Nuprl Lemma : nc-e'-lemma4
∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[k:{i1:ℕ| ¬i1 ∈ I+i} ]. ∀[l:{i:ℕ| ¬i ∈ J+j} ].
((i0) ⋅ s ⋅ g,i=j,k=l = g,i=j ⋅ (j0) ⋅ s ∈ J+j+l ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-0: (i0),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
names-hom: I ⟶ J,
member: t ∈ T,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
compose: f o g,
names: names(I),
nc-0: (i0),
nc-e': g,i=j,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
DeMorgan-algebra: DeMorganAlgebra,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
dM-lift: dM-lift(I;J;f),
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra-property,
free-dist-lattice-property,
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,
empty-fset: {},
nil: [],
lattice-0: 0,
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],
eq_atom: x =a y,
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),
dM0: 0,
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P)
Lemmas referenced :
names_wf,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
fset-member_wf,
nat_wf,
int-deq_wf,
istype-void,
names-hom_wf,
istype-nat,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
f-subset-add-name1,
f-subset-add-name,
eq_int_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,
not-added-name,
nh-comp-sq,
dM0-sq-empty,
equal_wf,
squash_wf,
true_wf,
istype-universe,
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,
dM-lift_wf2,
nc-e'_wf,
dM-lift-0,
nc-s_wf,
dM-lift-inc,
nc-0_wf,
trivial-member-add-name1,
subtype_rel_self,
iff_weakening_equal,
dM0_wf,
bool_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
dM_inc_wf,
dM-lift-is-id,
f-subset_wf,
int_subtype_base,
names-subtype,
dM-point-subtype,
dM-lift-s,
equal_functionality_wrt_subtype_rel2,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionExtensionality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
setIsType,
because_Cache,
functionIsType,
applyEquality,
intEquality,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
equalityIstype,
promote_hyp,
instantiate,
imageElimination,
universeEquality,
productEquality,
cumulativity,
isectEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} ]. \mforall{}[g:J {}\mrightarrow{} I]. \mforall{}[k:\{i1:\mBbbN{}| \mneg{}i1 \mmember{} I+i\} ].
\mforall{}[l:\{i:\mBbbN{}| \mneg{}i \mmember{} J+j\} ].
((i0) \mcdot{} s \mcdot{} g,i=j,k=l = g,i=j \mcdot{} (j0) \mcdot{} s)
Date html generated:
2020_05_20-PM-01_37_49
Last ObjectModification:
2020_01_15-PM-02_55_22
Theory : cubical!type!theory
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