Nuprl Lemma : first-member-cons
∀[T:Type]
∀P:T ⟶ 𝔹. ∀x,u:T. ∀L:T List. (first-member(T;x;[u / L];P) ⇐⇒ if P u then x = u ∈ T else first-member(T;x;L;P) fi )
Proof
Definitions occuring in Statement :
first-member: first-member(T;x;L;P),
cons: [a / b],
list: T List,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: 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,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
first-member: first-member(T;x;L;P),
exists: ∃x:A. B[x],
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
select: L[n],
cons: [a / b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
top: Top,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtract: n - m,
nat_plus: ℕ+,
true: True
Lemmas referenced :
first-member_wf,
cons_wf,
ifthenelse_wf,
equal_wf,
list_wf,
bool_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
assert_elim,
and_wf,
not_assert_elim,
btrue_neq_bfalse,
select-cons-tl,
length_of_cons_lemma,
int_seg_properties,
length_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
false_wf,
lelt_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
assert_wf,
select_wf,
int_seg_wf,
all_wf,
not_wf,
add-member-int_seg2,
add-subtract-cancel,
bool_cases,
assert_of_bnot,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
instantiate,
universeEquality,
functionEquality,
productElimination,
dependent_functionElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
equalityElimination,
dependent_pairFormation,
promote_hyp,
voidElimination,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
applyLambdaEquality,
isect_memberEquality,
voidEquality,
addEquality,
lambdaEquality,
int_eqEquality,
computeAll,
productEquality,
imageElimination,
imageMemberEquality,
baseClosed,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}x,u:T. \mforall{}L:T List.
(first-member(T;x;[u / L];P) \mLeftarrow{}{}\mRightarrow{} if P u then x = u else first-member(T;x;L;P) fi )
Date html generated:
2018_05_21-PM-06_33_48
Last ObjectModification:
2017_07_26-PM-04_52_15
Theory : general
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