Nuprl Lemma : ocmon_properties
∀g:OCMon
(UniformLinorder(|g|;x,y.↑(x ≤b y))
∧ Cancel(|g|;|g|;*)
∧ (∀[z:|g|]. monot(|g|;x,y.↑(x ≤b y);λw.(z * w)))
∧ IsEqFun(|g|;=b))
Proof
Definitions occuring in Statement :
ocmon: OCMon,
grp_op: *,
grp_le: ≤b,
grp_eq: =b,
grp_car: |g|,
monot: monot(T;x,y.R[x; y];f),
cancel: Cancel(T;S;op),
ulinorder: UniformLinorder(T;x,y.R[x; y]),
eqfun_p: IsEqFun(T;eq),
assert: ↑b,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
all: ∀x:A. B[x],
and: P ∧ Q,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
ocmon: OCMon,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
prop: ℙ,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
infix_ap: x f y,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
monot: monot(T;x,y.R[x; y];f),
cancel: Cancel(T;S;op),
uimplies: b supposing a,
squash: ↓T,
cand: A c∧ B,
omon: OMon,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
ulinorder: UniformLinorder(T;x,y.R[x; y]),
uorder: UniformOrder(T;x,y.R[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y])
Lemmas referenced :
sq_stable__and,
ulinorder_wf,
grp_car_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
sq_stable__equal,
grp_eq_wf,
squash_wf,
sq_stable__cancel,
sq_stable__uall,
sq_stable__monot,
sq_stable_from_decidable,
decidable__assert,
assert_witness,
omon_properties,
eqtt_to_assert,
ocmon_wf,
urefl_wf,
utrans_wf,
uanti_sym_wf,
connex_wf,
isect_wf,
sq_stable__urefl,
sq_stable__utrans,
sq_stable__uanti_sym,
sq_stable__connex
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
productElimination,
productEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
functionEquality,
functionExtensionality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
isect_memberFormation,
independent_pairEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
independent_isectElimination
Latex:
\mforall{}g:OCMon
(UniformLinorder(|g|;x,y.\muparrow{}(x \mleq{}\msubb{} y))
\mwedge{} Cancel(|g|;|g|;*)
\mwedge{} (\mforall{}[z:|g|]. monot(|g|;x,y.\muparrow{}(x \mleq{}\msubb{} y);\mlambda{}w.(z * w)))
\mwedge{} IsEqFun(|g|;=\msubb{}))
Date html generated:
2017_10_01-AM-08_14_32
Last ObjectModification:
2017_02_28-PM-01_59_39
Theory : groups_1
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