hcju/isabelleps huggingface dataset:
eqvtRelI: fixes Rel :: "('a \<times> 'a) set"   and P :: "'a"   and Q :: "'a"   and perm :: "(name \<times> name) list" assumes "eqvt Rel"   and "(P, Q) \<in> Rel" shows "(perm \<bullet> P, perm \<bullet> Q) \<in> Rel"
length_negGorbit_list_sublist: fixes fs :: "(('f, 'g) aezfun \<Rightarrow> 'v) list"   and gs :: "'g list"   and T :: "'a \<Rightarrow> ('f, 'g) aezfun \<Rightarrow> 'v"   and as :: "'a list" assumes "fs \<in> set (negHorbit_list gs T as)" shows "length fs = length as"
c_is_sub_fun_lm_1: fixes ls :: "nat"   and f :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "c_is_sub_fun ls f"   and "c_assoc_have_key ls x = 0" shows "c_assoc_value ls x = f x"
token_run_merge_Suc: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "token_run x n = token_run y n" shows "token_run x (Suc n) = token_run y (Suc n)"
dim_diag_block_mat_2: fixes As :: "'a mat list" shows "dim_col (diag_block_mat As) = sum_list (map dim_col As)"
\<alpha>_def: fixes i :: "nat" shows "\<alpha> i = f (i - 1) ^ Suc (\<delta> (i - 2))"
card_of_Un2: fixes A :: "'a set"   and B :: "'a set" shows "|A| \<le>o |B \<union> A|"
apply_inverse: fixes x :: "'a perm" shows "Perm ((\<langle>$\<rangle>) x) = x"
cut_le_less_conv_if: fixes I :: "'b set"   and t :: "'b" shows "I \<down>\<le> t = (if t \<in> I then insert t (I \<down>< t) else I \<down>< t)"
defQ: shows "\<tau> \<Turnstile> \<delta> Q"
iTL_defs_7: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iRelease I P Q \<equiv> (\<box> t I. Q t) \<or> (\<diamond> t I. P t \<and> (\<box> t' I \<down>\<le> t. Q t'))"
hd_in_set: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "hd xs \<in> set xs"
assms_5: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)"
K_cfg_rept_in: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and st :: "'s \<times> ('c \<Rightarrow> real)"   and cfg' :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg"   and "abss st = state cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "(THE s'. s' \<in> set_pmf (rept st (action cfg)) \<and> abss s' = state cfg') \<in> set_pmf (rept st (action cfg))"
safe: fixes s :: "event list"   and r :: "room"   and g :: "guest" assumes "hotel s"   and "Trace.safe s r"   and "g \<in> Trace.isin s r" shows "Trace.owns s r = \<lfloor>g\<rfloor>"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D local.map cmp"
preprocess_def: fixes rs :: "'i common_primitive rule list" shows "(preprocess::'i common_primitive rule list \<Rightarrow> 'i common_primitive rule list) (rs::'i common_primitive rule list) \<equiv> upper_closure (packet_assume_new rs)"
lift_to_add2: fixes z :: "'b"   and q :: "'b"   and s :: "'b"   and x :: "'b"   and y :: "'b" assumes "Sep_Eq.add2 z q = Some s"   and "Sep_Eq.add2 x y = Some q" shows "total_partial.add2 (Some z) (total_partial.add2 (Some x) (Some y)) = Some s"
SCnotConc: fixes a :: "('c, 'd) Combinators"   and b :: "('c, 'd) Combinators"   and p :: "('c, 'd) Combinators list" assumes "a \<oplus> b \<in> set p"   and "singleCombinators p" shows "False"
det_interchange_different_rows: fixes i :: "'d"   and j :: "'d"   and A :: "(('e, 'd) vec, 'd) vec" assumes "i \<noteq> j" shows "det (interchange_rows A i j) = - det A"
lfinite_llist_of: fixes xs :: "'b list" shows "lfinite (llist_of xs)"
assms_2: shows "AE x in M. (\<lambda>xa. ereal (birkhoff_sum f xa x)) \<longlonglongrightarrow> \<infinity>"
