Nuprl Lemma : bound-term-induction
∀[opr:Type]. ∀[P:bound-term(opr) ⟶ ℙ].
((∀vs:varname() List. ∀v:varname(). ((¬(v = nullvar() ∈ varname())) ⇒ P[<vs, varterm(v)>]))
⇒ (∀bts:bound-term(opr) List
((∀bt:bound-term(opr). ((bt ∈ bts) ⇒ P[bt])) ⇒ (∀f:opr. ∀vs:varname() List. P[<vs, mkterm(f;bts)>])))
⇒ (∀bt:bound-term(opr). P[bt]))
Proof
Definitions occuring in Statement :
bound-term: bound-term(opr),
mkterm: mkterm(opr;bts),
varterm: varterm(v),
nullvar: nullvar(),
varname: varname(),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
bound-term: bound-term(opr),
not: ¬A,
false: False,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bound-term-size: bound-term-size(bt),
pi2: snd(t),
ge: i ≥ j ,
l_member: (x ∈ l),
cand: A c∧ B,
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b)
Lemmas referenced :
bound-term_wf,
list_wf,
l_member_wf,
subtype_rel_self,
varname_wf,
mkterm_wf,
nullvar_wf,
istype-void,
varterm_wf,
istype-universe,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
term-cases,
iff_weakening_equal,
term-size_wf,
le_wf,
bound-term-size_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
subtype_rel_list,
less_than_wf,
select_wf,
squash_wf,
true_wf,
easy-member-int_seg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
functionIsType,
applyEquality,
instantiate,
universeEquality,
independent_pairEquality,
because_Cache,
equalityIstype,
independent_isectElimination,
setElimination,
rename,
productElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
voidElimination,
unionElimination,
cumulativity,
intEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
promote_hyp,
hypothesis_subsumption,
functionEquality,
setIsType,
addEquality,
setEquality,
imageElimination,
imageMemberEquality,
baseClosed,
closedConclusion
Latex:
\mforall{}[opr:Type]. \mforall{}[P:bound-term(opr) {}\mrightarrow{} \mBbbP{}].
((\mforall{}vs:varname() List. \mforall{}v:varname(). ((\mneg{}(v = nullvar())) {}\mRightarrow{} P[<vs, varterm(v)>]))
{}\mRightarrow{} (\mforall{}bts:bound-term(opr) List
((\mforall{}bt:bound-term(opr). ((bt \mmember{} bts) {}\mRightarrow{} P[bt]))
{}\mRightarrow{} (\mforall{}f:opr. \mforall{}vs:varname() List. P[<vs, mkterm(f;bts)>])))
{}\mRightarrow{} (\mforall{}bt:bound-term(opr). P[bt]))
Date html generated:
2020_05_19-PM-09_54_22
Last ObjectModification:
2020_03_09-PM-04_08_32
Theory : terms
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