Nuprl Lemma : bfs-rm0-equiv
∀K:RngSig. ∀S:Type. ∀b:basic-formal-sum(K;S). ∀eq:EqDecider(|K|). (↓bfs-equiv(K;S;bfs-rm0(K;eq;b);b))
Proof
Definitions occuring in Statement :
bfs-rm0: bfs-rm0(K;eq;b),
bfs-equiv: bfs-equiv(K;S;fs1;fs2),
basic-formal-sum: basic-formal-sum(K;S),
deq: EqDecider(T),
all: ∀x:A. B[x],
squash: ↓T,
universe: Type,
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
basic-formal-sum: basic-formal-sum(K;S),
uall: ∀[x:A]. B[x],
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
decidable: Dec(P),
subtype_rel: A ⊆r B,
bfs-rm0: bfs-rm0(K;eq;b),
bag-filter: [x∈b|p[x]],
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-bag: {},
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
bag-append: as + bs,
append: as @ bs,
single-bag: {x},
pi1: fst(t),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
eqof: eqof(d),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sym: Sym(T;x,y.E[x; y]),
listp: A List+,
bfs-reduce: bfs-reduce(K;S;as;bs),
infix_ap: x f y,
bag: bag(T),
quotient: x,y:A//B[x; y],
true: True,
zero-bfs: 0 * ss,
bag-map: bag-map(f;bs),
map: map(f;as),
formal-sum-add: x + y
Lemmas referenced :
bfs-equiv-rel,
bag_to_squash_list,
rng_car_wf,
equiv_rel_wf,
bag_wf,
bfs-equiv_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
istype-nat,
squash_wf,
bfs-rm0_wf,
deq_wf,
basic-formal-sum_wf,
istype-universe,
rng_sig_wf,
empty-bag_wf,
bag-filter-append,
filter_cons_lemma,
filter_nil_lemma,
rng_zero_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal_wf,
cons_wf,
nil_wf,
list-subtype-bag,
subtype_rel_self,
implies-bfs-equiv,
cons_wf_listp,
subtype_rel_set,
less_than_wf,
length_wf,
bag_qinc,
bag-append_wf,
formal-sum-mul_wf1,
rng_plus_wf,
single-bag_wf,
zero-bfs_wf,
true_wf,
iff_weakening_equal,
formal-sum-add_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
instantiate,
isectElimination,
productEquality,
hypothesis,
imageElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
rename,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
imageMemberEquality,
baseClosed,
functionIsTypeImplies,
unionElimination,
hypothesis_subsumption,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
equalityTransitivity,
baseApply,
closedConclusion,
applyEquality,
intEquality,
sqequalBase,
universeEquality,
equalityElimination,
cumulativity,
independent_pairEquality,
voidEquality,
inlFormation_alt,
productIsType
Latex:
\mforall{}K:RngSig. \mforall{}S:Type. \mforall{}b:basic-formal-sum(K;S). \mforall{}eq:EqDecider(|K|).
(\mdownarrow{}bfs-equiv(K;S;bfs-rm0(K;eq;b);b))
Date html generated:
2019_10_31-AM-06_28_52
Last ObjectModification:
2019_08_27-PM-03_46_32
Theory : linear!algebra
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