Nuprl Lemma : valuation-exists
∀x:formula(). ∀v0:{a:formula()| a ⊆ x ∧ (↑pvar?(a))} ⟶ 𝔹. (∃f:{a:formula()| a ⊆ x} ⟶ 𝔹 [valuation(v0;x;f)])
Proof
Definitions occuring in Statement :
valuation: valuation(v0;x;f),
psub: a ⊆ b,
pvar?: pvar?(v),
formula: formula(),
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
prop: ℙ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
bdd-val: bdd-val(v0;x;n),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
pimp: pimp(left;right),
por: por(left;right),
band: p ∧b q,
pand: pand(left;right),
squash: ↓T,
less_than: a < b,
pnot: pnot(sub),
bnot: ¬bb,
bfalse: ff,
true: True,
pi1: fst(t),
pvar?: pvar?(v),
assert: ↑b,
cand: A c∧ B,
formula_ind: formula_ind,
prank: prank(x),
extend-val: extend-val(v0;g;x),
formula_size: formula_size(p),
pvar: pvar(name),
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
ext-eq: A ≡ B,
sq_type: SQType(T),
psub: a ⊆ b,
istype: istype(T),
sq_exists: ∃x:A [B[x]],
valuation: valuation(v0;x;f)
Lemmas referenced :
formula_wf,
bool_wf,
pvar?_wf,
assert_wf,
psub_wf,
istype-false,
int_seg_subtype_nat,
int_seg_wf,
nat_wf,
bdd-val_wf,
uniform-comp-nat-induction,
prank_wf,
less_than_wf,
le_wf,
decidable__lt,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
subtract_nat_wf,
subtract_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
assert_of_le_int,
bool_cases,
iff_weakening_equal,
imax_unfold,
add_functionality_wrt_eq,
true_wf,
squash_wf,
not_wf,
le_int_wf,
ifthenelse_wf,
pimp_wf,
por_wf,
bor_wf,
imax_wf,
pand_wf,
pnot_wf,
false_wf,
add-is-int-iff,
bnot_wf,
int_term_value_add_lemma,
itermAdd_wf,
formula_size_wf,
neg_assert_of_eq_atom,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
pvar_wf,
atom_subtype_base,
assert_of_eq_atom,
eqtt_to_assert,
eq_atom_wf,
formula-ext,
subtype_rel_self,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
decidable__equal_int,
subtract-1-ge-0,
int_seg_properties,
ge_wf,
psub_transitivity,
psub_weakening,
equal_wf,
istype-less_than,
istype-le,
istype-assert,
psub-same,
band_wf,
subtype_rel_sets,
subtype_rel_dep_function,
prank_functionality,
extend-val_wf,
add_nat_wf,
valuation_wf,
set_wf
Rules used in proof :
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
universeIsType,
productIsType,
sqequalRule,
hypothesisEquality,
inhabitedIsType,
setIsType,
functionIsType,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
because_Cache,
independent_pairFormation,
independent_isectElimination,
applyEquality,
rename,
setElimination,
natural_numberEquality,
isectIsType,
isect_memberFormation_alt,
independent_functionElimination,
lambdaEquality_alt,
dependent_set_memberFormation_alt,
equalitySymmetry,
equalityTransitivity,
equalityIsType1,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
applyLambdaEquality,
unionElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
productElimination,
universeEquality,
imageMemberEquality,
intEquality,
addEquality,
pointwiseFunctionality,
imageElimination,
baseClosed,
closedConclusion,
baseApply,
equalityIsType2,
atomEquality,
cumulativity,
equalityElimination,
tokenEquality,
promote_hyp,
hypothesis_subsumption,
instantiate,
functionIsTypeImplies,
axiomEquality,
intWeakElimination,
inrFormation,
inlFormation,
inrFormation_alt,
Error :memTop,
isectIsTypeImplies,
equalityIstype,
inlFormation_alt,
unionIsType,
productEquality,
setEquality,
dependent_set_memberEquality,
lambdaEquality,
lambdaFormation,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
dependent_set_memberFormation
Latex:
\mforall{}x:formula(). \mforall{}v0:\{a:formula()| a \msubseteq{} x \mwedge{} (\muparrow{}pvar?(a))\} {}\mrightarrow{} \mBbbB{}.
(\mexists{}f:\{a:formula()| a \msubseteq{} x\} {}\mrightarrow{} \mBbbB{} [valuation(v0;x;f)])
Date html generated:
2020_05_20-AM-08_19_18
Last ObjectModification:
2020_01_27-PM-01_36_09
Theory : general
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