hcju/isabelleps huggingface dataset:
pair_is_static_def: fixes p :: "'a \<times> 'b" shows "is_static p = is_static (snd p)"
sel_4: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.LB (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x4"
kmp_correct: fixes s :: "'a list"   and t :: "'a list" assumes "s \<noteq> []" shows "kmp s t \<le> kmp_SPEC s t"
swap: fixes A :: "((real, 'n) vec, 'n) vec"   and m :: "'n"   and n :: "'n" assumes "P A"   and "m \<noteq> n" shows "P (\<chi>i j. A $ i $ Fun.swap m n id j)"
simps_1: fixes r :: "'a rexp" shows "pderivs [] r = {r}"
CONSTRAINTI: fixes f :: "'a"   and R :: "('b \<times> 'a) set" shows "CONSTRAINT f R"
reduction_correspondence_simulation_condition_trans: fixes Q :: "'b"   and Cal :: "'b processCalculus"   and P :: "'b"   and Rel :: "('b \<times> 'b) set"   and R :: "'b" assumes "\<forall>Q'. Q \<longmapsto>Cal* Q' \<longrightarrow> (\<exists>P'' Q''. P \<longmapsto>Cal* P'' \<and> Q' \<longmapsto>Cal* Q'' \<and> (P'', Q'') \<in> Rel)"   and "\<forall>R'. R \<longmapsto>Cal* R' \<longrightarrow> (\<exists>Q'' R''. Q \<longmapsto>Cal* Q'' \<and> R' \<longmapsto>Cal* R'' \<and> (Q'', R'') \<in> Rel)"   and "weak_reduction_simulation Rel Cal"   and "trans Rel" shows "\<forall>R'. R \<longmapsto>Cal* R' \<longrightarrow> (\<exists>P'' R''. P \<longmapsto>Cal* P'' \<and> R' \<longmapsto>Cal* R'' \<and> (P'', R'') \<in> Rel)"
Field_converse: fixes r :: "('a \<times> 'a) set" shows "Field (r\<inverse>) = Field r"
smult_closed: fixes H :: "'e set"   and R :: "('f, 'g) ring_scheme"   and M :: "('f, 'e, 'h) module_scheme"   and a :: "'f"   and x :: "'e" assumes "submodule H R M"   and "a \<in> carrier R"   and "x \<in> H" shows "a \<odot>\<^bsub>M\<^esub> x \<in> H"
prefix_match_semantics_ipset_from_netmask2: fixes pfx :: "'b prefix_match"   and a :: "'b word" assumes "valid_prefix pfx" shows "prefix_match_semantics pfx a = (a \<in> ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx))"
psp: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "F \<in> B co B'" shows "F \<in> A \<inter> B' \<longmapsto> A' \<inter> B \<union> (B' - B)"
not_learn_total_V: fixes s :: "nat \<Rightarrow> nat option" assumes "total_cons s" shows "\<not> learn_total \<phi> V_constotal s"
test_inf_closed_sup_complement: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y"   and "\<forall>u v. test u \<and> test v \<longrightarrow> test (u \<sqinter> v)" shows "! x \<sqinter> ! y \<sqinter> (x \<squnion> y) = bot"
simps_4: fixes b :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and X :: "char list set" shows "bury (IF b THEN c\<^sub>1 ELSE c\<^sub>2) X = IF b THEN bury c\<^sub>1 X ELSE bury c\<^sub>2 X"
confluentp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "confluentp r \<equiv> \<forall>x y1 y2. r\<^sup>*\<^sup>* x y1 \<longrightarrow> r\<^sup>*\<^sup>* x y2 \<longrightarrow> (\<exists>z. r\<^sup>*\<^sup>* y1 z \<and> r\<^sup>*\<^sup>* y2 z)"
evenperm_swap: fixes a :: "'b"   and b :: "'b" shows "evenperm (Fun.swap a b id) = (a = b)"
weak_is_transitive: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier \<and> y \<in> carrier \<and> z \<in> carrier"   and "x \<succeq> y"   and "y \<succeq> z" shows "x \<succeq> z"
kcomp_isor: fixes f :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b set"   and h :: "'b \<Rightarrow> 'c set" assumes "f \<le> g" shows "f \<circ>\<^sub>K h \<le> g \<circ>\<^sub>K h"
cnf_cnf: fixes F :: "'b formula" assumes "is_nnf F" shows "cnf (cnf_form_of F) = cnf F"
no_defection_empty: fixes s :: "v_state"   and r' :: "nat" shows "no_defection s Map.empty r'"
