Nuprl Lemma : grp_lt_complement
∀[g:OCMon]. ∀[a,b:|g|]. uiff(¬(b < a);a ≤ b)
Proof
Definitions occuring in Statement :
grp_lt: a < b,
grp_leq: a ≤ b,
ocmon: OCMon,
grp_car: |g|,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
ocmon: OCMon,
omon: OMon,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
infix_ap: x f y,
so_apply: x[s],
cand: A c∧ B,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
set_leq: a ≤ b,
set_le: ≤b,
pi2: snd(t),
grp_lt: a < b,
grp_leq: a ≤ b,
not: ¬A,
false: False
Lemmas referenced :
set_lt_complement,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
grp_car_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
assert_witness,
not_wf,
grp_lt_wf,
grp_leq_wf,
ocmon_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
hypothesis,
because_Cache,
lambdaEquality,
productEquality,
setElimination,
rename,
cumulativity,
universeEquality,
functionEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
setEquality,
independent_pairFormation,
isect_memberEquality,
independent_pairEquality,
voidElimination
Latex:
\mforall{}[g:OCMon]. \mforall{}[a,b:|g|]. uiff(\mneg{}(b < a);a \mleq{} b)
Date html generated:
2017_10_01-AM-08_15_01
Last ObjectModification:
2017_02_28-PM-01_59_58
Theory : groups_1
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