To Shape append
CWSSStructure.toShape_append
Plain-language statement
The shape induced by appended CWSS data is the generic append of the component shapes.
Exact Lean statement
theorem toShape_append (D₁ : CWSSStructure pSpec₁) (D₂ : CWSSStructure pSpec₂) :
CWSSStructure.toShape (append D₁ D₂) =
(CWSSStructure.toShape D₁).append (CWSSStructure.toShape D₂)Formal artifact
Lean source
theorem toShape_append (D₁ : CWSSStructure pSpec₁) (D₂ : CWSSStructure pSpec₂) : CWSSStructure.toShape (append D₁ D₂) = (CWSSStructure.toShape D₁).append (CWSSStructure.toShape D₂) := by refine ChallengeTreeShape.ext rfl (heq_of_eq ?_) funext i challenges simp only [CWSSStructure.toShape, ChallengeTreeShape.append] split · rename_i i₁ heq obtain rfl : i = ChallengeIdx.inl i₁ := by have := (Equiv.symm_apply_eq ChallengeIdx.sumEquiv).mp heq simpa [ChallengeIdx.sumEquiv_apply] using this have hell : (append D₁ D₂).ell (ChallengeIdx.inl i₁) = D₁.ell i₁ := congrArg Subtype.val (append_coordIndex_inl D₁ D₂ i₁) have hk : (append D₁ D₂).k (ChallengeIdx.inl i₁) = D₁.k i₁ := congrArg Subtype.val (append_soundnessParam_inl D₁ D₂ i₁) have halpha : (append D₁ D₂).alphabet (ChallengeIdx.inl i₁) = D₁.alphabet i₁ := append_alphabet_inl D₁ D₂ i₁ unfold CWSSStructure.nodeOk congr 1 refine Function.hfunext (by rw [hell, hk]) (fun j j' hj => ?_) refine heq_equiv_apply (by simp [ProtocolSpec.append, ChallengeIdx.inl]) (by rw [append_coordIndex_inl, append_alphabet_inl]) (append_decompose_inl D₁ D₂ i₁) ?_ refine HEq.trans (heq_of_eq (congrArg challenges (Fin.ext ?_))) (cast_heq _ _).symm change j.val = j'.val exact (Fin.heq_ext_iff (by rw [hell, hk])).mp hj · rename_i i₂ heq obtain rfl : i = ChallengeIdx.inr i₂ := by have := (Equiv.symm_apply_eq ChallengeIdx.sumEquiv).mp heq simpa [ChallengeIdx.sumEquiv_apply] using this have hell : (append D₁ D₂).ell (ChallengeIdx.inr i₂) = D₂.ell i₂ := congrArg Subtype.val (append_coordIndex_inr D₁ D₂ i₂) have hk : (append D₁ D₂).k (ChallengeIdx.inr i₂) = D₂.k i₂ := congrArg Subtype.val (append_soundnessParam_inr D₁ D₂ i₂) have halpha : (append D₁ D₂).alphabet (ChallengeIdx.inr i₂) = D₂.alphabet i₂ := append_alphabet_inr D₁ D₂ i₂ unfold CWSSStructure.nodeOk congr 1 refine Function.hfunext (by rw [hell, hk]) (fun j j' hj => ?_) refine heq_equiv_apply (by simp [ProtocolSpec.append, ChallengeIdx.inr]) (by rw [append_coordIndex_inr, append_alphabet_inr]) (append_decompose_inr D₁ D₂ i₂) ?_ refine HEq.trans (heq_of_eq (congrArg challenges (Fin.ext ?_))) (cast_heq _ _).symm change j.val = j'.val exact (Fin.heq_ext_iff (by rw [hell, hk])).mp hj- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/OracleReduction/Security/CoordinateWiseSpecialSoundness/Composition.lean:222-266
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Person-level attribution pending.
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Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
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