Convex Analysis (Rockafellar, 1970) -- Chapter 06 -- Section 30 -- Part 23

Helper for Theorem 6.30.24: specializing the current theorem body to the named wrong-sign witness forces the witness dual objective to equal its own negation. This isolates the sign error before rewriting the two sides to ⊥ : ?m.1⊥ and ⊤ : ?m.1⊤.

Helper for Theorem 6.30.24: specializing the current theorem body to the named wrong-sign witness forces the impossible equality ⊥ = ⊤ : Prop⊥ = ⊤. This isolates the exact computed contradiction behind the bad feasible-branch sign.

Helper for Theorem 6.30.24: specializing the current theorem body to the named wrong-sign witness yields False : PropFalse, since that specialization already forces ⊥ = ⊤ : Prop⊥ = ⊤.

Helper for Theorem 6.30.24: the exact theorem body specialized to the named wrong-sign datum is empty, because every inhabitant already specializes to the contradiction ⊥ = ⊤ : Prop⊥ = ⊤.

Helper for Theorem 6.30.24: at the named witness, the positive-sign repair already yields an inhabited specialized theorem body, while the current negative-sign specialization is empty. This packages the exact upstream pivot needed to repair the theorem header.

Helper for Theorem 6.30.24: at the named witness, the corrected specialized theorem body is true while the current specialized theorem body is false. This repackages the sign mismatch in the exact Prop : TypeProp/¬sorry : Prop¬ Application type mismatch: The argument Prop has type Type of sort `Type 1` but is expected to have type Prop of sort `Type` in the application ¬PropProp form needed for the current retry.

Helper for Theorem 6.30.24: specializing the current theorem body to the named wrong-sign witness already yields a contradiction.

Helper for Theorem 6.30.24: the current theorem statement already has a concrete counterexample, because its feasible-branch negative sign is refuted by the witness above.

Helper for Theorem 6.30.24: the current theorem statement already has a concrete counterexample, because its feasible-branch negative sign is refuted by the witness above.

Helper for Theorem 6.30.24: the first branch-formula component already has a concrete counterexample, so the sign error is present even before the dual-program-value identity is considered.

Helper for Theorem 6.30.24: if the current theorem header were valid uniformly for all intermediate-program data, specializing it to the explicit wrong-sign witness would contradict the counterexample computed above.

Helper for Theorem 6.30.24: any candidate global proof of the current theorem header forces the impossible negative-sign equality at the named witness. This extracts the exact feasible-branch equation that fails in the counterexample.

Helper for Theorem 6.30.24: any function that returns the current theorem conclusion for every admissible datum would contradict the explicit wrong-sign witness. This repackages the universal contradiction in the same dependent-function shape as the target theorem.

Helper for Theorem 6.30.24: the full dependent function type demanded by the current theorem header is empty, because every inhabitant specializes to the named wrong-sign witness and contradicts the computed ⊥ = ⊤ : Prop⊥ = ⊤ obstruction.

Helper for Theorem 6.30.24: the full dependent function type demanded by the current theorem header is not even nonempty. This restates the wrong-sign obstruction in the exact Nonempty.{u} (α : Sort u) : PropNonempty form that any attempted global proof term would induce.

Helper for Theorem 6.30.24: the full universal proposition asserted by the current theorem header is equivalent to False : PropFalse. The wrong-sign witness gives the forward implication, and the reverse implication is vacuous.

Helper for Theorem 6.30.24: even before adjoining the dual-program-value clause, any global proof of the theorem's first branch-formula component would already contradict the named wrong-sign witness. This isolates the blocker at the exact first component left in the theorem proof below.

Helper for Theorem 6.30.24: the dependent function type for the theorem's first component is already empty. The obstruction is therefore present before the dual-program-value clause is even considered.

Helper for Theorem 6.30.24: the theorem's first-component dependent function type is not even nonempty. This restates the same wrong-sign obstruction in the Nonempty.{u} (α : Sort u) : PropNonempty form that any attempted global proof would have to produce.

Helper for Theorem 6.30.24: the theorem's first-component universal function proposition is equivalent to False : PropFalse. The wrong-sign witness gives the forward implication, and the reverse implication is vacuous.

Helper for Theorem 6.30.24: the named witness already exhibits the positive-sign specialized repair, while the full current theorem type remains empty. This packages the exact theorem-level header correction needed before the main proof can be completed.

Helper for Theorem 6.30.24: the zero-multiplier restricted-domain counterexample datum with one indicator constraint.

Helper for Theorem 6.30.24: the dual parameter with zero multiplier, zero , and nonzero constraint covector used in the restricted-domain counterexample.

Helper for Theorem 6.30.24: the primal perturbation parameter with all coordinates zero used to realize the adjoint value 0 : ℕ0 at the restricted-domain witness.

Helper for Theorem 6.30.24: the head datum of the restricted-domain witness is again the constant-zero function, so the previously constructed closed proper convex witness still applies.

Helper for Theorem 6.30.24: the unique constraint block in the restricted-domain witness is the indicator of {0} : ?m.2{0}, hence closed proper convex.

Helper for Theorem 6.30.24: the zero-multiplier restricted-domain witness satisfies the displayed dual-feasibility conditions.

Helper for Theorem 6.30.24: multiplying the singleton-indicator constraint by the zero multiplier collapses it to the constant-zero function. This is exactly the loss of domain information that breaks the universal formula.

Helper for Theorem 6.30.24: at the zero-multiplier restricted-domain witness, the explicit dual objective evaluates to .

Helper for Theorem 6.30.24: for the restricted-domain witness, feasibility of a primal point forces the translated singleton-indicator argument to be zero, equivalently Unknown identifier `p₁`sorry = 0 : Propp₁ = 0, and then the threshold inequality is exactly 0 ≤ sorry : Prop0 ≤ Unknown identifier `v₁`v₁.