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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