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+(* exists *******************************************************************)
+
+notation "hvbox(∃ _ break . p)"
+ with precedence 20
+for @{'exists $p }.
+
+notation < "hvbox(\exists ident i : ty break . p)"
+ right associative with precedence 20
+for @{'exists (\lambda ${ident i} : $ty. $p) }.
+
+notation < "hvbox(\exists ident i break . p)"
+  with precedence 20
+for @{'exists (\lambda ${ident i}. $p) }.
+
+(*
+notation < "hvbox(\exists ident i opt (: ty) break . p)"
+  right associative with precedence 20
+for @{ 'exists ${default
+  @{\lambda ${ident i} : $ty. $p}
+  @{\lambda ${ident i} . $p}}}.
+*)
+
+notation > "\exists list1 ident x sep , opt (: T). term 19 Px"
+  with precedence 20
+  for ${ default
+          @{ ${ fold right @{$Px} rec acc @{'exists (λ${ident x}:$T.$acc)} } }
+          @{ ${ fold right @{$Px} rec acc @{'exists (λ${ident x}.$acc)} } }
+       }.
+
+(* sigma ********************************************************************)
+
+notation < "hvbox(\Sigma ident i : ty break . p)"
+ left associative with precedence 20
+for @{'sigma (\lambda ${ident i} : $ty. $p) }.
+
+notation < "hvbox(\Sigma ident i break . p)"
+ with precedence 20
+for @{'sigma (\lambda ${ident i}. $p) }.
+
+(*
+notation < "hvbox(\Sigma ident i opt (: ty) break . p)"
+  right associative with precedence 20
+for @{ 'sigma ${default
+  @{\lambda ${ident i} : $ty. $p}
+  @{\lambda ${ident i} . $p}}}.
+*)
+
+notation > "\Sigma list1 ident x sep , opt (: T). term 19 Px"
+  with precedence 20
+  for ${ default
+          @{ ${ fold right @{$Px} rec acc @{'sigma (λ${ident x}:$T.$acc)} } }
+          @{ ${ fold right @{$Px} rec acc @{'sigma (λ${ident x}.$acc)} } }
+       }.
+
+(* equality, conguence ******************************************************)
+
+notation > "hvbox(a break = b)" 
+  non associative with precedence 45
+for @{ 'eq ? $a $b }.
+
+notation < "hvbox(term 46 a break maction (=) (=\sub t) term 46 b)" 
+  non associative with precedence 45
+for @{ 'eq $t $a $b }.
+
+notation > "hvbox(n break (≅ \sub term 90 p) m)"
+  non associative with precedence 45
+for @{ 'congruent $n $m $p }.
+
+notation < "hvbox(term 46 n break ≅ \sub p term 46 m)"
+  non associative with precedence 45
+for @{ 'congruent $n $m $p }.
+
+(* other notations **********************************************************)
+
+notation "hvbox(\langle term 19 a, break term 19 b\rangle)" 
+with precedence 90 for @{ 'pair $a $b}.
+
+notation "hvbox(x break \times y)" with precedence 70
+for @{ 'product $x $y}.
+
+notation < "\fst \nbsp term 90 x" with precedence 69 for @{'pi1a $x}.
+notation < "\snd \nbsp term 90 x" with precedence 69 for @{'pi2a $x}.
+
+notation < "\fst \nbsp term 90 x \nbsp term 90 y" with precedence 69 for @{'pi1b $x $y}.
+notation < "\snd \nbsp term 90 x \nbsp term 90 y" with precedence 69 for @{'pi2b $x $y}.
+
+notation > "\fst" with precedence 90 for @{'pi1}.
+notation > "\snd" with precedence 90 for @{'pi2}.
+
+notation "hvbox(a break \to b)" 
+  right associative with precedence 20
+for @{ \forall $_:$a.$b }.
+
+notation < "hvbox(a break \to b)" 
+  right associative with precedence 20
+for @{ \Pi $_:$a.$b }.
+
+notation "hvbox(a break \leq b)" 
+  non associative with precedence 45
+for @{ 'leq $a $b }.
+
+notation "hvbox(a break \geq b)" 
+  non associative with precedence 45
+for @{ 'geq $a $b }.
+
+notation "hvbox(a break \lt b)" 
+  non associative with precedence 45
+for @{ 'lt $a $b }.
+
+notation "hvbox(a break \gt b)" 
+  non associative with precedence 45
+for @{ 'gt $a $b }.
+
+notation > "hvbox(a break \neq b)" 
+  non associative with precedence 45
+for @{ 'neq ? $a $b }.
+
+notation < "hvbox(a break maction (\neq) (\neq\sub t) b)" 
+  non associative with precedence 45
+for @{ 'neq $t $a $b }.
+
+notation "hvbox(a break \nleq b)" 
+  non associative with precedence 45
+for @{ 'nleq $a $b }.
+
+notation "hvbox(a break \ngeq b)" 
+  non associative with precedence 45
+for @{ 'ngeq $a $b }.
+
+notation "hvbox(a break \nless b)" 
+  non associative with precedence 45
+for @{ 'nless $a $b }.
+
+notation "hvbox(a break \ngtr b)" 
+  non associative with precedence 45
+for @{ 'ngtr $a $b }.
+
+notation "hvbox(a break \divides b)"
+  non associative with precedence 45
+for @{ 'divides $a $b }.
+
+notation "hvbox(a break \ndivides b)"
+  non associative with precedence 45
+for @{ 'ndivides $a $b }.
+
+notation "hvbox(a break + b)" 
+  left associative with precedence 50
+for @{ 'plus $a $b }.
+
+notation "hvbox(a break - b)" 
+  left associative with precedence 50
+for @{ 'minus $a $b }.