WT_red_external_aggr_imp_red_external: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and ta :: "('b, 'c, 'd) external_thread_action"   and va :: "'b extCallRet"   and h' :: "'d"   and addr2thread_id :: "'b \<Rightarrow> 'c"   and thread_id2addr :: "'c \<Rightarrow> 'b"   and spurious_wakeups :: "bool"   and empty_heap :: "'d"   and allocate :: "'d \<Rightarrow> htype \<Rightarrow> ('d \<times> 'b) set"   and typeof_addr :: "'d \<Rightarrow> 'b \<Rightarrow> htype option"   and heap_read :: "'d \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> bool"   and heap_write :: "'d \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c"   and a :: "'b"   and M :: "String.literal"   and vs :: "'b val list"   and h :: "'d"   and U :: "ty" assumes "wf_prog wf_md P"   and "(ta, va, h') \<in> heap_base.red_external_aggr addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t a M vs h"   and "heap_base.external_WT' typeof_addr P h a M vs U"   and "heap_base.tconf thread_id2addr typeof_addr P h t" shows "heap_base.red_external addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t h a M vs ta va h'"
a: shows "infinite {\<phi>'. \<exists>\<phi>''\<in>reachable g \<phi>. phiArg g \<phi>'' \<phi>'}"
path_independent_on_pseudo_rationalizable_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and Rs :: "('a \<times> 'a) set set" assumes "path_independent_on A f"   and "f_range_on A f"   and "Rs = (\<Union>C\<in>Pow A. linord_of_list ` mk_linear_orders f C A)" shows "pseudo_rationalizable_on A Rs f \<and> (\<forall>r\<in>Rs. (refl_on A r \<and> total_on A r) = decisive_on A f)"
reachable_snapshot_inv_white_root: fixes w :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and m :: "'f" assumes "white w s"   and "w \<in> roots (s (mutator m)) \<or> w \<in> ghost_honorary_root (s (mutator m))"   and "mut_m.reachable_snapshot_inv m s" shows "\<exists>g. (g grey_protects_white w) s"
simps_1: fixes x :: "'a" shows "0 \<times> x = x"
mult_1_left: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_algebra.comp_inf.covector a"
rel_cases: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "('d, 'f, 'h) gpv"   and b :: "('e, 'g, 'h) gpv" assumes "rel_gpv R1 R2 a b" obtains xa :: "('d, 'f, ('d, 'f, 'h) rpv) generat spmf" and ya :: "('e, 'g, ('e, 'g, 'h) rpv) generat spmf" where "a = GPV xa" and "b = GPV ya" and "rel_spmf (rel_generat R1 R2 (rel_rpv R1 R2)) xa ya"
card_face_cycle_sets_da: shows "card H.face_cycle_sets = Suc (card G.face_cycle_sets)"
G_face_cycle_succ_a_no_loop: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc_head_isolated G M a"   and "head G a \<noteq> tail G a" shows "pre_digraph_map.face_cycle_succ M a = edge_rev M a"
card_of_Un_infinite: fixes A :: "'b set"   and B :: "'b set" assumes "infinite A"   and "|B| \<le>o |A|" shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
Y'_AE: shows "AE \<omega> in restrict_space lborel {0<..<1}. Y' \<omega> = M.I \<omega>"
LNil_eq_lmap: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "([]\<^sub>l = lmap f xs) = (xs = []\<^sub>l)"
set_intros_25: fixes aac :: "'a"   and x71 :: "'a list"   and x72 :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and x73 :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "aac \<in> set x71" shows "aac \<in> vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
vsv_vimage_vinsert: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" shows "f `\<^sub>\<circ> vinsert a A = vinsert (f\<lparr>a\<rparr>) (f `\<^sub>\<circ> A)"
\<Q>SucEq: fixes k :: "nat" assumes "length Doc \<le> k" shows "\<Q> (Suc k) = \<Q> k"
subprob_space_measure_spmf: fixes p :: "'c spmf" shows "subprob_space (measure_spmf p)"
ereal_divide_eq_0_iff: fixes x :: "ereal"   and y :: "ereal" shows "(x / y = 0) = (x = 0 \<or> \<bar>y\<bar> = \<infinity>)"
assms_2: shows "dom subs \<subseteq> tvsT T"
src_simps_3: fixes F :: "('a E.term, 'a) arr" assumes "arr F" shows "Map (src F) = EVAL (E.Src (Dom F))"
x_power_aq_minus_1_rw: fixes x :: "nat"   and a :: "nat"   and b :: "'b"   and q :: "nat" assumes "1 < x"   and "0 < a"   and "(0::'b) < b" shows "x ^ (a * q) - 1 = (x ^ a - 1) * sum ((^) (x ^ a)) {..<q}"
