## Formalized Name Kripke Soundness Theorem for Propositional Intuitionistic Logic ## Formalized Statement Let $(K,\le,\Vdash)$ be a propositional Kripke model with persistent atomic forcing and the standard forcing clauses for $\bot$, $\top$, $\land$, $\lor$, and $\to$. Let $\mathrm{IPL}$ be the natural-deduction proof system for propositional intuitionistic logic over that same formula grammar. If $\vdash_{\mathrm{IPL}} A$, then for every world $w \in K$, one has $w \Vdash A$. ## 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 Kripke Soundness Theorem for 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]