Nuprl Lemma : wfd-tree-induction
∀[A:Type]. ∀[P:wfd-tree(A) ⟶ ℙ].
(P[w-nil()] ⇒ (∀f:A ⟶ wfd-tree(A). ((∀a:A. P[f a]) ⇒ P[mk-wfd-tree(f)])) ⇒ (∀w:wfd-tree(A). P[w]))
Proof
Definitions occuring in Statement :
mk-wfd-tree: mk-wfd-tree(f),
w-nil: w-nil(),
wfd-tree2: wfd-tree(A),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
wfd-tree2: wfd-tree(A),
squash: ↓T,
or: P ∨ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
co-w-select: w@s,
btrue: tt,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
assert: ↑b,
co-w-null: co-w-null(w),
isl: isl(x),
w-nil: w-nil(),
bfalse: ff,
wfd-subtrees: wfd-subtrees(w),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
w-bars: w-bars(w;p)
Lemmas referenced :
wfd-tree2_wf,
all_wf,
mk-wfd-tree_wf,
w-nil_wf,
bool-bar-induction,
co-w-select-wfd,
list_wf,
co-w-null_wf,
co-w-select_wf,
set_wf,
assert_wf,
append_wf,
cons_wf,
nil_wf,
not_wf,
nat_wf,
wfd-tree-cases,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
assert_elim,
not_assert_elim,
btrue_neq_bfalse,
list_induction,
isect_wf,
list_ind_nil_lemma,
null_nil_lemma,
reduce_tl_nil_lemma,
co_w_select_nil_lemma,
wfd-subtrees_wf,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert-co-w-null,
equal-wf-T-base,
co-w_wf,
iff_imp_equal_bool,
bfalse_wf,
false_wf,
assert_of_ff,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
functionEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
universeEquality,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
unionElimination,
independent_isectElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
productElimination,
addLevel,
voidElimination,
levelHypothesis,
isect_memberEquality,
voidEquality,
hyp_replacement,
applyLambdaEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
independent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[P:wfd-tree(A) {}\mrightarrow{} \mBbbP{}].
(P[w-nil()]
{}\mRightarrow{} (\mforall{}f:A {}\mrightarrow{} wfd-tree(A). ((\mforall{}a:A. P[f a]) {}\mRightarrow{} P[mk-wfd-tree(f)]))
{}\mRightarrow{} (\mforall{}w:wfd-tree(A). P[w]))
Date html generated:
2018_05_21-PM-10_18_20
Last ObjectModification:
2017_07_26-PM-06_36_36
Theory : bar!induction
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