Nuprl Lemma : mFOL-ext
mFOL() ≡ lbl:Atom × if lbl =a "atomic" then name:Atom × (ℤ List)
if lbl =a "connect" then knd:Atom × left:mFOL() × mFOL()
if lbl =a "quant" then isall:𝔹 × var:ℤ × mFOL()
else Void
fi
Proof
Definitions occuring in Statement :
mFOL: mFOL(),
list: T List,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bool: 𝔹,
ext-eq: A ≡ B,
product: x:A × B[x],
int: ℤ,
token: "$token",
atom: Atom,
void: Void
Definitions unfolded in proof :
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
mFOL: mFOL(),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
guard: {T},
eq_atom: x =a y,
mFOLco_size: mFOLco_size(p),
has-value: (a)↓,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
spreadn: spread3,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
mFOL_size: mFOL_size(p),
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A
Lemmas referenced :
mFOLco-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
list_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
int_subtype_base,
mFOLco_size_wf,
subtype_partial_sqtype_base,
nat_wf,
set_subtype_base,
le_wf,
base_wf,
value-type-has-value,
int-value-type,
has-value_wf-partial,
set-value-type,
mFOL_wf,
ifthenelse_wf,
mFOLco_wf,
add-nat,
false_wf,
mFOL_size_wf,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
hypothesis,
promote_hyp,
productElimination,
hypothesis_subsumption,
hypothesisEquality,
applyEquality,
sqequalRule,
dependent_pairEquality,
isectElimination,
tokenEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
because_Cache,
instantiate,
cumulativity,
atomEquality,
dependent_functionElimination,
independent_functionElimination,
intEquality,
dependent_pairFormation,
voidElimination,
dependent_set_memberEquality,
natural_numberEquality,
baseApply,
closedConclusion,
baseClosed,
callbyvalueAdd,
productEquality,
universeEquality,
voidEquality,
sqleReflexivity
Latex:
mFOL() \mequiv{} lbl:Atom \mtimes{} if lbl =a "atomic" then name:Atom \mtimes{} (\mBbbZ{} List)
if lbl =a "connect" then knd:Atom \mtimes{} left:mFOL() \mtimes{} mFOL()
if lbl =a "quant" then isall:\mBbbB{} \mtimes{} var:\mBbbZ{} \mtimes{} mFOL()
else Void
fi
Date html generated:
2018_05_21-PM-10_20_50
Last ObjectModification:
2017_07_26-PM-06_37_37
Theory : minimal-first-order-logic
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