Nuprl Lemma : not-all-int_seg
∀i,j:ℤ. ∀[P:{i..j-} ⟶ ℙ]. ((∀x:{i..j-}. Dec(P[x])) ⇒ (¬(∀x:{i..j-}. P[x]) ⇐⇒ ∃x:{i..j-}. (¬P[x])))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
not: ¬A,
false: False
Lemmas referenced :
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
int_seg_wf,
not_wf,
not-all-int_seg2,
decidable__lt,
decidable_wf,
exists_wf,
all_wf
Rules used in proof :
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
productElimination,
rename,
setElimination,
inrFormation,
inlFormation,
functionExtensionality,
unionElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
independent_functionElimination,
voidElimination,
functionEquality,
cumulativity,
universeEquality,
intEquality
Latex:
\mforall{}i,j:\mBbbZ{}.
\mforall{}[P:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:\{i..j\msupminus{}\}. Dec(P[x])) {}\mRightarrow{} (\mneg{}(\mforall{}x:\{i..j\msupminus{}\}. P[x]) \mLeftarrow{}{}\mRightarrow{} \mexists{}x:\{i..j\msupminus{}\}. (\mneg{}P[x])))
Date html generated:
2016_10_21-AM-09_59_29
Last ObjectModification:
2016_09_26-PM-01_38_58
Theory : int_2
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