Nuprl Lemma : first-member-iff
∀[T:Type]
∀L:T List. ∀P:T ⟶ 𝔹. ∀x:T.
(first-member(T;x;L;P) ⇐⇒ ∃K,J:T List. ((L = (K @ [x / J]) ∈ (T List)) ∧ (↑(P x)) ∧ (∀y∈K.¬↑(P y))))
Proof
Definitions occuring in Statement :
first-member: first-member(T;x;L;P),
l_all: (∀x∈L.P[x]),
append: as @ bs,
cons: [a / b],
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
first-member: first-member(T;x;L;P),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
cand: A c∧ B,
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nat: ℕ,
true: True,
sq_type: SQType(T),
select: L[n],
cons: [a / b]
Lemmas referenced :
first-member_wf,
append_wf,
cons_wf,
assert_wf,
l_all_wf2,
not_wf,
l_member_wf,
bool_wf,
list_wf,
firstn_wf,
nth_tl_wf,
add-commutes,
istype-void,
nth_tl_decomp_eq,
int_seg_subtype_nat,
length_wf,
istype-false,
int_seg_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
append_firstn_lastn_sq,
subtype_rel_list,
top_wf,
itermAdd_wf,
itermConstant_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
le_wf,
less_than_wf,
equal_wf,
l_all_iff,
member-firstn,
length-append,
length_of_cons_lemma,
non_neg_length,
intformle_wf,
int_formula_prop_le_lemma,
equal-wf-base,
int_seg_wf,
length_wf_nat,
set_subtype_base,
int_subtype_base,
select_wf,
decidable__le,
false_wf,
select_append_back,
iff_weakening_equal,
subtype_base_sq,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
squash_wf,
true_wf,
subtype_rel_self,
select_append_front,
select_member,
l_all_fwd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
productIsType,
inhabitedIsType,
equalityIsType1,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
setIsType,
functionIsType,
universeEquality,
dependent_pairFormation_alt,
addEquality,
because_Cache,
natural_numberEquality,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
imageElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
equalitySymmetry,
dependent_set_memberEquality_alt,
hyp_replacement,
applyLambdaEquality,
intEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
instantiate,
cumulativity,
productEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}x:T.
(first-member(T;x;L;P) \mLeftarrow{}{}\mRightarrow{} \mexists{}K,J:T List. ((L = (K @ [x / J])) \mwedge{} (\muparrow{}(P x)) \mwedge{} (\mforall{}y\mmember{}K.\mneg{}\muparrow{}(P y))))
Date html generated:
2019_10_15-AM-11_08_00
Last ObjectModification:
2018_10_16-AM-09_34_26
Theory : general
Home
Index