Nuprl Lemma : formula_ind_wf
∀[A:Type]. ∀[R:A ⟶ formula() ⟶ ℙ]. ∀[v:formula()]. ∀[var:name:Atom ⟶ {x:A| R[x;pvar(name)]} ].
∀[not:sub:formula() ⟶ {x:A| R[x;sub]} ⟶ {x:A| R[x;pnot(sub)]} ]. ∀[and:left:formula()
⟶ right:formula()
⟶ {x:A| R[x;left]}
⟶ {x:A| R[x;right]}
⟶ {x:A| R[x;pand(left;right)]} ].
∀[or:left:formula() ⟶ right:formula() ⟶ {x:A| R[x;left]} ⟶ {x:A| R[x;right]} ⟶ {x:A| R[x;por(left;right)]} ].
∀[imp:left:formula() ⟶ right:formula() ⟶ {x:A| R[x;left]} ⟶ {x:A| R[x;right]} ⟶ {x:A| R[x;pimp(left;right)]} ].
(formula_ind(v;
pvar(name)⇒ var[name];
pnot(sub)⇒ rec1.not[sub;rec1];
pand(left,right)⇒ rec2,rec3.and[left;right;rec2;rec3];
por(left,right)⇒ rec4,rec5.or[left;right;rec4;rec5];
pimp(left,right)⇒ rec6,rec7.imp[left;right;rec6;rec7]) ∈ {x:A| R[x;v]} )
Proof
Definitions occuring in Statement :
formula_ind: formula_ind,
pimp: pimp(left;right),
por: por(left;right),
pand: pand(left;right),
pnot: pnot(sub),
pvar: pvar(name),
formula: formula(),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
formula_ind: formula_ind,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
so_apply: x[s],
formula-definition,
formula-induction,
uniform-comp-nat-induction,
formula-ext,
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_lambda: λ2x.t[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
formula-definition,
has-value_wf_base,
is-exception_wf,
lifting-strict-atom_eq,
strict4-decide,
istype-atom,
pvar_wf,
pnot_wf,
pand_wf,
por_wf,
pimp_wf,
formula_wf,
all_wf,
set_wf,
subtype_rel_function,
subtype_rel_self,
istype-universe,
formula-induction,
uniform-comp-nat-induction,
formula-ext,
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,
atomEquality,
setElimination,
rename,
dependent_set_memberEquality_alt
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} formula() {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:formula()]. \mforall{}[var:name:Atom {}\mrightarrow{} \{x:A| R[x;pvar(name)]\} ].
\mforall{}[not:sub:formula() {}\mrightarrow{} \{x:A| R[x;sub]\} {}\mrightarrow{} \{x:A| R[x;pnot(sub)]\} ]. \mforall{}[and:left:formula()
{}\mrightarrow{} right:formula()
{}\mrightarrow{} \{x:A| R[x;left]\}
{}\mrightarrow{} \{x:A| R[x;right]\}
{}\mrightarrow{} \{x:A|
R[x;pand(left;right)]\} ]\000C.
\mforall{}[or:left:formula()
{}\mrightarrow{} right:formula()
{}\mrightarrow{} \{x:A| R[x;left]\}
{}\mrightarrow{} \{x:A| R[x;right]\}
{}\mrightarrow{} \{x:A| R[x;por(left;right)]\} ]. \mforall{}[imp:left:formula()
{}\mrightarrow{} right:formula()
{}\mrightarrow{} \{x:A| R[x;left]\}
{}\mrightarrow{} \{x:A| R[x;right]\}
{}\mrightarrow{} \{x:A| R[x;pimp(left;right)]\} ].
(formula\_ind(v;
pvar(name){}\mRightarrow{} var[name];
pnot(sub){}\mRightarrow{} rec1.not[sub;rec1];
pand(left,right){}\mRightarrow{} rec2,rec3.and[left;right;rec2;rec3];
por(left,right){}\mRightarrow{} rec4,rec5.or[left;right;rec4;rec5];
pimp(left,right){}\mRightarrow{} rec6,rec7.imp[left;right;rec6;rec7]) \mmember{} \{x:A| R[x;v]\} )
Date html generated:
2020_05_20-AM-08_18_38
Last ObjectModification:
2020_01_24-PM-02_36_12
Theory : general
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