Nuprl Lemma : s-comp-if-lemma1
∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀x:Point(dM(J)). (s ⋅ λj.if (j =z new-name(I)) then x else 1 ⋅ f j fi = f ∈ J ⟶ I)
Proof
Definitions occuring in Statement :
nc-s: s,
new-name: new-name(I),
add-name: I+i,
nh-comp: g ⋅ f,
nh-id: 1,
names-hom: I ⟶ J,
dM: dM(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
names-hom: I ⟶ J,
nc-s: s,
nh-id: 1,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
dM: dM(I),
dM-lift: dM-lift(I;J;f),
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
names: names(I),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
DeMorgan-algebra: DeMorganAlgebra,
label: ...$L... t,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
dM-lift-inc,
add-name_wf,
new-name_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
eq_int_wf,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
dM-lift_wf,
dma-hom_wf,
dM_wf,
all_wf,
names_wf,
dM_inc_wf,
not-added-name,
names-subtype,
f-subset-add-name,
iff_weakening_equal,
eqtt_to_assert,
assert_of_eq_int,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
names-hom_wf,
fset_wf,
deq_wf,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
new-name-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionExtensionality,
sqequalRule,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
setElimination,
rename,
setEquality,
intEquality,
independent_isectElimination,
natural_numberEquality,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
productEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}I,J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I. \mforall{}x:Point(dM(J)).
(s \mcdot{} \mlambda{}j.if (j =\msubz{} new-name(I)) then x else 1 \mcdot{} f j fi = f)
Date html generated:
2018_05_23-AM-08_30_46
Last ObjectModification:
2017_11_26-PM-03_40_32
Theory : cubical!type!theory
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