Difference between revisions of "Arithmetic Rewrite Rules"
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Jump to navigationJump to searchimported>Laurent m (Fixed typo in SIMP_CARD_POW) |
(Rules DEF_EXPN_STEP, SIMP_{MIN,MAX}_IN and SIMP_KBOOL_LIT_EQUAL_TRUE have been implemented in Rodin 3.9) |
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{{RRRow}}|*||{{Rulename|SIMP_MIN_UPTO}}||<math> \min (E \upto F) \;\;\defi\;\; E </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_MIN_UPTO}}||<math> \min (E \upto F) \;\;\defi\;\; E </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_MAX_UPTO}}||<math> \max (E \upto F) \;\;\defi\;\; F </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_MAX_UPTO}}||<math> \max (E \upto F) \;\;\defi\;\; F </math>|| || A | ||
+ | {{RRRow}}|*||{{Rulename|SIMP_MIN_IN}}||<math> \min (S) \in S \;\;\defi\;\; \btrue </math>|| || A | ||
+ | {{RRRow}}|*||{{Rulename|SIMP_MAX_IN}}||<math> \max (S) \in S \;\;\defi\;\; \btrue </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_LIT_MIN}}||<math> \min (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \min (\{ E, \ldots , i, \ldots , H\} ) </math>|| where <math>i</math> and <math>j</math> are literals and <math>i \leq j</math> || A | {{RRRow}}|*||{{Rulename|SIMP_LIT_MIN}}||<math> \min (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \min (\{ E, \ldots , i, \ldots , H\} ) </math>|| where <math>i</math> and <math>j</math> are literals and <math>i \leq j</math> || A | ||
{{RRRow}}|*||{{Rulename|SIMP_LIT_MAX}}||<math> \max (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \max (\{ E, \ldots , i, \ldots , H\} ) </math>|| where <math>i</math> and <math>j</math> are literals and <math>i \geq j</math> || A | {{RRRow}}|*||{{Rulename|SIMP_LIT_MAX}}||<math> \max (\{ E, \ldots , i, \ldots , j, \ldots , H\} ) \;\;\defi\;\; \max (\{ E, \ldots , i, \ldots , H\} ) </math>|| where <math>i</math> and <math>j</math> are literals and <math>i \geq j</math> || A | ||
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{{RRRow}}|*||{{Rulename|SIMP_CARD_POW}}||<math> \card (\pow (S)) \;\;\defi\;\; 2\expn{\card(S)} </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_CARD_POW}}||<math> \card (\pow (S)) \;\;\defi\;\; 2\expn{\card(S)} </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_CARD_BUNION}}||<math> \card (S \bunion T) \;\;\defi\;\; \card (S) + \card (T) - \card (S \binter T) </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_CARD_BUNION}}||<math> \card (S \bunion T) \;\;\defi\;\; \card (S) + \card (T) - \card (S \binter T) </math>|| || A | ||
+ | {{RRRow}}| ||{{Rulename|SIMP_CARD_SETMINUS}}||<math>\card(S\setminus T)\;\;\defi\;\;\card(S) - \card(T)</math>|| with hypotheses <math>T\subseteq S</math> and either <math>\finite(S)</math> or <math>\finite(T)</math>|| A | ||
+ | {{RRRow}}| ||{{Rulename|SIMP_CARD_SETMINUS_SETENUM}}||<math>\card(S\setminus\{E_1,\ldots,E_n\})\;\;\defi\;\;\card(S) - \card(\{E_1,\ldots,E_n\})</math>|| with hypotheses <math>E_i\in S</math> for all <math>i\in 1\upto n</math>|| A | ||
{{RRRow}}|*||{{Rulename|SIMP_CARD_CONVERSE}}||<math> \card (r^{-1} ) \;\;\defi\;\; \card (r) </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_CARD_CONVERSE}}||<math> \card (r^{-1} ) \;\;\defi\;\; \card (r) </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_CARD_ID}}||<math> \card (\id) \;\;\defi\;\; \card (S) </math>|| where <math>\id</math> has type <math>\pow (S \cprod S) </math>|| A | {{RRRow}}|*||{{Rulename|SIMP_CARD_ID}}||<math> \card (\id) \;\;\defi\;\; \card (S) </math>|| where <math>\id</math> has type <math>\pow (S \cprod S) </math>|| A | ||
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{{RRRow}}|*||{{Rulename|SIMP_LIT_EQUAL_KBOOL_TRUE}}||<math> \bool (P) = \True \;\;\defi\;\; P </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_LIT_EQUAL_KBOOL_TRUE}}||<math> \bool (P) = \True \;\;\defi\;\; P </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_LIT_EQUAL_KBOOL_FALSE}}||<math> \bool (P) = \False \;\;\defi\;\; \lnot\, P </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_LIT_EQUAL_KBOOL_FALSE}}||<math> \bool (P) = \False \;\;\defi\;\; \lnot\, P </math>|| || A | ||
− | {{RRRow}}|||{{Rulename|DEF_EQUAL_MIN}}||<math> E = \min (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \leq x) </math>|| where <math>x</math> non free in <math>S, E</math> || M | + | {{RRRow}}|*||{{Rulename|SIMP_KBOOL_LIT_EQUAL_TRUE}}||<math> \bool (B = \True) \;\;\defi\;\; B </math>|| || A |
− | {{RRRow}}|||{{Rulename|DEF_EQUAL_MAX}}||<math> E = \max (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \geq x) </math>|| where <math>x</math> non free in <math>S, E</math> || M | + | {{RRRow}}|*||{{Rulename|DEF_EQUAL_MIN}}||<math> E = \min (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \leq x) </math>|| where <math>x</math> non free in <math>S, E</math> || M |
