Nuprl Lemma : peval-pnegate
∀[x:formula()]. ∀[v:{a:formula()| a ⊆ x ∧ (↑pvar?(a))} ⟶ 𝔹]. peval(v;pnegate(x)) = ¬bpeval(v;x)
Proof
Definitions occuring in Statement :
pnegate: pnegate(x),
peval: peval(v0;x),
psub: a ⊆ b,
pvar?: pvar?(v),
formula: formula(),
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
pvar: pvar(name),
formula_size: formula_size(p),
peval: peval(v0;x),
valuation-exists-ext,
extend-val: extend-val(v0;g;x),
formula_ind: formula_ind,
pnegate: pnegate(x),
pnot: pnot(sub),
bnot: ¬bb,
bfalse: ff,
assert: ↑b,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
pand: pand(left;right),
por: por(left;right),
pimp: pimp(left;right),
psub: a ⊆ b,
so_lambda: λ2x.t[x],
so_apply: x[s],
band: p ∧b q,
bor: p ∨bq
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
formula_wf,
psub_wf,
assert_wf,
pvar?_wf,
bool_wf,
le_wf,
formula_size_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
formula-ext,
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
bnot_wf,
peval_wf,
pvar_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
pnot_wf,
pand_wf,
por_wf,
pimp_wf,
nat_wf,
subtype_rel_self,
set_wf,
bnot_thru_band,
bor_wf,
bnot_thru_bor,
subtype_rel_dep_function,
subtype_rel_sets,
peval-unroll,
btrue_wf,
valuation-exists-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
functionEquality,
setEquality,
productEquality,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
promote_hyp,
tokenEquality,
equalityElimination,
instantiate,
cumulativity,
atomEquality,
functionExtensionality,
imageElimination,
addEquality,
inrFormation,
inlFormation
Latex:
\mforall{}[x:formula()]. \mforall{}[v:\{a:formula()| a \msubseteq{} x \mwedge{} (\muparrow{}pvar?(a))\} {}\mrightarrow{} \mBbbB{}]. peval(v;pnegate(x)) = \mneg{}\msubb{}peval(v;x)
Date html generated:
2018_05_21-PM-08_54_42
Last ObjectModification:
2017_07_26-PM-06_18_20
Theory : general
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