Nuprl Lemma : K-forces_wf
∀[K:mKripkeStruct]. ∀[fmla:mFOL()]. (K-forces(K;fmla) ∈ i:World ⟶ FOAssignment(mFOL-freevars(fmla),Dom(i)) ⟶ ℙ)
Proof
Definitions occuring in Statement :
K-forces: K-forces(K;fmla),
K-dom: Dom(i),
K-world: World,
mFO-Kripke-struct: mKripkeStruct,
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
FOAssignment: FOAssignment(vs,Dom),
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
function: x:A ⟶ B[x]
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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
mFOatomic: name(vars),
mFOL_size: mFOL_size(p),
spreadn: spread3,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
mFOconnect: mFOconnect(knd;left;right),
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
mFOL-freevars: mFOL-freevars(fmla),
mFOL_ind: mFOL_ind,
mFOquant: mFOquant(isall;var;body),
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
l_contains: A ⊆ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
le_wf,
subtype_rel_self,
mFOL-ext,
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
atom_subtype_base,
K_forces_atomic_lemma,
K-sat_wf,
mFOatomic_wf,
FOAssignment_wf,
mFOL-freevars_wf,
K-dom_wf,
K-world_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
mFOL_size_wf,
itermAdd_wf,
int_term_value_add_lemma,
K_forces_connect_lemma,
subtype_rel_FOAssignment,
mFOconnect_wf,
val-union-l-union,
int-deq_wf,
int-valueall-type,
union-contains,
union-contains2,
or_wf,
all_wf,
K-le_wf,
K-assignment_subtype,
K_forces_quant_lemma,
update-assignment_wf,
mFOquant_wf,
btrue_wf,
exists_wf,
filter_wf5,
l_member_wf,
bnot_wf,
eq_int_wf,
bfalse_wf,
nat_wf,
mFOL_wf,
mFO-Kripke-struct_wf,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
promote_hyp,
tokenEquality,
equalityElimination,
cumulativity,
atomEquality,
equalityIsType2,
baseApply,
closedConclusion,
baseClosed,
imageElimination,
productEquality,
intEquality,
functionEquality,
equalityIsType1,
setIsType,
addEquality,
isectIsTypeImplies
Latex:
\mforall{}[K:mKripkeStruct]. \mforall{}[fmla:mFOL()].
(K-forces(K;fmla) \mmember{} i:World {}\mrightarrow{} FOAssignment(mFOL-freevars(fmla),Dom(i)) {}\mrightarrow{} \mBbbP{})
Date html generated:
2019_10_16-AM-11_45_23
Last ObjectModification:
2018_10_13-AM-10_19_50
Theory : minimal-first-order-logic
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