Nuprl Lemma : last_induction
∀[T:Type]. ∀[Q:(T List) ⟶ ℙ]. (Q[[]] ⇒ (∀ys:T List. ∀y:T. (Q[ys] ⇒ Q[ys @ [y]])) ⇒ {∀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 :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
set_wf,
lelt_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
less_than_wf,
decidable__assert,
null_wf,
all_wf,
list_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
append_wf,
cons_wf,
nil_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
last_lemma,
last_wf,
squash_wf,
true_wf,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
length-singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
cumulativity,
imageElimination,
independent_functionElimination,
functionEquality,
functionExtensionality,
addEquality,
universeEquality,
promote_hyp,
imageMemberEquality,
baseClosed,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[Q:(T List) {}\mrightarrow{} \mBbbP{}].
(Q[[]] {}\mRightarrow{} (\mforall{}ys:T List. \mforall{}y:T. (Q[ys] {}\mRightarrow{} Q[ys @ [y]])) {}\mRightarrow{} \{\mforall{}zs:T List. Q[zs]\})
Date html generated:
2016_10_21-AM-10_08_40
Last ObjectModification:
2016_07_12-AM-05_28_17
Theory : list_1
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