Penalties in milk price due to changes in milk quality and elevations in somatic cell count Reduced production for the remainder of the season due to permanent damage to the secretory tissue
It affects the farmer financially in several ways Mastitis is generally considered the most costly production disease of dairy cows. Put simply, mastitis refers to inflammation of the mammary gland (udder).
#Cambridge definition of lame update#
) " by ( simp add : succ_def mem_not_refl cons_fun_eq ) subsection ‹Function Updates› definition update :: " ⇒ i " where " update ( f, a, b ) ≡ λ x ∈ cons ( a, domain ( f ) ). x ∈ A ⟹ f ` x ∈ B ( x ) ⟧ ⟹ f ∈ Pi ( A, B ) " apply ( simp only : Pi_iff ) apply ( blast dest : function_apply_equality ) done (*Such functions arise in non-standard datatypes, ZF/ex/Ntree for instance*) lemma Pi_Collect_iff : " ( f ∈ Pi ( A, λ x. c = " apply ( frule fun_is_rel ) apply ( blast dest : apply_equality ) done lemma function_apply_Pair : " ⟦ function ( f ) a ∈ domain ( f ) ⟧ ⟹ : f " apply ( simp add : function_def, clarify ) apply ( subgoal_tac " f ` a = y ", blast ) apply ( simp add : apply_def, blast ) done lemma apply_Pair : " ⟦ f ∈ Pi ( A, B ) a ∈ A ⟧ ⟹ : f " apply ( simp add : Pi_iff ) apply ( blast intro : function_apply_Pair ) done (*Conclusion is flexible - use rule_tac or else apply_funtype below!*) lemma apply_type : " ⟦ f ∈ Pi ( A, B ) a ∈ A ⟧ ⟹ f ` a ∈ B ( a ) " by ( blast intro : apply_Pair dest : fun_is_rel ) (*This version is acceptable to the simplifier*) lemma apply_funtype : " ⟦ f ∈ A -> B a ∈ A ⟧ ⟹ f ` a ∈ B " by ( blast dest : apply_type ) lemma apply_iff : " f ∈ Pi ( A, B ) ⟹ ⟨ a, b ⟩ : f ⟷ a ∈ A ∧ f ` a = b " apply ( frule fun_is_rel ) apply ( blast intro ! : apply_Pair apply_equality ) done (*Refining one Pi type to another*) lemma Pi_type : " ⟦ f ∈ Pi ( A, C ) ⋀ x. Sigmas and Pis are abbreviated as * or -> *) (*Weakening one function type to another see also Pi_type*) lemma fun_weaken_type : " ⟦ f ∈ A -> B B D " by ( unfold Pi_def, best ) subsection ‹Function Application› lemma apply_equality2 : " ⟦ ⟨ a, b ⟩ : f ⟨ a, c ⟩ : f f ∈ Pi ( A, B ) ⟧ ⟹ b = c " by ( unfold Pi_def function_def, blast ) lemma function_apply_equality : " ⟦ ⟨ a, b ⟩ : f function ( f ) ⟧ ⟹ f ` a = b " by ( unfold apply_def function_def, blast ) lemma apply_equality : " ⟦ ⟨ a, b ⟩ : f f ∈ Pi ( A, B ) ⟧ ⟹ f ` a = b " unfolding Pi_def apply ( blast intro : function_apply_equality ) done (*Applying a function outside its domain yields 0*) lemma apply_0 : " a ∉ domain ( f ) ⟹ f ` a = 0 " by ( unfold apply_def, blast ) lemma Pi_memberD : " ⟦ f ∈ Pi ( A, B ) c ∈ f ⟧ ⟹ ∃ x ∈ A. x ∈ A' ⟹ B ( x ) = B' ( x ) ⟧ ⟹ Pi ( A, B ) = Pi ( A', B' ) " by ( simp add : Pi_def cong add : Sigma_cong ) (*Sigma_cong, Pi_cong NOT given to Addcongs: they causeįlex-flex pairs and the "Check your prover" error. ⟦ ⟨ x, y ⟩ : r : r ⟧ ⟹ y = y' ⟧ ⟹ function ( r ) " by ( simp add : function_def, blast ) (*Functions are relations*) lemma fun_is_rel : " f ∈ Pi ( A, B ) ⟹ f ⊆ Sigma ( A, B ) " by ( unfold Pi_def, blast ) lemma Pi_cong : " ⟦ A = A' ⋀ x. *) section ‹Functions, Function Spaces, Lambda-Abstraction› theory func imports equalities Sum begin subsection ‹The Pi Operator: Dependent Function Space› lemma subset_Sigma_imp_relation : " r ⊆ Sigma ( A, B ) ⟹ relation ( r ) " by ( simp add : relation_def, blast ) lemma relation_converse_converse : " relation ( r ) ⟹ converse ( converse ( r ) ) = r " by ( simp add : relation_def, blast ) lemma relation_restrict : " relation ( restrict ( r, A ) ) " by ( simp add : restrict_def relation_def, blast ) lemma Pi_iff : " f ∈ Pi ( A, B ) ⟷ function ( f ) ∧ f range ( f ) " by ( simp add : Pi_iff relation_def, blast ) lemma functionI : " ⟦ ⋀ x y y'. Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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