r_result_Suc_saturating: fixes t :: "nat"   and i :: "nat"   and x :: "nat"   and v :: "nat" assumes "eval r_result [t, i, x] \<down>= Suc v" shows "eval r_result [Suc t, i, x] \<down>= Suc v"
accObjectmheads_def: fixes G :: "prog"   and S :: "qtname"   and T :: "ref_ty" shows "accObjectmheads G S T = (if G \<turnstile> RefT T accessible_in pid S then Objectmheads G S else (\<lambda>sig. {}))"
assms_1: shows "(x has_vderiv_on (\<lambda>t. f t (x t))) K"
STA_drop_prefix: fixes ps :: "('b, 'c) fm list"   and qs :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and A :: "'c set"   and n :: "nat"   and ps' :: "('b, 'c) fm list" assumes "set ps \<subseteq> set qs"   and "(qs, a) \<in>. branch"   and "A, n \<turnstile> (ps @ ps', a) # branch" shows "A, n \<turnstile> (ps', a) # branch"
assms_1: shows "subst_domain \<delta> \<union> range_vars \<delta> \<subseteq> vars\<^sub>s\<^sub>t S"
diagonal_to_Smith_PQ_JNF_local_typedef: fixes m :: "nat"   and n :: "nat"   and A :: "'d mat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and P :: "'d mat"   and S :: "'d mat"   and Q :: "'d mat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<int m}"   and "\<exists>Rep Abs. type_definition Rep Abs {0..<int n}"   and "1 < m"   and "1 < n"   and "isDiagonal_mat A"   and "is_bezout_ext bezout"   and "A \<in> carrier_mat m n"   and "(P, S, Q) = diagonal_to_Smith_PQ_JNF A bezout" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S \<and> P \<in> carrier_mat m m \<and> S \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
open_cover_of_subset_axioms_def: fixes is_open :: "'a set \<Rightarrow> bool"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" shows "open_cover_of_subset_axioms is_open I C \<equiv> \<forall>i. i \<in> I \<longrightarrow> is_open (C i)"
less_SUP_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less a (Sup (f ` A)) = (\<exists>x\<in>A. less a (f x))"
ge2: shows "2 \<le> size (filter_mset ((\<subseteq>) {x, y}) \<B>)"
singleton: fixes x :: "'a" shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>n \<eta> x = x"
converse_tranclE: fixes x :: "'a"   and z :: "'a"   and r :: "('a \<times> 'a) set" assumes "(x, z) \<in> r\<^sup>+" obtains "(x, z) \<in> r"   | y :: "'a" where "(x, y) \<in> r" and "(y, z) \<in> r\<^sup>+"
lm028: fixes X :: "('b \<times> 'a set) set" assumes "{} \<notin> Range X" shows "inj_on omega X"
Un_commute: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B = B \<union> A"
mult_nonneg_nonpos2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times b a) zero"
simps_3: fixes A :: "hfm"   and i :: "name"   and x :: "fm" shows "(HNeg A)(i:::=x) = HNeg (A(i:::=x))"
serial_sas_plus_equivalent_to_serial_strips_ii: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list" shows "concat (\<phi>\<^sub>P \<Psi> List_Supplement.embed \<psi>) = map (sasp_op_to_strips \<Psi>) \<psi>"
closed_superdiagonal: shows "closed {(x, y) |x y. y \<le> x}"
Vs'_1: shows "set Vs' \<subseteq> set Vs"
elims_1: fixes x :: "('b, 'c, 'd, 'e) Tree"   and xa :: "('b, 'c, 'd, 'e) Tree"   and y :: "bool" assumes "(x =\<^sub>\<alpha> xa) = y" obtains tset1 :: "('b, 'c, 'd, 'e) Tree set['b]" and tset2 :: "('b, 'c, 'd, 'e) Tree set['b]" where "x = tConj tset1" and "xa = tConj tset2" and "y = rel_bset (=\<^sub>\<alpha>) tset1 tset2"   | t1 :: "('b, 'c, 'd, 'e) Tree" and t2 :: "('b, 'c, 'd, 'e) Tree" where "x = tNot t1__" and "xa = tNot t2__" and "y = (t1__ =\<^sub>\<alpha> t2__)"   | f1 :: "'e" and \<phi>1 :: "'c" and f2 :: "'e" and \<phi>2 :: "'c" where "x = tPred f1__ \<phi>1" and "xa = tPred f2__ \<phi>2" and "y = (f1__ = f2__ \<and> \<phi>1 = \<phi>2)"   | f1 :: "'e" and \<alpha>1 :: "'d" and t1 :: "('b, 'c, 'd, 'e) Tree" and f2 :: "'e" and \<alpha>2 :: "'d" and t2 :: "('b, 'c, 'd, 'e) Tree" where "x = tAct f1 \<alpha>1 t1" and "xa = tAct f2 \<alpha>2 t2" and "y = (f1 = f2 \<and> (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)))"   | v :: "('b, 'c, 'd, 'e) Tree" and va :: "('b, 'c, 'd, 'e) Tree set['b]" where "x = tNot v__" and "xa = tConj va__" and "\<not> y"   | v :: "('b, 'c, 'd, 'e) Tree" and va :: "'e" and vb :: "'c" where "x = tNot vb__" and "xa = tPred vaa__ vba__" and "\<not> y"   | v :: "('b, 'c, 'd, 'e) Tree" and va :: "'e" and vb :: "'d" and vc :: "('b, 'c, 'd, 'e) Tree" where "x = tNot vc__" and "xa = tAct vab__ vbb__ vca__" and "\<not> y"   | v :: "'e" and va :: "'c" and vb :: "('b, 'c, 'd, 'e) Tree set['b]" where "x = tPred vd__ vac__" and "xa = tConj vbc__" and "\<not> y"   | v :: "'e" and va :: "'c" and vb :: "('b, 'c, 'd, 'e) Tree" where "x = tPred ve__ vad__" and "xa = tNot vbd__" and "\<not> y"   | v :: "'e" and va :: "'c" and vb :: "'e" and vc :: "'d" and vd :: "('b, 'c, 'd, 'e) Tree" where "x = tPred vf__ vae__" and "xa = tAct vbe__ vcb__ vda__" and "\<not> y"   | v :: "'e" and va :: "'d" and vb :: "('b, 'c, 'd, 'e) Tree" and vc :: "('b, 'c, 'd, 'e) Tree set['b]" where "x = tAct vg__ vaf__ vbf__" and "xa = tConj vcc__" and "\<not> y"   | v :: "'e" and va :: "'d" and vb :: "('b, 'c, 'd, 'e) Tree" and vc :: "('b, 'c, 'd, 'e) Tree" where "x = tAct vh__ vag__ vbg__" and "xa = tNot vcd__" and "\<not> y"   | v :: "'e" and va :: "'d" and vb :: "('b, 'c, 'd, 'e) Tree" and vc :: "'e" and vd :: "'c" where "x = tAct vi__ vah__ vbh__" and "xa = tPred vce__ vd" and "\<not> y"   | va :: "('b, 'c, 'd, 'e) Tree set['b]" and v :: "('b, 'c, 'd, 'e) Tree" where "x = tConj vai__" and "xa = tNot vj__" and "\<not> y"   | vb :: "('b, 'c, 'd, 'e) Tree set['b]" and v :: "'e" and va :: "'c" where "x = tConj vbi__" and "xa = tPred vk__ vaj__" and "\<not> y"   | vc :: "('b, 'c, 'd, 'e) Tree set['b]" and v :: "'e" and va :: "'d" and vb :: "('b, 'c, 'd, 'e) Tree" where "x = tConj vc" and "xa = tAct v va vb" and "\<not> y"
conjugate_sup: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "conjugate f (x \<squnion> y) = conjugate f x \<squnion> conjugate f y"
wf\<^sub>s\<^sub>t\<^sub>s'_def: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list" shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A> \<equiv> (\<forall>S\<in>\<S>. wf\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t \<A>) (dual\<^sub>s\<^sub>t S)) \<and> (\<forall>S\<in>\<S>. \<forall>S'\<in>\<S>. fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S' = {}) \<and> (\<forall>S\<in>\<S>. fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>e\<^sub>s\<^sub>t \<A> = {}) \<and> (\<forall>S\<in>\<S>. fv\<^sub>e\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>s\<^sub>t S = {})"
Rep_mod_ring_mod: fixes a :: "'b mod_ring" shows "Rep_mod_ring a mod int CARD('b) = Rep_mod_ring a"
of_nat_diff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" assumes "class.semiring_1_cancel plus minus zero one times"   and "n \<le> m" shows "semiring_1.of_nat one plus zero (m - n) = minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
Units_1: shows "invertible u1 \<and> u1 \<in> M"
disjoint: shows "(vars1 \<union> vars2) \<inter> vars3 = {}"
reversepath_linepath: fixes a :: "'a"   and b :: "'a" shows "reversepath (linepath a b) = linepath b a"
comp_null_2: fixes f :: "'c" shows "f \<cdot> C.null = C.null"
