Nuprl Lemma : face-lattice-induction
∀T:Type. ∀eq:EqDecider(T).
∀[P:Point(face-lattice(T;eq)) ⟶ ℙ]
((∀x:Point(face-lattice(T;eq)). SqStable(P[x]))
⇒ P[0]
⇒ P[1]
⇒ (∀x,y:Point(face-lattice(T;eq)). (P[x] ⇒ P[y] ⇒ P[x ∨ y]))
⇒ (∀x:Point(face-lattice(T;eq)). (P[x] ⇒ (∀i:T. (P[(i=0) ∧ x] ∧ P[(i=1) ∧ x]))))
⇒ (∀x:Point(face-lattice(T;eq)). P[x]))
Proof
Definitions occuring in Statement :
face-lattice1: (x=1),
face-lattice0: (x=0),
face-lattice: face-lattice(T;eq),
lattice-0: 0,
lattice-1: 1,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
deq: EqDecider(T),
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
top: Top,
subtype_rel: A ⊆r B,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
bdd-distributive-lattice: BoundedDistributiveLattice,
face-lattice0: (x=0),
face-lattice1: (x=1),
guard: {T},
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
lattice-0: 0,
record-select: r.x,
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
not: ¬A,
false: False,
squash: ↓T,
bdd-lattice: BoundedLattice,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
lattice-fset-meet: /\(s),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b]
Lemmas referenced :
face-lattice-basis,
fl-point-sq,
istype-void,
deq-fset_wf,
fset_wf,
union-deq_wf,
strong-subtype-deq-subtype,
fset-all_wf,
fset-contains-none_wf,
strong-subtype-set2,
assert_wf,
fset-antichain_wf,
face-lattice-constraints_wf,
lattice-point_wf,
face-lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
subtype_rel_self,
face-lattice0_wf,
face-lattice1_wf,
lattice-1_wf,
lattice-0_wf,
sq_stable_wf,
deq_wf,
istype-universe,
fset-image_wf,
lattice-fset-meet_wf,
decidable__equal-fl-point,
fset-induction,
fset-member_wf,
lattice-fset-join_wf,
fset-subtype,
squash_wf,
decidable_wf,
bdd-lattice_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
fset-add-as-cons,
iff_weakening_equal,
reduce_cons_lemma,
member-fset-image-iff,
fset-subtype2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
isect_memberFormation_alt,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
unionEquality,
applyEquality,
setEquality,
because_Cache,
productEquality,
lambdaEquality_alt,
unionIsType,
universeIsType,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
instantiate,
cumulativity,
inhabitedIsType,
equalityTransitivity,
functionIsType,
universeEquality,
productIsType,
setElimination,
rename,
equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
productElimination,
unionElimination,
setIsType,
functionIsTypeImplies,
imageElimination,
functionEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T).
\mforall{}[P:Point(face-lattice(T;eq)) {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:Point(face-lattice(T;eq)). SqStable(P[x]))
{}\mRightarrow{} P[0]
{}\mRightarrow{} P[1]
{}\mRightarrow{} (\mforall{}x,y:Point(face-lattice(T;eq)). (P[x] {}\mRightarrow{} P[y] {}\mRightarrow{} P[x \mvee{} y]))
{}\mRightarrow{} (\mforall{}x:Point(face-lattice(T;eq)). (P[x] {}\mRightarrow{} (\mforall{}i:T. (P[(i=0) \mwedge{} x] \mwedge{} P[(i=1) \mwedge{} x]))))
{}\mRightarrow{} (\mforall{}x:Point(face-lattice(T;eq)). P[x]))
Date html generated:
2019_10_31-AM-07_22_17
Last ObjectModification:
2018_11_08-PM-06_00_29
Theory : lattices
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