Nuprl Lemma : formula_ind_wf_simple
∀[A:Type]. ∀[v:formula()]. ∀[var:name:Atom ⟶ A]. ∀[not:sub:formula() ⟶ A ⟶ A]. ∀[and,or,imp:left:formula()
⟶ right:formula()
⟶ A
⟶ A
⟶ A].
(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]) ∈ A)
Proof
Definitions occuring in Statement :
formula_ind: formula_ind,
formula: formula(),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
atom: Atom,
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],
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
formula_ind_wf,
true_wf,
formula_wf,
istype-true,
subtype_rel_dep_function,
istype-atom,
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,
functionExtensionality,
applyEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
closedConclusion,
atomEquality,
setEquality,
setIsType,
independent_isectElimination,
setElimination,
rename,
lambdaFormation_alt,
because_Cache,
functionEquality,
inhabitedIsType,
applyLambdaEquality,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[v:formula()]. \mforall{}[var:name:Atom {}\mrightarrow{} A]. \mforall{}[not:sub:formula() {}\mrightarrow{} A {}\mrightarrow{} A].
\mforall{}[and,or,imp:left:formula() {}\mrightarrow{} right:formula() {}\mrightarrow{} A {}\mrightarrow{} A {}\mrightarrow{} A].
(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{} A)
Date html generated:
2020_05_20-AM-08_18_53
Last ObjectModification:
2020_01_24-PM-00_51_06
Theory : general
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