Nuprl Lemma : tree_ind_wf_simple
∀[E,A:Type]. ∀[v:tree(E)]. ∀[leaf:value:E ⟶ A]. ∀[node:left:tree(E) ⟶ right:tree(E) ⟶ A ⟶ A ⟶ A].
(tree_ind(v;
tree_leaf(value)⇒ leaf[value];
tree_node(left,right)⇒ rec1,rec2.node[left;right;rec1;rec2]) ∈ A)
Proof
Definitions occuring in Statement :
tree_ind: tree_ind,
tree: tree(E),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3;s4],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
tree_ind_wf,
true_wf,
tree_wf,
istype-true,
subtype_rel_dep_function,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
universeIsType,
because_Cache,
functionExtensionality,
applyEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
functionEquality,
inhabitedIsType,
setEquality,
independent_isectElimination,
lambdaFormation_alt,
setIsType,
setElimination,
rename,
applyLambdaEquality,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[E,A:Type]. \mforall{}[v:tree(E)]. \mforall{}[leaf:value:E {}\mrightarrow{} A].
\mforall{}[node:left:tree(E) {}\mrightarrow{} right:tree(E) {}\mrightarrow{} A {}\mrightarrow{} A {}\mrightarrow{} A].
(tree\_ind(v;
tree\_leaf(value){}\mRightarrow{} leaf[value];
tree\_node(left,right){}\mRightarrow{} rec1,rec2.node[left;right;rec1;rec2]) \mmember{} A)
Date html generated:
2020_05_20-AM-07_48_02
Last ObjectModification:
2020_01_24-PM-02_47_50
Theory : tree_1
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