Nuprl Lemma : length-remove-first-le
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. (||remove-first(P;L)|| ≤ ||L||)
Proof
Definitions occuring in Statement :
remove-first: remove-first(P;L),
l_member: (x ∈ l),
length: ||as||,
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
le: A ≤ B,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
le: A ≤ B
Lemmas referenced :
bool_wf,
l_member_wf,
remove-first_wf,
less_than'_wf,
false_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
itermConstant_wf,
itermSubtract_wf,
subtract-is-int-iff,
int_subtype_base,
subtype_base_sq,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
length_wf,
decidable__le,
length-remove-first
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
introduction,
dependent_functionElimination,
unionElimination,
productElimination,
sqequalRule,
because_Cache,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
pointwiseFunctionality,
rename,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_pairEquality,
axiomEquality,
functionEquality,
setEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. (||remove-first(P;L)|| \mleq{} ||L||)
Date html generated:
2016_05_14-PM-02_46_36
Last ObjectModification:
2016_01_15-AM-07_35_11
Theory : list_1
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