Nuprl Lemma : FOL-subst-abstract
The evidence that fmla[x/y] in Dom, S, a is the same as the evidence
that fmla is true in Dom, S, a[y := a x].⋅
∀[Dom:Type]. ∀[S:FOStruct+{i:l}(Dom)]. ∀[fmla:mFOL()]. ∀[x,y:ℤ]. ∀[a:FOAssignment(mFOL-freevars(fmla[x/y]),Dom)].
(Dom,S,a +|= FOL-abstract(fmla[x/y]) = Dom,S,a[y := a x] +|= FOL-abstract(fmla) ∈ ℙ)
Proof
Definitions occuring in Statement :
FOL-subst: fmla[nw/old],
FOL-abstract: FOL-abstract(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
FOSatWith+: Dom,S,a +|= fmla,
update-assignment: a[x := v],
FOStruct+: FOStruct+{i:l}(Dom),
FOAssignment: FOAssignment(vs,Dom),
uall: ∀[x:A]. B[x],
prop: ℙ,
apply: f a,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
FOL-subst: fmla[nw/old],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
squash: ↓T,
true: True,
so_apply: x[s],
implies: P ⇒ Q,
uiff: uiff(P;Q),
FOAssignment: FOAssignment(vs,Dom),
update-assignment: a[x := v],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
rev_implies: P ⇐ Q,
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
not: ¬A
Lemmas referenced :
FOL-rename-bound-to-avoid_wf,
cons_wf,
nil_wf,
set_wf,
mFOL_wf,
equal_wf,
list_wf,
mFOL-freevars_wf,
length_wf,
FOL-abstract_wf,
subtype_rel-equal,
AbstractFOFormula+_wf,
l_disjoint_wf,
mFOL-boundvars_wf,
l_disjoint_singleton2,
FOAssignment_wf,
mFOL-rename_wf,
FOL-subst_wf,
FOStruct+_wf,
deq-member_wf,
int-deq_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
eq_int_wf,
assert_of_eq_int,
l_member_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
mFOL-freevars-of-rename,
equal-wf-base,
int_subtype_base,
not_wf,
FOL-abstract-rename,
squash_wf,
true_wf,
FOSatWith+_wf,
list_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
intEquality,
hypothesis,
instantiate,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
sqequalRule,
productEquality,
applyLambdaEquality,
because_Cache,
independent_isectElimination,
imageElimination,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
lambdaFormation,
setElimination,
rename,
productElimination,
equalityTransitivity,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
functionExtensionality,
unionElimination,
equalityElimination,
dependent_set_memberEquality,
dependent_pairFormation,
promote_hyp,
voidElimination,
inlFormation,
independent_pairFormation,
inrFormation,
setEquality,
hyp_replacement
Latex:
\mforall{}[Dom:Type]. \mforall{}[S:FOStruct+\{i:l\}(Dom)]. \mforall{}[fmla:mFOL()]. \mforall{}[x,y:\mBbbZ{}].
\mforall{}[a:FOAssignment(mFOL-freevars(fmla[x/y]),Dom)].
(Dom,S,a +|= FOL-abstract(fmla[x/y]) = Dom,S,a[y := a x] +|= FOL-abstract(fmla))
Date html generated:
2018_05_21-PM-10_24_39
Last ObjectModification:
2017_07_26-PM-06_38_37
Theory : minimal-first-order-logic
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