Nuprl Lemma : omon_properties
∀g:OMon. (UniformLinorder(|g|;x,y.↑(x ≤b y)) ∧ IsEqFun(|g|;=b))
Proof
Definitions occuring in Statement :
omon: OMon,
grp_le: ≤b,
grp_eq: =b,
grp_car: |g|,
ulinorder: UniformLinorder(T;x,y.R[x; y]),
eqfun_p: IsEqFun(T;eq),
assert: ↑b,
infix_ap: x f y,
all: ∀x:A. B[x],
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
omon: OMon,
uall: ∀[x:A]. B[x],
member: t ∈ T,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B,
ulinorder: UniformLinorder(T;x,y.R[x; y]),
uorder: UniformOrder(T;x,y.R[x; y]),
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s],
utrans: UniformlyTrans(T;x,y.E[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
eqfun_p: IsEqFun(T;eq),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
urefl: UniformlyRefl(T;x,y.E[x; y])
Lemmas referenced :
sq_stable__and,
ulinorder_wf,
grp_car_wf,
assert_wf,
grp_le_wf,
equal_wf,
bool_wf,
sq_stable__equal,
grp_eq_wf,
band_wf,
squash_wf,
omon_wf,
urefl_wf,
infix_ap_wf,
utrans_wf,
uanti_sym_wf,
connex_wf,
uall_wf,
isect_wf,
sq_stable__urefl,
sq_stable_from_decidable,
decidable__assert,
assert_witness,
sq_stable__utrans,
sq_stable__uanti_sym,
sq_stable__connex,
sq_stable__uall,
assert_of_band,
assert_elim,
subtype_base_sq,
bool_subtype_base,
eqfun_p_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
productElimination,
isect_memberEquality,
functionEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
productEquality,
isect_memberFormation,
independent_pairEquality,
independent_isectElimination,
instantiate,
cumulativity,
natural_numberEquality,
hyp_replacement,
Error :applyLambdaEquality,
dependent_set_memberEquality,
setEquality
Latex:
\mforall{}g:OMon. (UniformLinorder(|g|;x,y.\muparrow{}(x \mleq{}\msubb{} y)) \mwedge{} IsEqFun(|g|;=\msubb{}))
Date html generated:
2016_10_21-AM-11_25_34
Last ObjectModification:
2016_07_12-PM-01_08_15
Theory : groups_1
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