hcju/isabelleps huggingface dataset:
f_simp: fixes ts :: "checker_type" assumes "ts \<noteq> Bot" shows "f e ts = check_single \<C> e ts"
diff_fun_times: fixes a :: "'b \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'c" assumes "diff_fun k charts2 a"   and "diff_fun k charts2 b" shows "diff_fun k charts2 (a * b)"
elims_1: fixes x :: "'b formula"   and y :: "bool" assumes "is_nnf x = y" obtains F :: "'b formula" and G :: "'b formula" where "x = Fa__ \<^bold>\<rightarrow> G__" and "\<not> y"   | F :: "'b formula" and G :: "'b formula" where "x = Fb__ \<^bold>\<and> Ga__" and "y = (is_nnf Fb__ \<and> is_nnf Ga__)"   | F :: "'b formula" and G :: "'b formula" where "x = F \<^bold>\<or> G" and "y = (is_nnf F \<and> is_nnf G)"   | v :: "'b" where "x = Atom v__" and "y = is_lit_plus (Atom v__)"   | "x = \<bottom>" and "y = is_lit_plus \<bottom>"   | v :: "'b formula" where "x = \<^bold>\<not> v" and "y = is_lit_plus (\<^bold>\<not> v)"
measurable_case_enat: fixes f :: "'b \<Rightarrow> enat"   and M :: "'b measure"   and g :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'c measure"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"   and "\<And>i. g i \<in> M \<rightarrow>\<^sub>M N"   and "h \<in> M \<rightarrow>\<^sub>M N" shows "(\<lambda>x. case f x of enat i \<Rightarrow> g i x | \<infinity> \<Rightarrow> h x) \<in> M \<rightarrow>\<^sub>M N"
2: fixes U :: "'b set" assumes "negligible U \<and> U \<subseteq> T" shows "{x \<in> S. f x \<in> U} \<in> sets lebesgue"
refl: fixes a :: "'c multiset" shows "a \<subseteq># a"
alternating_list_append_2: fixes m :: "nat"   and s :: "'b"   and t :: "'b"   and n :: "nat" assumes "\<not> gcd_nat.greater_eq m 2" shows "alternating_list m s t @ alternating_list n t s = alternating_list (m + n) s t"
is_arr_def: fixes \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" shows "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b = (f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr> \<and> \<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a \<and> \<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b)"
simps_2: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and ys :: "'d list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
Cons_2: shows "a \<in> snd ` set (Atoms (preprocess' (hh # t) start))"
simps: fixes i :: "'a \<times> nat"   and m :: "('a, 'b) event" shows "event_at i m = event_pred i ((=) m)"
plus_eq_infty_iff_eint: fixes m :: "eint"   and n :: "eint" shows "(m + n = \<infinity>) = (m = \<infinity> \<or> n = \<infinity>)"
lookup_zero: fixes k :: "'f" shows "lookup 0 k = (0::'e)"
gt_imp_ge: fixes y :: "int"   and x :: "int" assumes "y < x" shows "y \<le> x"
l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs \<equiv> (l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel \<and> l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf) \<and> l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs type_wf \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<and> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes get_shadow_root get_tag_name heap_is_wellformed"
CFG_intra_edge_no_Def_equal: fixes a :: "'a"   and V :: "'c"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "valid_edge a"   and "V \<notin> Def (sourcenode a)"   and "intra_kind (kind a)"   and "local.CFG.pred (kind a) s" shows "local.CFG.state_val (local.CFG.transfer (kind a) s) V = local.CFG.state_val s V"
homeomorphic_convex_sets: fixes S :: "'a set"   and T :: "'b set" assumes "convex S"   and "convex T"   and "S homeomorphic T" shows "aff_dim S = aff_dim T"
vcomp_ide_cod: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd" assumes "natural_transformation A B F G \<tau>" shows "vertical_composite.map A B \<tau> G = \<tau>"
strict_iff_order: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less" shows "less a b = (less_eq a b \<and> a \<noteq> b)"
solution_simplex': fixes s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "solution_simplex (Simplex_State s) = solution_simplex' s"
DC_comm: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<Sqinter> b = b \<Sqinter> a"
inj_onI: fixes A :: "'k set"   and f :: "'k \<Rightarrow> 'l" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
dghm_comp_is_dghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
