Nuprl Lemma : MTree-induction2
∀[T:Type]. ∀[P:MultiTree(T) ⟶ ℙ].
((∀labels:{L:Atom List| 0 < ||L||} . ∀children:{a:Atom| (a ∈ labels)} ⟶ MultiTree(T).
((∀a∈labels.P[children a]) ⇒ P[MTree_Node(labels;children)]))
⇒ (∀val:T. P[MTree_Leaf(val)])
⇒ {∀x:MultiTree(T). P[x]})
Proof
Definitions occuring in Statement :
MTree_Leaf: MTree_Leaf(val),
MTree_Node: MTree_Node(labels;children),
MultiTree: MultiTree(T),
l_all: (∀x∈L.P[x]),
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,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
less_than: a < b,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
MTree-rank: MTree-rank(t),
MTree_Node: MTree_Node(labels;children),
MTree_Leaf?: MTree_Leaf?(v),
pi1: fst(t),
MTree_Node-children: MTree_Node-children(v),
pi2: snd(t),
MTree_Node-labels: MTree_Node-labels(v),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
cand: A c∧ B
Lemmas referenced :
l_member-settype,
member_map,
l_exists_iff,
map-length,
subtype_rel_list,
imax-list-ub,
map_wf,
list-subtype,
l_all_iff,
MultiTree-induction,
int_term_value_add_lemma,
itermAdd_wf,
nat_properties,
primrec-wf2,
set_wf,
decidable__lt,
nat_wf,
MTree-rank_wf,
guard_wf,
le_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
intformeq_wf,
itermSubtract_wf,
intformnot_wf,
decidable__le,
lelt_wf,
false_wf,
int_seg_subtype,
subtract_wf,
decidable__equal_int,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
MTree_Node_wf,
l_all_wf2,
MultiTree_wf,
l_member_wf,
length_wf,
less_than_wf,
list_wf,
MTree_Leaf_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
setEquality,
atomEquality,
natural_numberEquality,
because_Cache,
setElimination,
rename,
functionEquality,
cumulativity,
dependent_set_memberEquality,
universeEquality,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
equalityTransitivity,
equalitySymmetry,
levelHypothesis,
hypothesis_subsumption,
introduction,
addEquality,
independent_functionElimination,
equalityEquality,
productEquality
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{}a\mmember{}labels.P[children a]) {}\mRightarrow{} P[MTree\_Node(labels;children)]))
{}\mRightarrow{} (\mforall{}val:T. P[MTree\_Leaf(val)])
{}\mRightarrow{} \{\mforall{}x:MultiTree(T). P[x]\})
Date html generated:
2016_05_16-AM-08_54_10
Last ObjectModification:
2016_01_17-AM-09_43_19
Theory : C-semantics
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