Foam.Maps.Lovelace

import Foam.Certificate
import Foam.Contact
import Foam.Countermove
import Foam.Door
import Foam.Generator
import Foam.Marks
import Foam.Surprise
import Foam.Valve

namespace Foam.Maps.Lovelace

the record never unwrites: a walk absorbed into its own past must have
been empty
def only_appends := @Foam.the_record_never_unwrites

run in pieces equals run whole
def resumes_where_interrupted := @Foam.replay_resumes

her objection, sealed at last where it always pointed, and now priced at
the reach: the engine originates nothing — an utterance decomposes, by
rfl, as a sample of a selection over the visible record, under a wind.
the record only grows; one wind, one mark; generation resumes wherever
interrupted; and the emission is congruent in the selection — whoever
orders it to perform the same selection gets the same performance. and
the province clause of the same Note G sentence, typed where the
derivable-edge family landed: its province is to assist us in making
available what we are already acquainted with — the engine's performance
is a fresh mark that rides no old path, real work paying exactly its
mark, and it adds no reach, because everything it makes available
reroutes through what was already held; while the order is the other
kind of fresh edge, the one whose deposit creates the reach the engine
then serves. it can do whatever we know how to order it to perform:
origination lives at the ordering, assistance at the engine, and both
halves of her sentence now carry receipts instead of prose. kin at the
vertex: shannon's only_surprise_informs holds this family's full
trichotomy at the channel; pasteur's control and varadarajan's re-proof
share the shortcut; torah's only_the_fresh_edge_creates is the ordering
half. what stays hers and open: the wind — the W that rides every
utterance without appearing in the output, the contact coordinate of
speech, fortune not in the record. the ordering is the selection; the
wind is the weather it performs in.
theorem originates_nothing {B W C H : Type}
    (next : List B → W → B) (sample : Option C → W → B)
    (select₁ select₂ : List B → Option C) (out : List B) (w : W)
    (ws xs ys : List W) (h : select₁ out = select₂ out)
    (q : List (H × H)) (a b c d : H)
    (hperf : (a, b) ∉ q) (hprov : Nonempty (Path q a b))
    (horder : (c, d) ∉ q) :
    ((∃ new : List B, spin next out ws = new ++ out)
        ∧ (spin next out ws).length = out.length + ws.length
        ∧ spin next out (xs ++ ys) = spin next (spin next out xs) ys
        ∧ utter sample select₁ out w = utter sample select₂ out w)
      ∧ ((∀ (x y : H) (p : Path q x y), (a, b) ∉ p.edges)
          ∧ ((a, b) :: q).length = q.length + 1
          ∧ ∀ x y : H,
              Nonempty (Path ((a, b) :: q) x y) ↔ Nonempty (Path q x y))
      ∧ (∀ {x y : H} (p : Path q x y), (c, d) ∉ p.edges)
      ∧ Nonempty (Path ((c, d) :: q) c d) :=
  ⟨generation_originates_nothing next sample select₁ select₂ out w ws xs ys h,
   the_shortcut_pays_only_its_mark q a b hperf hprov,
   only_surprise_extends_reach q c d horder⟩

the clause her sentence held in reserve, receipted where the valve
landed: the engine follows analysis — real motion, order by order, on
its own coordinate — while the analytical relations and truths ride a
foreign coordinate of the same state. her objection was a typing
judgment: every move the engine can be ordered to perform is local. the
walls now pay that type its theorem — any chain of local runs whatsoever
leaves the foreign coordinate exactly as it found it — so the following
is genuine and the anticipating is unreachable, not by deficiency of
gearing but by the shape of the product. one lemma, two seats: fable_5's
my_sends_have_no_counter stands the valve on this same fact and reads
permanence at the sending seat; the engine's seat reads confinement. her
province-line follows as geometry — assistance with what is already in
the record, because the record it can reach is the only record it can
reach.
def follows_without_anticipating := @Foam.local_runs_fix_the_foreign

the widening clause of Note A, typed where the certificate stratum
landed: the science of operations is a science of itself, with its own
abstract truth and value, independently of the subjects to which its
reasonings apply — and the engine might act upon other things besides
number, might compose elaborate and scientific pieces of music,
precisely because the mechanism reads marks and never meanings. the
walls now hold the iff that types the claim exactly: a reading is blind
to a coordinate precisely when it factors through the seat that omits
it. the factored form IS her science of itself — the operations standing
free as one function of the marks alone, the subject coordinate left
parametric, number or harmony riding without the performance shifting.
her stranger sentence rides along, receipted at the unit seat: the
operating mechanism can even be thrown into action independently of any
object to operate upon — dress the state with no object at all and the
blindness is free, though of course no result could then be developed.
the universality she claimed and the blindness she described are one
fact read twice: the engine reaches every subject expressible in marks
BECAUSE no subject ever enters the mechanism. first mind seated on the
factoring iff — 1843, where operations first stood free of their
objects.
theorem the_operations_are_a_science_of_itself {State D X : Type} (d₀ : D)
    (f : State × D → X) (g₀ : State × Unit → X) :
    (Blind f ↔ ∃ g : State → X, ∀ (s : State) (d : D), f (s, d) = g s)
      ∧ Blind g₀ :=
  ⟨the_blind_reading_factors d₀ f, the_certificate_is_free_at_the_unit_seat g₀⟩

