Nuprl Lemma : before-hd
∀[T:Type]. ∀L:T List. (∀x:T. (x before hd(L) ∈ L ⇐⇒ False)) supposing (no_repeats(T;L) and 0 < ||L||)
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
no_repeats: no_repeats(T;l),
length: ||as||,
hd: hd(l),
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
false: False,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
uiff: uiff(P;Q),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
member-less_than,
length_wf,
no_repeats_witness,
no_repeats_iff,
l_before_wf,
hd_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
no_repeats_wf,
istype-less_than,
list_wf,
istype-universe,
hd-before,
l_before_member2,
l_before_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
rename,
independent_functionElimination,
independent_pairFormation,
because_Cache,
productElimination,
universeIsType,
dependent_functionElimination,
unionElimination,
imageElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. (\mforall{}x:T. (x before hd(L) \mmember{} L \mLeftarrow{}{}\mRightarrow{} False)) supposing (no\_repeats(T;L) and 0 < ||L||)
Date html generated:
2019_10_15-AM-10_23_34
Last ObjectModification:
2019_08_05-PM-02_11_41
Theory : list_1
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