Nuprl Lemma : better-FOL-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+{i:l}(Dom)].
∀a:FOAssignment(mFOL-sequent-freevars(<hyps, y>),Dom). (Dom,S,a +|= FOL-abstract(x) ⇒ Dom,S,a +|= FOL-abstract(\000Cy)))
⇒ FOL-sequent-evidence{i:l}(<hyps, x>)
⇒ FOL-sequent-evidence{i:l}(<hyps, y>))
Proof
Definitions occuring in Statement :
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s),
mFOL-sequent-freevars: mFOL-sequent-freevars(s),
FOL-abstract: FOL-abstract(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
FOSatWith+: Dom,S,a +|= fmla,
FOStruct+: FOStruct+{i:l}(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,
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s),
FO-uniform-evidence: FO-uniform-evidence(vs;fmla),
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
FOL-sequent-abstract: FOL-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,
FOL-hyps-meaning_wf,
FOAssignment_wf,
FOStruct+_wf,
FOL-sequent-evidence_wf,
list_wf,
mFOL_wf,
uall_wf,
all_wf,
FOSatWith+_wf,
mFOL-freevars_wf,
FOL-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+\{i:l\}(Dom)].
\mforall{}a:FOAssignment(mFOL-sequent-freevars(<hyps, y>),Dom). (Dom,S,a +|= FOL-abstract(x) {}\mRightarrow{} Dom,S\000C,a +|= FOL-abstract(y)))
{}\mRightarrow{} FOL-sequent-evidence\{i:l\}(<hyps, x>)
{}\mRightarrow{} FOL-sequent-evidence\{i:l\}(<hyps, y>))
Date html generated:
2016_07_08-PM-05_21_41
Last ObjectModification:
2015_12_27-PM-06_25_54
Theory : minimal-first-order-logic
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