Nuprl Lemma : FOL-sequent-evidence_and
From uniform evidence that hyps 
⇒ x and 
uniform evidence that (y ∧ hyps) 
⇒ y
we construct uniform evidence that hyps 
⇒ x ∧ y⋅
∀hyps:mFOL() List. ∀[x,y:mFOL()].  (FOL-sequent-evidence{i:l}(<hyps, x>) 
⇒ FOL-sequent-evidence{i:l}(<hyps, y>) 
⇒ FOL-\000Csequent-evidence{i:l}(<hyps, x ∧ y>))
Proof
Definitions occuring in Statement : 
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s)
, 
mFOconnect: mFOconnect(knd;left;right)
, 
mFOL: mFOL()
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
pair: <a, b>
, 
token: "$token"
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s)
, 
FO-uniform-evidence: FO-uniform-evidence(vs;fmla)
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
mFOL-freevars: mFOL-freevars(fmla)
, 
mFOconnect: mFOconnect(knd;left;right)
, 
mFOL_ind: mFOL_ind, 
FOL-sequent-abstract: FOL-sequent-abstract(s)
, 
FOSatWith+: Dom,S,a +|= fmla
, 
FOL-abstract: FOL-abstract(fmla)
, 
FOConnective+: FOConnective+(knd)
, 
eq_atom: x =a y
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
let: let, 
mFOL-sequent: mFOL-sequent()
, 
pi2: snd(t)
, 
prop: ℙ
, 
FOStruct+: FOStruct+{i:l}(Dom)
, 
FOStruct: FOStruct(Dom)
Lemmas referenced : 
subtype_rel_FOAssignment, 
mFOL-sequent-freevars_wf, 
mFOconnect_wf, 
mFOL-sequent-freevars-subset-3, 
val-union-l-union, 
int-deq_wf, 
mFOL-freevars_wf, 
int-valueall-type, 
union-contains, 
union-contains2, 
tuple-type_wf, 
FOL-hyps-meaning_wf, 
FOAssignment_wf, 
FOStruct+_wf, 
FOL-sequent-evidence_wf, 
list_wf, 
mFOL_wf, 
mFOL-sequent-freevars-subset-1, 
subtype_rel_b-union-left, 
FOSatWith+_wf, 
mFOL-sequent-freevars-contains-concl, 
FOL-abstract_wf, 
nil_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
isect_memberFormation, 
cut, 
sqequalHypSubstitution, 
hypothesis, 
isectElimination, 
thin, 
hypothesisEquality, 
dependent_functionElimination, 
applyEquality, 
lemma_by_obid, 
independent_pairEquality, 
tokenEquality, 
because_Cache, 
sqequalRule, 
independent_isectElimination, 
independent_functionElimination, 
intEquality, 
independent_pairFormation, 
productElimination, 
universeEquality, 
lambdaEquality, 
productEquality, 
introduction, 
cumulativity, 
setElimination, 
rename
Latex:
\mforall{}hyps:mFOL()  List
    \mforall{}[x,y:mFOL()].    (FOL-sequent-evidence\{i:l\}(<hyps,  x>)  {}\mRightarrow{}  FOL-sequent-evidence\{i:l\}(<hyps,  y>)  {}\mRightarrow{}  F\000COL-sequent-evidence\{i:l\}(<hyps,  x  \mwedge{}  y>))
Date html generated:
2016_07_08-PM-05_22_10
Last ObjectModification:
2015_12_27-PM-06_25_49
Theory : minimal-first-order-logic
Home
Index