normal_imp_not_denormal: fixes f :: "('a, 'b) IEEE.float" assumes "is_normal f" shows "\<not> is_denormal f"
comp_lunit_lunit'_1: fixes a :: "'d" assumes "D.ide a" shows "\<l>\<^sub>1[a] \<cdot>\<^sub>D D.C\<^sub>1.lunit' a = a"
refine: fixes G :: "('a, 'b) igb_graph_rec_scheme"   and Gi :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme"   and Re :: "('d \<times> 'b) set"   and Rv :: "('c \<times> 'a) set"   and x :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme \<Rightarrow> 'e"   and x' :: "('a, 'b) igb_graph_rec_scheme \<Rightarrow> 'f"   and Re' :: "('e \<times> 'f) set" assumes "igb_graph G"   and "(Gi, G) \<in> igbg_impl_rel_ext Re Rv"   and "(x, x') \<in> igbg_impl_rel_ext Re Rv \<rightarrow> Re'" shows "(degeneralize_ext_impl Gi x, igb_graph.degeneralize_ext G x') \<in> bg_impl_rel_ext Re' (Rv \<times>\<^sub>r nat_rel)"
supp_Abs_bset: fixes A :: "'a set" assumes "|A| <o natLeq +c |UNIV|" shows "supp (Abs_bset A) = supp A"
interval_cbox: fixes a :: "'b"   and b :: "'b" shows "{a..b} = cbox a b"
lunit_simps_1: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "arr \<l>[f]"
subdegree_fps_of_poly: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "subdegree (fps_of_poly p) = order (0::'b) p"
intro\<^sub>3: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'" shows "\<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"
Seq_transC_invert: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com"   and s :: "'b"   and c' :: "('a, 'c) com"   and s' :: "'b" assumes "PL.transC tval aval (c1 ;; c2, s) (c', s')" obtains c1' :: "('a, 'c) com" where "PL.transC tval aval (c1, s) (c1', s')" and "c' = c1' ;; c2"   | "PL.transT tval aval (c1, s) s'" and "c' = c2"
m2r_pred_def: fixes p :: "'d \<Rightarrow> 'd option \<Rightarrow> bool" shows "m2r_pred p \<equiv> \<lambda>x. case x of None \<Rightarrow> \<lambda>_. False | Some a \<Rightarrow> p a"
strict_mono_on_imp_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "SetInterval2.strict_mono_on f A" shows "inj_on f A"
inverse_p_def: fixes p :: "int"   and x :: "int" shows "inverse_p p x = (if x = 0 then 0 else power_p p x (nat (p - 2)))"
finite_S: shows "finite (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
perm_restrict_def: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and x :: "'e" shows "perm_restrict f S x \<equiv> if x \<in> S then f x else x"
Fourier_sum_limit_Dirichlet_kernel_half: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l = (\<lambda>n. LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f (t + x) + f (t - x) - 2 * l)) \<longlonglongrightarrow> 0"
poly_add_degree: fixes p1 :: "'a list list"   and p2 :: "'a list list" shows "degree (UP.poly_add p1 p2) \<le> max (degree p1) (degree p2)"
vinsertCI: fixes x :: "V"   and A :: "V"   and y :: "V" assumes "x \<notin>\<^sub>\<circ> A \<Longrightarrow> x = y" shows "x \<in>\<^sub>\<circ> vinsert y A"
assms_5: shows "card (selected_part Sel P2) = Suc n"
get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes element_ptr :: "(_) element_ptr"   and uu :: "unit" shows "get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr uu = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr child_nodes"
Par2B_1: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P \<parallel> Q'"
fake_mono_bad: fixes bad :: "agent set"   and bad' :: "agent set"   and IK :: "msg set"   and chan :: "chan set" assumes "bad \<subseteq> bad'" shows "fake bad IK chan \<subseteq> fake bad' IK chan"
prems_2: shows "set (x # xs) \<subseteq> space C"
measure_begin_def: shows "measure_begin = measures [measure_begin_state, measure_begin_step]"
random_oracle_def: fixes \<sigma> :: "'a \<Rightarrow> bool list option"   and x :: "'a" shows "PRF.random_oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV prf_clen) \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
