hcju/isabelleps huggingface dataset:
D: shows "\<exists>S A C. fst SK = Some S \<and> A \<in> fst (snd SK) \<and> C \<in> snd (snd SK) \<and> {PriKey S, PriKey A, PriKey C} \<subseteq> spied s"
\<sigma>_3: shows "subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<sigma>) \<subseteq> {t. {} \<turnstile>\<^sub>c t} - T"
ct: shows "r2.cond_action_oks (ls, (ts2, m2), ws, is) t cts"
assms_2: shows "\<guillemotleft>\<mu> : R f \<Rightarrow>\<^sub>S f\<guillemotright>"
above_decr: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "trans r"   and "(a, b) \<in> r" shows "above r b \<subseteq> above r a"
dghm_singleton_components_4: fixes j :: "V"   and \<CC> :: "V" shows "dghm_singleton j \<CC>\<lparr>HomCod\<rparr> = (\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)"
induct: fixes x1 :: "collection_unop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'a OCL_Types.type"   and P :: "collection_unop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "collection_unop_type x1 x2 x3"   and "\<And>\<tau> uu_. element_type \<tau> uu_ \<Longrightarrow> P CollectionSizeOp \<tau> Integer[1]"   and "\<And>\<tau> uv_. element_type \<tau> uv_ \<Longrightarrow> P IsEmptyOp \<tau> Boolean[1]"   and "\<And>\<tau> uw_. element_type \<tau> uw_ \<Longrightarrow> P NotEmptyOp \<tau> Boolean[1]"   and "\<And>\<tau> \<sigma>. \<lbrakk>element_type \<tau> \<sigma>; \<sigma> = UnlimitedNatural[1]\<midarrow>Real[1]\<rbrakk> \<Longrightarrow> P CollectionMaxOp \<tau> \<sigma>"   and "\<And>\<tau> \<sigma> oper. \<lbrakk>element_type \<tau> \<sigma>; operation \<sigma> STR ''max'' [\<sigma>] oper\<rbrakk> \<Longrightarrow> P CollectionMaxOp \<tau> \<sigma>"   and "\<And>\<tau> \<sigma>. \<lbrakk>element_type \<tau> \<sigma>; \<sigma> = UnlimitedNatural[1]\<midarrow>Real[1]\<rbrakk> \<Longrightarrow> P CollectionMinOp \<tau> \<sigma>"   and "\<And>\<tau> \<sigma> oper. \<lbrakk>element_type \<tau> \<sigma>; operation \<sigma> STR ''min'' [\<sigma>] oper\<rbrakk> \<Longrightarrow> P CollectionMinOp \<tau> \<sigma>"   and "\<And>\<tau> \<sigma>. \<lbrakk>element_type \<tau> \<sigma>; \<sigma> = UnlimitedNatural[1]\<midarrow>Real[1]\<rbrakk> \<Longrightarrow> P SumOp \<tau> \<sigma>"   and "\<And>\<tau> \<sigma> oper. \<lbrakk>element_type \<tau> \<sigma>; operation \<sigma> STR ''+'' [\<sigma>] oper\<rbrakk> \<Longrightarrow> P SumOp \<tau> \<sigma>"   and "\<And>\<tau> \<sigma>. element_type \<tau> \<sigma> \<Longrightarrow> P AsSetOp \<tau> (Set \<sigma>)"   and "\<And>\<tau> \<sigma>. element_type \<tau> \<sigma> \<Longrightarrow> P AsOrderedSetOp \<tau> (OrderedSet \<sigma>)"   and "\<And>\<tau> \<sigma>. element_type \<tau> \<sigma> \<Longrightarrow> P AsBagOp \<tau> (Bag \<sigma>)"   and "\<And>\<tau> \<sigma>. element_type \<tau> \<sigma> \<Longrightarrow> P AsSequenceOp \<tau> (Sequence \<sigma>)"   and "\<And>\<tau> \<sigma>. update_element_type \<tau> (to_single_type \<tau>) \<sigma> \<Longrightarrow> P FlattenOp \<tau> \<sigma>"   and "\<And>\<tau>. P FirstOp (OrderedSet \<tau>) \<tau>"   and "\<And>\<tau>. P FirstOp (Sequence \<tau>) \<tau>"   and "\<And>\<tau>. P LastOp (OrderedSet \<tau>) \<tau>"   and "\<And>\<tau>. P LastOp (Sequence \<tau>) \<tau>"   and "\<And>\<tau>. P ReverseOp (OrderedSet \<tau>) (OrderedSet \<tau>)"   and "\<And>\<tau>. P ReverseOp (Sequence \<tau>) (Sequence \<tau>)" shows "P x1 x2 x3"
seq: shows "\<forall>i. (f i, f (Suc i)) \<in> r"
eval_inverse_primfun: fixes f :: "primfun \<times> real"   and x :: "real" shows "eval_primfun (inverse_primfun f) x = inverse (eval_primfun f x)"
floor_float_intervalI: fixes x :: "real"   and X :: "float interval" assumes "x \<in>\<^sub>r X" shows "real_of_int \<lfloor>x\<rfloor> \<in>\<^sub>r floor_float_interval X"
