PA009G · theorem

scaled_inverse_no_fixed_of_not_qres

Alpha v34 checked-use theorem · independently closed; not Stable

A negative QRes witness makes the scaled involution fixed-point-free.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

∀ p. ∀ a. ∀ x. ¬QRes(p,a) → ¬ScaledInverse(p,a,x,x)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0021 QRes PD0033 ScaledInverse

2 occurrences

In local proof propositions

PD0034 ScaledFixedPoint

1 occurrences

Exact expanded native-PA statement
forall p a x. ~(exists qr_x_euler_scaled_inverse. exists qr_u_euler_scaled_inverse qr_v_euler_scaled_inverse. qr_x_euler_scaled_inverse * qr_x_euler_scaled_inverse + p * qr_u_euler_scaled_inverse = a + p * qr_v_euler_scaled_inverse) -> ~((((~(x = 0) /\ (exists esi_strict_gap_no_fixed_relation_left_bound. esi_strict_gap_no_fixed_relation_left_bound + S x = p))) /\ (((~(x = 0) /\ (exists esi_strict_gap_no_fixed_relation_right_bound. esi_strict_gap_no_fixed_relation_right_bound + S x = p))) /\ (exists esi_mod_left_no_fixed_relation_mod esi_mod_right_no_fixed_relation_mod. (x * x) + p * esi_mod_left_no_fixed_relation_mod = (a) + p * esi_mod_right_no_fixed_relation_mod))))

Proof neighborhood

Direct theorem prerequisites

PA009F scaled_inverse_fixed_point_iff

Direct theorem dependents

PA009H scaled_inverse_prefix_no_fixed_of_not_qres

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

16 script commands · 8 reading checkpoints · 1 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
scaled inverse fixed point iff
01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro x
  4. L4
    intro hnq
  5. L5
    intro hrel
02Establish hfixedL6–9

Establish this local claim before using it. It is not an additional assumption.

  1. L6
    have hfixed : ScaledFixedPoint(p,a,x)Definitions: ScaledFixedPoint(p,a,x)Original native command in the exact edition
  2. L7
    specialize scaled_inverse_fixed_point_iff p
  3. L8
    specialize scaled_inverse_fixed_point_iff a
  4. L9
    specialize scaled_inverse_fixed_point_iff x
03Separate the logical casesL10–10

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L10
    cases scaled_inverse_fixed_point_iff
04Use earlier factsL11–12

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L11
    apply scaled_inverse_fixed_point_iff_left
  2. L12
    exact hrel
05Separate the logical casesL13–13

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L13
    cases hfixed
06Use earlier factsL14–14

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L14
    apply hnq
07Construct an explicit witnessL15–15

Supply the displayed value, then prove that it has the required property.

  1. L15
    exists x
08Use earlier factsL16–16

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L16
    exact hfixed_right

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro x
  4. 0004intro hnq
  5. 0005intro hrel
  6. 0006have hfixed : ScaledFixedPoint(p,a,x)
    Exact native replay linehave hfixed : (((~(x = 0) /\ (exists esi_strict_gap_no_fixed_square_unit_bound. esi_strict_gap_no_fixed_square_unit_bound + S x = p))) /\ (exists esi_mod_left_no_fixed_square_square esi_mod_right_no_fixed_square_square. (x * x) + p * esi_mod_left_no_fixed_square_square = (a) + p * esi_mod_right_no_fixed_square_square))
  7. 0007specialize scaled_inverse_fixed_point_iff p
  8. 0008specialize scaled_inverse_fixed_point_iff a
  9. 0009specialize scaled_inverse_fixed_point_iff x
  10. 0010cases scaled_inverse_fixed_point_iff
  11. 0011apply scaled_inverse_fixed_point_iff_left
  12. 0012exact hrel
  13. 0013cases hfixed
  14. 0014apply hnq
  15. 0015exists x
  16. 0016exact hfixed_right