vsubset_vsingleton_leftI: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "ZFC_in_HOL.set {a} \<subseteq>\<^sub>\<circ> A"
sub_fun: fixes T1 :: "ty"   and T1' :: "ty"   and T2 :: "ty"   and T2' :: "ty" assumes "T1 <: T1'"   and "T1' <: T1"   and "T2 <: T2'"   and "T2' <: T2" shows "T1 \<rightarrow> T2 <:: T1' \<rightarrow> T2'"
multisetUnionLeMono: fixes r :: "('b \<times> 'b) set"   and a1 :: "'b multiset"   and a2 :: "'b multiset"   and b1 :: "'b multiset"   and b2 :: "'b multiset" assumes "trans r"   and "multiset_le a1 a2 r"   and "multiset_le b1 b2 r" shows "multiset_le (a1 + b1) (a2 + b2) r"
pair_fun_components: fixes C :: "'a \<Rightarrow> real \<times> real" shows "C = (\<lambda>x. (C x \<bullet> i, C x \<bullet> j))"
pow_cong: fixes x :: "int"   and n :: "nat" shows "(x mod m) [^] n = x ^ n mod m"
submanifold_axioms_def: fixes S :: "'c set" shows "submanifold_axioms S \<equiv> open S"
Texp_subst: fixes e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var"   and a :: "Arity" shows "Texp e[y::=x]\<cdot>a \<sqsubseteq> many_calls x \<otimes>\<otimes> without y (Texp e\<cdot>a)"
successively_if_sorted_wrt: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" assumes "sorted_wrt P xs" shows "successively P xs"
inj_term_simps_2: fixes e :: "expr" shows "\<langle>e\<rangle> = In1l e"
cone: shows "D.cone (C.dom e') (D.mkCone e')"
bspec: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
circline_equation: fixes H :: "circline"   and A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex"   and z :: "complex" assumes "H = mk_circline A B C D"   and "(A, B, C, D) \<in> hermitean_nonzero" shows "(of_complex z \<in> circline_set H) = circline_equation A B C D z 1"
isSuccOrd_def: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "isSuccOrd r \<equiv> \<exists>j\<in>Field r. \<forall>i\<in>Field r. (i, j) \<in> r"
bij_f_o_inf_f: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "f \<circ> inv f = id"
list_slice_div_eq_1: fixes xs :: "'b list"   and k :: "nat" assumes "length xs div k = Suc 0" shows "list_slice xs k = [xs \<down> k]"
nrcons_inv_1: fixes n :: "nat"   and t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_leveln_ftree n t"   and "is_leveln_node n nd" shows "is_leveln_ftree n (nrcons t nd)"
consistent_mergeSibling: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t" shows "consistent (mergeSibling t a)"
subset_box_complex_1: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" shows "(cbox a b \<subseteq> cbox c d) = (Re a \<le> Re b \<and> Im a \<le> Im b \<longrightarrow> Re c \<le> Re a \<and> Im c \<le> Im a \<and> Re b \<le> Re d \<and> Im b \<le> Im d)"
simps_1: fixes \<alpha> :: "'b \<Rightarrow> 'c set \<times> 'c set"   and w :: "'b list" shows "[] \<otimes>\<^bsub>\<alpha>\<^esub> w = {w}"
linear_construct: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "vector_space_pair s1 s2"   and "\<not> module.dependent s1 B" shows "linear s1 s2 (vector_space_pair.construct s1 s2 B f)"
domains_char': fixes f :: "Arr" shows "domains f = (if f = Null then {} else {Dom f})"
empty: shows "Lcm\<^sub>f\<^sub>i\<^sub>n {} = (1::'b)"
has_integral_eq: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and k :: "'c" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"   and "(f has_integral k) s" shows "(g has_integral k) s"
neg_divide_less_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less (divide b c) a = less (times a c) b"
set_cases_3: fixes e :: "'d"   and a :: "('e, 'f, 'd) generat" assumes "e \<in> generat_conts a" obtains z1 :: "'f" where "a = IO z1 e"
parallel_append: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list"   and d :: "'c list" assumes "a \<parallel> b" shows "a @ c \<parallel> b @ d"
