Nuprl Lemma : term-induction
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ].
((∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . P[varterm(v)])
⇒ (∀bts:bound-term(opr) List. ((∀bt:bound-term(opr). ((bt ∈ bts) ⇒ P[snd(bt)])) ⇒ (∀f:opr. P[mkterm(f;bts)])))
⇒ {∀t:term(opr). P[t]})
Proof
Definitions occuring in Statement :
bound-term: bound-term(opr),
mkterm: mkterm(opr;bts),
varterm: varterm(v),
term: term(opr),
nullvar: nullvar(),
varname: varname(),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
not: ¬A,
false: False,
so_apply: x[s],
so_lambda: so_lambda3,
subtype_rel: A ⊆r B,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
spreadn: spread3,
cand: A c∧ B,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bound-term: bound-term(opr),
pi2: snd(t),
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_apply: x[s1;s2;s3]
Lemmas referenced :
term-ind_wf,
varname_wf,
not_wf,
equal-wf-T-base,
nullvar_wf,
istype-void,
list_wf,
bound-term_wf,
int_seg_wf,
length_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
subtype_rel-equal,
l_member_wf,
select_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
mkterm_wf,
varterm_wf,
term_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
applyEquality,
functionExtensionality,
setEquality,
baseClosed,
setIsType,
universeIsType,
functionIsType,
equalityIstype,
inhabitedIsType,
because_Cache,
productElimination,
closedConclusion,
natural_numberEquality,
dependent_set_memberEquality_alt,
setElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
voidElimination,
productIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
universeEquality,
instantiate
Latex:
\mforall{}[opr:Type]. \mforall{}[P:term(opr) {}\mrightarrow{} \mBbbP{}].
((\mforall{}v:\{v:varname()| \mneg{}(v = nullvar())\} . P[varterm(v)])
{}\mRightarrow{} (\mforall{}bts:bound-term(opr) List
((\mforall{}bt:bound-term(opr). ((bt \mmember{} bts) {}\mRightarrow{} P[snd(bt)])) {}\mRightarrow{} (\mforall{}f:opr. P[mkterm(f;bts)])))
{}\mRightarrow{} \{\mforall{}t:term(opr). P[t]\})
Date html generated:
2020_05_19-PM-09_54_30
Last ObjectModification:
2020_03_09-PM-04_08_36
Theory : terms
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