Nuprl Lemma : mFOLRule_ind_wf_simple
∀[A:Type]. ∀[v:mFOLRule()]. ∀[andI,impI:A]. ∀[allI,existsI:var:ℤ ⟶ A]. ∀[orI:left:𝔹 ⟶ A]. ∀[hyp:A].
∀[andE,orE,impE:hypnum:ℕ ⟶ A]. ∀[allE,existsE:hypnum:ℕ ⟶ var:ℤ ⟶ A].
(case(v)
andI => andI
impI => impI
allI with var => allI[var]
existsI with var => existsI[var]
orI (left?left) => orI[left]
hyp => hyp
andE @hypnum => andE[hypnum]
orE @hypnum => orE[hypnum]
impE @hypnum => impE[hypnum]
allE @hypnum with var => allE[hypnum;var]
existsE @hypnum with var => existsE[hypnum;var] ∈ A)
Proof
Definitions occuring in Statement :
mFOLRule_ind: mFOLRule_ind,
mFOLRule: mFOLRule(),
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a
Lemmas referenced :
mFOLRule_ind_wf,
true_wf,
mFOLRule_wf,
istype-true,
bool_wf,
nat_wf,
subtype_rel_function,
subtype_rel_self,
istype-nat,
istype-int,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
universeIsType,
dependent_set_memberEquality_alt,
natural_numberEquality,
functionExtensionality,
applyEquality,
closedConclusion,
intEquality,
because_Cache,
setEquality,
independent_isectElimination,
applyLambdaEquality,
setElimination,
rename,
inhabitedIsType,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[v:mFOLRule()]. \mforall{}[andI,impI:A]. \mforall{}[allI,existsI:var:\mBbbZ{} {}\mrightarrow{} A]. \mforall{}[orI:left:\mBbbB{} {}\mrightarrow{} A].
\mforall{}[hyp:A]. \mforall{}[andE,orE,impE:hypnum:\mBbbN{} {}\mrightarrow{} A]. \mforall{}[allE,existsE:hypnum:\mBbbN{} {}\mrightarrow{} var:\mBbbZ{} {}\mrightarrow{} A].
(case(v)
andI => andI
impI => impI
allI with var => allI[var]
existsI with var => existsI[var]
orI (left?left) => orI[left]
hyp => hyp
andE @hypnum => andE[hypnum]
orE @hypnum => orE[hypnum]
impE @hypnum => impE[hypnum]
allE @hypnum with var => allE[hypnum;var]
existsE @hypnum with var => existsE[hypnum;var] \mmember{} A)
Date html generated:
2020_05_20-AM-09_09_20
Last ObjectModification:
2020_02_04-PM-02_14_22
Theory : minimal-first-order-logic
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