hyps: fixes u :: "'a" assumes "awalk u es v"   and "awalk v q w" shows "awalk u (es @ q) w"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Assign (VE (Loc l)) (VE v)])"
fmlookup_of_list: fixes m :: "('d \<times> 'e) list" shows "fmlookup (fmap_of_list m) = map_of m"
induct: fixes x :: "'a"   and H :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> generate H"   and "P \<one>"   and "\<And>a. a \<in> H \<Longrightarrow> P a"   and "\<And>a. a \<in> H \<Longrightarrow> P (local.inverse a)"   and "\<And>a b. \<lbrakk>a \<in> generate H; P a; b \<in> generate H; P b\<rbrakk> \<Longrightarrow> P (a \<cdot> b)" shows "P x"
lin_independent_imp_distinct: fixes ms :: "'v list" assumes "set ms \<subseteq> V"   and "R_lin_independent ms" shows "distinct ms"
row_mat_plus: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list"   and i :: "nat"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr nc m1"   and "mat nr nc m2"   and "i < nr" shows "row (mat_plusI pl m1 m2) i = vec_plusI pl (row m1 i) (row m2 i)"
det30_uminus3: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "coll 0 a (- b) = coll 0 a b"
lambda_predicates_4_0: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>"   and \<psi> :: "\<nu> \<Rightarrow> \<o>"   and \<chi> :: "\<kappa> \<Rightarrow> \<o>" assumes "\<And>x. [\<^bold>\<A>(\<phi> x \<^bold>\<equiv> \<psi> x) in v]" shows "[[\<^bold>\<lambda>\<^sup>0 (\<chi> (\<^bold>\<iota>x. \<phi> x)) \<^bold>= \<^bold>\<lambda>\<^sup>0 (\<chi> (\<^bold>\<iota>x. \<psi> x))]]"
full_span: shows "local.span (a ` {0..<m}) = carrier_vec n"
id_subs: fixes x :: "name"   and t :: "trm" shows "t[x::=Var x] = t"
intros_1: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel" shows "(!P, !Q) \<in> bangRel Rel"
polys_rel_full_def: shows "polys_rel_full = {((\<V>, A), \<V>', B). (A, B) \<in> polys_rel \<and> \<V> = \<V>'}"
gcd_poly_code_def: fixes p :: "'b poly"   and q :: "'b poly" shows "gcd_poly_code p q = (if p = 0 then normalize q else if q = 0 then normalize p else smult (gcd (content p) (content q)) (gcd_poly_code_aux (primitive_part p) (primitive_part q)))"
mono_gamma_c: fixes ca :: "'av Abs_State.st option acom"   and ca' :: "'av Abs_State.st option acom" assumes "ca \<sqsubseteq> ca'" shows "\<gamma>\<^sub>c ca \<le> \<gamma>\<^sub>c ca'"
abs_prob_valid: shows "is_valid_problem_sas_plus abs_prob"
contra_subsetD: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "A \<subseteq> B"   and "c \<notin> B" shows "c \<notin> A"
to_tree_order_parent_child_rel: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and child_ptr :: "(_) object_ptr" assumes "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to" shows "((ptr, child_ptr) \<in> (parent_child_rel h)\<^sup>*) = (child_ptr \<in> set to)"
pre2: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "I l s"   and "\<not> bval b s" shows "Q l s \<and> 1 \<le> E s \<and> S s = s on \<Union> (funStar SS ` (LVE \<union> LVQ))"
SN_inv_image: fixes R :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a" assumes "SN R" shows "SN (inv_image R f)"
path_to_left_lang: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M"   and "q \<in> dfa.accessible M" shows "dfa.path_to M q \<in> dfa.left_lang M q"
prod_upto_nat_unfold: fixes f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "prod f {m..n} = (if n < m then 1::'c else if n = 0 then f 0 else f n * prod f {m..n - 1})"
Diff1_foldSetD: fixes A :: "'b set"   and x :: "'b"   and y :: "'a"   and e :: "'a" assumes "(A - {x}, y) \<in> foldSetD D (\<cdot>) e"   and "x \<in> A"   and "A \<subseteq> B" shows "(A, x \<cdot> y) \<in> foldSetD D (\<cdot>) e"
