Nuprl Lemma : FOL-sequent-evidence_transitivity2
From uniform evidence that hyps ⇒ y and
uniform evidence that (y ∧ hyps) ⇒ x
we construct uniform evidence that hyps ⇒ x.⋅
∀hyps:mFOL() List. ∀[x,y:mFOL()]. (mFOL-freevars(y) ⊆ mFOL-sequent-freevars(<hyps, x>) ⇒ FOL-sequent-evidence{i:l}(<hy\000Cps, y>) ⇒ FOL-sequent-evidence{i:l}(<[y / hyps], x>) ⇒ FOL-sequent-evidence{i:l}(<hyps, x>))
Proof
Definitions occuring in Statement :
FOL-sequent-evidence: FOL-sequent-evidence{i:l}(s),
mFOL-sequent-freevars: mFOL-sequent-freevars(s),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
l_contains: A ⊆ B,
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
pair: <a, b>,
int: ℤ
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),
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
pi2: snd(t),
pi1: fst(t),
or: P ∨ Q,
prop: ℙ,
mFOL-sequent: mFOL-sequent(),
FOL-sequent-abstract: FOL-sequent-abstract(s),
FOSatWith+: Dom,S,a +|= fmla,
FOL-hyps-meaning: FOL-hyps-meaning(Dom;S;a;hyps),
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
cand: A c∧ B
Lemmas referenced :
subtype_rel_FOAssignment,
mFOL-sequent-freevars_wf,
cons_wf,
mFOL_wf,
mFOL-sequent-freevars-contained,
mFOL-sequent-freevars-subset-1,
cons_member,
l_contains_wf,
mFOL-freevars_wf,
mFOL-sequent-freevars-subset-2,
l_member_wf,
FOAssignment_wf,
FOStruct+_wf,
FOL-sequent-evidence_wf,
list_wf,
mFOL-sequent-freevars-subset-4,
map_cons_lemma,
tupletype_cons_lemma,
null-map,
null_wf3,
subtype_rel_list,
top_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
tuple-type_wf,
FOL-hyps-meaning_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
sqequalHypSubstitution,
hypothesis,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
introduction,
extract_by_obid,
because_Cache,
independent_pairEquality,
sqequalRule,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
unionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
intEquality,
cumulativity,
universeEquality,
lambdaEquality,
productEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityElimination,
equalityTransitivity,
dependent_pairFormation,
promote_hyp,
instantiate,
baseClosed
Latex:
\mforall{}hyps:mFOL() List
\mforall{}[x,y:mFOL()].
(mFOL-freevars(y) \msubseteq{} mFOL-sequent-freevars(<hyps, x>) {}\mRightarrow{} FOL-sequent-evidence\{i:l\}(<hyps, y>) {}\mRightarrow{} \000CFOL-sequent-evidence\{i:l\}(<[y / hyps], x>) {}\mRightarrow{} FOL-sequent-evidence\{i:l\}(<hyps, x>))
Date html generated:
2018_05_21-PM-10_29_43
Last ObjectModification:
2017_07_26-PM-06_41_51
Theory : minimal-first-order-logic
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