Arithmetic Rewrite Rules

Rules that are marked with a * in the first column are implemented in the latest version of Rodin. Rules without a * are planned to be implemented in future versions. Other conventions used in these tables are described in The_Proving_Perspective_(Rodin_User_Manual)#Rewrite_Rules.

	Name	Rule	Side Condition	A/M
*	SIMP_SPECIAL_MOD_0	$0 \,\bmod\, E \;\;\defi\;\; 0$		A
*	SIMP_SPECIAL_MOD_1	$E \,\bmod\, 1 \;\;\defi\;\; 0$		A
*	SIMP_MIN_SING	$\min (\{ E\} ) \;\;\defi\;\; E$	where $E$ is a single expression	A
*	SIMP_MAX_SING	$\max (\{ E\} ) \;\;\defi\;\; E$	where $E$ is a single expression	A
*	SIMP_MIN_NATURAL	$\min (\nat ) \;\;\defi\;\; 0$		A
*	SIMP_MIN_NATURAL1	$\min (\natn ) \;\;\defi\;\; 1$		A
*	SIMP_MIN_BUNION_SING	$\begin{array}{cl} & \min (S \bunion \ldots \bunion \{ \min (T)\} \bunion \ldots \bunion U) \\ \defi & \min (S \bunion \ldots \bunion T \bunion \ldots \bunion U) \\ \end{array}$		A
*	SIMP_MAX_BUNION_SING	$\begin{array}{cl} & \max (S \bunion \ldots \bunion \{ \max (T)\} \bunion \ldots \bunion U) \\ \defi & \max (S \bunion \ldots \bunion T \bunion \ldots \bunion U) \\ \end{array}$		A
*	SIMP_MIN_UPTO	$\min (E \upto F) \;\;\defi\;\; E$		A
*	SIMP_MAX_UPTO	$\max (E \upto F) \;\;\defi\;\; F$		A
	SIMP_MIN_IN	$\min (S) \in S \;\;\defi\;\; \btrue$		A
	SIMP_MAX_IN	$\max (S) \in S \;\;\defi\;\; \btrue$		A
*	SIMP_LIT_MIN	$\min (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \min (\{ E, \ldots , i, \ldots , H\} )$	where $i$ and $j$ are literals and $i \leq j$	A
*	SIMP_LIT_MAX	$\max (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \max (\{ E, \ldots , i, \ldots , H\} )$	where $i$ and $j$ are literals and $i \geq j$	A
*	SIMP_SPECIAL_CARD	$\card (\emptyset ) \;\;\defi\;\; 0$		A
*	SIMP_CARD_SING	$\card (\{ E\} ) \;\;\defi\;\; 1$	where $E$ is a single expression	A
*	SIMP_SPECIAL_EQUAL_CARD	$\card (S) = 0 \;\;\defi\;\; S = \emptyset$		A
*	SIMP_CARD_POW	$\card (\pow (S)) \;\;\defi\;\; 2\expn{\card(S)}$		A
*	SIMP_CARD_BUNION	$\card (S \bunion T) \;\;\defi\;\; \card (S) + \card (T) - \card (S \binter T)$		A
	SIMP_CARD_SETMINUS	$\card(S\setminus T)\;\;\defi\;\;\card(S) - \card(T)$	with hypotheses $T\subseteq S$ and either $\finite(S)$ or $\finite(T)$	A
	SIMP_CARD_SETMINUS_SETENUM	$\card(S\setminus\{E_1,\ldots,E_n\})\;\;\defi\;\;\card(S) - \card(\{E_1,\ldots,E_n\})$	with hypotheses $E_i\in S$ for all $i\in 1\upto n$	A
*	SIMP_CARD_CONVERSE	$\card (r^{-1} ) \;\;\defi\;\; \card (r)$		A
*	SIMP_CARD_ID	$\card (\id) \;\;\defi\;\; \card (S)$	where $\id$ has type $\pow (S \cprod S)$	A
*	SIMP_CARD_ID_DOMRES	$\card (S\domres\id) \;\;\defi\;\; \card (S)$		A
*	SIMP_CARD_PRJ1	$\card (\prjone) \;\;\defi\;\; \card (S \cprod T)$	where $\prjone$ has type $\pow(S \cprod T \cprod S)$	A
*	SIMP_CARD_PRJ2	$\card (\prjtwo) \;\;\defi\;\; \card (S \cprod T)$	where $\prjtwo$ has type $\pow(S \cprod T \cprod T)$	A
*	SIMP_CARD_PRJ1_DOMRES	$\card (E \domres \prjone) \;\;\defi\;\; \card (E)$		A
*	SIMP_CARD_PRJ2_DOMRES	$\card (E \domres \prjtwo) \;\;\defi\;\; \card (E)$		A
