Nuprl Lemma : combination-decomp
∀[A:Type]. ∀[n:ℕ+]. ∀[L:Combination(n;A)]. {(hd(L) ∈ A) ∧ (tl(L) ∈ Combination(n - 1;{a:A| ¬(a = hd(L) ∈ A)} ))}
Proof
Definitions occuring in Statement :
combination: Combination(n;T),
hd: hd(l),
tl: tl(l),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
guard: {T},
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
combination: Combination(n;T),
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
and: P ∧ Q,
uimplies: b supposing a,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
cons: [a / b],
uiff: uiff(P;Q),
nat: ℕ,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
list-cases,
reduce_tl_nil_lemma,
length_of_nil_lemma,
hd_wf,
nil_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
product_subtype_list,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
length_of_cons_lemma,
combination_wf,
nat_plus_wf,
no_repeats_cons,
nat_properties,
intformle_wf,
int_formula_prop_le_lemma,
ge_wf,
less_than_wf,
not_wf,
l_member_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
equal_wf,
spread_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
cons_wf,
cons_member,
list_wf,
no_repeats-subtype,
add-is-int-iff,
false_wf,
no_repeats_wf,
length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
extract_by_obid,
isectElimination,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_pairFormation,
productElimination,
cumulativity,
because_Cache,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
hypothesis_subsumption,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaFormation,
intWeakElimination,
independent_functionElimination,
applyEquality,
setEquality,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
inlFormation,
hyp_replacement,
inrFormation,
pointwiseFunctionality,
baseApply,
closedConclusion,
productEquality
Latex:
\mforall{}[A:Type]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[L:Combination(n;A)].
\{(hd(L) \mmember{} A) \mwedge{} (tl(L) \mmember{} Combination(n - 1;\{a:A| \mneg{}(a = hd(L))\} ))\}
Date html generated:
2018_05_21-PM-08_08_25
Last ObjectModification:
2017_07_26-PM-05_44_07
Theory : general
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