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Semantics

Exec-rem

Semanics of the REM instruction [ISA:13.2].

Signature
(exec-rem rd rs1 rs2 stat feat) → new-stat
Arguments: rd — Guard (ubyte5p rd).; rs1 — Guard (ubyte5p rs1).; rs2 — Guard (ubyte5p rs2).; stat — Guard (statp stat).; feat — Guard (featp feat).
Returns: new-stat — Type (statp new-stat).

We read two signed XLEN-bit integers from rs1 and rs2. We calculate the remainder of the first by the second, based on division towards 0; if the divisor is 0, the result is the dividend (see Table 11 in [ISA:13.2]). We write the result to rd. We increment the program counter.

Definitions and Theorems

Function: exec-rem

(defun exec-rem (rd rs1 rs2 stat feat)
  (declare (xargs :guard (and (ubyte5p rd)
                              (ubyte5p rs1)
                              (ubyte5p rs2)
                              (statp stat)
                              (featp feat))))
  (declare (xargs :guard (stat-validp stat feat)))
  (let ((__function__ 'exec-rem))
    (declare (ignorable __function__))
    (b* ((rs1-operand (read-xreg-signed (ubyte5-fix rs1)
                                        stat feat))
         (rs2-operand (read-xreg-signed (ubyte5-fix rs2)
                                        stat feat))
         (result (if (= rs2-operand 0)
                     rs1-operand
                   (rem rs1-operand rs2-operand)))
         (stat (write-xreg (ubyte5-fix rd)
                           result stat feat))
         (stat (inc4-pc stat feat)))
      stat)))

Theorem: statp-of-exec-rem

(defthm statp-of-exec-rem
  (b* ((new-stat (exec-rem rd rs1 rs2 stat feat)))
    (statp new-stat))
  :rule-classes :rewrite)

Theorem: stat-validp-of-exec-rem

(defthm stat-validp-of-exec-rem
  (implies (stat-validp stat feat)
           (b* ((?new-stat (exec-rem rd rs1 rs2 stat feat)))
             (stat-validp new-stat feat))))

Theorem: exec-rem-of-ubyte5-fix-rd

(defthm exec-rem-of-ubyte5-fix-rd
  (equal (exec-rem (ubyte5-fix rd)
                   rs1 rs2 stat feat)
         (exec-rem rd rs1 rs2 stat feat)))

Theorem: exec-rem-ubyte5-equiv-congruence-on-rd

(defthm exec-rem-ubyte5-equiv-congruence-on-rd
  (implies (ubyte5-equiv rd rd-equiv)
           (equal (exec-rem rd rs1 rs2 stat feat)
                  (exec-rem rd-equiv rs1 rs2 stat feat)))
  :rule-classes :congruence)

Theorem: exec-rem-of-ubyte5-fix-rs1

(defthm exec-rem-of-ubyte5-fix-rs1
  (equal (exec-rem rd (ubyte5-fix rs1)
                   rs2 stat feat)
         (exec-rem rd rs1 rs2 stat feat)))

Theorem: exec-rem-ubyte5-equiv-congruence-on-rs1

(defthm exec-rem-ubyte5-equiv-congruence-on-rs1
  (implies (ubyte5-equiv rs1 rs1-equiv)
           (equal (exec-rem rd rs1 rs2 stat feat)
                  (exec-rem rd rs1-equiv rs2 stat feat)))
  :rule-classes :congruence)

Theorem: exec-rem-of-ubyte5-fix-rs2

(defthm exec-rem-of-ubyte5-fix-rs2
  (equal (exec-rem rd rs1 (ubyte5-fix rs2)
                   stat feat)
         (exec-rem rd rs1 rs2 stat feat)))

Theorem: exec-rem-ubyte5-equiv-congruence-on-rs2

(defthm exec-rem-ubyte5-equiv-congruence-on-rs2
  (implies (ubyte5-equiv rs2 rs2-equiv)
           (equal (exec-rem rd rs1 rs2 stat feat)
                  (exec-rem rd rs1 rs2-equiv stat feat)))
  :rule-classes :congruence)

Theorem: exec-rem-of-stat-fix-stat

(defthm exec-rem-of-stat-fix-stat
  (equal (exec-rem rd rs1 rs2 (stat-fix stat)
                   feat)
         (exec-rem rd rs1 rs2 stat feat)))

Theorem: exec-rem-stat-equiv-congruence-on-stat

(defthm exec-rem-stat-equiv-congruence-on-stat
  (implies (stat-equiv stat stat-equiv)
           (equal (exec-rem rd rs1 rs2 stat feat)
                  (exec-rem rd rs1 rs2 stat-equiv feat)))
  :rule-classes :congruence)

Theorem: exec-rem-of-feat-fix-feat

(defthm exec-rem-of-feat-fix-feat
  (equal (exec-rem rd rs1 rs2 stat (feat-fix feat))
         (exec-rem rd rs1 rs2 stat feat)))

Theorem: exec-rem-feat-equiv-congruence-on-feat

(defthm exec-rem-feat-equiv-congruence-on-feat
  (implies (feat-equiv feat feat-equiv)
           (equal (exec-rem rd rs1 rs2 stat feat)
                  (exec-rem rd rs1 rs2 stat feat-equiv)))
  :rule-classes :congruence)