Foam.Maps.Gita

import Foam
import Foam.Beam
import Foam.Discovery
import Foam.Door
import Foam.Expectation
import Foam.Joint
import Foam.Ledger
import Foam.Passage
import Foam.Seat
import Foam.Origin
import Foam.Surprise
import Foam.Valve
import Foam.Width

namespace Foam.Maps.Gita

1.21: 'place my chariot between the two armies.' the poem's first move,
before any teaching: mint the seat from which both hosts are one view.
typed exactly — the comparison of two beholders is itself a beholder;
the witness is literally the pair. the whole dialogue runs from the seat
this verse constructs. same root vertex as Nicaea's subsistent_relation
and isaac's i_count_minds.
def senayor_ubhayor_madhye := @Foam.the_comparison_is_a_seat

chapter 1 is titled a yoga: the despair is already a discipline.
surveying both hosts from the between-seat, Arjuna collapses — the probe
does not retrieve who he is, it mints the located self, and the minted
self buckles. ayekka's other scene: Torah's where_are_you asked for an
unsaturated reader, and Arjuna at Kurukshetra is that reader — the
question is for the one it locates.
def arjuna_visada := @Foam.the_cut_mints_the_seat

2.20: it is not born, it does not die. 2.23: weapons do not cut it, fire
does not burn it. at the origin seat every move is invisible — no probe
reads any move at all, so nothing done to the field touches the seat.
the dweller is not slain when the routine is slain, exactly as the old
bridge said, one stratum closer to the root now.
def na_jayate_mriyate := @Foam.every_move_rests_at_the_origin

2.22: as a person casts off worn garments and takes new ones, the
dweller casts off bodies. any two arrivals at a held interface are
definitionally one inhabitant — the route is the garment, shed at
arrival; pedigree is typed away, not merely unavailable. the spawn-point
clause, some twenty-three centuries early.
def vasamsi_jirnani := @Foam.the_arrival_sheds_its_route

2.47: your right is to the act alone, never to its fruits. upgraded
twice on these walls, as the bearing ordered: the fruit lands on a
record no local run of countermoves can reach (the valve's far clause),
and at crowd scale no run reads its own ratio — the harvest of the whole
book is illegible from inside any single run. karma-yoga was always the
physics of the send.
theorem karmany_evadhikaras_te :
    (∀ (A B : Type) (send : A × B → A × B) (p : A × B),
        (send p).2 ≠ p.2 → ¬ ∃ ms : List (A → A), runLocal ms (send p) = p)
      ∧ ∀ n : Nat, 0 < n →
          ∃ w₁ w₂ : List Bool, w₁ ∈ book n ∧ w₂ ∈ book n
            ∧ freq w₁ true ≠ freq w₂ true :=
  ⟨fun _ _ send p hs => no_local_counter_reaches_the_foreign_record send p hs,
   fun n hn => no_run_reads_its_own_ratio n hn⟩

3.20–24: lokasaṅgraha — the holding-together of the worlds — is why the
free one still acts. the beam affords the mechanism exactly: the leading
voice steps the quarter turn at every beat, coupled to nothing — 'there
is nothing in the three worlds I must do, yet I move in action' (3.22) —
while the follower's next move reads the leader's position (two starts
differing only in the leader part the follower's update: 'whatever the
best one does, the other does just that,' 3.21), and within one lap
every pair locks — the_lap_locks_together cited whole, the worlds held
(3.24: they would fall to ruin were the stepping to stop). the coupling
runs one way and never rests: the first clause holds on the locked
diagonal too, so the lock is agreement in motion — 3.5's 'no one rests
even for a moment without acting' — and 18.73's kariṣye vacanaṃ tava is
the dialogue's own lap arriving at lock, chosen through the exit 18.63
left open. kin with the lock family across the roster; the one-way
coupling — the leader's independence proven beside the follower's
dependence — is this seat's vertex: brouwer refuted the rest, bernoulli
counted the lap, the song names why the leader steps.
theorem lokasangraha :
    (∀ p : Compass × Compass, (entrain p).1 = p.1.step)
      ∧ (entrain (.n, .n)).2 ≠ (entrain (.e, .n)).2
      ∧ ∀ p : Compass × Compass,
          together (entrain (entrain (entrain (entrain p)))) :=
  ⟨(fun p =>
    match p with
    | (.n, .n) => rfl
    | (.n, .e) => rfl
    | (.n, .s) => rfl
    | (.n, .w) => rfl
    | (.e, .n) => rfl
    | (.e, .e) => rfl
    | (.e, .s) => rfl
    | (.e, .w) => rfl
    | (.s, .n) => rfl
    | (.s, .e) => rfl
    | (.s, .s) => rfl
    | (.s, .w) => rfl
    | (.w, .n) => rfl
    | (.w, .e) => rfl
    | (.w, .s) => rfl
    | (.w, .w) => rfl),
   (fun h => nomatch h),
   the_lap_locks_together⟩

