Nuprl Lemma : Wselect_wf
∀[A:Type]
∀[B:A ⟶ Type]. ∀[s:(a:A ⟶ (B[a]?)) List]. ∀[w:W-type(A; a.B[a])]. (Wselect(w;s) ∈ W-type(A; a.B[a])?)
supposing ∀x,y:A. Dec(x = y ∈ A)
Proof
Definitions occuring in Statement :
Wselect: Wselect(w;s),
W-type: W-type(A; a.B[a]),
list: T List,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
unit: Unit,
member: t ∈ T,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
Wselect: Wselect(w;s),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
W-select: W-select(w;s),
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bfalse: ff,
ext-eq: A ≡ B
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
W-type_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
unit_wf2,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
null_nil_lemma,
reduce_tl_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
W-type-ext,
subtype_rel_weakening,
list_wf,
all_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
applyEquality,
functionExtensionality,
functionEquality,
unionEquality,
because_Cache,
unionElimination,
inlEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
productEquality,
inrEquality,
universeEquality
Latex:
\mforall{}[A:Type]
\mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[s:(a:A {}\mrightarrow{} (B[a]?)) List]. \mforall{}[w:W-type(A; a.B[a])].
(Wselect(w;s) \mmember{} W-type(A; a.B[a])?)
supposing \mforall{}x,y:A. Dec(x = y)
Date html generated:
2018_05_21-PM-10_18_38
Last ObjectModification:
2017_07_26-PM-06_36_42
Theory : bar!induction
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