Nuprl Lemma : subst-term_functionality
∀[opr:Type]
  ∀t1,t2:term(opr). ∀s1,s2:(varname() × term(opr)) List.
    (alpha-eq-terms(opr;t1;t2) 
⇒ equiv-substs(opr;s1;s2) 
⇒ alpha-eq-terms(opr;subst-term(s1;t1);subst-term(s2;t2)))
Proof
Definitions occuring in Statement : 
subst-term: subst-term(s;t)
, 
equiv-substs: equiv-substs(opr;s1;s2)
, 
alpha-eq-terms: alpha-eq-terms(opr;a;b)
, 
term: term(opr)
, 
varname: varname()
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
product: x:A × B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
subst-term: subst-term(s;t)
, 
member: t ∈ T
, 
guard: {T}
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
not: ¬A
, 
false: False
, 
alpha-aux: alpha-aux(opr;vs;ws;a;b)
, 
varterm: varterm(v)
, 
mkterm: mkterm(opr;bts)
, 
bound-term: bound-term(opr)
, 
pi2: snd(t)
, 
alpha-eq-terms: alpha-eq-terms(opr;a;b)
, 
replace-vars: replace-vars(s;t)
, 
vars-of-subst: vars-of-subst(s)
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
assert: ↑b
, 
l_disjoint: l_disjoint(T;l1;l2)
, 
squash: ↓T
, 
true: True
, 
sq_stable: SqStable(P)
, 
equiv-substs: equiv-substs(opr;s1;s2)
, 
isl: isl(x)
, 
outl: outl(x)
, 
free-vars: free-vars(t)
, 
append: as @ bs
, 
list_ind: list_ind, 
nil: []
, 
so_lambda: so_lambda3, 
so_apply: x[s1;s2;s3]
, 
nat: ℕ
, 
free-vars-aux: free-vars-aux(bnds;t)
, 
l_member: (x ∈ l)
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
select: L[n]
, 
cons: [a / b]
, 
less_than: a < b
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
colength: colength(L)
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
same-binding: same-binding(vs;ws;v;w)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
band: p ∧b q
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
pi1: fst(t)
, 
reverse: rev(as)
, 
respects-equality: respects-equality(S;T)
, 
has-value: (a)↓
, 
binders-disjoint: binders-disjoint(opr;L;t)
Lemmas referenced : 
subst-frame-alpha, 
alpha-eq-terms_inversion, 
subst-frame_wf, 
alpha-eq-terms_transitivity, 
subst-frame-binders-disjoint, 
term-induction, 
term_wf, 
list_wf, 
varname_wf, 
l_disjoint_wf, 
vars-of-subst_wf, 
alpha-aux_wf, 
binders-disjoint_wf, 
replace-vars_wf, 
varterm_wf, 
subtype_rel_list, 
not_wf, 
equal-wf-T-base, 
nullvar_wf, 
mkterm_wf, 
bound-term_wf, 
l_member_wf, 
istype-void, 
nil_wf, 
l_disjoint_nil, 
alpha-eq-terms_wf, 
equiv-substs_wf, 
istype-universe, 
apply-alist_wf, 
var-deq_wf, 
apply-alist-inl, 
member-l-union-list, 
map_wf, 
ifthenelse_wf, 
eq_var_wf, 
free-vars_wf, 
insert_wf, 
eqtt_to_assert, 
assert-eq_var, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
iff_weakening_uiff, 
assert_wf, 
member-map, 
same-binding-not-bound, 
member-insert, 
squash_wf, 
true_wf, 
equal_wf, 
unit_wf2, 
deq_wf, 
iff_weakening_equal, 
inl-one-one, 
not-0-eq-1, 
inr-one-one, 
sq_stable__not, 
subtype_rel_self, 
btrue_wf, 
bfalse_wf, 
outl_wf, 
isl_wf, 
btrue_neq_bfalse, 
length_wf, 
list_ind_nil_lemma, 
append_back_nil, 
equal-wf-base, 
length_wf_nat, 
set_subtype_base, 
le_wf, 
istype-int, 
int_subtype_base, 
append_wf, 
free-vars-aux_wf, 
deq-member_wf, 
assert-deq-member, 
istype-le, 
length_of_cons_lemma, 
length_of_nil_lemma, 
cons_wf, 
istype-less_than, 
select_wf, 
nat_properties, 
decidable__le, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
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, 
intformless_wf, 
int_formula_prop_less_lemma, 
ge_wf, 
assert_witness, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
list-cases, 
product_subtype_list, 
colength-cons-not-zero, 
same-binding_wf, 
subtract-1-ge-0, 
spread_cons_lemma, 
list_ind_cons_lemma, 
non_neg_length, 
itermAdd_wf, 
int_term_value_add_lemma, 
colength_wf_list, 
decidable__equal_int, 
subtract_wf, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
istype-nat, 
istype-assert, 
member-implies-null-eq-bfalse, 
null_nil_lemma, 
cons_member, 
add-is-int-iff, 
false_wf, 
alpha-aux-mkterm, 
rev-append-as-reverse, 
append_assoc, 
int_seg_wf, 
int_seg_properties, 
decidable__lt, 
select_member, 
reverse_wf, 
length-append, 
length-reverse, 
rev-append-property, 
rev-append_wf, 
respects-equality-list, 
respects-equality-set-trivial2, 
same-binding-symm, 
product-value-type, 
value-type-has-value, 
list-value-type, 
select-map, 
top_wf, 
eager-map-is-map, 
map-length, 
int_seg_subtype_nat, 
istype-false, 
l_disjoint_append2, 
l_all_iff, 
member-reverse
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
dependent_functionElimination, 
hypothesis, 
independent_functionElimination, 
inhabitedIsType, 
sqequalRule, 
lambdaEquality_alt, 
functionEquality, 
applyEquality, 
because_Cache, 
universeIsType, 
setElimination, 
rename, 
independent_isectElimination, 
voidElimination, 
setEquality, 
baseClosed, 
equalityTransitivity, 
equalitySymmetry, 
equalityIstype, 
functionIsType, 
productElimination, 
setIsType, 
productEquality, 
instantiate, 
universeEquality, 
unionElimination, 
spreadEquality, 
productIsType, 
dependent_pairFormation_alt, 
independent_pairEquality, 
independent_pairFormation, 
equalityElimination, 
promote_hyp, 
cumulativity, 
inlFormation_alt, 
imageElimination, 
unionEquality, 
natural_numberEquality, 
imageMemberEquality, 
applyLambdaEquality, 
dependent_set_memberEquality_alt, 
unionIsType, 
sqequalBase, 
inrFormation_alt, 
voidEquality, 
Error :memTop, 
intEquality, 
approximateComputation, 
int_eqEquality, 
intWeakElimination, 
functionIsTypeImplies, 
hypothesis_subsumption, 
addEquality, 
baseApply, 
closedConclusion, 
hyp_replacement, 
pointwiseFunctionality, 
callbyvalueReduce
Latex:
\mforall{}[opr:Type]
    \mforall{}t1,t2:term(opr).  \mforall{}s1,s2:(varname()  \mtimes{}  term(opr))  List.
        (alpha-eq-terms(opr;t1;t2)
        {}\mRightarrow{}  equiv-substs(opr;s1;s2)
        {}\mRightarrow{}  alpha-eq-terms(opr;subst-term(s1;t1);subst-term(s2;t2)))
Date html generated:
2020_05_19-PM-09_58_06
Last ObjectModification:
2020_03_09-PM-04_10_09
Theory : terms
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