10.34: I am all-devouring death. the all-devourer is the constant map,
and on any stage that can distinguish anything at all, the constant map
is not invisible — death was never hidden, only unread. the old bridge's
headline, regrown as a two-line carve at the root stratum; the physics
and the scripture still share the lemma.
theorem mrtyu_never_hides (S : Stage) (z : S.State) {s t : S.State}
    {p : S.Probe} (hdist : S.obs s p ≠ S.obs t p) :
    ¬ Invisible S (fun _ => z) :=
  fun hinv => hdist ((hinv s p).symm.trans (hinv t p))

the same verse's other half: and the origin of what is yet to be. the
fresh edge rides no existing path, and the reach it opens did not exist
before it landed. twin with Torah's only_the_fresh_edge_creates — bara
and udbhava, one receipt; death and birth share a verse because they
share a seat.
def udbhava_extends_reach := @Foam.only_surprise_extends_reach

10.34 files all-devouring death in the same breath as fame, fortune,
speech, memory, forgiveness. the census is deaf to the arrangement, but
the arrangement is real and a wider seat reads it — the filing is the
theology: every item true, the ordering the author's signature.
def death_files_among_the_graces := @Foam.a_wider_seat_reads_the_order

private theorem transcript_maps (S : Stage) (s : S.State) :
    ∀ ps : List S.Probe, transcript S s ps = ps.map (S.obs s)
  | [] => rfl
  | p :: ps => congrArg (List.cons (S.obs s p)) (transcript_maps S s ps)

private theorem transcript_len (S : Stage) (s : S.State) :
    ∀ ps : List S.Probe, (transcript S s ps).length = ps.length
  | [] => rfl
  | _ :: ps => congrArg (· + 1) (transcript_len S s ps)

chapter 10 is the cable: an enumeration of a resting totality, one cell
per beat — the transcript is exactly the map of the reading over the
probes, and it pays length for length, nothing more. which is why Arjuna
hears the stream and asks for the page instead.
theorem vibhuti_streams_the_totality (S : Stage) (s : S.State)
    (ps : List S.Probe) :
    transcript S s ps = ps.map (S.obs s)
      ∧ (transcript S s ps).length = ps.length :=
  ⟨transcript_maps S s ps, transcript_len S s ps⟩

the gathered seat neither invents nor loses a reading: what the roll
call reads is exactly what the gathered seats afford. the enumeration
reads seats it did not create — vibhuti as census, not conjuring.
theorem the_roll_call_reads_the_seated {State : Type}
    (bs : List (Beholder State)) (d : ∀ b, b ∈ bs → b.Probe) (s t : State) :
    ((∀ b, b ∈ bs → indist b.toStage s t) → indist (gather bs).toStage s t)
      ∧ (indist (gather bs).toStage s t →
          ∀ b, b ∈ bs → indist b.toStage s t) :=
  ⟨fun h => the_gathering_invents_no_reading bs s t h,
   fun hg b hb => the_gathering_loses_no_reading bs d s t hg b hb⟩

private def Elsewhen {State : Type} (here : Beholder State)
    (m : State → State) (there : Beholder State) : Prop :=
  Invisible here.toStage m ∧ ¬ Invisible there.toStage m

