155 lines
6.4 KiB
Plaintext
155 lines
6.4 KiB
Plaintext
theory StarForth_Loop4_Pipeline
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imports StarForth_Q48_16 StarForth_Base
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begin
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(* =========================================================================
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StarForth_Loop4_Pipeline — Prefetch / Pipelining Metrics (Physics Loop #4)
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Mirrors: src/physics_pipelining_metrics.c
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include/physics_pipelining_metrics.h
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The pipelining subsystem maintains word-to-word transition probabilities
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(per-word wt_transition_heat arrays) and global pipeline_metrics. On
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every word execution, the most-likely next word is pre-fetched from the
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dictionary; the prefetch is scored as a hit or miss.
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This theory proves:
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• Prefetch accuracy (hits / (hits + misses)) ∈ [0, 1] as a Q48.16 value.
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• Transition recording increments the correct counter and preserves totals.
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• Global pipeline_metrics totals are consistent with per-step deltas.
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======================================================================== *)
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(* =========================================================================
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Section 1: Pipeline accuracy
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======================================================================== *)
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definition pm_accuracy_q48 :: "pipeline_metrics_state \<Rightarrow> nat" where
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"pm_accuracy_q48 pm =
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(if pm_prefetch_attempts pm = 0
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then 0
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else (pm_prefetch_hits pm * Q48_SCALE) div pm_prefetch_attempts pm)"
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lemma pm_accuracy_zero_attempts [simp]:
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"pm_accuracy_q48 (pm\<lparr>pm_prefetch_attempts := 0\<rparr>) = 0"
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by (simp add: pm_accuracy_q48_def)
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lemma pm_accuracy_upper_bound:
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assumes "pm_prefetch_hits pm \<le> pm_prefetch_attempts pm"
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shows "pm_accuracy_q48 pm \<le> Q48_ONE"
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proof (cases "pm_prefetch_attempts pm = 0")
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case True thus ?thesis by (simp add: pm_accuracy_q48_def Q48_ONE_def)
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next
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case False
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have "pm_prefetch_hits pm * Q48_SCALE \<le> pm_prefetch_attempts pm * Q48_SCALE"
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using assms by (simp add: mult_le_mono1)
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hence "(pm_prefetch_hits pm * Q48_SCALE) div pm_prefetch_attempts pm \<le> Q48_SCALE"
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using False by (simp add: div_le_iff_le_mult)
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thus ?thesis by (simp add: pm_accuracy_q48_def Q48_ONE_def False)
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qed
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lemma pm_accuracy_non_negative:
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"pm_accuracy_q48 pm \<ge> 0"
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by simp
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(* =========================================================================
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Section 2: Pipeline metrics well-formedness
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======================================================================== *)
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definition pm_wf :: "pipeline_metrics_state \<Rightarrow> bool" where
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"pm_wf pm \<longleftrightarrow>
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pm_prefetch_hits pm \<le> pm_prefetch_attempts pm \<and>
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(pm_prefetch_attempts pm = 0 \<longrightarrow>
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pm_last_accuracy_num pm = 0) \<and>
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(pm_prefetch_attempts pm > 0 \<longrightarrow>
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pm_last_accuracy_den pm > 0)"
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lemma pm_wf_accuracy_range:
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assumes "pm_wf pm"
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shows "pm_accuracy_q48 pm \<le> Q48_ONE"
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using assms by (simp add: pm_accuracy_upper_bound pm_wf_def)
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(* =========================================================================
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Section 3: Recording a prefetch hit or miss
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======================================================================== *)
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definition pm_record_hit :: "pipeline_metrics_state \<Rightarrow> pipeline_metrics_state" where
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"pm_record_hit pm =
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pm\<lparr>pm_prefetch_attempts := pm_prefetch_attempts pm + 1,
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pm_prefetch_hits := pm_prefetch_hits pm + 1\<rparr>"
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definition pm_record_miss :: "pipeline_metrics_state \<Rightarrow> pipeline_metrics_state" where
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"pm_record_miss pm =
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pm\<lparr>pm_prefetch_attempts := pm_prefetch_attempts pm + 1\<rparr>"
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lemma pm_record_hit_attempts:
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"pm_prefetch_attempts (pm_record_hit pm) = pm_prefetch_attempts pm + 1"
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by (simp add: pm_record_hit_def)
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lemma pm_record_hit_hits:
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"pm_prefetch_hits (pm_record_hit pm) = pm_prefetch_hits pm + 1"
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by (simp add: pm_record_hit_def)
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lemma pm_record_miss_attempts:
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"pm_prefetch_attempts (pm_record_miss pm) = pm_prefetch_attempts pm + 1"
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by (simp add: pm_record_miss_def)
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lemma pm_record_miss_hits_unchanged:
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"pm_prefetch_hits (pm_record_miss pm) = pm_prefetch_hits pm"
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by (simp add: pm_record_miss_def)
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lemma pm_record_hit_preserves_wf:
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assumes "pm_wf pm"
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shows "pm_wf (pm_record_hit pm)"
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using assms by (simp add: pm_wf_def pm_record_hit_def)
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lemma pm_record_miss_preserves_wf:
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assumes "pm_wf pm"
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shows "pm_wf (pm_record_miss pm)"
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using assms by (simp add: pm_wf_def pm_record_miss_def)
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(* After a hit, hits ≤ attempts still holds. *)
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lemma pm_record_hit_hits_le_attempts:
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assumes "pm_prefetch_hits pm \<le> pm_prefetch_attempts pm"
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shows "pm_prefetch_hits (pm_record_hit pm) \<le> pm_prefetch_attempts (pm_record_hit pm)"
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using assms by (simp add: pm_record_hit_def)
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(* After a miss, hits ≤ attempts still holds. *)
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lemma pm_record_miss_hits_le_attempts:
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assumes "pm_prefetch_hits pm \<le> pm_prefetch_attempts pm"
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shows "pm_prefetch_hits (pm_record_miss pm) \<le> pm_prefetch_attempts (pm_record_miss pm)"
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using assms by (simp add: pm_record_miss_def)
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(* =========================================================================
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Section 4: Word transition metrics (per-word hot-path transition table)
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======================================================================== *)
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(* Record a transition from the current context to word_id next. *)
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definition wt_record_transition :: "nat \<Rightarrow> word_transition_metrics \<Rightarrow> word_transition_metrics" where
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"wt_record_transition next wt =
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wt\<lparr>wt_transition_heat :=
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(wt_transition_heat wt)(next := wt_transition_heat wt next + 1),
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wt_total_transitions := wt_total_transitions wt + 1\<rparr>"
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lemma wt_transition_heat_updated:
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"wt_transition_heat (wt_record_transition n wt) n = wt_transition_heat wt n + 1"
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by (simp add: wt_record_transition_def)
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lemma wt_transition_heat_other_unchanged:
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assumes "m \<noteq> n"
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shows "wt_transition_heat (wt_record_transition n wt) m = wt_transition_heat wt m"
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by (simp add: wt_record_transition_def assms)
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lemma wt_total_transitions_increases:
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"wt_total_transitions (wt_record_transition n wt) = wt_total_transitions wt + 1"
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by (simp add: wt_record_transition_def)
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(* =========================================================================
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Section 5: Pipeline fields are preserved by pure data-stack word execution
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======================================================================== *)
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lemma ds_word_preserves_pipeline:
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"pipeline_metrics (vm\<lparr>data_stack := xs\<rparr>) = pipeline_metrics vm"
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by simp
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end
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