Nuprl Lemma : csm-ap-interval-1-adjoin-lemma
∀H:j⊢. ∀I:fset(ℕ). ∀v:H(I). ∀j:{i:ℕ| ¬i ∈ I} . ((j1)((s(v);<j>)) = ([1(𝕀)])v ∈ H.𝕀(I))
Proof
Definitions occuring in Statement :
interval-1: 1(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
csm-ap: (s)x,
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-1: (i1),
nc-s: s,
add-name: I+i,
dM_inc: <x>,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
false: False,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
true: True,
squash: ↓T,
and: P ∧ Q,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
interval-1: 1(𝕀),
DeMorgan-algebra: DeMorganAlgebra,
names: names(I),
fset-singleton: {x},
cons: [a / b],
dM1: 1,
lattice-1: 1,
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
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),
btrue: tt
Lemmas referenced :
cubical_set_wf,
fset_wf,
I_cube_wf,
istype-void,
strong-subtype-self,
istype-int,
le_wf,
strong-subtype-set3,
strong-subtype-deq-subtype,
int-deq_wf,
nat_wf,
fset-member_wf,
istype-nat,
cubical-type_wf,
istype-cubical-type-at,
true_wf,
squash_wf,
cc-adjoin-cube_wf,
interval-type_wf,
cc-adjoin-cube-restriction,
csm-id-adjoin-ap,
cube-set-restriction-id,
nc-1_wf,
f-subset-add-name,
nc-s_wf,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
add-name_wf,
equal_wf,
istype-universe,
cube-set-restriction-comp,
subtype_rel_self,
iff_weakening_equal,
cube-set-restriction_wf,
names-hom_wf,
s-comp-nc-1,
interval-type-at-is-point,
lattice-point_wf,
dM_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,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
interval-type-ap-inc,
trivial-member-add-name1,
dM1_wf,
nc-1-lemma2
Rules used in proof :
instantiate,
hypothesisEquality,
natural_numberEquality,
lambdaEquality_alt,
because_Cache,
independent_isectElimination,
intEquality,
applyEquality,
thin,
isectElimination,
sqequalHypSubstitution,
universeIsType,
functionIsType,
sqequalRule,
extract_by_obid,
introduction,
setIsType,
hypothesis,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
imageElimination,
Error :memTop,
voidElimination,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
rename,
setElimination,
dependent_set_memberEquality_alt,
universeEquality,
productElimination,
inhabitedIsType,
productEquality,
cumulativity,
isectEquality
Latex:
\mforall{}H:j\mvdash{}. \mforall{}I:fset(\mBbbN{}). \mforall{}v:H(I). \mforall{}j:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} . ((j1)((s(v);<j>)) = ([1(\mBbbI{})])v)
Date html generated:
2020_05_20-PM-02_36_40
Last ObjectModification:
2020_04_04-PM-01_35_06
Theory : cubical!type!theory
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