Nuprl Lemma : W-sup_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[a:A]. ∀[f:B[a] ⟶ W-type(A; a.B[a])]. (W-sup(a;f) ∈ W-type(A; a.B[a]))
Proof
Definitions occuring in Statement :
W-sup: W-sup(a;f),
W-type: W-type(A; a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
W-type: W-type(A; a.B[a]),
W-sup: W-sup(a;f),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
ext-eq: A ≡ B,
outl: outl(x),
uimplies: b supposing a,
isl: isl(x),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
W-bars: W-bars(w;p),
W-select: W-select(w;s),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
isr: isr(x),
assert: ↑b,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
compose: f o g,
upto: upto(n),
from-upto: [n, m),
lt_int: i <z j
Lemmas referenced :
W-type_wf,
co-W_wf,
co-W-ext,
decidable__assert,
isl_wf,
unit_wf2,
nat_wf,
false_wf,
le_wf,
all_wf,
W-bars_wf,
assert_elim,
bfalse_wf,
and_wf,
equal_wf,
btrue_neq_bfalse,
sq_stable__W-bars,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
null-map,
null-upto,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upto_decomp2,
decidable__lt,
not-lt-2,
not-equal-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
map_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
assert_wf,
isr_wf,
W-select_wf,
map_wf,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat,
subtype_rel_self,
upto_wf,
int_seg_properties,
true_wf,
map-map,
add-subtract-cancel,
null_cons_lemma,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalRule,
dependent_pairEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
functionEquality,
because_Cache,
lambdaFormation,
dependent_functionElimination,
cumulativity,
natural_numberEquality,
independent_pairFormation,
unionElimination,
unionEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
universeEquality,
productElimination,
independent_isectElimination,
applyLambdaEquality,
independent_functionElimination,
voidElimination,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityElimination,
promote_hyp,
instantiate,
minusEquality,
callbyvalueReduce,
sqleReflexivity
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[a:A]. \mforall{}[f:B[a] {}\mrightarrow{} W-type(A; a.B[a])]. (W-sup(a;f) \mmember{} W-type(A; a.B[a]))
Date html generated:
2019_10_16-AM-11_37_51
Last ObjectModification:
2018_08_22-AM-10_05_31
Theory : bar!induction
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