### Nuprl Lemma : poset_functor_extend_id

`∀C:SmallCategory. ∀I:Cname List. ∀L:name-morph(I;[]) ⟶ cat-ob(C). ∀E:i:nameset(I)`
`                                                                      ⟶ c:{c:name-morph(I;[])| (c i) = 0 ∈ ℕ2} `
`                                                                      ⟶ (cat-arrow(C) (L c) (L flip(c;i))).`
`∀x:cat-ob(poset-cat(I)).`
`  (poset_functor_extend(C;I;L;E;x;x) = (cat-id(C) (L x)) ∈ (cat-arrow(C) (L x) (L x)))`

Proof

Definitions occuring in Statement :  poset_functor_extend: `poset_functor_extend(C;I;L;E;c1;c2)` poset-cat: `poset-cat(J)` name-morph-flip: `flip(f;y)` name-morph: `name-morph(I;J)` nameset: `nameset(L)` coordinate_name: `Cname` cat-id: `cat-id(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` nil: `[]` list: `T List` int_seg: `{i..j-}` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` name-morph: `name-morph(I;J)` subtype_rel: `A ⊆r B` prop: `ℙ` poset-cat: `poset-cat(J)` cat-ob: `cat-ob(C)` pi1: `fst(t)` poset_functor_extend: `poset_functor_extend(C;I;L;E;c1;c2)` nameset: `nameset(L)` uimplies: `b supposing a` nil: `[]` it: `⋅` ifthenelse: `if b then t else f fi ` null: `null(as)` btrue: `tt` cat-id: `cat-id(C)` pi2: `snd(t)` l_all: `(∀x∈L.P[x])` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` iff: `P `⇐⇒` Q` uiff: `uiff(P;Q)` rev_implies: `P `` Q`
Lemmas referenced :  cat-ob_wf poset-cat_wf nameset_wf name-morph_wf nil_wf coordinate_name_wf equal-wf-T-base int_seg_wf extd-nameset-nil cat-arrow_wf name-morph-flip_wf list_wf small-category_wf filter_is_nil band_wf eq_int_wf extd-nameset_subtype_int l_member_wf list-subtype cat-id_wf select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf 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 decidable__lt intformless_wf int_formula_prop_less_lemma extd-nameset_wf equal_wf length_wf intformeq_wf int_formula_prop_eq_lemma assert_wf not_wf iff_transitivity iff_weakening_uiff assert_of_band assert_of_eq_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality functionEquality setEquality natural_numberEquality applyEquality setElimination rename sqequalRule baseClosed functionExtensionality because_Cache lambdaEquality equalityTransitivity equalitySymmetry independent_isectElimination callbyvalueReduce sqleReflexivity productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination productEquality addLevel impliesFunctionality

Latex:
\mforall{}C:SmallCategory.  \mforall{}I:Cname  List.  \mforall{}L:name-morph(I;[])  {}\mrightarrow{}  cat-ob(C).  \mforall{}E:i:nameset(I)
{}\mrightarrow{}  c:\{c:name-morph(I;[])|
(c  i)  =  0\}
{}\mrightarrow{}  (cat-arrow(C)  (L  c)
(L  flip(c;i))).
\mforall{}x:cat-ob(poset-cat(I)).
(poset\_functor\_extend(C;I;L;E;x;x)  =  (cat-id(C)  (L  x)))

Date html generated: 2017_10_05-AM-10_30_51
Last ObjectModification: 2017_07_28-AM-11_24_32

Theory : cubical!sets

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