Nuprl Lemma : gcd-reduce-eq-constraints_wf2
∀[n:ℕ]. ∀[LL,sat:{L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List].
(gcd-reduce-eq-constraints(sat;LL) ∈ {L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List?)
Proof
Definitions occuring in Statement :
gcd-reduce-eq-constraints: gcd-reduce-eq-constraints(sat;LL),
length: ||as||,
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
unit: Unit,
member: t ∈ T,
set: {x:A| B[x]} ,
union: left + right,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
gcd-reduce-eq-constraints: gcd-reduce-eq-constraints(sat;LL),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
or: P ∨ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
uiff: uiff(P;Q),
and: P ∧ Q,
guard: {T},
subtract: n - m,
top: Top,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False,
cons: [a / b],
decidable: Dec(P),
nil: [],
it: ⋅,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
eager-map: eager-map(f;as),
list_ind: list_ind,
has-value: (a)↓,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
listp: A List+,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
less_than: a < b,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
accumulate_abort_wf,
list_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
set_subtype_base,
le_wf,
istype-int,
istype-nat,
unit_wf2,
list-cases,
length_of_nil_lemma,
sq_stable__le,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
product_subtype_list,
decidable__assert,
null_wf,
eager_map_cons_lemma,
null_cons_lemma,
null_nil_lemma,
istype-true,
eager_map_nil_lemma,
cons_wf,
it_wf,
value-type-has-value,
nat_wf,
set-value-type,
int-value-type,
absval_wf,
gcd-list_wf,
length_of_cons_lemma,
length_wf_nat,
decidable__lt,
istype-false,
not-lt-2,
istype-less_than,
length_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
remainder_wfa,
not-equal-2,
decidable__le,
istype-le,
not-le-2,
less-iff-le,
add-swap,
le-add-cancel2,
nequal_wf,
eq_int_wf,
assert_of_eq_int,
list-value-type,
eager-map_wf,
divide_wfa,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
iff_weakening_uiff,
assert_wf,
less_than_wf,
map-length,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
eager-map-is-map,
list-valueall-type,
set-valueall-type,
int-valueall-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
Error :lambdaEquality_alt,
independent_isectElimination,
hypothesis,
because_Cache,
Error :universeIsType,
setEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
natural_numberEquality,
Error :inlEquality_alt,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_functionElimination,
imageMemberEquality,
imageElimination,
addEquality,
productElimination,
Error :isect_memberEquality_alt,
voidElimination,
minusEquality,
promote_hyp,
hypothesis_subsumption,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
Error :equalityIstype,
callbyvalueReduce,
sqleReflexivity,
Error :functionIsType,
int_eqEquality,
Error :dependent_set_memberEquality_alt,
sqequalBase,
equalitySymmetry,
Error :inrEquality_alt,
equalityTransitivity,
independent_pairFormation,
equalityElimination,
lessCases,
axiomSqEquality,
Error :isectIsTypeImplies,
Error :inlFormation_alt,
Error :inrFormation_alt,
int_eqReduceTrueSq,
Error :dependent_pairFormation_alt,
instantiate,
cumulativity,
int_eqReduceFalseSq,
universeEquality,
Error :setIsType
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[LL,sat:\{L:\mBbbZ{} List| ||L|| = (n + 1)\} List].
(gcd-reduce-eq-constraints(sat;LL) \mmember{} \{L:\mBbbZ{} List| ||L|| = (n + 1)\} List?)
Date html generated:
2019_06_20-PM-00_51_01
Last ObjectModification:
2019_03_06-PM-05_17_23
Theory : omega
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