Nuprl Lemma : accum_induction
∀[T:Type]. ∀[Q:(T List) ⟶ ℙ]. ((∀[ys:T List]. (Q[ys] ⇒ (∀y:T. Q[ys @ [y]]))) ⇒ Q[[]] ⇒ {∀zs:T List. Q[zs]})
Proof
Definitions occuring in Statement :
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
le: A ≤ B,
less_than: a < b,
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
bfalse: ff,
int_iseg: {i...j},
cand: A c∧ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
subtract: n - m,
less_than': less_than'(a;b)
Lemmas referenced :
nil_wf,
subtype_rel_self,
istype-universe,
append_wf,
cons_wf,
list_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
le_wf,
length_wf,
non_neg_length,
decidable__assert,
null_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
length_wf_nat,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
list_accum_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
list_accum_cons_lemma,
last-lemma-sq,
list_accum_append,
firstn_wf,
subtype_rel_list,
top_wf,
squash_wf,
true_wf,
length_firstn_eq,
subtract_nat_wf,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
subtract-is-int-iff,
false_wf,
iff_weakening_equal,
last_wf
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
Error :isect_memberEquality_alt,
Error :lambdaEquality_alt,
Error :universeIsType,
because_Cache,
cut,
applyEquality,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
Error :isectIsType,
Error :functionIsType,
instantiate,
universeEquality,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
voidElimination,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
imageElimination,
Error :inhabitedIsType,
addEquality,
promote_hyp,
imageMemberEquality,
baseClosed,
minusEquality,
Error :equalityIsType1,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[Q:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}[ys:T List]. (Q[ys] {}\mRightarrow{} (\mforall{}y:T. Q[ys @ [y]]))) {}\mRightarrow{} Q[[]] {}\mRightarrow{} \{\mforall{}zs:T List. Q[zs]\})
Date html generated:
2019_06_20-PM-01_30_16
Last ObjectModification:
2018_10_05-PM-04_46_57
Theory : list_1
Home
Index