### Nuprl Lemma : refl-path_wf

`∀X:CubicalSet. ∀A:{X ⊢ _}. ∀a:{X ⊢ _:A}. ∀I:Cname List. ∀alpha:X(I).  (refl-path(A;a;I;alpha) ∈ I-path(X;A;a;a;I;alpha))`

Proof

Definitions occuring in Statement :  refl-path: `refl-path(A;a;I;alpha)` I-path: `I-path(X;A;a;b;I;alpha)` cubical-term: `{X ⊢ _:AF}` cubical-type: `{X ⊢ _}` I-cube: `X(I)` cubical-set: `CubicalSet` coordinate_name: `Cname` list: `T List` all: `∀x:A. B[x]` member: `t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` refl-path: `refl-path(A;a;I;alpha)` I-path: `I-path(X;A;a;b;I;alpha)` uall: `∀[x:A]. B[x]` named-path: `named-path(X;A;a;b;I;alpha;z)` cubical-term: `{X ⊢ _:AF}` subtype_rel: `A ⊆r B` cubical-type-at: `A(a)` uimplies: `b supposing a` prop: `ℙ` not: `¬A` implies: `P `` Q` false: `False` add-fresh-cname: `I+` has-value: `(a)↓` so_lambda: `λ2x.t[x]` so_apply: `x[s]` squash: `↓T` iota: `iota(x)` iota': `iota'(I)` true: `True` name-path-endpoints: `name-path-endpoints(X;A;a;b;I;alpha;z;omega)` and: `P ∧ Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` less_than: `a < b` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  fresh-cname_wf cubical-type-ap-morph_wf add-fresh-cname_wf iota'_wf subtype_rel_self cubical-type-at_wf name-path-endpoints_wf fresh-cname-not-member2 named-path_wf l_member_wf coordinate_name_wf I-cube_wf list_wf cubical-term_wf cubical-type_wf cubical-set_wf value-type-has-value not_wf set-value-type coordinate_name-value-type cube-set-restriction_wf face-map_wf_fresh-cname istype-false istype-le istype-less_than equal_wf cubical-type-ap-morph-id name-comp_wf iota'-identity iff_weakening_equal squash_wf true_wf istype-universe cubical-type-ap-morph-comp subtype_rel-equal cube-set-restriction-id
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut sqequalRule dependent_pairEquality_alt introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis dependent_set_memberEquality_alt applyEquality setElimination rename because_Cache equalityTransitivity equalitySymmetry lambdaEquality_alt hyp_replacement universeIsType independent_isectElimination dependent_functionElimination independent_functionElimination voidElimination callbyvalueReduce setEquality imageElimination imageMemberEquality baseClosed natural_numberEquality independent_pairFormation productIsType productElimination inhabitedIsType instantiate universeEquality

Latex:
\mforall{}X:CubicalSet.  \mforall{}A:\{X  \mvdash{}  \_\}.  \mforall{}a:\{X  \mvdash{}  \_:A\}.  \mforall{}I:Cname  List.  \mforall{}alpha:X(I).
(refl-path(A;a;I;alpha)  \mmember{}  I-path(X;A;a;a;I;alpha))

Date html generated: 2019_11_06-PM-00_53_38
Last ObjectModification: 2018_11_10-PM-03_16_57

Theory : cubical!sets

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