Nuprl Lemma : hd-before
∀[T:Type]. ∀L:T List. ∀x:T. ((x ∈ L) ⇒ hd(L) before x ∈ L supposing ¬(x = hd(L) ∈ T)) supposing 0 < ||L||
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
l_member: (x ∈ l),
hd: hd(l),
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
less_than': less_than'(a;b),
cons: [a / b],
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q
Lemmas referenced :
member-less_than,
length_wf,
equal_wf,
hd_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_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,
list-cases,
length_of_nil_lemma,
product_subtype_list,
reduce_hd_cons_lemma,
length_of_cons_lemma,
not_wf,
l_member_wf,
less_than_wf,
list_wf,
cons_member,
l_before_wf,
cons_before
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_isectElimination,
rename,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
voidElimination,
because_Cache,
unionElimination,
imageElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
hypothesis_subsumption,
universeEquality,
independent_functionElimination,
inlFormation
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} hd(L) before x \mmember{} L supposing \mneg{}(x = hd(L))) supposing 0 < ||L||
Date html generated:
2017_04_17-AM-08_47_53
Last ObjectModification:
2017_02_27-PM-05_06_24
Theory : list_1
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