Nuprl Lemma : add-poly-prepend-sq1
∀[p,q,l:(ℤ × (ℤ List)) List]. (add-ipoly-prepend(p;q;l) ~ rev(l) + add-ipoly(p;q))
Proof
Definitions occuring in Statement :
add-ipoly-prepend: add-ipoly-prepend(p;q;l),
add-ipoly: add-ipoly(p;q),
rev-append: rev(as) + bs,
list: T List,
uall: ∀[x:A]. B[x],
product: x:A × B[x],
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
add-ipoly-prepend: add-ipoly-prepend(p;q;l),
add-ipoly: add-ipoly(p;q),
null: null(as),
btrue: tt,
ifthenelse: if b then t else f fi ,
has-value: (a)↓,
bfalse: ff,
imonomial-le: imonomial-le(m1;m2),
bool: 𝔹,
unit: Unit,
pi1: fst(t),
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev-append: rev(as) + bs
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
list_wf,
equal-wf-base,
nat_wf,
list-cases,
product_subtype_list,
spread_cons_lemma,
colength_wf_list,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
equal-wf-T-base,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
equal_wf,
list_subtype_base,
product_subtype_base,
int_subtype_base,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
bool_wf,
bool_subtype_base,
btrue_wf,
value-type-has-value,
list-value-type,
rev-append_wf,
bfalse_wf,
intlex_wf,
pi2_wf,
add-ipoly-wf1,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
int-value-type,
valueall-type-has-valueall,
list-valueall-type,
product-valueall-type,
int-valueall-type,
evalall-reduce,
eq_int_wf,
assert_of_eq_int,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
has-value_wf_base,
is-exception_wf,
cons_wf,
decidable__equal_int,
rev_app_cons_lemma,
list_accum_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
productEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
voidEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
callbyvalueReduce,
sqleReflexivity,
independent_pairEquality,
equalityElimination,
int_eqReduceTrueSq,
dependent_pairFormation,
int_eqReduceFalseSq,
divergentSqle,
int_eqEquality
Latex:
\mforall{}[p,q,l:(\mBbbZ{} \mtimes{} (\mBbbZ{} List)) List]. (add-ipoly-prepend(p;q;l) \msim{} rev(l) + add-ipoly(p;q))
Date html generated:
2017_09_29-PM-05_52_59
Last ObjectModification:
2017_05_11-PM-06_42_14
Theory : omega
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