rev_sub_to_poly: fixes g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "g \<in> carrier P"   and "a \<in> carrier R" shows "rev_compose R g (to_polynomial R a) = to_polynomial R a"
bidel_arc_axioms_def: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" shows "bidel_arc_axioms G a \<equiv> a \<in> arcs G"
pm_def: shows "pm = root (traverse_tree path (stern_brocot_iterate_aux I))"
strips_state_to_state_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<times> 'd \<Rightarrow> bool option" shows "\<phi>\<^sub>S\<inverse> \<Psi> s \<equiv> map_of (filter (\<lambda>(v, a). s (v, a) = Some True) (all_possible_assignments_for \<Psi>))"
conforms_newG: fixes s :: "st"   and oref :: "oref"   and x :: "abopt"   and G :: "prog"   and L :: "lenv"   and oi :: "obj_tag"   and vs :: "vn \<Rightarrow> val option" assumes "globs s oref = None"   and "(x, s)\<Colon>\<preceq>(G, L)"   and "wf_prog G"   and "case oref of Inl a \<Rightarrow> is_type G (obj_ty \<lparr>tag = oi, values = vs\<rparr>) | Inr C \<Rightarrow> is_class G C" shows "(x, init_obj G oi oref s)\<Colon>\<preceq>(G, L)"
hermitian_testN: fixes k :: "nat" shows "hermitian (testN k)"
sum_in_Reals: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "sum f s \<in> \<real>"
simps_2: fixes l :: "('b \<times> 'c) tree"   and a :: "'b"   and uu :: "'c"   and r :: "('b \<times> 'c) tree" shows "Tree2.inorder \<langle>l, (a, uu), r\<rangle> = Tree2.inorder l @ a # Tree2.inorder r"
dom_m_fmrestrict_set': fixes xs :: "'a set"   and N :: "('a, 'b) fmap" shows "dom_m (fmrestrict_set xs N) = mset_set (xs \<inter> set_mset (dom_m N))"
geod_upper: fixes a :: "nat"   and b :: "nat"   and n :: "nat" assumes "a \<le> b"   and "b \<le> 2 ^ n" shows "dist (geod n a) (geod n b) \<le> (real b - real a) * dist x0 y0 / 2 ^ n"
assms_2: assumes "\<gamma> t n" shows "\<gamma>'' t n"
four_distinct_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[Contingent F \<^bold>\<rightarrow> \<^bold>\<not>(\<^bold>\<exists>G. NonContingent G \<^bold>& G \<^bold>= F) in v]"
g1: shows "(g1 \<longlongrightarrow> b1) (at s within S)"
subtype_refl': fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile> T <: T"
assms: fixes p :: "real poly" assumes "p \<in> set fs" shows "poly p x \<noteq> 0"
lmeasurable_cball: fixes a :: "'b"   and r :: "real" shows "cball a r \<in> lmeasurable"
mat_mult: fixes m :: "(('c, 'e) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and m' :: "(('c, 'd) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "($h) (m ** m') = mat_mult_row m m'"
CFG_SSA_axioms_def: fixes \<alpha>n :: "'f \<Rightarrow> 'g list"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and phis :: "'f \<Rightarrow> 'g \<times> 'i \<Rightarrow> 'i list option" shows "CFG_SSA_axioms \<alpha>n inEdges' defs phis \<equiv> ((\<forall>g. finite (dom (phis g))) \<and> (\<forall>g n v vs. phis g (n, v) = Some vs \<longrightarrow> n \<in> set (\<alpha>n g))) \<and> (\<forall>g n v args. phis g (n, v) = Some args \<longrightarrow> length (graph_path_base.predecessors inEdges' g n) = length args) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> defs g n \<inter> CFG_SSA_base.phiDefs phis g n = {}) \<and> (\<forall>n g m. n \<in> set (\<alpha>n g) \<longrightarrow> m \<in> set (\<alpha>n g) \<longrightarrow> n \<noteq> m \<longrightarrow> CFG_SSA_base.allDefs defs phis g n \<inter> CFG_SSA_base.allDefs defs phis g m = {})"
ah_update_cons: fixes e :: "'a set \<times> 'a set"   and w :: "('a set \<times> 'a set) list" shows "\<alpha>ah (e # w) = ah_update (\<alpha>ah w) e (mon_pl w)"
Body_no_jump: fixes G :: "prog"   and s0 :: "state"   and D :: "qtname"   and c :: "stmt"   and v :: "val"   and s1 :: "state"   and j :: "jump" assumes "G\<turnstile>s0 \<midarrow>Body D c-\<succ>v\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)" shows "fst s1 \<noteq> Some (Jump j)"
emeasure_uniform_measure: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"