nn_integral_conv_infsetsum: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" assumes "f abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space A = ennreal (infsetsum f A)"
floor_exists1: fixes x :: "'b" shows "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
map_of_is_SomeI: fixes xys :: "('c \<times> 'd) list"   and x :: "'c"   and y :: "'d" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = Some y"
simps_2: fixes T :: "typ"   and body :: "Core.term"   and lev :: "nat"   and args :: "Core.term list" shows "subst_bvs1 (Abs T body) lev args = Abs T (subst_bvs1 body (lev + 1) args)"
partitioning_nottail_equi: fixes ss :: "'b set list"   and ts :: "'b set set" shows "partitioning_nontail ss ts = partitioning ss ts"
sel: fixes x :: "(('e, 'f) T\<^sub>m, 'f T\<^sub>h, 'g, 'h) F\<^sub>m" shows "the_G\<^sub>m (G\<^sub>m x) = x"
lemma_iod: fixes S :: "(nat \<Rightarrow> real) set"   and T :: "(nat \<Rightarrow> real) set"   and n :: "nat" assumes "S \<subseteq> T"   and "S \<noteq> {}"   and "T \<subseteq> topspace (Euclidean_space n)"   and "\<And>a b u. \<lbrakk>a \<in> S; b \<in> T; 0 < u; u < 1\<rbrakk> \<Longrightarrow> (\<lambda>i. (1 - u) * a i + u * b i) \<in> S" shows "path_connectedin (Euclidean_space n) T"
ht: fixes P :: "'s \<Rightarrow> real" assumes "unitary P" shows "unitary (t P)"
between_back: fixes v :: "'a"   and us :: "'a list"   and u :: "'a"   and vs :: "'a list" assumes "v \<notin> set us"   and "u \<notin> set vs"   and "v \<noteq> u" shows "between (v # vs @ u # us) u v = us"
assms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F G \<tau>"
PiE_cancel_merge: fixes I :: "'b set"   and J :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c set" assumes "I \<inter> J = {}" shows "(merge I J (x, y) \<in> Pi\<^sub>E (I \<union> J) B) = (x \<in> Pi I B \<and> y \<in> Pi J B)"
set_bn_to_atom_domA: fixes as :: "assn" shows "set (bn as) = atom ` domA (asToHeap as)"
order_sym_trans: fixes y :: "'b"   and x :: "'b"   and z :: "'b" assumes "y \<le> x"   and "sympartp (\<le>) y z" shows "z \<le> x"
wf_iff_no_infinite_down_chain: fixes r :: "('b \<times> 'b) set" shows "wf r = (\<nexists>f. \<forall>i. (f (Suc i), f i) \<in> r)"
lin: shows "linear (\<lambda>x. \<chi>i. if i = m then x $ m + x $ n else x $ i)"
well_valued0_t1: shows "well_valued0 (take j as) t1"
elims_2: fixes x :: "('a, 'b) Term.term literal list"   and xa :: "('a, 'c) Term.term literal list"   and xb :: "'b \<Rightarrow> ('a, 'c) Term.term option" assumes "subsumes_list x xa xb" obtains Ks :: "('a, 'c) Term.term literal list" and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option" where "x = []" and "xa = Ks__" and "xb = \<sigma>__"   | L :: "('a, 'b) Term.term literal" and Ls :: "('a, 'b) Term.term literal list" and Ks :: "('a, 'c) Term.term literal list" and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option" where "x = L # Ls" and "xa = Ks" and "xb = \<sigma>" and "\<exists>K\<in>set Ks. is_pos K = is_pos L \<and> (case match_term_list [(atm_of L, atm_of K)] \<sigma> of None \<Rightarrow> False | Some x \<Rightarrow> subsumes_list Ls (remove1 K Ks) x)"
divide_divide_eq_left': fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (divide a b) c = divide a (times c b)"
llast_singleton: fixes x :: "'b" shows "llast (LCons x LNil) = x"
dual_map_def: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b \<Rightarrow> real" shows "dual_map f y = restrict0 S (\<lambda>x. y (f x))"