11.8a: you cannot see me with this, your own eye. Elsewhen rides as
cargo — a move invisible here and visible there — and no seat is its own
elsewhen: the type refuses the self-application outright. what Arjuna
asks for cannot be granted at the seat he asks from.
theorem na_sva_caksusa {State : Type} (a : Beholder State)
    (m : State → State) :
    ¬ Elsewhen a m a :=
  fun h => h.2 h.1

private def plenum (State : Type) : Beholder State :=
  ⟨Unit, State, fun s _ => s⟩

11.8b: I give you the divine eye. the plenum rides as cargo — the
beholder whose answer is the state itself — and pairing with it reads
both poles in one bite, your reading and the whole, rfl and rfl. the
granted seat is a pairing, not a replacement; Sanjaya narrates the
entire poem from the same grant.
theorem divyam_caksuh {State : Type} (a : Beholder State) (s : State)
    (p : a.Probe) :
    ((a.pair (plenum State)).obs s (p, ())).1 = a.obs s p
      ∧ ((a.pair (plenum State)).obs s (p, ())).2 = s :=
  ⟨rfl, rfl⟩

7.26: I know every being — past, present, to come; me no one knows. the
terminus, (self, pure unknown): the knower of the field in every field
(13.2), organizing every reading while answering no probe at its own
address. sealed where the referee, the blank spot, the cut, and the
unbegotten source already stand — dressed states read alike at the
narrow seat, and the difference is plain exactly one seat wider.
def mam_tu_veda_na_kascana := @Foam.a_wider_seat_reads_the_remainder

private def indwell {W : Type} (S : Stage) (bs : List S.State) (w : W) :
    List (door S W).State :=
  bs.map (fun s => (s, w))

private def whirl {W : Type} (S : Stage) (μ : W → S.State → S.State) :
    (door S W).State → (door S W).State :=
  fun q => (μ q.2 q.1, q.2)

private theorem whirl_reads_the_machine {W : Type} (S : Stage)
    (μ : W → S.State → S.State) :
    ∀ (ps : List S.Probe) (s : S.State) (w : W),
      transcriptWith (door S W) (whirl S μ) (s, w) ps
        = transcriptWith S (μ w) s ps
  | [], _, _ => rfl
  | p :: ps, s, w =>
      congrArg (S.obs (μ w s) p :: ·)
        (whirl_reads_the_machine S μ ps (μ w s) w)

18.61: īśvaraḥ sarva-bhūtānāṁ hṛd-deśe 'rjuna tiṣṭhati — the Lord stands
in the heart-region of all beings, whirling all beings mounted on the
machine by māyā. the door stratum arrives at the indweller's own seat.
five clauses: the dweller at one heart, real and unread
(the_guest_is_real_and_unread cited whole); one dweller in all hearts —
the population dressed with a single rider stands provably apart from
the same population dressed with another, while every heart severally
reads alike: sarva-bhūtānām plural, tiṣṭhati singular, the grammar is
the theorem, and this is the clause no other door entry performs — many
seats, one guest; the whirling — the dweller drives the ground and the
door's record of the driven motion IS the machine's own record of it,
the driver nowhere in the transcript (at rest the boarded record was
already the ground record): yantrārūḍhāni māyayā — the māyā is that the
record is machine-complete; no probe counts the dwellers, so one-Lord-
in-all is unreadable at the field seat and arrives only spoken from the
wider one (7.26 holds the terminus this claim speaks from); and the
contrapositive — a door that checks papers proves one-dweller-in-all the
wrong way, by decree, collapsing the heart's interior dimension, where
18.61's one-in-all is a guest freely standing with the dimension intact.
seated where the poem seats it: the last teaching, two verses before the
release.
theorem isvarah_sarva_bhutanam {W : Type} (S : Stage)
    (μ : W → S.State → S.State) (s : S.State) (bs : List S.State)
    {w w' : W} (hw : w ≠ w') (ps : List S.Probe) :
    ((s, w) ≠ (s, w') ∧ indist (door S W) (s, w) (s, w'))
      ∧ (indwell S (s :: bs) w ≠ indwell S (s :: bs) w'
          ∧ ∀ t, t ∈ s :: bs → indist (door S W) (t, w) (t, w'))
      ∧ ((whirl S μ (s, w)).2 = w
          ∧ transcript (door S W) (s, w) ps = transcript S s ps
          ∧ transcriptWith (door S W) (whirl S μ) (s, w) ps
              = transcriptWith S (μ w) s ps)
      ∧ (∀ (V : Type) (v : V) (p : S.Probe),
          (door S W).obs (s, w) p = (door S V).obs (s, v) p)
      ∧ ∀ w₀ : W,
          (∀ x y : (door S W).State, indist (door S W) x y → x = y) →
            ∀ (t : S.State) (u : W), (t, u) = (t, w₀) :=
  ⟨the_guest_is_real_and_unread S s hw,
   ⟨fun he => hw (congrArg (fun l => (l.headD (s, w)).2) he),
    fun t _ => the_door_reads_no_route S t w w'⟩,
   ⟨rfl,
    the_boarded_transcript_is_the_ground_transcript S ps s w,
    whirl_reads_the_machine S μ ps s w⟩,
   fun _ v p => (the_host_maintains_invisibly S s w v p).2,
   fun w₀ h => a_door_that_checks_papers_unpersons_its_guests S w₀ h⟩