star_of_le: fixes x :: "'b"   and y :: "'b" shows "(star_of x \<le> star_of y) = (x \<le> y)"
after_def: fixes f :: "'b \<Rightarrow> 'c" shows "f$ \<equiv> %s. f (second s)"
subst_box_simp: fixes p :: "('a, 'b, 'c) hp"   and q :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst ([[p]]q) \<sigma> = ([[Psubst p \<sigma>]]Fsubst q \<sigma>)"
set_of_add_inc: fixes A :: "'a interval"   and A' :: "'a interval"   and B :: "'a interval"   and B' :: "'a interval" assumes "set_of A \<subseteq> set_of A'"   and "set_of B \<subseteq> set_of B'" shows "set_of (A + B) \<subseteq> set_of (A' + B')"
min_less_iff_conj: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and z :: "'g"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less" shows "less z (ord.min less_eq x y) = (less z x \<and> less z y)"
ipurge_unwinding: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd" assumes "ref_union_closed P" shows "secure P I D = weakly_future_consistent P I D (rel_ipurge P I D)"
right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(y \<otimes> x = z \<otimes> x) = (y = z)"
case_uprod_simps: fixes f :: "('c, 'b) commute"   and x :: "'c"   and y :: "'c" shows "case_uprod f (Upair x y) = apply_commute f x y"
assms_2: shows "(s, s') \<in> \<langle>Rk\<rangle>Rs"
h': shows "h2 \<turnstile> (if owner_document \<noteq> old_document then Heap_Error_Monad.bind (get_disconnected_nodes old_document) (\<lambda>old_disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes old_document (remove1 node_ptr old_disc_nodes)) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. set_disconnected_nodes owner_document (node_ptr # disc_nodes)))) else return ()) \<rightarrow>\<^sub>h h'"
inverse_inverse: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "Groups.group f z inverse" shows "inverse (inverse a) = a"
formulas_var_iff: fixes i :: "'a"   and S :: "'a set" shows "(var i \<in> formulas S) = (i \<in> S)"
f: shows "f \<midarrow>a\<rightarrow> f a"
ccw_convex_axioms_def: fixes ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and oriented :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ccw_convex_axioms ccw oriented \<equiv> \<forall>u v a b c d. 0 \<le> u \<longrightarrow> 0 \<le> v \<longrightarrow> u + v = 1 \<longrightarrow> ccw a b c \<longrightarrow> ccw a b d \<longrightarrow> oriented a b \<longrightarrow> ccw a b (u *\<^sub>R c + v *\<^sub>R d)"
mcont_if_bot: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bound :: "'b"   and f :: "'b \<Rightarrow> 'c"   and bot :: "'c" assumes "class.ccpo Sup less_eq less"   and "class.ccpo lub ord lt"   and "\<And>x y. \<lbrakk>less_eq x y; \<not> less_eq x bound\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>Y. \<lbrakk>Complete_Partial_Order.chain less_eq Y; Y \<noteq> {}; \<And>x. x \<in> Y \<Longrightarrow> \<not> less_eq x bound\<rbrakk> \<Longrightarrow> f (Sup Y) = lub (f ` Y)"   and "\<And>x. \<not> less_eq x bound \<Longrightarrow> ord bot (f x)" shows "mcont Sup less_eq lub ord (\<lambda>x. if less_eq x bound then bot else f x)"
mem_eq_un_eq: fixes ts' :: "'b list"   and ts :: "'c list"   and P :: "'b \<Rightarrow> 'd set"   and Q :: "'c \<Rightarrow> 'd set" assumes "length ts' = length ts"   and "\<forall>i<length ts'. P (ts' ! i) = Q (ts ! i)" shows "\<Union> (P ` set ts') = \<Union> (Q ` set ts)"
length_list_of: fixes xs :: "'a llist" assumes "lfinite xs" shows "enat (length (list_of xs)) = llength xs"
Collector_def: shows "Collector \<equiv> \<lbrace>\<acute>Proper\<rbrace>  WHILE True INV \<lbrace>\<acute>Proper\<rbrace>  DO Blacken_Roots;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M\<rbrace> \<acute>obc := {};; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc = {}\<rbrace> \<acute>bc := Roots;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc = {} \<and> \<acute>bc = Roots\<rbrace> \<acute>Ma := M_init;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc = {} \<and> \<acute>bc = Roots \<and> \<acute>Ma = M_init\<rbrace>  WHILE \<acute>obc \<noteq> \<acute>bc INV \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> \<acute>bc \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace>  DO \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M\<rbrace> \<acute>obc := \<acute>bc;; Propagate_Black;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> (\<acute>obc \<subset> Blacks \<acute>M \<or> \<acute>Safe)\<rbrace> \<acute>Ma := \<acute>M;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe)\<rbrace> \<acute>bc := {};; Count OD;; Append OD"