wf_Union: fixes R :: "('b \<times> 'b) set set" assumes "\<forall>r\<in>R. wf r"   and "\<forall>r\<in>R. \<forall>s\<in>R. r \<noteq> s \<longrightarrow> Domain r \<inter> Range s = {}" shows "wf (\<Union> R)"
nat_pow_one: fixes n :: "nat" shows "(1::'b) ^ n = (1::'b)"
l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and insert_before_locs :: "(_) object_ptr \<Rightarrow> (_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_child_nodes set_child_nodes set_child_nodes_locs get_ancestors adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes get_owner_document insert_before insert_before_locs \<equiv> insert_before = l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before get_parent get_child_nodes set_child_nodes get_ancestors adopt_node set_disconnected_nodes get_disconnected_nodes get_owner_document \<and> insert_before_locs = l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before_locs set_child_nodes_locs adopt_node_locs set_disconnected_nodes_locs"
orthogonal_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner x y = (inner x y = 0)"
x_1: shows "x = a + u *\<^sub>R (b - a)"
closure_ball: fixes e :: "real"   and x :: "'a" assumes "0 < e" shows "closure (ball x e) = cball x e"
path_component_set: fixes S :: "'a set"   and x :: "'a" shows "path_component_set S x = {y. \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y}"
finter_fempty_right: fixes A :: "'c fset" shows "A |\<inter>| {||} = {||}"
Collect_mem_eq: fixes A :: "'d set" shows "{x. x \<in> A} = A"
iT_Div_empty_iff: fixes I :: "nat set"   and k :: "nat" shows "(I \<oslash> k = {}) = (I = {})"
convex_hull_Int: fixes S :: "'a set"   and T :: "'a set" assumes "\<not> affine_dependent (S \<union> T)" shows "convex hull (S \<inter> T) = convex hull S \<inter> convex hull T"
Units_r_inv: fixes x :: "'a" assumes "x \<in> Units G" shows "x \<otimes> inv x = \<one>"
locallyE: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and w :: "'c set"   and x :: "'c" assumes "locally P S"   and "openin (top_of_set S) w"   and "x \<in> w" obtains u :: "'c set" and v :: "'c set" where "openin (top_of_set S) u" and "P v" and "x \<in> u" and "u \<subseteq> v" and "v \<subseteq> w"
brec_brw_invar_cons: fixes \<Sigma> :: "('a, 'b tree) HashMap.hashmap \<times> ('a list \<times> 'a list) \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap \<times> 'a option"   and Qi :: "('a, 'b) ta_rule set" assumes "brec_\<alpha> \<Sigma> \<in> brw_invar Qi" shows "set (fifo.\<alpha> (fst (snd \<Sigma>))) \<subseteq> dom (hm.\<alpha> (fst \<Sigma>))"
east_def: shows "east = proj2_abs (vector [1, 0, 1])"
comp_inj_on_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and f' :: "'d \<Rightarrow> 'e" assumes "inj_on f A" shows "inj_on f' (f ` A) = inj_on (f' \<circ> f) A"
create_element_is_strongly_scdom_component_safe_step: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)|\<^sub>r"   and "ptr \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> create_element document_ptr tag|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
disj_comm: fixes P :: "bool"   and Q :: "bool" assumes "P \<or> Q" shows "Q \<or> P"
cl_eq_Collect_relcl: fixes L :: "'b set set"   and X :: "'b set" assumes "lattice L" shows "cl L X = {t. \<exists>s. s \<in> X \<and> (s, t) \<in> relcl L}"
fold_coeffs_pCons_coeff_not_0_eq: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "'b poly" assumes "a \<noteq> (0::'b)" shows "fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
uniformly_continuous_onE: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and e :: "real" assumes "uniformly_continuous_on s f"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>x x'. \<lbrakk>x \<in> s; x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