*	SIMP_CARD_LAMBDA	$\card(\{x\qdot P\mid E\mapsto F\}) \;\;\defi\;\; \card(\{x\qdot P\mid E\} )$	where $E$ is a maplet combination of bound identifiers and expressions that are not bound by the comprehension set (i.e., $E$ is syntactically injective) and all identifiers bound by the comprehension set that occur in $F$ also occur in $E$	A
*	SIMP_LIT_CARD_UPTO	$\card (i \upto j) \;\;\defi\;\; j-i+1$	where $i$ and $j$ are literals and $i \leq j$	A
	SIMP_TYPE_CARD	$\card (\mathit{Tenum}) \;\;\defi\;\; N$	where $\mathit{Tenum}$ is a carrier set containing $N$ elements	A
*	SIMP_LIT_GE_CARD_1	$\card (S) \geq 1 \;\;\defi\;\; \lnot\, S = \emptyset$		A
*	SIMP_LIT_LE_CARD_1	$1 \leq \card (S) \;\;\defi\;\; \lnot\, S = \emptyset$		A
*	SIMP_LIT_LE_CARD_0	$0 \leq \card (S) \;\;\defi\;\; \btrue$		A
*	SIMP_LIT_GE_CARD_0	$\card (S) \geq 0 \;\;\defi\;\; \btrue$		A
*	SIMP_LIT_GT_CARD_0	$\card (S) > 0 \;\;\defi\;\; \lnot\, S = \emptyset$		A
*	SIMP_LIT_LT_CARD_0	$0 < \card (S) \;\;\defi\;\; \lnot\, S = \emptyset$		A
*	SIMP_LIT_EQUAL_CARD_1	$\card (S) = 1 \;\;\defi\;\; \exists x \qdot S = \{ x\}$		A
*	SIMP_CARD_NATURAL	$\card (S) \in \nat \;\;\defi\;\; \btrue$		A
*	SIMP_CARD_NATURAL1	$\card (S) \in \natn \;\;\defi\;\; \lnot\, S = \emptyset$		A
*	SIMP_LIT_IN_NATURAL	$i \in \nat \;\;\defi\;\; \btrue$	where $i$ is a non-negative literal	A
*	SIMP_SPECIAL_IN_NATURAL1	$0 \in \natn \;\;\defi\;\; \bfalse$		A
*	SIMP_LIT_IN_NATURAL1	$i \in \natn \;\;\defi\;\; \btrue$	where $i$ is a positive literal	A
*	SIMP_LIT_UPTO	$i \upto j \;\;\defi\;\; \emptyset$	where $i$ and $j$ are literals and $j < i$	A
*	SIMP_LIT_IN_MINUS_NATURAL	$-i \in \nat \;\;\defi\;\; \bfalse$	where $i$ is a positive literal	A
*	SIMP_LIT_IN_MINUS_NATURAL1	$-i \in \natn \;\;\defi\;\; \bfalse$	where $i$ is a non-negative literal	A
*	DEF_IN_NATURAL	$x \in \nat \;\;\defi\;\; 0 \leq x$		M
*	DEF_IN_NATURAL1	$x \in \natn \;\;\defi\;\; 1 \leq x$		M
*	SIMP_LIT_EQUAL_KBOOL_TRUE	$\bool (P) = \True \;\;\defi\;\; P$		A
*	SIMP_LIT_EQUAL_KBOOL_FALSE	$\bool (P) = \False \;\;\defi\;\; \lnot\, P$		A
	SIMP_KBOOL_LIT_EQUAL_TRUE	$\bool (B = \True) \;\;\defi\;\; B$		A
*	DEF_EQUAL_MIN	$E = \min (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \leq x)$	where $x$ non free in $S, E$	M
*	DEF_EQUAL_MAX	$E = \max (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \geq x)$	where $x$ non free in $S, E$	M
*	SIMP_SPECIAL_PLUS	$E + \ldots + 0 + \ldots + F \;\;\defi\;\; E + \ldots + F$		A
*	SIMP_SPECIAL_MINUS_R	$E - 0 \;\;\defi\;\; E$		A
*	SIMP_SPECIAL_MINUS_L	$0 - E \;\;\defi\;\; -E$		A
*	SIMP_MINUS_MINUS	$- (- E) \;\;\defi\;\; E$		A
*	SIMP_MINUS_UNMINUS	$E - (- F) \;\;\defi\;\; E + F$	where $(-F)$ is a unary minus expression or a negative literal	M
*	SIMP_MULTI_MINUS	$E - E \;\;\defi\;\; 0$		A
*	SIMP_MULTI_MINUS_PLUS_L	$(A + \ldots + C + \ldots + B) - C \;\;\defi\;\; A + \ldots + B$		M
*	SIMP_MULTI_MINUS_PLUS_R	$C - (A + \ldots + C + \ldots + B) \;\;\defi\;\; -(A + \ldots + B)$		M
