Nuprl Lemma : omon_lt_mono_imp_leq_mono
∀[g:OMon]. {∀[a:|g|]. monot(|g|;x,y.x ≤ y;λz.(a * z))} supposing ∀a:|g|. monot(|g|;x,y.x < y;λz.(a * z))
Proof
Definitions occuring in Statement :
grp_lt: a < b,
grp_leq: a ≤ b,
omon: OMon,
grp_op: *,
grp_car: |g|,
monot: monot(T;x,y.R[x; y];f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
infix_ap: x f y,
all: ∀x:A. B[x],
lambda: λx.A[x]
Definitions unfolded in proof :
monot: monot(T;x,y.R[x; y];f),
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
omon: OMon,
abmonoid: AbMon,
mon: Mon,
grp_leq: a ≤ b,
infix_ap: x f y,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
grp_leq_wf,
grp_car_wf,
assert_witness,
grp_le_wf,
grp_op_wf,
all_wf,
grp_lt_wf,
infix_ap_wf,
omon_wf,
grp_leq_iff_lt_or_eq,
grp_leq_weakening_lt,
grp_leq_weakening_eq,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
independent_isectElimination,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[g:OMon]
\{\mforall{}[a:|g|]. monot(|g|;x,y.x \mleq{} y;\mlambda{}z.(a * z))\} supposing \mforall{}a:|g|. monot(|g|;x,y.x < y;\mlambda{}z.(a * z))
Date html generated:
2017_10_01-AM-08_14_52
Last ObjectModification:
2017_02_28-PM-01_59_09
Theory : groups_1
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