File: Mathlib/Algebra/Homology/Embedding/Boundary.lean

full_name: ComplexShape.Embedding.BoundaryGE
code: /-- The lower boundary of an embedding `e : Embedding c c'`, as a predicate on `ι`.
It is satisfied by `j : ι` when there exists `i' : ι'` not in the image of `e.f`
such that `c'.Rel i' (e.f j)`. -/
def BoundaryGE (j : ι) : Prop 
goal: Prop

full_name: ComplexShape.Embedding.boundaryGE
code: lemma boundaryGE {i' : ι'} {j : ι} (hj : c'.Rel i' (e.f j)) (hi' : ∀ i, e.f i ≠ i') :
    e.BoundaryGE j 
goal: e.BoundaryGE j

full_name: ComplexShape.Embedding.not_boundaryGE_next
code: lemma not_boundaryGE_next [e.IsRelIff] {j k : ι} (hk : c.Rel j k) :
    ¬ e.BoundaryGE k 
goal: ¬e.BoundaryGE k

full_name: ComplexShape.Embedding.not_boundaryGE_next'
code: lemma not_boundaryGE_next' [e.IsRelIff] {j k : ι} (hj : ¬ e.BoundaryGE j) (hk : c.next j = k) :
    ¬ e.BoundaryGE k 
goal: ¬e.BoundaryGE k

full_name: ComplexShape.Embedding.BoundaryGE.not_mem
code: lemma BoundaryGE.not_mem {j : ι} (hj : e.BoundaryGE j) {i' : ι'} (hi' : c'.Rel i' (e.f j))
    (a : ι) : e.f a ≠ i' 
goal: e.f a ≠ i'

full_name: ComplexShape.Embedding.prev_f_of_not_boundaryGE
code: lemma prev_f_of_not_boundaryGE [e.IsRelIff] {i j : ι} (hij : c.prev j = i)
    (hj : ¬ e.BoundaryGE j) :
    c'.prev (e.f j) = e.f i 
goal: c'.prev (e.f j) = e.f i

full_name: ComplexShape.Embedding.BoundaryGE.false_of_isTruncLE
code: lemma BoundaryGE.false_of_isTruncLE {j : ι} (hj : e.BoundaryGE j) [e.IsTruncLE] : False 
goal: False

full_name: ComplexShape.Embedding.BoundaryLE
code: /-- The upper boundary of an embedding `e : Embedding c c'`, as a predicate on `ι`.
It is satisfied by `j : ι` when there exists `k' : ι'` not in the image of `e.f`
such that `c'.Rel (e.f j) k'`. -/
def BoundaryLE (j : ι) : Prop 
goal: Prop

full_name: ComplexShape.Embedding.boundaryLE
code: lemma boundaryLE {k' : ι'} {j : ι} (hj : c'.Rel (e.f j) k') (hk' : ∀ i, e.f i ≠ k') :
    e.BoundaryLE j 
goal: e.BoundaryLE j

full_name: ComplexShape.Embedding.not_boundaryLE_prev
code: lemma not_boundaryLE_prev [e.IsRelIff] {i j : ι} (hi : c.Rel i j) :
    ¬ e.BoundaryLE i 
goal: ¬e.BoundaryLE i

full_name: ComplexShape.Embedding.not_boundaryLE_prev'
code: lemma not_boundaryLE_prev' [e.IsRelIff] {i j : ι} (hj : ¬ e.BoundaryLE j) (hk : c.prev j = i) :
    ¬ e.BoundaryLE i 
goal: ¬e.BoundaryLE i

full_name: ComplexShape.Embedding.BoundaryLE.not_mem
code: lemma BoundaryLE.not_mem {j : ι} (hj : e.BoundaryLE j) {k' : ι'} (hk' : c'.Rel (e.f j) k')
    (a : ι) : e.f a ≠ k' 
goal: e.f a ≠ k'

full_name: ComplexShape.Embedding.next_f_of_not_boundaryLE
code: lemma next_f_of_not_boundaryLE [e.IsRelIff] {j k : ι} (hjk : c.next j = k)
    (hj : ¬ e.BoundaryLE j) :
    c'.next (e.f j) = e.f k 
goal: c'.next (e.f j) = e.f k

full_name: ComplexShape.Embedding.BoundaryLE.false_of_isTruncGE
code: lemma BoundaryLE.false_of_isTruncGE {j : ι} (hj : e.BoundaryLE j) [e.IsTruncGE] : False 
goal: False

full_name: ComplexShape.boundaryGE_embeddingUpIntGE_iff
code: lemma boundaryGE_embeddingUpIntGE_iff (p : ℤ) (n : ℕ) :
    (embeddingUpIntGE p).BoundaryGE n ↔ n = 0 
goal: (ComplexShape.embeddingUpIntGE p).BoundaryGE n ↔ n = 0

full_name: ComplexShape.boundaryLE_embeddingUpIntLE_iff
code: lemma boundaryLE_embeddingUpIntLE_iff (p : ℤ) (n : ℕ) :
    (embeddingUpIntGE p).BoundaryGE n ↔ n = 0 
goal: (ComplexShape.embeddingUpIntGE p).BoundaryGE n ↔ n = 0