Nuprl Lemma : wfd-tree-cases
∀[A:Type]
∀w:wfd-tree(A). ((w = w-nil() ∈ wfd-tree(A)) ∨ ((¬↑co-w-null(w)) ∧ (w = mk-wfd-tree(wfd-subtrees(w)) ∈ wfd-tree(A))))
Proof
Definitions occuring in Statement :
wfd-subtrees: wfd-subtrees(w),
mk-wfd-tree: mk-wfd-tree(f),
w-nil: w-nil(),
wfd-tree2: wfd-tree(A),
co-w-null: co-w-null(w),
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
wfd-tree2: wfd-tree(A),
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
co-w-null: co-w-null(w),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
unit: Unit,
it: ⋅,
w-nil: w-nil(),
cand: A c∧ B,
wfd-subtrees: wfd-subtrees(w),
mk-wfd-tree: mk-wfd-tree(f),
outr: outr(x),
true: True
Lemmas referenced :
decidable__assert,
co-w-null_wf,
wfd-tree2_wf,
not_wf,
assert_wf,
equal_wf,
mk-wfd-tree_wf,
w-nil_wf,
co-w-ext,
subtype_rel_transitivity,
co-w_wf,
unit_wf2,
subtype_rel_weakening,
true_wf,
false_wf,
wfd-subtrees_wf,
equal-wf-T-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
cumulativity,
hypothesisEquality,
setElimination,
rename,
hypothesis,
unionElimination,
universeEquality,
inlFormation,
productEquality,
functionExtensionality,
applyEquality,
independent_functionElimination,
voidElimination,
lambdaEquality,
because_Cache,
unionEquality,
functionEquality,
independent_isectElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
inrFormation,
independent_pairFormation,
baseClosed,
natural_numberEquality
Latex:
\mforall{}[A:Type]. \mforall{}w:wfd-tree(A). ((w = w-nil()) \mvee{} ((\mneg{}\muparrow{}co-w-null(w)) \mwedge{} (w = mk-wfd-tree(wfd-subtrees(w)))))
Date html generated:
2018_05_21-PM-10_18_10
Last ObjectModification:
2017_07_26-PM-06_36_34
Theory : bar!induction
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