Nuprl Lemma : MultiTree_ind_wf
∀[T,A:Type]. ∀[R:A ⟶ MultiTree(T) ⟶ ℙ]. ∀[v:MultiTree(T)]. ∀[Node:labels:{L:Atom List| 0 < ||L||}
⟶ children:({a:Atom| (a ∈ labels)} ⟶ MultiTree(T)\000C)
⟶ (u:{a:Atom| (a ∈ labels)} ⟶ {x:A| R[x;children \000Cu]} )
⟶ {x:A| R[x;MTree_Node(labels;children)]} ].
∀[Leaf:val:T ⟶ {x:A| R[x;MTree_Leaf(val)]} ].
(MultiTree_ind(v;
MTree_Node(labels,children)⇒ rec1.Node[labels;children;rec1];
MTree_Leaf(val)⇒ Leaf[val]) ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement :
MultiTree_ind: MultiTree_ind,
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,
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
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],
member: t ∈ T,
MultiTree_ind: MultiTree_ind,
so_apply: x[s],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
MultiTree-definition,
MultiTree-induction,
uniform-comp-nat-induction,
MultiTree-ext,
eq_atom: x =a y,
bool_cases_sqequal,
eqff_to_assert,
any: any x,
btrue: tt,
bfalse: ff,
it: ⋅,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
subtype_rel: A ⊆r B
Lemmas referenced :
MultiTree-definition,
MultiTree-induction,
uniform-comp-nat-induction,
MultiTree-ext,
bool_cases_sqequal,
eqff_to_assert,
set_wf,
all_wf,
MTree_Leaf_wf,
MTree_Node_wf,
l_member_wf,
length_wf,
less_than_wf,
list_wf,
MultiTree_wf,
base_wf,
lifting-strict-atom_eq,
is-exception_wf,
has-value_wf_base,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
thin,
lemma_by_obid,
hypothesis,
lambdaFormation,
because_Cache,
sqequalSqle,
divergentSqle,
callbyvalueDecide,
sqequalHypSubstitution,
unionEquality,
unionElimination,
sqleReflexivity,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
decideExceptionCases,
axiomSqleEquality,
exceptionSqequal,
baseApply,
closedConclusion,
baseClosed,
isectElimination,
independent_isectElimination,
independent_pairFormation,
inrFormation,
imageMemberEquality,
imageElimination,
inlFormation,
instantiate,
extract_by_obid,
applyEquality,
lambdaEquality,
isectEquality,
universeEquality,
functionEquality,
cumulativity,
setEquality,
atomEquality,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
axiomEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[R:A {}\mrightarrow{} MultiTree(T) {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:MultiTree(T)].
\mforall{}[Node:labels:\{L:Atom List| 0 < ||L||\}
{}\mrightarrow{} children:(\{a:Atom| (a \mmember{} labels)\} {}\mrightarrow{} MultiTree(T))
{}\mrightarrow{} (u:\{a:Atom| (a \mmember{} labels)\} {}\mrightarrow{} \{x:A| R[x;children u]\} )
{}\mrightarrow{} \{x:A| R[x;MTree\_Node(labels;children)]\} ]. \mforall{}[Leaf:val:T {}\mrightarrow{} \{x:A| R[x;MTree\_Leaf(val)]\} ].
(MultiTree\_ind(v;
MTree\_Node(labels,children){}\mRightarrow{} rec1.Node[labels;children;rec1];
MTree\_Leaf(val){}\mRightarrow{} Leaf[val]) \mmember{} \{x:A| R[x;v]\} )
Date html generated:
2016_05_16-AM-08_53_46
Last ObjectModification:
2016_01_17-AM-09_42_34
Theory : C-semantics
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