# Library iris.bi.notation

Reserved Notation "P ⊢ Q" (at level 99, Q at level 200, right associativity).

Reserved Notation "P '⊢@{' PROP } Q" (at level 99, Q at level 200, right associativity).

Reserved Notation "(⊢)".

Reserved Notation "'(⊢@{' PROP } )".

Reserved Notation "( P ⊣⊢.)".

Reserved Notation "(.⊣⊢ Q )".

Reserved Notation "P ⊣⊢ Q" (at level 95, no associativity).

Reserved Notation "P '⊣⊢@{' PROP } Q" (at level 95, no associativity).

Reserved Notation "(⊣⊢)".

Reserved Notation "'(⊣⊢@{' PROP } )".

Reserved Notation "(.⊢ Q )".

Reserved Notation "( P ⊢.)".

Reserved Notation "⊢ Q" (at level 20, Q at level 200).

Reserved Notation "'⊢@{' PROP } Q" (at level 20, Q at level 200).

Reserved Notation "P '⊢@{' PROP } Q" (at level 99, Q at level 200, right associativity).

Reserved Notation "(⊢)".

Reserved Notation "'(⊢@{' PROP } )".

Reserved Notation "( P ⊣⊢.)".

Reserved Notation "(.⊣⊢ Q )".

Reserved Notation "P ⊣⊢ Q" (at level 95, no associativity).

Reserved Notation "P '⊣⊢@{' PROP } Q" (at level 95, no associativity).

Reserved Notation "(⊣⊢)".

Reserved Notation "'(⊣⊢@{' PROP } )".

Reserved Notation "(.⊢ Q )".

Reserved Notation "( P ⊢.)".

Reserved Notation "⊢ Q" (at level 20, Q at level 200).

Reserved Notation "'⊢@{' PROP } Q" (at level 20, Q at level 200).

The definition must coincide with "'⊢@{' PROP } Q".

Reserved Notation "'(⊢@{' PROP } Q )".

Rationale:
Notation ( '⊢@{' PROP } ) prevents parsing (⊢@{PROP} Q) using the
⊢@{PROP} Q notation; since the latter parse arises from composing two
notations, it is missed by the automatic left-factorization.
To fix that, we force left-factorization by explicitly composing parentheses with
'⊢@{' PROP } Q into the new notation ( '⊢@{' PROP } Q ),
which successfully undergoes automatic left-factoring.

# BI connectives

Reserved Notation "'emp'".

Reserved Notation "'⌜' φ '⌝'" (at level 1, φ at level 200, format "⌜ φ ⌝").

Reserved Notation "P ∗ Q" (at level 80, right associativity, format "P ∗ '/' Q").

Reserved Notation "P -∗ Q"

(at level 99, Q at level 200, right associativity,

format "'[' P -∗ '/' '[' Q ']' ']'").

Reserved Notation "⎡ P ⎤".

Reserved Notation "'⌜' φ '⌝'" (at level 1, φ at level 200, format "⌜ φ ⌝").

Reserved Notation "P ∗ Q" (at level 80, right associativity, format "P ∗ '/' Q").

Reserved Notation "P -∗ Q"

(at level 99, Q at level 200, right associativity,

format "'[' P -∗ '/' '[' Q ']' ']'").

Reserved Notation "⎡ P ⎤".

Modalities

Reserved Notation "'<pers>' P" (at level 20, right associativity).

Reserved Notation "'<pers>?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'<pers>?' p P").

Reserved Notation "▷ P" (at level 20, right associativity).

Reserved Notation "▷? p P" (at level 20, p at level 9, P at level 20,

format "▷? p P").

Reserved Notation "▷^ n P" (at level 20, n at level 9, P at level 20,

format "▷^ n P").

Reserved Infix "∗-∗" (at level 95, no associativity).

Reserved Notation "'<affine>' P" (at level 20, right associativity).

Reserved Notation "'<affine>?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'<affine>?' p P").

Reserved Notation "'<absorb>' P" (at level 20, right associativity).

Reserved Notation "'<absorb>?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'<absorb>?' p P").

Reserved Notation "□ P" (at level 20, right associativity).

Reserved Notation "'□?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'□?' p P").

Reserved Notation "◇ P" (at level 20, right associativity).

Reserved Notation "■ P" (at level 20, right associativity).

Reserved Notation "■? p P" (at level 20, p at level 9, P at level 20,

right associativity, format "■? p P").

Reserved Notation "'<obj>' P" (at level 20, right associativity).

Reserved Notation "'<subj>' P" (at level 20, right associativity).

Reserved Notation "'<pers>?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'<pers>?' p P").

