Nuprl Lemma : FOLRule_ind_wf
∀[A:Type]. ∀[R:A ⟶ FOLRule() ⟶ ℙ]. ∀[v:FOLRule()]. ∀[andI:{x:A| R[x;andI]} ]. ∀[impI:{x:A| R[x;impI]} ].
∀[allI:var:ℤ ⟶ {x:A| R[x;allI with var]} ]. ∀[existsI:var:ℤ ⟶ {x:A| R[x;existsI with var]} ].
∀[orI:left:𝔹 ⟶ {x:A| R[x;fRuleorI(left)]} ]. ∀[hyp:{x:A| R[x;hyp]} ]. ∀[andE:hypnum:ℕ ⟶ {x:A| R[x;andE on hypnum]} ].
∀[orE:hypnum:ℕ ⟶ {x:A| R[x;orE on hypnum]} ]. ∀[impE:hypnum:ℕ ⟶ {x:A| R[x;impE on hypnum]} ].
∀[allE:hypnum:ℕ ⟶ var:ℤ ⟶ {x:A| R[x;allE on hypnum with var]} ]. ∀[existsE:hypnum:ℕ
⟶ var:ℤ
⟶ {x:A| R[x;existsE on hypnum with var]} ].
∀[falseE:hypnum:ℕ ⟶ {x:A| R[x;falseE on hypnum]} ].
(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]
falseE @hypnum => falseE[hypnum] ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement :
FOLRule_ind: FOLRule_ind,
fRulefalseE: falseE on hypnum,
fRuleexistsE: existsE on hypnum with var,
fRuleallE: allE on hypnum with var,
fRuleimpE: impE on hypnum,
fRuleorE: orE on hypnum,
fRuleandE: andE on hypnum,
fRulehyp: hyp,
fRuleorI: fRuleorI(left),
fRuleexistsI: existsI with var,
fRuleallI: allI with var,
fRuleimpI: impI,
fRuleandI: andI,
FOLRule: FOLRule(),
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
FOLRule_ind: FOLRule_ind,
so_apply: x[s],
so_apply: x[s1;s2],
FOLRule-definition,
FOLRule-induction,
FOLRule-ext,
ext-eq_weakening,
eq_atom: x =a y,
btrue: tt,
it: ⋅,
bfalse: ff,
bool_cases_sqequal,
eqff_to_assert,
any: any x,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
FOLRule-definition,
has-value_wf_base,
is-exception_wf,
lifting-strict-atom_eq,
strict4-decide,
fRuleandI_wf,
fRuleimpI_wf,
istype-int,
fRuleallI_wf,
fRuleexistsI_wf,
bool_wf,
fRuleorI_wf,
fRulehyp_wf,
istype-nat,
fRuleandE_wf,
fRuleorE_wf,
fRuleimpE_wf,
fRuleallE_wf,
fRuleexistsE_wf,
fRulefalseE_wf,
FOLRule_wf,
subtype_rel_function,
nat_wf,
subtype_rel_self,
istype-universe,
FOLRule-induction,
FOLRule-ext,
ext-eq_weakening,
bool_cases_sqequal,
eqff_to_assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
Error :memTop,
inhabitedIsType,
hypothesis,
lambdaFormation_alt,
thin,
sqequalSqle,
divergentSqle,
callbyvalueDecide,
sqequalHypSubstitution,
hypothesisEquality,
unionElimination,
sqleReflexivity,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
introduction,
decideExceptionCases,
axiomSqleEquality,
exceptionSqequal,
baseApply,
closedConclusion,
baseClosed,
extract_by_obid,
isectElimination,
independent_isectElimination,
lambdaEquality_alt,
isectIsType,
functionIsType,
universeIsType,
universeEquality,
setIsType,
because_Cache,
applyEquality,
functionEquality,
setEquality,
instantiate,
functionExtensionality,
intEquality
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} FOLRule() {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:FOLRule()]. \mforall{}[andI:\{x:A| R[x;andI]\} ]. \mforall{}[impI:\{x:A|
R[x;impI]\} ].
\mforall{}[allI:var:\mBbbZ{} {}\mrightarrow{} \{x:A| R[x;allI with var]\} ]. \mforall{}[existsI:var:\mBbbZ{} {}\mrightarrow{} \{x:A| R[x;existsI with var]\} ].
\mforall{}[orI:left:\mBbbB{} {}\mrightarrow{} \{x:A| R[x;fRuleorI(left)]\} ]. \mforall{}[hyp:\{x:A| R[x;hyp]\} ].
\mforall{}[andE:hypnum:\mBbbN{} {}\mrightarrow{} \{x:A| R[x;andE on hypnum]\} ]. \mforall{}[orE:hypnum:\mBbbN{} {}\mrightarrow{} \{x:A| R[x;orE on hypnum]\} ].
\mforall{}[impE:hypnum:\mBbbN{} {}\mrightarrow{} \{x:A| R[x;impE on hypnum]\} ]. \mforall{}[allE:hypnum:\mBbbN{}
{}\mrightarrow{} var:\mBbbZ{}
{}\mrightarrow{} \{x:A| R[x;allE on hypnum with var]\} ].
\mforall{}[existsE:hypnum:\mBbbN{} {}\mrightarrow{} var:\mBbbZ{} {}\mrightarrow{} \{x:A| R[x;existsE on hypnum with var]\} ].
\mforall{}[falseE:hypnum:\mBbbN{} {}\mrightarrow{} \{x:A| R[x;falseE on hypnum]\} ].
(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]
falseE @hypnum => falseE[hypnum] \mmember{} \{x:A| R[x;v]\} )
Date html generated:
2020_05_20-AM-09_10_02
Last ObjectModification:
2020_02_03-PM-02_54_00
Theory : minimal-first-order-logic
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