Nuprl Lemma : FOL-sequent-evidence-false-hyp
Uniform evidence for a trivially true sequent where the conclusion is
a given member of the hypotheses.⋅
∀hyps:mFOL() List. ∀concl:mFOL(). ∀i:ℕ||hyps||.
((↑mFOatomic?(hyps[i]))
⇒ (mFOatomic-name(hyps[i]) = "false" ∈ Atom)
⇒ (mFOatomic-vars(hyps[i]) = [] ∈ (ℤ List))
⇒ FOL-sequent-evidence{i:l}(<hyps, concl>))
Proof
Definitions occuring in Statement :
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s),
mFOatomic-vars: mFOatomic-vars(v),
mFOatomic-name: mFOatomic-name(v),
mFOatomic?: mFOatomic?(v),
mFOL: mFOL(),
select: L[n],
length: ||as||,
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pair: <a, b>,
natural_number: $n,
int: ℤ,
token: "$token",
atom: Atom,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s),
FO-uniform-evidence: FO-uniform-evidence(vs;fmla),
FOL-sequent-abstract: FOL-sequent-abstract(s),
FOSatWith+: Dom,S,a +|= fmla,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
mFOL-sequent: mFOL-sequent(),
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
FOL-hyps-meaning: FOL-hyps-meaning(Dom;S;a;hyps),
so_lambda: λ2x.t[x],
so_apply: x[s],
mFOatomic: name(vars),
mFOatomic?: mFOatomic?(v),
pi1: fst(t),
mFOatomic-name: mFOatomic-name(v),
pi2: snd(t),
mFOatomic-vars: mFOatomic-vars(v),
eq_atom: x =a y,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
mFOconnect: mFOconnect(knd;left;right),
bfalse: ff,
mFOquant: mFOquant(isall;var;body),
FOL-abstract: FOL-abstract(fmla),
mFOL_ind: mFOL_ind,
AbstractFOAtomic+: AbstractFOAtomic+(n;L),
FOStruct+: FOStruct+{i:l}(Dom),
FOStruct: FOStruct(Dom),
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
mFOL-freevars: mFOL-freevars(fmla),
remove-repeats: remove-repeats(eq;L),
list_ind: list_ind,
nil: [],
it: ⋅,
sq_type: SQType(T)
Lemmas referenced :
FOStruct-false-subtype-evidence,
subtype_rel_FOAssignment,
mFOL-sequent-freevars_wf,
mFOL-freevars_wf,
mFOL-sequent-freevars-subset-1,
tuple-type_wf,
FOL-hyps-meaning_wf,
FOAssignment_wf,
list_wf,
mFOL_wf,
FOStruct+_wf,
equal-wf-T-base,
mFOatomic-vars_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
mFOatomic-name_wf,
assert_wf,
mFOatomic?_wf,
int_seg_wf,
select-tuple_wf,
int_seg_subtype_nat,
length-map,
select-map,
subtype_rel_list,
top_wf,
equal_wf,
mFOL-induction,
squash_wf,
true_wf,
mFOatomic_wf,
iff_weakening_equal,
equal-wf-base,
list_subtype_base,
int_subtype_base,
atom_subtype_base,
false_wf,
bool_wf,
map_nil_lemma,
nil_wf,
b-union_wf,
subtype_rel_wf,
FOSatWith+_wf,
FOL-abstract_wf,
subtype_base_sq,
l_contains_nil
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalRule,
introduction,
lambdaEquality,
cut,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_pairEquality,
hypothesis,
because_Cache,
independent_isectElimination,
cumulativity,
productEquality,
universeEquality,
setElimination,
rename,
productElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionEquality,
atomEquality,
imageMemberEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
tokenEquality,
instantiate,
equalityElimination
Latex:
\mforall{}hyps:mFOL() List. \mforall{}concl:mFOL(). \mforall{}i:\mBbbN{}||hyps||.
((\muparrow{}mFOatomic?(hyps[i]))
{}\mRightarrow{} (mFOatomic-name(hyps[i]) = "false")
{}\mRightarrow{} (mFOatomic-vars(hyps[i]) = [])
{}\mRightarrow{} FOL-sequent-evidence\{i:l\}(<hyps, concl>))
Date html generated:
2018_05_21-PM-10_34_44
Last ObjectModification:
2017_07_26-PM-06_42_07
Theory : minimal-first-order-logic
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