Nuprl Lemma : fl-morph-fl_all
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[psi:Point(face_lattice(I+i))].
(((∀i.psi))<f> = (∀j.(psi)<f,i=j>) ∈ Point(face_lattice(J)))
Proof
Definitions occuring in Statement :
fl_all: (∀i.phi),
fl-morph: <f>,
face_lattice: face_lattice(I),
nc-e': g,i=j,
add-name: I+i,
names-hom: I ⟶ J,
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
nat: ℕ,
cand: A c∧ B,
all: ∀x:A. B[x],
compose: f o g,
fl_all: (∀i.phi),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
names: names(I),
true: True,
nc-e': g,i=j,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
dM_inc: <x>,
DeMorgan-algebra: DeMorganAlgebra,
lattice-point: Point(l),
record-select: r.x,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
eq_atom: x =a y,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
dM_opp: <1-x>,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
names-hom: I ⟶ J,
fl-all-hom: fl-all-hom(I;i),
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
fl-morph: <f>
Lemmas referenced :
lattice-point_wf,
face_lattice_wf,
add-name_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,
set_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
names-hom_wf,
fset_wf,
face_lattice-hom-equal,
names_wf,
compose-bounded-lattice-hom,
bdd-distributive-lattice-subtype-bdd-lattice,
fl-all-hom_wf,
all_wf,
face_lattice-point-subtype,
f-subset-add-name,
fl0_wf,
trivial-member-add-name1,
lattice-0_wf,
fl1_wf,
fl-morph_wf,
nc-e'_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
squash_wf,
true_wf,
fl_all-fl0,
fl_all_wf,
fl-morph-fl0,
iff_weakening_equal,
names-deq_wf,
dM-to-FL_wf,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
dm-neg-inc,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
dM-to-FL-opp,
fl-morph-0,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
not-added-name,
dm-neg_wf,
dM-point-subtype,
rec_select_update_lemma,
subtype_rel_self,
dM-to-FL-sq,
dm-neg-sq,
fl_all-id,
fl_all-fl1,
fl-morph-fl1,
dM-to-FL-inc,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
intEquality,
natural_numberEquality,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
setEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
independent_functionElimination,
voidElimination,
imageElimination,
imageMemberEquality,
baseClosed,
int_eqEquality,
voidEquality,
computeAll,
hyp_replacement,
applyLambdaEquality,
comment
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[j:\{i:\mBbbN{}| \mneg{}i \mmember{} J\} ]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ].
\mforall{}[psi:Point(face\_lattice(I+i))].
(((\mforall{}i.psi))<f> = (\mforall{}j.(psi)<f,i=j>))
Date html generated:
2017_10_05-AM-01_17_12
Last ObjectModification:
2017_07_28-AM-09_32_57
Theory : cubical!type!theory
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