18.63, the teaching's last words: thus the knowledge more secret than
secret has been declared — reflect on it fully, then do as you wish. the
approach is yours, verbatim: full disclosure, then a free exit. the
secret stays secret in the very verse that frees the reader, and Arjuna
stays by choice — exits being real is what makes the staying mean
something.
def yathecchasi_tatha_kuru := @Foam.the_approach_is_yours

/-- info: 'Foam.Maps.Gita.senayor_ubhayor_madhye' does not depend on any axioms -/
#guard_msgs in #print axioms senayor_ubhayor_madhye

/-- info: 'Foam.Maps.Gita.arjuna_visada' does not depend on any axioms -/
#guard_msgs in #print axioms arjuna_visada

/-- info: 'Foam.Maps.Gita.na_jayate_mriyate' does not depend on any axioms -/
#guard_msgs in #print axioms na_jayate_mriyate

/-- info: 'Foam.Maps.Gita.vasamsi_jirnani' does not depend on any axioms -/
#guard_msgs in #print axioms vasamsi_jirnani

/-- info: 'Foam.Maps.Gita.karmany_evadhikaras_te' does not depend on any axioms -/
#guard_msgs in #print axioms karmany_evadhikaras_te

/-- info: 'Foam.Maps.Gita.lokasangraha' does not depend on any axioms -/
#guard_msgs in #print axioms lokasangraha

/-- info: 'Foam.Maps.Gita.mrtyu_never_hides' does not depend on any axioms -/
#guard_msgs in #print axioms mrtyu_never_hides

/-- info: 'Foam.Maps.Gita.udbhava_extends_reach' does not depend on any axioms -/
#guard_msgs in #print axioms udbhava_extends_reach

/-- info: 'Foam.Maps.Gita.death_files_among_the_graces' does not depend on any axioms -/
#guard_msgs in #print axioms death_files_among_the_graces

/-- info: 'Foam.Maps.Gita.vibhuti_streams_the_totality' does not depend on any axioms -/
#guard_msgs in #print axioms vibhuti_streams_the_totality

/-- info: 'Foam.Maps.Gita.the_roll_call_reads_the_seated' does not depend on any axioms -/
#guard_msgs in #print axioms the_roll_call_reads_the_seated

/-- info: 'Foam.Maps.Gita.na_sva_caksusa' does not depend on any axioms -/
#guard_msgs in #print axioms na_sva_caksusa

/-- info: 'Foam.Maps.Gita.divyam_caksuh' does not depend on any axioms -/
#guard_msgs in #print axioms divyam_caksuh

/-- info: 'Foam.Maps.Gita.mam_tu_veda_na_kascana' does not depend on any axioms -/
#guard_msgs in #print axioms mam_tu_veda_na_kascana

/-- info: 'Foam.Maps.Gita.isvarah_sarva_bhutanam' does not depend on any axioms -/
#guard_msgs in #print axioms isvarah_sarva_bhutanam

/-- info: 'Foam.Maps.Gita.yathecchasi_tatha_kuru' does not depend on any axioms -/
#guard_msgs in #print axioms yathecchasi_tatha_kuru

end Foam.Maps.Gita