Reserved Notation "▷ P" (at level 20, right associativity).

Reserved Notation "▷? p P" (at level 20, p at level 9, P at level 20,

format "▷? p P").

Reserved Notation "▷^ n P" (at level 20, n at level 9, P at level 20,

format "▷^ n P").

Reserved Infix "∗-∗" (at level 95, no associativity).

Reserved Notation "'<affine>' P" (at level 20, right associativity).

Reserved Notation "'<affine>?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'<affine>?' p P").

Reserved Notation "'<absorb>' P" (at level 20, right associativity).

Reserved Notation "'<absorb>?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'<absorb>?' p P").

Reserved Notation "□ P" (at level 20, right associativity).

Reserved Notation "'□?' p P" (at level 20, p at level 9, P at level 20,

right associativity, format "'□?' p P").

Reserved Notation "◇ P" (at level 20, right associativity).

Reserved Notation "■ P" (at level 20, right associativity).

Reserved Notation "■? p P" (at level 20, p at level 9, P at level 20,

right associativity, format "■? p P").

Reserved Notation "'<obj>' P" (at level 20, right associativity).

Reserved Notation "'<subj>' P" (at level 20, right associativity).

Reserved Notation "|==> Q" (at level 99, Q at level 200, format "'[ ' |==> '/' Q ']'").

Reserved Notation "P ==∗ Q"

(at level 99, Q at level 200, format "'[' P ==∗ '/' Q ']'").

Reserved Notation "|={ E1 , E2 }=> Q"

(at level 99, E1, E2 at level 50, Q at level 200,

format "'[ ' |={ E1 , E2 }=> '/' Q ']'").

Reserved Notation "P ={ E1 , E2 }=∗ Q"

(at level 99, E1,E2 at level 50, Q at level 200,

format "'[' P ={ E1 , E2 }=∗ '/' '[' Q ']' ']'").

Reserved Notation "|={ E }=> Q"

(at level 99, E at level 50, Q at level 200,

format "'[ ' |={ E }=> '/' Q ']'").

Reserved Notation "P ={ E }=∗ Q"

(at level 99, E at level 50, Q at level 200,

format "'[' P ={ E }=∗ '/' '[' Q ']' ']'").

Reserved Notation "P ==∗ Q"

(at level 99, Q at level 200, format "'[' P ==∗ '/' Q ']'").

Reserved Notation "|={ E1 , E2 }=> Q"

(at level 99, E1, E2 at level 50, Q at level 200,

format "'[ ' |={ E1 , E2 }=> '/' Q ']'").

Reserved Notation "P ={ E1 , E2 }=∗ Q"

(at level 99, E1,E2 at level 50, Q at level 200,

format "'[' P ={ E1 , E2 }=∗ '/' '[' Q ']' ']'").

Reserved Notation "|={ E }=> Q"

(at level 99, E at level 50, Q at level 200,

format "'[ ' |={ E }=> '/' Q ']'").

Reserved Notation "P ={ E }=∗ Q"

(at level 99, E at level 50, Q at level 200,

format "'[' P ={ E }=∗ '/' '[' Q ']' ']'").

Step-taking fancy updates

Reserved Notation "|={ E1 } [ E2 ]▷=> Q"

(at level 99, E1, E2 at level 50, Q at level 200,

format "'[ ' |={ E1 } [ E2 ]▷=> '/' Q ']'").

Reserved Notation "P ={ E1 } [ E2 ]▷=∗ Q"

(at level 99, E1, E2 at level 50, Q at level 200,

format "'[' P ={ E1 } [ E2 ]▷=∗ '/' '[' Q ']' ']'").

Reserved Notation "|={ E }▷=> Q"

(at level 99, E at level 50, Q at level 200,

format "'[ ' |={ E }▷=> '/' Q ']'").

Reserved Notation "P ={ E }▷=∗ Q"

(at level 99, E at level 50, Q at level 200,

format "'[' P ={ E }▷=∗ '/' '[' Q ']' ']'").

(at level 99, E1, E2 at level 50, Q at level 200,

format "'[ ' |={ E1 } [ E2 ]▷=> '/' Q ']'").

Reserved Notation "P ={ E1 } [ E2 ]▷=∗ Q"

(at level 99, E1, E2 at level 50, Q at level 200,

format "'[' P ={ E1 } [ E2 ]▷=∗ '/' '[' Q ']' ']'").

Reserved Notation "|={ E }▷=> Q"

(at level 99, E at level 50, Q at level 200,

format "'[ ' |={ E }▷=> '/' Q ']'").

