Nuprl Lemma : binary-tree_ind_wf
∀[A:Type]. ∀[R:A ⟶ binary-tree() ⟶ ℙ]. ∀[v:binary-tree()]. ∀[Leaf:val:ℤ ⟶ {x:A| R[x;btr_Leaf(val)]} ].
∀[Node:left:binary-tree()
⟶ right:binary-tree()
⟶ {x:A| R[x;left]}
⟶ {x:A| R[x;right]}
⟶ {x:A| R[x;btr_Node(left;right)]} ].
(binary-tree_ind(v;
btr_Leaf(val)⇒ Leaf[val];
btr_Node(left,right)⇒ rec1,rec2.Node[left;right;rec1;rec2]) ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement :
binary-tree_ind: binary-tree_ind,
btr_Node: btr_Node(left;right),
btr_Leaf: btr_Leaf(val),
binary-tree: binary-tree(),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
binary-tree_ind: binary-tree_ind,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s],
so_apply: x[s1;s2],
binary-tree-definition,
binary-tree-induction,
uniform-comp-nat-induction,
binary-tree-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_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 :
binary-tree-definition,
binary-tree-induction,
uniform-comp-nat-induction,
binary-tree-ext,
bool_cases_sqequal,
eqff_to_assert,
set_wf,
all_wf,
guard_wf,
btr_Node_wf,
btr_Leaf_wf,
binary-tree_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,
functionEquality,
cumulativity,
universeEquality,
intEquality,
setEquality,
setElimination,
rename,
dependent_set_memberEquality,
axiomEquality
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} binary-tree() {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:binary-tree()]. \mforall{}[Leaf:val:\mBbbZ{} {}\mrightarrow{} \{x:A|
R[x;btr\_Leaf(val)]\} ].
\mforall{}[Node:left:binary-tree()
{}\mrightarrow{} right:binary-tree()
{}\mrightarrow{} \{x:A| R[x;left]\}
{}\mrightarrow{} \{x:A| R[x;right]\}
{}\mrightarrow{} \{x:A| R[x;btr\_Node(left;right)]\} ].
(binary-tree\_ind(v;
btr\_Leaf(val){}\mRightarrow{} Leaf[val];
btr\_Node(left,right){}\mRightarrow{} rec1,rec2.Node[left;right;rec1;rec2]) \mmember{} \{x:A| R[x;v]\} )
Date html generated:
2016_05_16-AM-09_07_18
Last ObjectModification:
2016_01_17-AM-09_42_45
Theory : C-semantics
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