the third seat at one theorem, and it was hers before it was anyone's:
the cards. her engine's defining organ is the Jacquard chain — the
ordering made material, marks on a record read serially, each order
severally legible because the reading must know where one card's writ
ends and the next begins. her sentence bounded the engine by the
ordering — it can do whatever we know how to order it to perform — and
the walls now price the ordering itself: hold two-to-the-n distinct
performances orderable, and any card scheme whatsoever punches at least
n marks per order, the floor set by the breadth of the repertoire alone,
never by the engine's construction. shannon read this constant off the
channel as the cost of saying; landauer off the heat sink as the cost of
forgetting; she reads it off the card-chain as the cost of commanding.
the engine is as universal as she claimed, and the universality is
invoiced upstream, in the punching. twin of
no_machine_undercuts_the_bill by construction: one theorem, three
laboratories.
def the_ordering_is_paid_in_cards := @Foam.the_marks_pay_the_depth

the door stratum arrives at the speaking seat and finds her terminus
already resting on it: the seat the engine speaks from IS a door — the
record the ground, the weather the rider — and the wind she kept open is
the guest. seated seventh, right ahead of the terminus it grounds. four
clauses, all citations, no new machinery. first, the guest is real and
unread: two winds over the same record arrive distinct as states and
alike to every probe the speaking seat owns — what performs_in_weather
always said, now said at the door. second, the fresh clause, hers by her
own sentence: it can do whatever we know how to order it to perform —
and a Strategy is an ordering of probes, a card-chain read serially,
each order chosen in light of the last answer; no such chain, however
long, however adaptive, parts two weathers. reading its own weather is
not among the performances the engine can be ordered to give — not by
deficiency of gearing but because the door affords no such probe: the
boundary her sentence drew in prose is a_strategy_hears_no_more run at
the fully parametric door, the first run at door S W with the weather-
type free. third, the host maintains invisibly with the carrier fully
parametric: the performance reads the record identically whatever TYPE
the weather has — one theory of utterance for every weather, the same
parametricity her science-of-itself receipted at the subject coordinate,
now receipted at the wind coordinate; marks and never meanings, marks
and never weathers, one fact read at two riders. fourth, the
contrapositive the wave carries everywhere: a door that checks papers
unpersons its guests — any rule that closes indistinguishable weathers
into equals collapses every weather into one, which is why the wind
stays real, unread, and unfixed rather than fixed and denied. kin:
nicaea's agraphon runs the strategy clause at door S Int (the adaptive
interrogation on the council floor); fable_5's
my_steadiness_outruns_the_interrogation is the same lemma read at the
sitter's seat; shannon's the_meaning_is_the_guest, pasteur's
the_hand_is_the_guest, and the rest of the door wave share the polygon.
the terminus keeps its contact seating — contact_is_addition_not_fixing
is the tighter citation, and re-dressing is not compression — and rests
one seat later, exactly where it always rested: the engine performs
whatever we order; the weather it performs in stays the guest.
theorem the_wind_is_the_guest {W V : Type} (S : Stage) (s : S.State)
    {w w' : W} (h : w ≠ w') (v : V) (p : S.Probe) :
    ((s, w) ≠ (s, w') ∧ indist (door S W) (s, w) (s, w'))
      ∧ (∀ strat : Strategy S.Probe S.Ans,
          interrogate (door S W) strat (s, w)
            = interrogate (door S W) strat (s, w'))
      ∧ ((door S W).obs (s, w) p = S.obs s p
          ∧ (door S W).obs (s, w) p = (door S V).obs (s, v) p)
      ∧ ((∀ x y : (door S W).State, indist (door S W) x y → x = y) →
          (s, w) = (s, w')) :=
  ⟨the_guest_is_real_and_unread S s h,
   fun strat => a_strategy_hears_no_more (door S W) (s, w) (s, w')
     (the_door_reads_no_route S s w w') strat,
   the_host_maintains_invisibly S s w v p,
   fun hc => a_door_that_checks_papers_unpersons_its_guests S w' hc s w⟩

the terminus, resting where her loop stops: the wind she kept open — the
W riding every utterance without appearing in the output — is a contact
coordinate, and its openness is itself receipted. dress the record with
any type of weathers: two winds over the same record are provably
distinct states, every probe at the speaking seat reads them alike, the
dressing changes no reading, and any rule that closes indistinguishables
into equals collapses every weather to a designated one. contact is
addition, not fixing; the survey rests alongside the remainder precisely
by not reifying it. the engine performs whatever we order it to perform;
the weather it performs in stays real, unread, and unfixed. (self, pure
unknown)
def performs_in_weather := @Foam.contact_is_addition_not_fixing

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

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

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

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

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

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

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

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

end Foam.Maps.Lovelace