isDiagonal_Diagonal_to_Smith_row_i: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "isDiagonal A"   and "i < min (nrows A) (ncols A)" shows "isDiagonal (Diagonal_to_Smith_row_i A i bezout)"
prm_eq_def: fixes pi1 :: "('a \<times> 'a) list"   and pi2 :: "('a \<times> 'a) list" shows " pi1 \<triangleq> pi2  = (\<forall>a. pi1 \<bullet> a = pi2 \<bullet> a)"
\<iota>_def: shows "\<iota> = Infer (CAs0 @ [DA0]) E0"
lookup_except_eq_idI: fixes t :: "'b"   and S :: "'b set"   and p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "t \<notin> S" shows "lookup (except p S) t = lookup p t"
order_less_le_subst1: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c"   and c :: "'c" assumes "a < f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a < f c"
equivalence_one_closed: shows "(top \<le> top \<and> top \<sqinter> top \<le> top) \<and> local.comp_inf.symmetric top"
multiset_filter_mono2: fixes P :: "'c \<Rightarrow> bool"   and A :: "'c multiset"   and Q :: "'c \<Rightarrow> bool" shows "(filter_mset P A \<subseteq># filter_mset Q A) = (\<forall>a\<in>#A. P a \<longrightarrow> Q a)"
vconn_implies_hpar_4: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>S g\<guillemotright>" shows "S.trg g = S.trg f"
nvC_def: fixes c :: "('a, 'b) lit list" shows "nvC c \<equiv> \<Union> (set (map nvL c))"
Vr_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant" shows "Vr K v = Sr K {x \<in> carrier K. 0 \<le> v x}"
isomorphic_group_triviality1: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme" assumes "G \<cong> H"   and "Group.group H"   and "trivial_group G" shows "trivial_group H"
is_OK_if_return_1: fixes b :: "bool"   and x :: "'b"   and m :: "'a + 'b" shows "isOK (if b then Inr x else m) = (b \<or> isOK m)"
Limsup_obtain: fixes c :: "'b"   and F :: "'c filter"   and u :: "'c \<Rightarrow> 'b" assumes "c < Limsup F u" shows "\<exists>i. c < u i"
properfactorI: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a divides\<^bsub>G\<^esub> b"   and "\<not> b divides\<^bsub>G\<^esub> a" shows "properfactor G a b"
subgroup_generated_by_image: fixes S :: "'a set" assumes "S \<subseteq> carrier G" shows "carrier (subgroup_generated H (h ` S)) = h ` carrier (subgroup_generated G S)"
fun_mult_closed: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "f \<in> carrier (function_ring (carrier R) R)"   and "g \<in> carrier (function_ring (carrier R) R)" shows "f \<otimes>\<^bsub>function_ring (carrier R) R\<^esub> g \<in> carrier (function_ring (carrier R) R)"
univ_commute: fixes A :: "'d set"   and r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "equiv A r"   and "f respects r"   and "x \<in> A" shows "univ f (Equiv_Relations.proj r x) = f x"
conjugate_conjugate_circline: fixes H :: "circline" shows "conjugate_circline (conjugate_circline H) = H"
h'h: shows "\<forall>x\<in>T. h' x = h x"
list_update_code_1: fixes i :: "nat"   and y :: "'c" shows "[][i := y] = []"
show_strict_partial_order: fixes r :: "('a \<times> 'a) set" assumes "trans r"   and "irrefl r" shows "strict_partial_order r"
wt_subst: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and arOf :: "'f \<Rightarrow> 'e list"   and resOf :: "'f \<Rightarrow> 'e"   and \<pi> :: "Preliminaries.var \<Rightarrow> 'f trm"   and T :: "'f trm" assumes "Signature TYPE('e) wtFsym wtPsym"   and "Signature.wtSB wtFsym arOf resOf \<pi>"   and "Signature.wt wtFsym arOf resOf T" shows "Signature.wt wtFsym arOf resOf (subst \<pi> T)"
push_down_state_rank_configuration_None: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = None"   and "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q" shows "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n = {}"
