Nuprl Lemma : stump'-inductive
∀T:Type. ∀t:wfd-tree(T).
(stump'(t)
= wfd-tree-rec(λn,s. (n =z 0);r.λn,s. if (n =z 0) then ff else r (s 0) (n - 1) (λi.(s (i + 1))) fi ;t)
∈ (n:ℕ ⟶ (ℕn ⟶ T) ⟶ 𝔹))
Proof
Definitions occuring in Statement :
stump': stump'(t),
wfd-tree-rec: wfd-tree-rec(b;r.F[r];t),
wfd-tree: wfd-tree(T),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bfalse: ff,
bool: 𝔹,
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
top: Top,
true: True,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
so_apply: x[s],
stump': stump'(t),
stump: stump(t),
empty-wfd-tree: empty-wfd-tree(t),
band: p ∧b q,
wfd-tree-rec: wfd-tree-rec(b;r.F[r];t),
W-rec: W-rec(a,f,r.F[a; f; r];w),
Wsup: Wsup(a;b),
less_than: a < b,
nequal: a ≠ b ∈ T
Lemmas referenced :
wfd-tree-induction,
equal_wf,
nat_wf,
int_seg_wf,
bool_wf,
stump'_wf,
wfd-tree-rec_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
bfalse_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel,
lelt_wf,
subtract_wf,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-member-int_seg2,
add-zero,
le-add-cancel2,
wfd-tree_wf,
all_wf,
wfd_tree_rec_leaf_lemma,
btrue_wf,
wfd_tree_rec_node_lemma,
int_subtype_base,
stump-nil,
le_antisymmetry_iff,
minus-zero,
le-add-cancel-alt,
less_than_transitivity1,
less_than_irreflexivity,
bnot_wf,
stump_wf,
not-equal-2,
less-iff-le,
int_upper_wf,
empty-wfd-tree_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
natural_numberEquality,
setElimination,
rename,
because_Cache,
cumulativity,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
applyEquality,
functionExtensionality,
isect_memberEquality,
voidEquality,
intEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
minusEquality,
universeEquality,
applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}t:wfd-tree(T).
(stump'(t)
= wfd-tree-rec(\mlambda{}n,s. (n =\msubz{} 0);r.\mlambda{}n,s. if (n =\msubz{} 0) then ff else r (s 0) (n - 1) (\mlambda{}i.(s (i + 1))) fi\000C ;t))
Date html generated:
2017_04_14-AM-07_45_37
Last ObjectModification:
2017_02_27-PM-03_16_57
Theory : co-recursion
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