Nuprl Lemma : path-term-1
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}. ∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}.
(path-term(psi;w;a;b;1(𝕀)) = b ∈ {X ⊢ _:T})
Proof
Definitions occuring in Statement :
path-term: path-term(phi;w;a;b;r),
path-type: (Path_A a b),
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
path-term: path-term(phi;w;a;b;r),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
interval-1: 1(𝕀),
face-zero: (i=0),
case-term: (u ∨ v),
cubical-term-at: u(a),
ifthenelse: if b then t else f fi ,
bfalse: ff,
dm-neg: ¬(x),
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,
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],
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),
btrue: tt,
fset-singleton: {x},
cons: [a / b],
nil: [],
it: ⋅,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
lattice-join: a ∨ b,
opposite-lattice: opposite-lattice(L),
so_lambda: λ2x y.t[x; y],
lattice-meet: a ∧ b,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
lattice-fset-meet: /\(s),
empty-fset: {},
lattice-0: 0,
dM0: 0,
fl-eq: (x==y),
free-dml-deq: free-dml-deq(T;eq),
deq-fset: deq-fset(eq),
isl: isl(x),
decidable__equal_fset,
decidable_functionality,
iff_preserves_decidability,
decidable__and2,
decidable__f-subset,
decidable__all_fset,
iff_weakening_uiff,
fset-all-iff,
decidable__assert,
fset-null: fset-null(s),
null: null(as),
dM-to-FL: dM-to-FL(I;z),
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
decidable__and,
bnot: ¬bb,
decidable__fset-member,
assert-deq-fset-member,
deq-fset-member: a ∈b s,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
same-cubical-term: X ⊢ u=v:A,
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-path-app-1,
context-subset_wf,
thin-context-subset,
context-subset-term-subtype,
subset-cubical-term2,
sub_cubical_set_self,
path-type_wf,
subset-cubical-term,
context-subset-is-subset,
path-type-subset,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
cubical-term-at_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
bfalse_wf,
case-term-same2,
face-1_wf,
equal_functionality_wrt_subtype_rel2,
face-or_wf,
sub_cubical_set_wf,
squash_wf,
true_wf,
face-or-1,
iff_weakening_equal,
context-1-subset,
decidable__equal_fset,
decidable_functionality,
iff_preserves_decidability,
decidable__and2,
decidable__f-subset,
decidable__all_fset,
iff_weakening_uiff,
fset-all-iff,
decidable__assert,
decidable__and,
decidable__fset-member,
assert-deq-fset-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
Error :memTop,
universeIsType,
instantiate,
equalitySymmetry,
functionExtensionality,
equalityTransitivity,
cumulativity,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality_alt,
imageElimination,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}X:j\mvdash{}. \mforall{}psi:\{X \mvdash{} \_:\mBbbF{}\}. \mforall{}T:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:T\}. \mforall{}w:\{X, psi \mvdash{} \_:(Path\_T a b)\}.
(path-term(psi;w;a;b;1(\mBbbI{})) = b)
Date html generated:
2020_05_20-PM-05_10_17
Last ObjectModification:
2020_04_10-AM-11_42_17
Theory : cubical!type!theory
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