r: shows "Vagree (stateinterpol \<nu> \<nu>' V) \<nu>' W \<and> Vagree (stateinterpol \<nu> \<nu>' V) \<nu>' W"
tau_inj0: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "\<not> zeroring R"   and "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "inj_on (tau R M N P) (M \<times>\<^sub>c N)"
fps_mod_eq_zero: fixes g :: "'a fps"   and f :: "'a fps" assumes "g \<noteq> 0"   and "subdegree g \<le> subdegree f" shows "f mod g = 0"
prime_factorization_unique'': fixes A :: "'a multiset"   and x :: "'a" assumes "\<And>p. p \<in># A \<Longrightarrow> prime p"   and "\<Prod>\<^sub># A = normalize x" shows "prime_factorization x = A"
comp_ide_self: fixes a :: "'d \<times> 'c" assumes "DopxC.ide a" shows "DopxC.comp a a = a"
distinct_17: fixes x31 :: "'b aexp"   and x32 :: "'b aexp"   and x51 :: "'b gexp"   and x52 :: "'b gexp" shows "Gt x31 x32 \<noteq> Nor x51 x52"
list_last: fixes xs :: "'d list" shows "xs = [] \<or> (\<exists>y ys. xs = ys @ [y])"
Rep_int_poly_factorization_algorithm: fixes x :: "int_poly_factorization_algorithm" shows "Rep_int_poly_factorization_algorithm x \<in> {alg. \<forall>f fs. square_free f \<longrightarrow> 0 < degree f \<longrightarrow> alg f = fs \<longrightarrow> f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)}"
module_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "module scale \<equiv> ((\<forall>a x y. scale a (x + y) = scale a x + scale a y) \<and> (\<forall>a b x. scale (a + b) x = scale a x + scale b x)) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'a) x = x)"
finite_lin_dep: fixes S :: "'e set" assumes "finite S"   and "W.module.lin_dep S"   and "S \<subseteq> carrier W" shows "\<exists>a v. a \<in> S \<rightarrow> carrier K \<and> W.module.lincomb a S = \<zero>\<^bsub>W\<^esub> \<and> v \<in> S \<and> a v \<noteq> \<zero>\<^bsub>K\<^esub>"
add_diff_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "c \<oplus>\<^sub>a a \<ominus>\<^sub>a (c \<oplus>\<^sub>a b) = a \<ominus>\<^sub>a b"
A: shows "A \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}"
subresultant_prs_locale_def: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly" shows "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2 \<equiv> primitive_remainder_sequence F n \<delta> f k \<beta> \<and> subresultant_prs_locale_axioms F G1 G2"
alpha_abs_set_stronger1: fixes as :: "atom set"   and x :: "'a"   and p' :: "perm"   and as' :: "atom set"   and x' :: "'a" assumes "(as, x) \<approx>set (=) supp p' (as', x')" shows "\<exists>p. (as, x) \<approx>set (=) supp p (as', x') \<and> supp p \<subseteq> as \<union> as'"
rsp: shows "eq_onp comp_fun_idem (\<lambda>b a. a - {b}) (\<lambda>b a. a - {b})"
abs_neg_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (uminus one) = one"
antisym_conv3: fixes x :: "'a"   and y :: "'a" assumes "\<not> x < y" shows "(\<not> y < x) = (x = y)"
le_conv_fun_ord: shows "(\<le>) = fun_ord (\<le>)"
TransVar_subst: fixes a :: "'a"   and v :: "Vars.var"   and x :: "Vars.var"   and y :: "Vars.var" shows "(TransVar a v)[x::=y] = TransVar a v[x::v=y]"
of_weak_ranking_altdef: fixes xs :: "'b set list"   and x :: "'b"   and y :: "'b" assumes "is_weak_ranking xs"   and "x \<in> \<Union> (set xs)"   and "y \<in> \<Union> (set xs)" shows "of_weak_ranking xs x y = (find_index ((\<in>) y) xs \<le> find_index ((\<in>) x) xs)"
smooth_on_minus: fixes k :: "enat"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x - g x)"
set_cases_1: fixes e :: "'d"   and a :: "('d, 'e) nondetT"   and thesis :: "bool" assumes "e \<in> phantom_nondetT a" shows "thesis"
simps_27: fixes f :: "'e \<Rightarrow> 'd"   and x31 :: "'e rexp"   and x32 :: "'e rexp" shows "map_rexp f (Conc x31 x32) = Conc (map_rexp f x31) (map_rexp f x32)"