Nuprl Lemma : first-iseg
∀[T:Type]. ∀[P:(T List) ⟶ ℙ].
((∀L:T List. Dec(P[L]))
⇒ (∀L:T List
(P[L] ⇒ (∃L':T List. (L' ≤ L ∧ P[L'] ∧ (∀L'':T List. (L'' ≤ L' ⇒ P[L''] ⇒ (L'' = L' ∈ (T List)))))))))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
true: True,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
bfalse: ff
Lemmas referenced :
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
lelt_wf,
less_than_wf,
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
list_wf,
all_wf,
exists_wf,
iseg_wf,
equal_wf,
set_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
decidable_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
nil_wf,
equal-wf-T-base,
assert_of_null,
assert_wf,
true_wf,
iseg_nil,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
last_lemma,
decidable-exists-iseg,
iseg_length,
squash_wf,
length_append,
cons_wf,
last_wf,
iff_weakening_equal,
iseg_transitivity,
iseg_append0,
iseg_weakening,
iseg_append_iff,
append_wf,
cons_iseg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
cumulativity,
imageElimination,
independent_functionElimination,
functionExtensionality,
functionEquality,
universeEquality,
productEquality,
addEquality,
baseClosed,
allFunctionality,
impliesFunctionality,
promote_hyp,
imageMemberEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}L:T List. Dec(P[L]))
{}\mRightarrow{} (\mforall{}L:T List
(P[L]
{}\mRightarrow{} (\mexists{}L':T List. (L' \mleq{} L \mwedge{} P[L'] \mwedge{} (\mforall{}L'':T List. (L'' \mleq{} L' {}\mRightarrow{} P[L''] {}\mRightarrow{} (L'' = L'))))))))
Date html generated:
2018_05_21-PM-07_21_46
Last ObjectModification:
2017_07_26-PM-05_05_22
Theory : general
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