obj_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>C a = a"
conc_epsilon_1: fixes A :: "'b list set" shows "{[]} @@ A = A"
beh_immediate_prefix_closed: fixes s :: "'e"   and b :: "'e list"   and T :: "('e, 'f) TS_scheme" assumes "s # b \<in> beh T" shows "b \<in> beh T"
i: assumes "inner x x = (0::'a)" shows "x = 0"
a_inv_def: fixes R :: "('a, 'b) ring_scheme" shows "a_inv R = m_inv (add_monoid R)"
t_utm_halt_eq: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "nat list"   and stp :: "nat"   and m :: "nat"   and rs :: "nat"   and n :: "nat"   and i :: "nat" assumes "tm_wf0 tp"   and "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"   and "0 < rs" shows "\<exists>stp m n. steps0 (Suc 0, [Bk], <[code tp, bl2wc (<lm>)]> @ Bk \<up> i) t_utm stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"
list_all_stake_least: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" assumes "\<exists>n. P (xs !! n)" shows "list_all (Not \<circ> P) (stake (LEAST n. P (xs !! n)) xs)"
reduce_basis_def: shows "local.reduce_basis = local.basis_reduction_main (True, 0, fs_init)"
minim_inField: fixes r :: "('d \<times> 'd) set"   and B :: "'d set" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "B \<noteq> {}" shows "minim r B \<in> Field r"
prj_cfun_def: shows "PRJ('e \<rightarrow> 'f) = decode_cfun oo PRJ('e\<^sub>\<bottom> \<rightarrow>! 'f)"
axioms_1: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "limit_cone J C D a \<chi>" shows "category C"
simps_2: fixes v :: "'a tree23"   and va :: "'a"   and vb :: "'a tree23s" shows "T_join_all (TTs v va vb) = T_join_adj (TTs v va vb) + T_join_all (join_adj (TTs v va vb)) + 1"
cases: fixes a :: "'d stream"   and A :: "'d set" assumes "a \<in> streams A" obtains aa :: "'d" and s :: "'d stream" where "a = aa ## s" and "aa \<in> A" and "s \<in> streams A"
fixing_swapidseq_decrease: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "swapidseq n p"   and "a \<noteq> b"   and "(Fun.swap a b id \<circ> p) a = a" shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)"
seqE: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "CopxC.seq g f" obtains "Cop.seq (fst g) (fst f)" and "C.seq (snd g) (snd f)"
sheaf_spec_morphisms_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set"   and V :: "'c set set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.sheaf_spec_morphisms R addition multiplication zero unit U V \<equiv> \<lambda>s\<in>comm_ring.sheaf_spec R addition multiplication zero unit U. restrict s V"
polyfun_extremal: fixes c :: "nat \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat"   and B :: "real" assumes "c k \<noteq> (0::'b)"   and "1 \<le> k"   and "k \<le> n" shows "\<forall>\<^sub>F z in at_infinity. B \<le> norm (\<Sum>i\<le>n. c i * z ^ i)"
sup2E: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "(A \<squnion> B) x y" obtains "A x y"   | "B x y"
neg_mono: fixes e :: "'a \<times> 'a"   and E :: "('a \<times> 'a) set" assumes "e \<in> E"   and "E \<subseteq> edges G - set (filter_IFS_no_violations G M edgesList)"   and "E \<noteq> {}"   and "\<not> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = {e} \<union> set (filter_IFS_no_violations G M edgesList)\<rparr>)" shows "\<not> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = E \<union> set (filter_IFS_no_violations G M edgesList)\<rparr>)"
SUP_cong_simp: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd"   and D :: "'c \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
