Nuprl Lemma : not-all-int_seg2
∀i,j:ℤ. ∀[P,Q:{i..j-} ⟶ ℙ]. ((∀x:{i..j-}. (P[x] ∨ Q[x])) ⇒ (¬(∀x:{i..j-}. P[x])) ⇒ (∃x:{i..j-}. Q[x]))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
label: ...$L... t,
guard: {T},
decidable: Dec(P),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
not: ¬A,
nat: ℕ,
or: P ∨ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
le_wf,
iff_weakening_equal,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
subtype_rel_self,
int_seg_subtype,
subtype_rel_dep_function,
lelt_wf,
decidable__le,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
nat_wf,
primrec-wf2,
set_wf,
exists_wf,
uall_wf,
subtract_wf,
less_than_wf,
or_wf,
int_seg_wf,
all_wf,
not_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
unionElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
addEquality,
because_Cache,
instantiate,
setElimination,
rename,
intEquality,
universeEquality,
cumulativity,
functionEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}i,j:\mBbbZ{}.
\mforall{}[P,Q:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:\{i..j\msupminus{}\}. (P[x] \mvee{} Q[x])) {}\mRightarrow{} (\mneg{}(\mforall{}x:\{i..j\msupminus{}\}. P[x])) {}\mRightarrow{} (\mexists{}x:\{i..j\msupminus{}\}. Q[x]))
Date html generated:
2016_10_21-AM-09_59_24
Last ObjectModification:
2016_09_26-PM-01_35_50
Theory : int_2
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