fix(init.path): a couple of small simplifications
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1 changed files with 6 additions and 6 deletions
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@ -573,19 +573,19 @@ namespace eq
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-- Whiskering and identity paths.
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-- Whiskering and identity paths.
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definition whisker_right_idp_right {p q : x = y} (h : p = q) :
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definition whisker_right_idp {p q : x = y} (h : p = q) :
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whisker_right h idp = h :=
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whisker_right h idp = h :=
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eq.rec_on h (eq.rec_on p idp)
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eq.rec_on h (eq.rec_on p idp)
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definition whisker_right_idp_left (p : x = y) (q : y = z) :
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definition whisker_right_idp_left (p : x = y) (q : y = z) :
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whisker_right idp q = idp :> (p ⬝ q = p ⬝ q) :=
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whisker_right idp q = idp :> (p ⬝ q = p ⬝ q) :=
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eq.rec_on q idp
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idp
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definition whisker_left_idp_right (p : x = y) (q : y = z) :
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definition whisker_left_idp_right (p : x = y) (q : y = z) :
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whisker_left p idp = idp :> (p ⬝ q = p ⬝ q) :=
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whisker_left p idp = idp :> (p ⬝ q = p ⬝ q) :=
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eq.rec_on q idp
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idp
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definition whisker_left_idp_left {p q : x = y} (h : p = q) :
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definition whisker_left_idp {p q : x = y} (h : p = q) :
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(idp_con p) ⁻¹ ⬝ whisker_left idp h ⬝ idp_con q = h :=
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(idp_con p) ⁻¹ ⬝ whisker_left idp h ⬝ idp_con q = h :=
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eq.rec_on h (eq.rec_on p idp)
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eq.rec_on h (eq.rec_on p idp)
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@ -624,11 +624,11 @@ namespace eq
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eq.rec_on q (eq.rec_on p idp)
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eq.rec_on q (eq.rec_on p idp)
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definition eckmann_hilton {x:A} (p q : idp = idp :> x = x) : p ⬝ q = q ⬝ p :=
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definition eckmann_hilton {x:A} (p q : idp = idp :> x = x) : p ⬝ q = q ⬝ p :=
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(!whisker_right_idp_right ◾ !whisker_left_idp_left)⁻¹
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(!whisker_right_idp ◾ !whisker_left_idp)⁻¹
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⬝ whisker_left _ !idp_con
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⬝ whisker_left _ !idp_con
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⬝ !whisker_right_con_whisker_left
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⬝ !whisker_right_con_whisker_left
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⬝ whisker_right !idp_con⁻¹ _
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⬝ whisker_right !idp_con⁻¹ _
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⬝ (!whisker_left_idp_left ◾ !whisker_right_idp_right)
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⬝ (!whisker_left_idp ◾ !whisker_right_idp)
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-- The action of functions on 2-dimensional paths
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-- The action of functions on 2-dimensional paths
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definition ap02 [unfold 8] [reducible] (f : A → B) {x y : A} {p q : x = y} (r : p = q)
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definition ap02 [unfold 8] [reducible] (f : A → B) {x y : A} {p q : x = y} (r : p = q)
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