Nuprl Lemma : bag-val_wf
∀[T:Type]. ∀[+:T ⟶ T ⟶ T]. ∀[zero:T]. (Comm(T;+) ⇒ Assoc(T;+) ⇒ Ident(T;+;zero) ⇒ (bag-val(zero;+) ∈ bag(T) ⟶ T))
Proof
Definitions occuring in Statement :
bag-val: bag-val(zero;+),
bag: bag(T),
comm: Comm(T;op),
assoc: Assoc(T;op),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
ident: Ident(T;op;id)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bag-val: bag-val(zero;+),
prop: ℙ,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
false: False,
cons: [a / b],
combine-list: combine-list(x,y.f[x; y];L),
top: Top,
guard: {T},
true: True,
ident: Ident(T;op;id),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
assoc: Assoc(T;op),
comm: Comm(T;op),
le: A ≤ B
Lemmas referenced :
bag_wf,
ident_wf,
assoc_wf,
comm_wf,
list_wf,
permutation-length,
decidable__lt,
permutation_wf,
equal_wf,
equal-wf-base,
list_accum_wf,
list-cases,
product_subtype_list,
list_accum_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
infix_ap_wf,
squash_wf,
true_wf,
combine-list_wf,
length_of_nil_lemma,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
combine-list-permutation,
nil_wf,
length_of_cons_lemma,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
intformnot_wf,
int_formula_prop_not_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
functionExtensionality,
applyEquality,
sqequalRule,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
rename,
independent_isectElimination,
natural_numberEquality,
unionElimination,
independent_functionElimination,
productEquality,
hyp_replacement,
applyLambdaEquality,
imageElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
voidEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[+:T {}\mrightarrow{} T {}\mrightarrow{} T]. \mforall{}[zero:T].
(Comm(T;+) {}\mRightarrow{} Assoc(T;+) {}\mRightarrow{} Ident(T;+;zero) {}\mRightarrow{} (bag-val(zero;+) \mmember{} bag(T) {}\mrightarrow{} T))
Date html generated:
2017_10_01-AM-08_46_16
Last ObjectModification:
2017_07_26-PM-04_31_12
Theory : bags
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