Nuprl Lemma : subterm-mkterm
∀[opr:Type]
∀s:term(opr). ∀f:opr. ∀bts:bound-term(opr) List.
(s << mkterm(f;bts) ⇐⇒ ∃i:ℕ||bts||. ((s = (snd(bts[i])) ∈ term(opr)) ∨ s << snd(bts[i])))
Proof
Definitions occuring in Statement :
subterm: s << t,
bound-term: bound-term(opr),
mkterm: mkterm(opr;bts),
term: term(opr),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
bound-term: bound-term(opr),
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
pi2: snd(t),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
immediate-subterm: s < t,
true: True,
cand: A c∧ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q)
Lemmas referenced :
subterm_wf,
mkterm_wf,
int_seg_wf,
length_wf,
bound-term_wf,
select_wf,
int_seg_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
list_wf,
term_wf,
istype-universe,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
istype-le,
istype-less_than,
subtype_rel_self,
subterm-cases,
term-size_wf,
le_wf,
equal_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
subterm-size,
term_size_mkterm_lemma,
squash_wf,
true_wf,
mkterm-one-one,
iff_weakening_equal,
immediate-subterm-size,
easy-member-int_seg,
subtract-is-int-iff,
false_wf,
immediate-is-subterm,
subterm_transitivity,
trivial-subterm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
hypothesis,
productIsType,
natural_numberEquality,
unionIsType,
equalityIstype,
inhabitedIsType,
setElimination,
rename,
because_Cache,
independent_isectElimination,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
voidElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
applyEquality,
applyLambdaEquality,
dependent_set_memberEquality_alt,
promote_hyp,
hypothesis_subsumption,
functionIsType,
functionEquality,
productEquality,
unionEquality,
setIsType,
addEquality,
imageMemberEquality,
baseClosed,
inlFormation_alt,
closedConclusion,
pointwiseFunctionality,
baseApply,
inrFormation_alt,
hyp_replacement
Latex:
\mforall{}[opr:Type]
\mforall{}s:term(opr). \mforall{}f:opr. \mforall{}bts:bound-term(opr) List.
(s << mkterm(f;bts) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||bts||. ((s = (snd(bts[i]))) \mvee{} s << snd(bts[i])))
Date html generated:
2020_05_19-PM-09_54_20
Last ObjectModification:
2020_03_10-PM-03_45_47
Theory : terms
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