Reserved Notation "P ={ E }▷=∗ Q"

(at level 99, E at level 50, Q at level 200,

format "'[' P ={ E }▷=∗ '/' '[' Q ']' ']'").

Multi-step-taking fancy updates

Reserved Notation "|={ E1 } [ E2 ]▷=>^ n Q"

(at level 99, E1, E2 at level 50, n at level 9, Q at level 200,

format "'[ ' |={ E1 } [ E2 ]▷=>^ n '/' Q ']'").

Reserved Notation "P ={ E1 } [ E2 ]▷=∗^ n Q"

(at level 99, E1, E2 at level 50, n at level 9, Q at level 200,

format "'[' P ={ E1 } [ E2 ]▷=∗^ n '/' '[' Q ']' ']'").

Reserved Notation "|={ E }▷=>^ n Q"

(at level 99, E at level 50, n at level 9, Q at level 200,

format "'[ ' |={ E }▷=>^ n '/' Q ']'").

Reserved Notation "P ={ E }▷=∗^ n Q"

(at level 99, E at level 50, n at level 9, Q at level 200,

format "'[' P ={ E }▷=∗^ n '/' '[' Q ']' ']'").

(at level 99, E1, E2 at level 50, n at level 9, Q at level 200,

format "'[ ' |={ E1 } [ E2 ]▷=>^ n '/' Q ']'").

Reserved Notation "P ={ E1 } [ E2 ]▷=∗^ n Q"

(at level 99, E1, E2 at level 50, n at level 9, Q at level 200,

format "'[' P ={ E1 } [ E2 ]▷=∗^ n '/' '[' Q ']' ']'").

Reserved Notation "|={ E }▷=>^ n Q"

(at level 99, E at level 50, n at level 9, Q at level 200,

format "'[ ' |={ E }▷=>^ n '/' Q ']'").

Reserved Notation "P ={ E }▷=∗^ n Q"

(at level 99, E at level 50, n at level 9, Q at level 200,

format "'[' P ={ E }▷=∗^ n '/' '[' Q ']' ']'").

Reserved Notation "'[∗' 'list]' k ↦ x ∈ l , P"

(at level 200, l at level 10, k, x at level 1, right associativity,

format "[∗ list] k ↦ x ∈ l , P").

Reserved Notation "'[∗' 'list]' x ∈ l , P"

(at level 200, l at level 10, x at level 1, right associativity,

format "[∗ list] x ∈ l , P").

Reserved Notation "'[∗' 'list]' k ↦ x1 ; x2 ∈ l1 ; l2 , P"

(at level 200, l1, l2 at level 10, k, x1, x2 at level 1, right associativity,

format "[∗ list] k ↦ x1 ; x2 ∈ l1 ; l2 , P").

Reserved Notation "'[∗' 'list]' x1 ; x2 ∈ l1 ; l2 , P"

(at level 200, l1, l2 at level 10, x1, x2 at level 1, right associativity,

format "[∗ list] x1 ; x2 ∈ l1 ; l2 , P").

Reserved Notation "'[∗]' Ps" (at level 20).

Reserved Notation "'[∧' 'list]' k ↦ x ∈ l , P"

(at level 200, l at level 10, k, x at level 1, right associativity,

format "[∧ list] k ↦ x ∈ l , P").

Reserved Notation "'[∧' 'list]' x ∈ l , P"

(at level 200, l at level 10, x at level 1, right associativity,

format "[∧ list] x ∈ l , P").

Reserved Notation "'[∧]' Ps" (at level 20).

Reserved Notation "'[∨' 'list]' k ↦ x ∈ l , P"

(at level 200, l at level 10, k, x at level 1, right associativity,

format "[∨ list] k ↦ x ∈ l , P").

Reserved Notation "'[∨' 'list]' x ∈ l , P"

(at level 200, l at level 10, x at level 1, right associativity,

format "[∨ list] x ∈ l , P").

Reserved Notation "'[∨]' Ps" (at level 20).

Reserved Notation "'[∗' 'map]' k ↦ x ∈ m , P"

(at level 200, m at level 10, k, x at level 1, right associativity,

format "[∗ map] k ↦ x ∈ m , P").

Reserved Notation "'[∗' 'map]' x ∈ m , P"

(at level 200, m at level 10, x at level 1, right associativity,

format "[∗ map] x ∈ m , P").

Reserved Notation "'[∗' 'map]' k ↦ x1 ; x2 ∈ m1 ; m2 , P"

(at level 200, m1, m2 at level 10, k, x1, x2 at level 1, right associativity,

format "[∗ map] k ↦ x1 ; x2 ∈ m1 ; m2 , P").

