Nuprl Lemma : add-polynom-length
∀[n:ℕ]. ∀[p,q:polyform(n) List]. (||add-polynom(n + 1;ff;p;q)|| = imax(||p||;||q||) ∈ ℤ)
Proof
Definitions occuring in Statement :
add-polynom: add-polynom(n;rmz;p;q),
polyform: polyform(n),
length: ||as||,
list: T List,
imax: imax(a;b),
nat: ℕ,
bfalse: ff,
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
subtract: n - m,
polyform: polyform(n),
less_than': less_than'(a;b),
less_than: a < b,
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
decidable: Dec(P),
le: A ≤ B,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
has-valueall: has-valueall(a),
has-value: (a)↓,
list_ind: list_ind,
length: ||as||,
nil: [],
evalall: evalall(t),
callbyvalueall: callbyvalueall,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
add-polynom: add-polynom(n;rmz;p;q),
or: P ∨ Q,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
btrue_wf,
false_wf,
add-is-int-iff,
le_int_wf,
ifthenelse_wf,
add_functionality_wrt_eq,
imax_wf,
bfalse_wf,
zero-add,
add-commutes,
add-swap,
add-associates,
add-subtract-cancel,
subtype_rel-equal,
add-polynom_wf1,
top_wf,
decidable__lt,
int-value-type,
value-type-has-value,
null_cons_lemma,
cons_wf,
length_of_cons_lemma,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
le_wf,
spread_cons_lemma,
product_subtype_list,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
non_neg_length,
assert_of_le_int,
ite_rw_true,
iff_weakening_equal,
imax_unfold,
length_wf,
true_wf,
squash_wf,
evalall-reduce,
valueall-type-polyform,
list-valueall-type,
list_wf,
valueall-type-has-valueall,
null_nil_lemma,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
length_of_nil_lemma,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
list-cases,
less_than_irreflexivity,
less_than_transitivity1,
polyform_wf,
colength_wf_list,
nat_wf,
equal-wf-T-base,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
closedConclusion,
baseApply,
pointwiseFunctionality,
minusEquality,
sqequalAxiom,
lessCases,
dependent_set_memberEquality,
applyLambdaEquality,
hypothesis_subsumption,
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination,
sqleReflexivity,
callbyvalueReduce,
cumulativity,
instantiate,
promote_hyp,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
addEquality,
unionElimination,
applyEquality,
because_Cache,
axiomEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polyform(n) List]. (||add-polynom(n + 1;ff;p;q)|| = imax(||p||;||q||))
Date html generated:
2017_04_20-AM-07_09_19
Last ObjectModification:
2017_04_17-AM-10_47_22
Theory : list_1
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