Nuprl Lemma : better-mFOL-sequent-evidence_transitivity1
From uniform evidence that hyps 
⇒ x and uniform evidence that x 
⇒ y
we construct uniform evidence that hyps 
⇒ y.⋅
∀[hyps:mFOL() List]. ∀[x,y:mFOL()].
  (mFOL-freevars(x) ⊆ mFOL-sequent-freevars(<hyps, y>)
  
⇒ (∀[Dom:Type]. ∀[S:FOStruct(Dom)].
        ∀a:FOAssignment(mFOL-sequent-freevars(<hyps, y>),Dom). (Dom,S,a |= mFOL-abstract(x) 
⇒ Dom,S,a |= mFOL-abstract(\000Cy)))
  
⇒ mFOL-sequent-evidence(<hyps, x>)
  
⇒ mFOL-sequent-evidence(<hyps, y>))
Proof
Definitions occuring in Statement : 
mFOL-sequent-evidence: mFOL-sequent-evidence(s)
, 
mFOL-sequent-freevars: mFOL-sequent-freevars(s)
, 
mFOL-abstract: mFOL-abstract(fmla)
, 
mFOL-freevars: mFOL-freevars(fmla)
, 
mFOL: mFOL()
, 
FOSatWith: Dom,S,a |= fmla
, 
FOStruct: FOStruct(Dom)
, 
FOAssignment: FOAssignment(vs,Dom)
, 
l_contains: A ⊆ B
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
pair: <a, b>
, 
int: ℤ
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
mFOL-sequent-evidence: mFOL-sequent-evidence(s)
, 
mFO-uniform-evidence: mFO-uniform-evidence(vs;fmla)
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
mFOL-sequent-abstract: mFOL-sequent-abstract(s)
, 
FOSatWith: Dom,S,a |= fmla
, 
mFOL-sequent: mFOL-sequent()
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
guard: {T}
Lemmas referenced : 
subtype_rel_FOAssignment, 
mFOL-sequent-freevars_wf, 
mFOL-sequent-freevars-subset-4, 
tuple-type_wf, 
mFOL-hyps-meaning_wf, 
FOAssignment_wf, 
FOStruct_wf, 
mFOL-sequent-evidence_wf, 
list_wf, 
mFOL_wf, 
uall_wf, 
all_wf, 
FOSatWith_wf, 
mFOL-freevars_wf, 
mFOL-abstract_wf, 
mFOL-sequent-freevars-subset-1, 
l_contains_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
sqequalHypSubstitution, 
hypothesis, 
isectElimination, 
thin, 
hypothesisEquality, 
dependent_functionElimination, 
applyEquality, 
lemma_by_obid, 
independent_pairEquality, 
because_Cache, 
sqequalRule, 
independent_isectElimination, 
independent_functionElimination, 
universeEquality, 
lambdaEquality, 
productEquality, 
instantiate, 
cumulativity, 
functionEquality, 
intEquality
Latex:
\mforall{}[hyps:mFOL()  List].  \mforall{}[x,y:mFOL()].
    (mFOL-freevars(x)  \msubseteq{}  mFOL-sequent-freevars(<hyps,  y>)
    {}\mRightarrow{}  (\mforall{}[Dom:Type].  \mforall{}[S:FOStruct(Dom)].
                \mforall{}a:FOAssignment(mFOL-sequent-freevars(<hyps,  y>),Dom).  (Dom,S,a  |=  mFOL-abstract(x)  {}\mRightarrow{}  Dom,S\000C,a  |=  mFOL-abstract(y)))
    {}\mRightarrow{}  mFOL-sequent-evidence(<hyps,  x>)
    {}\mRightarrow{}  mFOL-sequent-evidence(<hyps,  y>))
Date html generated:
2016_07_08-PM-05_21_29
Last ObjectModification:
2015_12_27-PM-06_26_12
Theory : minimal-first-order-logic
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