Reserved Notation "'[∗' 'map]' x1 ; x2 ∈ m1 ; m2 , P"

(at level 200, m1, m2 at level 10, x1, x2 at level 1, right associativity,

format "[∗ map] x1 ; x2 ∈ m1 ; m2 , P").

Reserved Notation "'[∧' 'map]' k ↦ x ∈ m , P"

(at level 200, m at level 10, k, x at level 1, right associativity,

format "[∧ map] k ↦ x ∈ m , P").

Reserved Notation "'[∧' 'map]' x ∈ m , P"

(at level 200, m at level 10, x at level 1, right associativity,

format "[∧ map] x ∈ m , P").

Reserved Notation "'[∗' 'set]' x ∈ X , P"

(at level 200, X at level 10, x at level 1, right associativity,

format "[∗ set] x ∈ X , P").

Reserved Notation "'[∗' 'mset]' x ∈ X , P"

(at level 200, X at level 10, x at level 1, right associativity,

format "[∗ mset] x ∈ X , P").

(at level 200, l at level 10, k, x at level 1, right associativity,

format "[∗ list] k ↦ x ∈ l , P").

Reserved Notation "'[∗' 'list]' x ∈ l , P"

(at level 200, l at level 10, x at level 1, right associativity,

format "[∗ list] x ∈ l , P").

Reserved Notation "'[∗' 'list]' k ↦ x1 ; x2 ∈ l1 ; l2 , P"

(at level 200, l1, l2 at level 10, k, x1, x2 at level 1, right associativity,

format "[∗ list] k ↦ x1 ; x2 ∈ l1 ; l2 , P").

Reserved Notation "'[∗' 'list]' x1 ; x2 ∈ l1 ; l2 , P"

(at level 200, l1, l2 at level 10, x1, x2 at level 1, right associativity,

format "[∗ list] x1 ; x2 ∈ l1 ; l2 , P").

Reserved Notation "'[∗]' Ps" (at level 20).

Reserved Notation "'[∧' 'list]' k ↦ x ∈ l , P"

(at level 200, l at level 10, k, x at level 1, right associativity,

format "[∧ list] k ↦ x ∈ l , P").

Reserved Notation "'[∧' 'list]' x ∈ l , P"

(at level 200, l at level 10, x at level 1, right associativity,

format "[∧ list] x ∈ l , P").

Reserved Notation "'[∧]' Ps" (at level 20).

Reserved Notation "'[∨' 'list]' k ↦ x ∈ l , P"

(at level 200, l at level 10, k, x at level 1, right associativity,

format "[∨ list] k ↦ x ∈ l , P").

Reserved Notation "'[∨' 'list]' x ∈ l , P"

(at level 200, l at level 10, x at level 1, right associativity,

format "[∨ list] x ∈ l , P").

Reserved Notation "'[∨]' Ps" (at level 20).

Reserved Notation "'[∗' 'map]' k ↦ x ∈ m , P"

(at level 200, m at level 10, k, x at level 1, right associativity,

format "[∗ map] k ↦ x ∈ m , P").

Reserved Notation "'[∗' 'map]' x ∈ m , P"

(at level 200, m at level 10, x at level 1, right associativity,

format "[∗ map] x ∈ m , P").

Reserved Notation "'[∗' 'map]' k ↦ x1 ; x2 ∈ m1 ; m2 , P"

(at level 200, m1, m2 at level 10, k, x1, x2 at level 1, right associativity,

format "[∗ map] k ↦ x1 ; x2 ∈ m1 ; m2 , P").

Reserved Notation "'[∗' 'map]' x1 ; x2 ∈ m1 ; m2 , P"

(at level 200, m1, m2 at level 10, x1, x2 at level 1, right associativity,

format "[∗ map] x1 ; x2 ∈ m1 ; m2 , P").

Reserved Notation "'[∧' 'map]' k ↦ x ∈ m , P"

(at level 200, m at level 10, k, x at level 1, right associativity,

format "[∧ map] k ↦ x ∈ m , P").

Reserved Notation "'[∧' 'map]' x ∈ m , P"

(at level 200, m at level 10, x at level 1, right associativity,

format "[∧ map] x ∈ m , P").

Reserved Notation "'[∗' 'set]' x ∈ X , P"

(at level 200, X at level 10, x at level 1, right associativity,

format "[∗ set] x ∈ X , P").

Reserved Notation "'[∗' 'mset]' x ∈ X , P"

(at level 200, X at level 10, x at level 1, right associativity,

format "[∗ mset] x ∈ X , P").

Define the scope

Declare Scope bi_scope.

Delimit Scope bi_scope with I.

Delimit Scope bi_scope with I.