streamsp_streams_eq: fixes A :: "'b set" shows "streamsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> streams A)"
emeasure_PiM_emb': fixes J :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set" assumes "J \<subseteq> UNIV"   and "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>i. M))" shows "emeasure (Pi\<^sub>M UNIV (\<lambda>i. M)) (emb UNIV J X) = emeasure (Pi\<^sub>M J (\<lambda>i. M)) X"
lambda_swap_Galois: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec"   and m :: "'b"   and n :: "'b" shows "(x = (\<chi>i. y $ Fun.swap m n id i)) = ((\<chi>i. x $ Fun.swap m n id i) = y)"
fp_ub_nonneg: fixes p :: "nat" shows "0 \<le> fp_ub p"
order: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "[:- a, 1::'a:] ^ order a p dvd p \<and> \<not> [:- a, 1::'a:] ^ Suc (order a p) dvd p"
start_of_suc_inc: fixes k :: "nat"   and ap :: "abc_inst list"   and n :: "nat" assumes "k < length ap"   and "ap ! k = Inc n" shows "start_of (layout_of ap) (Suc k) = start_of (layout_of ap) k + 2 * n + 9"
alg_dvd_triv_right_iff: fixes y :: "'b"   and x :: "'b" shows "(y alg_dvd x * y) = (y = (0::'b) \<or> algebraic_int x)"
perm_9: fixes pi :: "(name \<times> name) list"   and x1 :: "name"   and x2 :: "pi" shows "pi \<bullet> <\<nu>x1>x2 = <\<nu>(pi \<bullet> x1)>(pi \<bullet> x2)"
LIMSEQ_const_iff: fixes k :: "'c"   and l :: "'c" shows "(\<lambda>n. k) \<longlonglongrightarrow> l = (k = l)"
assms_1: shows "(B, B') \<in> (Ra \<rightarrow> \<langle>Rr\<rangle>nres_rel) \<rightarrow> Ra \<rightarrow> \<langle>Rr\<rangle>nres_rel"
ord_le_eq_trans: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "a \<sqsubseteq> b"   and "b = c" shows "a \<sqsubseteq> c"
simps_1: fixes g :: "('a, 'b) graph" shows "rem_unPath [] g = g"
wt: shows "P,E,h,sh \<turnstile> {V:T; e} : T'"
subgroup_carrier_complementary: fixes H :: "'a set"   and J :: "'a set"   and I :: "'a set"   and K :: "'a set" assumes "complementary H J"   and "subgroup I (G\<lparr>carrier := H\<rparr>)"   and "subgroup K (G\<lparr>carrier := J\<rparr>)" shows "complementary I K"
echelon_echelon_form_column_k: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" assumes "is_bezout_ext bezout"   and "echelon_form_upt_k A k"   and "k < ncols A" shows "echelon_form_upt_k (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k)) (Suc k)"
S_def: shows "S = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = dflt}"
PiE_I: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x"   and "\<And>x. x \<notin> A \<Longrightarrow> f x = undefined" shows "f \<in> Pi\<^sub>E A B"
IH: shows "sigma_array (IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (if i = mm then Suc j else Suc i)) (Suc mm)) (IArray.of_fun (\<mu>' mm) (Suc j)) (IArray.of_fun (\<mu>' (Suc j)) (if Suc j = mm then Suc j else Suc (Suc j))) (\<mu>' l l) l = \<sigma>s l mm (Suc j)"
Quotient3_rep_reflp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep a)"
simple_function_ennreal: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "simple_function M f" shows "simple_function M (\<lambda>x. ennreal (f x))"
assms: fixes n :: "nat" shows "n < length cl \<and> finished (cl ! n) \<or> n < length dl \<and> finished (dl ! n)"
env_restr_useless: fixes m :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "edom m \<subseteq> S" shows "m f|` S = m"
numeral_le_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "num"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (numeral.numeral one plus m) (numeral.numeral one plus n) = (m \<le> n)"
cong_l_filter: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters" shows "cong_l F a b = (a r\<rightarrow> b \<in> F \<and> b r\<rightarrow> a \<in> F)"
poly_dvd_antisym: fixes p :: "'c poly"   and q :: "'c poly" assumes "lead_coeff p = lead_coeff q"   and "p dvd q"   and "q dvd p" shows "p = q"
less_add_same_cancel2: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subset># b + a) = ({#} \<subset># b)"
normal_intersection_quotient_isom: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and S :: "'c set" assumes "second_isomorphism_grp H G S" shows "(\<lambda>X. the_elem ((#>\<^bsub>G\<^esub>) H ` X)) \<in> Group.iso (G\<lparr>carrier := S\<rparr> Mod H \<inter> S) (G\<lparr>carrier := H <#>\<^bsub>G\<^esub> S\<rparr> Mod H)"
