Nuprl Lemma : W-type-ext
∀[A:Type]. ∀[B:A ⟶ Type]. W-type(A; a.B[a]) ≡ a:A × (B[a] ⟶ W-type(A; a.B[a])) supposing ∀x,y:A. Dec(x = y ∈ A)
Proof
Definitions occuring in Statement :
W-type: W-type(A; a.B[a]),
ext-eq: A ≡ B,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
W-type: W-type(A; a.B[a]),
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ,
W-sup: W-sup(a;f),
nat: ℕ,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
deq: EqDecider(T),
eqof: eqof(d),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
W-bars: W-bars(w;p),
squash: ↓T,
upto: upto(n),
from-upto: [n, m),
lt_int: i <z j,
isr: isr(x),
nat_plus: ℕ+,
true: True,
subtract: n - m,
eq_int: (i =z j),
W-select: W-select(w;s),
compose: f o g,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nil: [],
less_than: a < b
Lemmas referenced :
co-W-ext,
subtype_rel_weakening,
co-W_wf,
deq-exists,
nat_wf,
unit_wf2,
all_wf,
W-bars_wf,
W-type_wf,
W-sup_wf,
decidable_wf,
equal_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
safe-assert-deq,
subtype_rel-equal,
and_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
it_wf,
neg_assert_of_eq_int,
upper_subtype_nat,
false_wf,
nat_properties,
nequal-le-implies,
zero-add,
le_wf,
subtract_wf,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__equal_int,
int_subtype_base,
map_nil_lemma,
W_select_nil_lemma,
intformeq_wf,
int_formula_prop_eq_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,
upto_decomp2,
decidable__lt,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
map_cons_lemma,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
map-map,
bnot_wf,
eqof_wf,
not_wf,
member_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
subtype_rel_list,
list_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
equal-wf-T-base,
colength_wf_list,
list-cases,
product_subtype_list,
spread_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
set_subtype_base,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis_subsumption,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
applyEquality,
because_Cache,
hypothesis,
productEquality,
functionEquality,
independent_isectElimination,
productElimination,
dependent_pairEquality,
functionExtensionality,
dependent_set_memberEquality,
lambdaFormation,
independent_functionElimination,
unionEquality,
dependent_functionElimination,
cumulativity,
independent_pairEquality,
axiomEquality,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
unionElimination,
equalityElimination,
inlEquality,
applyLambdaEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
inrEquality,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality,
imageElimination,
imageMemberEquality,
baseClosed,
addEquality,
minusEquality,
impliesFunctionality,
intWeakElimination,
axiomSqEquality
Latex:
\mforall{}[A:Type]
\mforall{}[B:A {}\mrightarrow{} Type]. W-type(A; a.B[a]) \mequiv{} a:A \mtimes{} (B[a] {}\mrightarrow{} W-type(A; a.B[a]))
supposing \mforall{}x,y:A. Dec(x = y)
Date html generated:
2019_10_16-AM-11_37_56
Last ObjectModification:
2018_08_22-AM-10_05_28
Theory : bar!induction
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