Nuprl Lemma : cons_functionality_wrt_permr
∀T:Type. ∀a,b:T. ∀as,bs:T List. ((a = b ∈ T) ⇒ (as ≡(T) bs) ⇒ ([a / as] ≡(T) [b / bs]))
Proof
Definitions occuring in Statement :
permr: as ≡(T) bs,
cons: [a / b],
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
permr: as ≡(T) bs,
top: Top,
cand: A c∧ B,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
and: P ∧ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
ge: i ≥ j ,
sym_grp: Sym(n),
perm: Perm(T),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
conj_perm: conj{p}(q),
comp_perm: comp_perm,
mk_perm: mk_perm(f;b),
perm_f: p.f,
pi1: fst(t),
rev_perm: ↔p{n},
extend_perm: ↑{n}(p),
inv_perm: inv_perm(p),
compose: f o g,
perm_b: p.b,
pi2: snd(t),
rev_permf: rev_permf(n),
extend_permf: extend_permf(pf;n),
uiff: uiff(P;Q),
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
permr_wf,
list_wf,
istype-universe,
length_of_cons_lemma,
istype-void,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
conj_perm_wf,
add_nat_wf,
length_wf_nat,
istype-false,
le_wf,
nat_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rev_perm_wf,
extend_perm_wf,
int_seg_wf,
length_wf,
select_wf,
cons_wf,
perm_f_wf,
non_neg_length,
int_seg_properties,
less_than_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
itermMinus_wf,
itermSubtract_wf,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
add-member-int_seg1,
subtract_wf,
eq_int_wf,
equal-wf-T-base,
bool_wf,
assert_wf,
equal-wf-base-T,
int_subtype_base,
add-associates,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-add,
zero-mul,
add-zero,
bnot_wf,
not_wf,
set_subtype_base,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
itermMultiply_wf,
int_term_value_mul_lemma,
equal_wf,
squash_wf,
true_wf,
select_cons_hd,
subtype_rel_self,
iff_weakening_equal,
select_cons_tl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
equalityIsType1,
inhabitedIsType,
isectElimination,
universeEquality,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
productElimination,
because_Cache,
unionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
applyLambdaEquality,
setElimination,
rename,
functionIsType,
addEquality,
applyEquality,
productIsType,
imageElimination,
multiplyEquality,
minusEquality,
baseClosed,
intEquality,
equalityIsType4,
equalityElimination,
imageMemberEquality,
instantiate
Latex:
\mforall{}T:Type. \mforall{}a,b:T. \mforall{}as,bs:T List. ((a = b) {}\mRightarrow{} (as \mequiv{}(T) bs) {}\mRightarrow{} ([a / as] \mequiv{}(T) [b / bs]))
Date html generated:
2019_10_16-PM-01_00_34
Last ObjectModification:
2018_10_08-AM-10_57_39
Theory : perms_2
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