Nuprl Lemma : mul_poly-sq
∀[p,q:iMonomial() List]. (mul_ipoly(p;q) ~ mul-ipoly(p;q))
Proof
Definitions occuring in Statement :
mul_ipoly: mul_ipoly(p;q),
mul-ipoly: mul-ipoly(p;q),
iMonomial: iMonomial(),
list: T List,
uall: ∀[x:A]. B[x],
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,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
mul-ipoly: mul-ipoly(p;q),
mul_ipoly: mul_ipoly(p;q),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
has-value: (a)↓,
iMonomial: iMonomial(),
int_nzero: ℤ-o,
has-valueall: has-valueall(a),
bfalse: ff,
add_ipoly: add_ipoly(p;q)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
list_wf,
iMonomial_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
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,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
null_nil_lemma,
value-type-has-value,
list-value-type,
valueall-type-has-valueall,
list-valueall-type,
product-valueall-type,
int_nzero_wf,
sorted_wf,
subtype_rel_self,
set-valueall-type,
nequal_wf,
int-valueall-type,
cons_wf,
evalall-reduce,
null_cons_lemma,
add-ipoly-prepend_wf,
nil_wf,
cbv_list_accum-is-list_accum,
mul-mono-poly_wf1,
eager-accum-list_accum,
add-ipoly_wf1,
list_accum_wf,
squash_wf,
true_wf,
add_ipoly-sq,
list_subtype_base,
product_subtype_base
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,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
voidEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
instantiate,
cumulativity,
callbyvalueReduce,
sqleReflexivity,
setEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[p,q:iMonomial() List]. (mul\_ipoly(p;q) \msim{} mul-ipoly(p;q))
Date html generated:
2017_09_29-PM-05_53_48
Last ObjectModification:
2017_05_04-PM-03_44_13
Theory : omega
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