+ | {{RRRow}}|*||{{Rulename|DEF_EQUAL_MAX}}||<math> E = \max (S) \;\;\defi\;\; E \in S \land (\forall x \qdot x \in S \limp E \geq x) </math>|| where <math>x</math> non free in <math>S, E</math> || M | ||
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_PLUS}}||<math> E + \ldots + 0 + \ldots + F \;\;\defi\;\; E + \ldots + F </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_SPECIAL_PLUS}}||<math> E + \ldots + 0 + \ldots + F \;\;\defi\;\; E + \ldots + F </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_MINUS_R}}||<math> E - 0 \;\;\defi\;\; E </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_SPECIAL_MINUS_R}}||<math> E - 0 \;\;\defi\;\; E </math>|| || A | ||
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{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EXPN_1_L}}||<math> 1 ^ E \;\;\defi\;\; 1 </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EXPN_1_L}}||<math> 1 ^ E \;\;\defi\;\; 1 </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EXPN_0}}||<math> E ^ 0 \;\;\defi\;\; 1 </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_SPECIAL_EXPN_0}}||<math> E ^ 0 \;\;\defi\;\; 1 </math>|| || A | ||
+ | {{RRRow}}|*||{{Rulename|DEF_EXPN_STEP}}||<math> E ^ P \;\;\defi\;\; E * E ^{(P - 1)} </math>|| with an additional PO <math>\lnot\, P = 0</math> || M | ||
{{RRRow}}|*||{{Rulename|SIMP_MULTI_LE}}||<math> E \leq E \;\;\defi\;\; \btrue </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_MULTI_LE}}||<math> E \leq E \;\;\defi\;\; \btrue </math>|| || A | ||
{{RRRow}}|*||{{Rulename|SIMP_MULTI_LT}}||<math> E < E \;\;\defi\;\; \bfalse </math>|| || A | {{RRRow}}|*||{{Rulename|SIMP_MULTI_LT}}||<math> E < E \;\;\defi\;\; \bfalse </math>|| || A |
Latest revision as of 15:11, 3 June 2024
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 |
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A | |
* | SIMP_SPECIAL_MOD_1 |
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A | |
* | SIMP_MIN_SING |
![]() |
where ![]() |
A |
* | SIMP_MAX_SING |
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where ![]() |
A |
* | SIMP_MIN_NATURAL |
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A | |
* | SIMP_MIN_NATURAL1 |
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A | |
* | SIMP_MIN_BUNION_SING |
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A | |
* | SIMP_MAX_BUNION_SING |
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A | |
* | SIMP_MIN_UPTO |
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A | |
* | SIMP_MAX_UPTO |
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A | |
* | SIMP_MIN_IN |
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A | |
* | SIMP_MAX_IN |
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A | |
* | SIMP_LIT_MIN |
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where ![]() ![]() ![]() |
A |
* | SIMP_LIT_MAX |
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where ![]() ![]() ![]() |
A |
* | SIMP_SPECIAL_CARD |
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A | |
* | SIMP_CARD_SING |
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where ![]() |
A |
* | SIMP_SPECIAL_EQUAL_CARD |
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A | |
* | SIMP_CARD_POW |
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A | |
* | SIMP_CARD_BUNION |
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A | |
SIMP_CARD_SETMINUS |
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with hypotheses ![]() ![]() ![]() |
A | |
SIMP_CARD_SETMINUS_SETENUM |
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with hypotheses ![]() ![]() |
A | |
* | SIMP_CARD_CONVERSE |
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A | |
* | SIMP_CARD_ID |
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where ![]() ![]() |
A |
* | SIMP_CARD_ID_DOMRES |
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A | |
* | SIMP_CARD_PRJ1 |
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where ![]() ![]() |
A |
* | SIMP_CARD_PRJ2 |
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where ![]() ![]() |
A |
* | SIMP_CARD_PRJ1_DOMRES |
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A | |
* | SIMP_CARD_PRJ2_DOMRES |
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A | |
* | SIMP_CARD_LAMBDA |
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where ![]() ![]() ![]() ![]() |
A |
* | SIMP_LIT_CARD_UPTO |
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where ![]() ![]() ![]() |
A |
SIMP_TYPE_CARD |
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where ![]() ![]() |
A | |
* | SIMP_LIT_GE_CARD_1 |
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A | |
* | SIMP_LIT_LE_CARD_1 |
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A | |
* | SIMP_LIT_LE_CARD_0 |
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A | |
* | SIMP_LIT_GE_CARD_0 |
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A | |
* | SIMP_LIT_GT_CARD_0 |
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A | |