*	SIMP_MULTI_MINUS_PLUS_PLUS	$(A + \ldots + E + \ldots + B) - (C + \ldots + E + \ldots + D) \;\;\defi\;\; (A + \ldots + B) - (C + \ldots + D)$		M
*	SIMP_MULTI_PLUS_MINUS	$(A + \ldots + D + \ldots + (C - D) + \ldots + B) \;\;\defi\;\; A + \ldots + C + \ldots + B$		M
*	SIMP_MULTI_ARITHREL_PLUS_PLUS	$A + \ldots + E + \ldots + B < C + \ldots + E + \ldots + D \;\;\defi\;\; A + \ldots + B < C + \ldots + D$	where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$	M
*	SIMP_MULTI_ARITHREL_PLUS_R	$C < A + \ldots + C \ldots + B \;\;\defi\;\; 0 < A + \ldots + B$	where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$	M
*	SIMP_MULTI_ARITHREL_PLUS_L	$A + \ldots + C \ldots + B < C \;\;\defi\;\; A + \ldots + B < 0$	where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$	M
*	SIMP_MULTI_ARITHREL_MINUS_MINUS_R	$A - C < B - C \;\;\defi\;\; A < B$	where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$	M
*	SIMP_MULTI_ARITHREL_MINUS_MINUS_L	$C - A < C - B \;\;\defi\;\; B < A$	where the root relation ($<$ here) is one of $\{=, <, \leq, >, \geq\}$	M
*	SIMP_SPECIAL_PROD_0	$E * \ldots * 0 * \ldots * F \;\;\defi\;\; 0$		A
*	SIMP_SPECIAL_PROD_1	$E * \ldots * 1 * \ldots * F \;\;\defi\;\; E * \ldots * F$		A
*	SIMP_SPECIAL_PROD_MINUS_EVEN	$(-E) * \ldots * (-F) \;\;\defi\;\; E * \ldots * F$	if an even number of $-$	A
*	SIMP_SPECIAL_PROD_MINUS_ODD	$(-E) * \ldots * (-F) \;\;\defi\;\; -(E * \ldots * F)$	if an odd number of $-$	A
*	SIMP_LIT_MINUS	$- (i) \;\;\defi\;\; (-i)$	where $i$ is a literal	A
*	SIMP_LIT_EQUAL	$i = j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$	where $i$ and $j$ are literals	A
*	SIMP_LIT_LE	$i \leq j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$	where $i$ and $j$ are literals	A
*	SIMP_LIT_LT	$i < j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$	where $i$ and $j$ are literals	A
*	SIMP_LIT_GE	$i \geq j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$	where $i$ and $j$ are literals	A
*	SIMP_LIT_GT	$i > j \;\;\defi\;\; \btrue \;or\; \bfalse \;\;(computation)$	where $i$ and $j$ are literals	A
*	SIMP_DIV_MINUS	$(- E) \div (-F) \;\;\defi\;\; E \div F$		A
*	SIMP_SPECIAL_DIV_1	$E \div 1 \;\;\defi\;\; E$		A
*	SIMP_SPECIAL_DIV_0	$0 \div E \;\;\defi\;\; 0$		A
*	SIMP_SPECIAL_EXPN_1_R	$E ^ 1 \;\;\defi\;\; E$		A
*	SIMP_SPECIAL_EXPN_1_L	$1 ^ E \;\;\defi\;\; 1$		A
*	SIMP_SPECIAL_EXPN_0	$E ^ 0 \;\;\defi\;\; 1$		A
	DEF_EXPN_STEP	$E ^ P \;\;\defi\;\; E * E ^{(P - 1)}$	with an additional PO $\lnot\, P = 0$	M
*	SIMP_MULTI_LE	$E \leq E \;\;\defi\;\; \btrue$		A
*	SIMP_MULTI_LT	$E < E \;\;\defi\;\; \bfalse$		A
*	SIMP_MULTI_GE	$E \geq E \;\;\defi\;\; \btrue$		A
*	SIMP_MULTI_GT	$E > E \;\;\defi\;\; \bfalse$		A
*	SIMP_MULTI_DIV	$E \div E \;\;\defi\;\; 1$		A
*	SIMP_MULTI_DIV_PROD	$(X * \ldots * E * \ldots * Y) \div E \;\;\defi\;\; X * \ldots * Y$		A
*	SIMP_MULTI_MOD	$E \,\bmod\, E \;\;\defi\;\; 0$		A
	DISTRI_PROD_PLUS	$a * (b + c) \;\;\defi\;\; (a * b) + (a * c)$		M
	DISTRI_PROD_MINUS	$a * (b - c) \;\;\defi\;\; (a * b) - (a * c)$		M
	DERIV_NOT_EQUAL	$\lnot E = F \;\;\defi\;\; E < F \lor E > F$	$E$ and $F$ must be of Integer type	M