∀s:fset(T). SqStable(P[s]))
⇒ P[{}]
⇒ (∀s:fset(T). ∀x:T. (P[s] ⇒ P[fset-add(eq;x;s)] supposing ¬x ∈ s))
⇒ {∀s:fset(T). P[s]})
Proof
Definitions occuring in Statement :
empty-fset: {},
fset-add: fset-add(eq;x;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
sq_stable: SqStable(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
sq_stable: SqStable(P),
fset: fset(T),
quotient: x,y:A//B[x; y],
squash: ↓T,
true: True,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
fset-size: ||s||,
ge: i ≥ j ,
fset-member: a ∈ s,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than: a < b,
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff
Lemmas referenced :
fset_wf,
subtype_rel_self,
not_wf,
fset-member_wf,
fset-add_wf,
empty-fset_wf,
sq_stable_wf,
deq_wf,
le_wf,
fset-size_wf,
subtract_wf,
istype-int,
less_than_wf,
primrec-wf2,
all_wf,
nat_wf,
fset-size-empty,
decidable__le,
squash_wf,
list_wf,
set-equal_wf,
set-equal-reflex,
equal-wf-base,
member_wf,
true_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
length-remove-repeats-le,
hd_wf,
length_wf,
assert-deq-member,
hd_member,
decidable__lt,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
assert_wf,
null_wf,
fset-remove_wf,
fset-add-remove,
iff_weakening_equal,
fset-size-remove,
iff_weakening_uiff,
member-fset-remove
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
Error :universeIsType,
introduction,
extract_by_obid,
isectElimination,
Error :functionIsType,
applyEquality,
instantiate,
universeEquality,
Error :isectIsType,
because_Cache,
natural_numberEquality,
rename,
setElimination,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
Error :setIsType,
cumulativity,
functionEquality,
functionExtensionality,
productElimination,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
unionElimination,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
productEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
hypothesis_subsumption,
Error :equalityIsType1,
Error :productIsType
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T)
\mforall{}[P:fset(T) {}\mrightarrow{} \mBbbP{}]
((\mforall{}s:fset(T). SqStable(P[s]))
{}\mRightarrow{} P[\{\}]
{}\mRightarrow{} (\mforall{}s:fset(T). \mforall{}x:T. (P[s] {}\mRightarrow{} P[fset-add(eq;x;s)] supposing \mneg{}x \mmember{} s))
{}\mRightarrow{} \{\mforall{}s:fset(T). P[s]\})
Date html generated:
2019_06_20-PM-02_00_01
Last ObjectModification:
2018_09_30-PM-02_47_12
Theory : finite!sets
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