* | SIMP_LIT_LT_CARD_0 |
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A | |
* | SIMP_LIT_EQUAL_CARD_1 |
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A | |
* | SIMP_CARD_NATURAL |
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A | |
* | SIMP_CARD_NATURAL1 |
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A | |
* | SIMP_LIT_IN_NATURAL |
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where ![]() |
A |
* | SIMP_SPECIAL_IN_NATURAL1 |
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A | |
* | SIMP_LIT_IN_NATURAL1 |
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where ![]() |
A |
* | SIMP_LIT_UPTO |
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where ![]() ![]() ![]() |
A |
* | SIMP_LIT_IN_MINUS_NATURAL |
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where ![]() |
A |
* | SIMP_LIT_IN_MINUS_NATURAL1 |
![]() |
where ![]() |
A |
* | DEF_IN_NATURAL |
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M | |
* | DEF_IN_NATURAL1 |
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M | |
* | SIMP_LIT_EQUAL_KBOOL_TRUE |
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A | |
* | SIMP_LIT_EQUAL_KBOOL_FALSE |
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A | |
* | SIMP_KBOOL_LIT_EQUAL_TRUE |
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A | |
* | DEF_EQUAL_MIN |
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where ![]() ![]() |
M |
* | DEF_EQUAL_MAX |
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where ![]() ![]() |
M |
* | SIMP_SPECIAL_PLUS |
![]() |
A | |
* | SIMP_SPECIAL_MINUS_R |
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A | |
* | SIMP_SPECIAL_MINUS_L |
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A | |
* | SIMP_MINUS_MINUS |
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A | |
* | SIMP_MINUS_UNMINUS |
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where ![]() |
M |
* | SIMP_MULTI_MINUS |
![]() |
A | |
* | SIMP_MULTI_MINUS_PLUS_L |
![]() |
M | |
* | SIMP_MULTI_MINUS_PLUS_R |
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M | |
* | SIMP_MULTI_MINUS_PLUS_PLUS |
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M | |
* | SIMP_MULTI_PLUS_MINUS |
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M | |
* | SIMP_MULTI_ARITHREL_PLUS_PLUS |
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where the root relation (![]() ![]() |
M |
* | SIMP_MULTI_ARITHREL_PLUS_R |
![]() |
where the root relation (![]() ![]() |
M |
* | SIMP_MULTI_ARITHREL_PLUS_L |
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where the root relation (![]() ![]() |
M |
* | SIMP_MULTI_ARITHREL_MINUS_MINUS_R |
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where the root relation (![]() ![]() |
M |
* | SIMP_MULTI_ARITHREL_MINUS_MINUS_L |
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where the root relation (![]() ![]() |
M |
* | SIMP_SPECIAL_PROD_0 |
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A | |
* | SIMP_SPECIAL_PROD_1 |
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A | |
* | SIMP_SPECIAL_PROD_MINUS_EVEN |
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if an even number of ![]() |
A |
* | SIMP_SPECIAL_PROD_MINUS_ODD |
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if an odd number of ![]() |
A |
* | SIMP_LIT_MINUS |
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where ![]() |
A |
* | SIMP_LIT_EQUAL |
![]() |
where ![]() ![]() |
A |
* | SIMP_LIT_LE |
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where ![]() ![]() |
A |
* | SIMP_LIT_LT |
![]() |
where ![]() ![]() |
A |
* | SIMP_LIT_GE |
![]() |
where ![]() ![]() |
A |
* | SIMP_LIT_GT |
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where ![]() ![]() |
A |
* | SIMP_DIV_MINUS |
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A | |
* | SIMP_SPECIAL_DIV_1 |
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A | |
* | SIMP_SPECIAL_DIV_0 |
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A | |
* | SIMP_SPECIAL_EXPN_1_R |
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A | |
* | SIMP_SPECIAL_EXPN_1_L |
![]() |
A | |
* | SIMP_SPECIAL_EXPN_0 |
![]() |
A | |
* | DEF_EXPN_STEP |
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with an additional PO ![]() |
M |
* | SIMP_MULTI_LE |
![]() |
A | |
* | SIMP_MULTI_LT |
![]() |
A | |
* | SIMP_MULTI_GE |
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A | |
* | SIMP_MULTI_GT |
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A | |
* | SIMP_MULTI_DIV |
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A | |
* | SIMP_MULTI_DIV_PROD |
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A | |
* | SIMP_MULTI_MOD |
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A | |
DISTRI_PROD_PLUS |
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M | ||
DISTRI_PROD_MINUS |
![]() |
M | ||
DERIV_NOT_EQUAL |
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![]() ![]() |
M |