Nuprl Lemma : last_induction_accum
∀[T:Type]. ∀[Q:(T List) ⟶ ℙ]. (Q[[]] ⇒ (∀[ys:T List]. (Q[ys] ⇒ (∀y:T. 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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
cand: A c∧ B
Lemmas referenced :
list_wf,
uall_wf,
all_wf,
append_wf,
cons_wf,
nil_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
le_wf,
length_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
lelt_wf,
decidable__assert,
null_wf,
list-cases,
list_accum_nil_lemma,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
length_wf_nat,
list_accum_append,
subtype_rel_list,
top_wf,
list_accum_cons_lemma,
last_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
universeEquality,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productElimination,
unionElimination,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
imageElimination,
promote_hyp,
baseClosed,
impliesFunctionality,
productEquality,
addEquality,
isectEquality
Latex:
\mforall{}[T:Type]. \mforall{}[Q:(T List) {}\mrightarrow{} \mBbbP{}].
(Q[[]] {}\mRightarrow{} (\mforall{}[ys:T List]. (Q[ys] {}\mRightarrow{} (\mforall{}y:T. Q[ys @ [y]]))) {}\mRightarrow{} \{\mforall{}zs:T List. Q[zs]\})
Date html generated:
2017_04_17-AM-07_33_25
Last ObjectModification:
2017_02_27-PM-04_11_10
Theory : list_1
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