Nuprl Lemma : MultiTree-induction
∀[T:Type]. ∀[P:MultiTree(T) ⟶ ℙ].
((∀labels:{L:Atom List| 0 < ||L||} . ∀children:{a:Atom| (a ∈ labels)} ⟶ MultiTree(T).
((∀u:{a:Atom| (a ∈ labels)} . P[children u]) ⇒ P[MTree_Node(labels;children)]))
⇒ (∀val:T. P[MTree_Leaf(val)])
⇒ {∀v:MultiTree(T). P[v]})
Proof
Definitions occuring in Statement :
MTree_Leaf: MTree_Leaf(val),
MTree_Node: MTree_Node(labels;children),
MultiTree: MultiTree(T),
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
MTree_Node: MTree_Node(labels;children),
MultiTree_size: MultiTree_size(p),
pi1: fst(t),
pi2: snd(t),
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
sq_stable: SqStable(P),
l_member: (x ∈ l),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
MTree_Leaf: MTree_Leaf(val)
Lemmas referenced :
and_wf,
equal-wf-base-T,
MTree_Node_wf,
less_than_wf,
list_wf,
MTree_Leaf_wf,
uall_wf,
neg_assert_of_eq_atom,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
set_wf,
sq_stable__le,
lelt_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
sum-nat-less,
int_term_value_add_lemma,
itermAdd_wf,
length_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
int_seg_properties,
list-subtype,
l_member_wf,
select_wf,
length_wf_nat,
sum-nat,
atom_subtype_base,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
bool_wf,
eq_atom_wf,
MultiTree-ext,
less_than'_wf,
nat_wf,
MultiTree_size_wf,
le_wf,
isect_wf,
MultiTree_wf,
all_wf,
uniform-comp-nat-induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
setElimination,
rename,
independent_functionElimination,
introduction,
productElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
tokenEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
instantiate,
cumulativity,
atomEquality,
setEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
equalityEquality,
functionEquality,
universeEquality,
productEquality,
addLevel,
levelHypothesis,
substitution
Latex:
\mforall{}[T:Type]. \mforall{}[P:MultiTree(T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}labels:\{L:Atom List| 0 < ||L||\} . \mforall{}children:\{a:Atom| (a \mmember{} labels)\} {}\mrightarrow{} MultiTree(T).
((\mforall{}u:\{a:Atom| (a \mmember{} labels)\} . P[children u]) {}\mRightarrow{} P[MTree\_Node(labels;children)]))
{}\mRightarrow{} (\mforall{}val:T. P[MTree\_Leaf(val)])
{}\mRightarrow{} \{\mforall{}v:MultiTree(T). P[v]\})
Date html generated:
2016_05_16-AM-08_53_34
Last ObjectModification:
2016_01_17-AM-09_42_59
Theory : C-semantics
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