distance_def: fixes A :: "'a"   and B :: "'a" shows "distance A B = dist A B"
thetaIfZOT_ZOretrT: shows "thetaIfZOT \<subseteq> ZOretrT (thetaIfZOT \<union> ZObisT)"
P2: shows "Bet D E D' \<and> Cong E D' D E"
assms_3: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and m :: "nat"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd"   and h :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "dickson_grading d"   and "set gs \<union> set bs \<union> set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "(set gs \<union> set bs) \<inter> set hs = {}"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "h \<noteq> g"   and "fst h \<noteq> 0"   and "fst g \<noteq> 0"   and "(h, g) \<in> set ps -\<^sub>p set (ap gs bs ps hs data)"   and "\<And>a b. \<lbrakk>(a, b) \<in>\<^sub>p set (ap gs bs ps hs data); (a, b) \<in>\<^sub>p set hs \<times> (set gs \<union> set bs \<union> set hs); fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)" shows "crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)"
is_monomial_monomial: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "is_monomial p" obtains c :: "'b" and t :: "'a" where "c \<noteq> (0::'b)" and "p = monomial c t"
intro: fixes \<Gamma> :: "'b inference set"   and I_of :: "'b literal multiset set \<Rightarrow> 'b set" assumes "counterex_reducing_inference_system \<Gamma> I_of" shows "standard_redundancy_criterion_counterex_reducing \<Gamma> I_of"
xy_2: shows "y\<^sub>1 = - (- y\<^sub>2) - l' * (x\<^sub>1 - x\<^sub>2)"
filter_eqI: fixes vs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>v. v \<in> set vs \<Longrightarrow> P v = Q v" shows "filter P vs = filter Q vs"
M_plus_2: fixes x :: "int"   and y :: "int" shows "M (x + M y) = M (x + y)"
ldrop_enat: fixes n :: "nat"   and xs :: "'c llist" shows "ldrop (enat n) xs = ldropn n xs"
assms_6: shows "bounded (m - 1) r\<^sub>0 i"
abort_def: fixes uu :: "String.literal"   and f :: "unit \<Rightarrow> 'c" shows "Code.abort uu f = f ()"
UP_l_zero: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
inv_eq_1_iff: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "(\<ominus> x = \<zero>) = (x = \<zero>)"
pick_insert3: fixes a :: "nat"   and I :: "nat set"   and i :: "nat"   and a' :: "nat" assumes "a \<notin> I"   and "i < card I"   and "pick (insert a I) a' = a"   and "a' \<le> i"   and "a' < card I + 1" shows "pick (insert a I) (Suc i) = pick I i"
drive: fixes ts :: "traffic"   and ts' :: "traffic"   and ts'' :: "traffic" assumes "\<exists>x\<ge>0.  ts \<^bold>\<midarrow> x \<^bold>\<rightarrow> ts'"   and "ts' \<^bold>\<leadsto> ts''" shows "ts \<^bold>\<leadsto> ts''"
min_height_le_height: fixes t :: "'b tree" shows "min_height t \<le> height t"
pwd_owner_iv_def: fixes n :: "nat" shows "pwd_owner_iv n \<equiv> stage_owner_iv n (pwd_spy_iii n)"
subcls_notin_has_fields: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and F :: "char list"   and D :: "char list"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "((F, D), T) \<in> set FDTs" shows "(D, C) \<notin> (subcls1 P)\<^sup>+"
distinct_21: fixes x21 :: "'a"   and x22 :: "addr_loc"   and x23 :: "'a val"   and x41 :: "'a"   and x42 :: "htype" shows "ReadMem x21 x22 x23 \<noteq> NewHeapElem x41 x42"
bounded_bilinear_bounded_uniform_limit_intros_2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and X :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'd" assumes "uniform_limit X f l F"   and "uniform_limit X g m F"   and "bounded (m ` X)"   and "bounded (l ` X)" shows "uniform_limit X (\<lambda>a b. f a b * g a b) (\<lambda>a. l a * m a) F"
integral_f: fixes k :: "nat" shows "integral {c..} (f k) = (if c \<le> real k then exp (- a * c) / a - exp (- a * real k) / a else 0)"
idom_mult_cancel_r: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "c \<noteq> \<zero>"   and "a \<cdot>\<^sub>r c = b \<cdot>\<^sub>r c" shows "a = b"
eval_poly_add: fixes hom :: "'b \<Rightarrow> 'c"   and p :: "'b poly"   and q :: "'b poly"   and x :: "'c" assumes "comm_semiring_hom hom" shows "eval_poly hom (p + q) x = eval_poly hom p x + eval_poly hom q x"
permute_plus: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "perm"   and q :: "perm"   and x :: "'e" assumes "class.pt permute" shows "permute (p + q) x = permute p (permute q x)"