monoid_isomorphism_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" shows "monoid_isomorphism \<eta> M composition unit M' composition' unit' \<equiv> (bijective_map \<eta> M M' \<and> monoid M composition unit) \<and> monoid M' composition' unit' \<and> monoid_isomorphism_axioms \<eta> M composition unit composition' unit'"
hyps: assumes "(f, g) \<in> set ps" shows "monom_mult ((1::'b) / lc (fst g)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst g)) (fst g) \<in> set (pdata_pairs_to_list ps)"
suminf_mult2: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "summable f" shows "suminf f * c = (\<Sum>n. f n * c)"
remove_vertex_walk_add: fixes G :: "('a, 'b) Graph_scheme"   and x :: "'a"   and xs :: "'a list" assumes "Digraph G"   and "Digraph.walk (Digraph.remove_vertex G x) xs" shows "Digraph.walk G xs"
list_all2_upt_append: fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "nat"   and b :: "nat"   and xs :: "'a list"   and c :: "nat"   and ys :: "'a list" assumes "list_all2 P [a..<b] xs"   and "list_all2 P [b..<c] ys"   and "a \<le> b"   and "b \<le> c" shows "list_all2 P [a..<c] (xs @ ys)"
eq: fixes x :: "'a" assumes "x \<in> Basis" shows "frechet_derivative (\<lambda>x. \<Sum>i\<in>Basis. v \<bullet> i * frechet_derivative f (at x) i) (at a) x = (\<Sum>j\<in>Basis. frechet_derivative (\<lambda>a. frechet_derivative f (at a) j) (at a) x * (v \<bullet> j))"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list"   and a2 :: "char list list" assumes "Subobjs\<^sub>R P a1 a2" obtains C :: "char list" where "a1 = Ca__" and "a2 = [Ca__]" and "is_class P Ca__"   | C :: "char list" and D :: "char list" and Cs :: "char list list" where "a1 = C" and "a2 = C # Cs" and "P \<turnstile> C \<prec>\<^sub>R D" and "Subobjs\<^sub>R P D Cs"
prems_5: shows "x < (a # l) ! (length (a # l) - 1)"
norm_imp_pos_and_ge: fixes x :: "'a"   and n :: "real" assumes "norm x \<equiv> n" shows "0 \<le> norm x \<and> norm x \<le> n"
axioms_2: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "well_ordered_set A less_eq" shows "well_related_set A less_eq"
extend_in_C_Neg_Forall: fixes C :: "(nat, 'a) form set set"   and S :: "(nat, 'a) form set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat" assumes "alt_consistency C"   and "infinite (- \<Union> (params ` S))"   and "extend S C f n \<union> {f n} \<in> C"   and "\<forall>p. f n \<noteq> Exists p"   and "\<exists>p. f n = Neg (Forall p)" shows "extend S C f (Suc n) \<in> C"
assms_2: fixes I :: "'b"   and a :: "'a"   and b :: "'a"   and J :: "'b"   and c :: "'a"   and d :: "'a" assumes "Q I a b"   and "Q J c d"   and "[[a b c d]]" shows "P I J"
llcp_ltake2: fixes xs :: "'c llist"   and n :: "enat"   and ys :: "'c llist" shows "llcp xs (ltake n ys) = min n (llcp xs ys)"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (logical_instr_sub1 instr_name result s1)) \<and> t2 = snd (fst (logical_instr_sub1 instr_name result s2))"
rmge2: fixes xs :: "nat list"   and e :: "nat" assumes "xs = Partial_Recursive.list_decode e" shows "rmge2 e = (if \<forall>i<length xs. xs ! i < 2 then Some 0 else Some (xs ! (GREATEST i. i < length xs \<and> 2 \<le> xs ! i)))"
bind_rule: fixes P :: "assn"   and f :: "'a Heap"   and R :: "'a \<Rightarrow> assn"   and g :: "'a \<Rightarrow> 'b Heap"   and Q :: "'b \<Rightarrow> assn" assumes "<P> f <R>"   and "\<And>x. <R x> g x <Q>" shows "<P> f \<bind> g <Q>"
simps_3: fixes h :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and a :: "'a"   and xs :: "'b list" shows "Logic.interpret h (Atom a) xs = h a xs"