Nuprl Lemma : tree_ind_wf
∀[E,A:Type]. ∀[R:A ⟶ tree(E) ⟶ ℙ]. ∀[v:tree(E)]. ∀[leaf:value:E ⟶ {x:A| R[x;tree_leaf(value)]} ].
∀[node:left:tree(E) ⟶ right:tree(E) ⟶ {x:A| R[x;left]} ⟶ {x:A| R[x;right]} ⟶ {x:A| R[x;tree_node(left;right)]} ].
(tree_ind(v;
tree_leaf(value)⇒ leaf[value];
tree_node(left,right)⇒ rec1,rec2.node[left;right;rec1;rec2]) ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement :
tree_ind: tree_ind,
tree_node: tree_node(left;right),
tree_leaf: tree_leaf(value),
tree: tree(E),
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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
tree_ind: tree_ind,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s],
so_apply: x[s1;s2],
tree-definition,
tree-induction,
uniform-comp-nat-induction,
tree-ext,
eq_atom: x =a y,
btrue: tt,
it: ⋅,
bfalse: ff,
bool_cases_sqequal,
eqff_to_assert,
any: any x,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
so_lambda: so_lambda4,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
tree-definition,
has-value_wf_base,
is-exception_wf,
lifting-strict-atom_eq,
strict4-decide,
tree_wf,
tree_leaf_wf,
tree_node_wf,
subtype_rel_function,
subtype_rel_self,
istype-universe,
tree-induction,
uniform-comp-nat-induction,
tree-ext,
bool_cases_sqequal,
eqff_to_assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
Error :memTop,
inhabitedIsType,
hypothesis,
lambdaFormation_alt,
thin,
sqequalSqle,
divergentSqle,
callbyvalueDecide,
sqequalHypSubstitution,
hypothesisEquality,
unionElimination,
sqleReflexivity,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
introduction,
decideExceptionCases,
axiomSqleEquality,
exceptionSqequal,
baseApply,
closedConclusion,
baseClosed,
extract_by_obid,
isectElimination,
independent_isectElimination,
lambdaEquality_alt,
isectIsType,
because_Cache,
functionIsType,
universeIsType,
universeEquality,
setIsType,
applyEquality,
functionEquality,
setEquality,
instantiate,
functionExtensionality
Latex:
\mforall{}[E,A:Type]. \mforall{}[R:A {}\mrightarrow{} tree(E) {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:tree(E)]. \mforall{}[leaf:value:E {}\mrightarrow{} \{x:A| R[x;tree\_leaf(value)]\} ].
\mforall{}[node:left:tree(E)
{}\mrightarrow{} right:tree(E)
{}\mrightarrow{} \{x:A| R[x;left]\}
{}\mrightarrow{} \{x:A| R[x;right]\}
{}\mrightarrow{} \{x:A| R[x;tree\_node(left;right)]\} ].
(tree\_ind(v;
tree\_leaf(value){}\mRightarrow{} leaf[value];
tree\_node(left,right){}\mRightarrow{} rec1,rec2.node[left;right;rec1;rec2]) \mmember{} \{x:A| R[x;v]\} )
Date html generated:
2020_05_20-AM-07_48_00
Last ObjectModification:
2020_01_24-PM-03_06_20
Theory : tree_1
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