+
+notation "hvbox(a break * b)" 
+  left associative with precedence 55
+for @{ 'times $a $b }.
+
+notation "hvbox(a break \middot b)" 
+  left associative with precedence 55
+  for @{ 'middot $a $b }.
+
+notation "hvbox(a break \mod b)" 
+  left associative with precedence 55
+for @{ 'module $a $b }.
+
+notation < "a \frac b" 
+  non associative with precedence 90
+for @{ 'divide $a $b }.
+
+notation "a \over b" 
+  left associative with precedence 55
+for @{ 'divide $a $b }.
+
+notation "hvbox(a break / b)" 
+  left associative with precedence 55
+for @{ 'divide $a $b }.
+
+notation "- term 60 a" with precedence 60 
+for @{ 'uminus $a }.
+
+notation "a !"
+  non associative with precedence 80
+for @{ 'fact $a }.
+
+notation "\sqrt a" 
+  non associative with precedence 60
+for @{ 'sqrt $a }.
+
+notation "hvbox(a break \lor b)" 
+  left associative with precedence 30
+for @{ 'or $a $b }.
+
+notation "hvbox(a break \land b)" 
+  left associative with precedence 35
+for @{ 'and $a $b }.
+
+notation "hvbox(\lnot a)" 
+  non associative with precedence 40
+for @{ 'not $a }.
+
+notation > "hvbox(a break \liff b)" 
+  left associative with precedence 25
+for @{ 'iff $a $b }.
+
+notation "hvbox(a break \leftrightarrow b)" 
+  left associative with precedence 25
+for @{ 'iff $a $b }.
+
+
+notation "hvbox(\Omega \sup term 90 A)" non associative with precedence 90
+for @{ 'powerset $A }.
+notation > "hvbox(\Omega ^ term 90 A)" non associative with precedence 90
+for @{ 'powerset $A }.
+
+notation < "hvbox({ ident i | term 19 p })" with precedence 90
+for @{ 'subset (\lambda ${ident i} : $nonexistent . $p)}.
+
+notation > "hvbox({ ident i | term 19 p })" with precedence 90
+for @{ 'subset (\lambda ${ident i}. $p)}.
+
+notation < "hvbox({ ident i ∈ s | term 19 p })" with precedence 90
+for @{ 'comprehension $s (\lambda ${ident i} : $nonexistent . $p)}.
+
+notation > "hvbox({ ident i ∈ s | term 19 p })" with precedence 90
+for @{ 'comprehension $s (\lambda ${ident i}. $p)}.
+
+notation "hvbox(a break ∈ b)" non associative with precedence 45
+for @{ 'mem $a $b }.
+
+notation "hvbox(a break ≬ b)" non associative with precedence 45
+for @{ 'overlaps $a $b }. (* \between *)
+
+notation "hvbox(a break ⊆ b)" non associative with precedence 45
+for @{ 'subseteq $a $b }. (* \subseteq *)
+
+notation "hvbox(a break ∩ b)" left associative with precedence 55
+for @{ 'intersects $a $b }. (* \cap *)
+
+notation "hvbox(a break ∪ b)" left associative with precedence 50
+for @{ 'union $a $b }. (* \cup *)
+
+notation "hvbox({ term 19 a })" with precedence 90 for @{ 'singl $a}.
+
+notation "hvbox(a break \approx b)" non associative with precedence 45 
+  for @{ 'napart $a $b}.
+        
+notation "hvbox(a break # b)" non associative with precedence 45 
+  for @{ 'apart $a $b}.
+    
+notation "hvbox(a break \circ b)" 
+  left associative with precedence 55
+for @{ 'compose $a $b }.
+
+notation < "↓ \ensp a" with precedence 55 for @{ 'downarrow $a }.
+notation > "↓ a" with precedence 55 for @{ 'downarrow $a }.
+
+notation "hvbox(U break ↓ V)" non associative with precedence 55 for @{ 'fintersects $U $V }.
+
+notation "↑a" with precedence 55 for @{ 'uparrow $a }.
+
+notation "hvbox(a break ↑ b)" with precedence 55 for @{ 'funion $a $b }.
+
+notation < "term 76 a \sup term 90 b" non associative with precedence 75 for @{ 'exp $a $b}.
+notation > "a \sup term 90 b" non associative with precedence 75 for @{ 'exp $a $b}.
+notation > "a ^ term 90 b"  non associative with precedence 75 for @{ 'exp $a $b}.
+notation "s \sup (-1)" non associative with precedence 75 for @{ 'invert $s }.
+notation > "s ^ (-1)" non associative with precedence 75 for @{ 'invert $s }.
+notation < "s \sup (-1) x" non associative with precedence 90 for @{ 'invert_appl $s $x}. 
+
+notation "| term 19 C |" with precedence 70 for @{ 'card $C }.
+
+notation "\naturals" non associative with precedence 90 for @{'N}.
+notation "\rationals" non associative with precedence 90 for @{'Q}.
+notation "\reals" non associative with precedence 90 for @{'R}.
+notation "\integers" non associative with precedence 90 for @{'Z}.
+notation "\complexes" non associative with precedence 90 for @{'C}.
+
+notation "\ee" with precedence 90 for @{ 'neutral }. (* ⅇ *)
+
+notation > "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y ?}.
+notation > "x ⊩⎽ term 90 c y" with precedence 45 for @{'Vdash2 $x $y $c}.
+notation "x (⊩ \sub term 90 c) y" with precedence 45 for @{'Vdash2 $x $y $c}.
+notation > "⊩ " with precedence 60 for @{'Vdash ?}.
+notation "(⊩ \sub term 90 c) " with precedence 60 for @{'Vdash $c}.
+
+notation < "maction (mstyle color #ff0000 (­…­)) (t)" 
+non associative with precedence 90 for @{'hide $t}.