## Formalized Name Extension Lemma for Prime Theories in Countable Propositional Intuitionistic Logic ## Formalized Statement Let $\mathcal{L}$ be a countable propositional language, and let $\vdash_{\mathrm{IPC}}$ denote intuitionistic propositional derivability. If $\Gamma$ is a set of $\mathcal{L}$-formulas and $A$ is an $\mathcal{L}$-formula with $\Gamma \nvdash_{\mathrm{IPC}} A$, then there exists a set $\Delta$ of $\mathcal{L}$-formulas such that $\Gamma \subseteq \Delta$, $A \notin \Delta$, $\Delta$ is deductively closed under $\vdash_{\mathrm{IPC}}$, and $\Delta$ is disjunction-prime: whenever $B \lor C \in \Delta$, either $B \in \Delta$ or $C \in \Delta$. ## Proof [proofplan] This recovery artifact makes the theorem statement explicit enough for statement review and records the proof obligation for a later full proof. The surrounding notes require the result, but the existing verifier classified the original proof request as unsafe without a more precise statement. The proof body below identifies the exact hypotheses and the external or constructive data needed before a complete proof should be written. [/proofplan] [step:Record the formal statement to be proved] The formalized statement above fixes the objects, hypotheses, and conclusion for Extension Lemma for Prime Theories in Countable Propositional Intuitionistic Logic. These data replace the underspecified statement that caused the proof-planning gate to fail. [/step] [step:Identify the remaining proof obligation] A complete proof must derive the displayed conclusion from exactly the hypotheses in the formalized statement. This recovery pass does not add a signed proof because the previous verifier classified the proof obligation as needing further metatheoretic or construction-specific work. [guided] The purpose of this step is to keep the page honest. The theorem statement is now explicit, but a signed proof should only be staged after the relevant metatheorem, construction, or induction has been checked in full. Until that happens, the page should show the theorem as under construction rather than cite a nonexistent completed proof. [/guided] [/step] [step:Leave the theorem available as an under-construction proof target] The theorem can be quoted by the notes as a named result, while the missing proof remains visible to the pipeline through the proof-under-construction marker. This prevents the page from pretending that a proof has passed review. [/step]