Nuprl Lemma : count_functionality
∀s:DSet. ∀a,a':|s|. ∀bs,bs':|s| List. ((a = a' ∈ |s|) ⇒ (bs ≡(|s|) bs') ⇒ ((a #∈ bs) = (a' #∈ bs') ∈ ℤ))
Proof
Definitions occuring in Statement :
count: a #∈ as,
permr: as ≡(T) bs,
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
count: a #∈ as,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
dset: DSet,
subtype_rel: A ⊆r B,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
iabmonoid: IAbMonoid,
imon: IMonoid,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
infix_ap: x f y,
grp_car: |g|,
pi1: fst(t),
int_add_grp: <ℤ+>,
guard: {T},
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
permr_wf,
set_car_wf,
equal_wf,
int_add_grp_wf,
subtype_rel_sets,
grp_sig_wf,
monoid_p_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
b2i_wf,
set_eq_wf,
subtype_rel_self,
mon_subtype_grp_sig,
grp_subtype_mon,
abgrp_subtype_grp,
subtype_rel_transitivity,
abgrp_wf,
grp_wf,
mon_wf,
mem_f_wf,
mon_for_wf,
squash_wf,
true_wf,
mon_for_functionality_wrt_permr,
bool_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesisEquality,
intEquality,
applyEquality,
sqequalRule,
instantiate,
setEquality,
cumulativity,
lambdaEquality,
independent_isectElimination,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}s:DSet. \mforall{}a,a':|s|. \mforall{}bs,bs':|s| List. ((a = a') {}\mRightarrow{} (bs \mequiv{}(|s|) bs') {}\mRightarrow{} ((a \#\mmember{} bs) = (a' \#\mmember{} bs')))
Date html generated:
2018_05_22-AM-07_45_20
Last ObjectModification:
2018_05_19-AM-08_32_28
Theory : list_2
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