hcju/isabelleps huggingface dataset:
inject: fixes x :: "(((common_metadata, common_metadata) blindable\<^sub>m \<times> (participant_metadata, participant_metadata) blindable\<^sub>m) \<times> view\<^sub>m list, (common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>m"   and ya :: "(((common_metadata, common_metadata) blindable\<^sub>m \<times> (participant_metadata, participant_metadata) blindable\<^sub>m) \<times> view\<^sub>m list, (common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>m" shows "(Transaction\<^sub>m x = Transaction\<^sub>m ya) = (x = ya)"
openin_subtopology_alt: fixes X :: "'b topology"   and U :: "'b set"   and S :: "'b set" shows "openin (subtopology X U) S = (S \<in> (\<inter>) U ` Collect (openin X))"
assms_2: shows "(c, d) \<in> Rs \<rightarrow> bool_rel"
primitive_extractor_correct_9: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)"   and "has_disc disc m" shows "as \<noteq> []"
in_hhom_def: fixes \<mu> :: "'c"   and a :: "'c"   and b :: "'c" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C b\<guillemotright> \<equiv> C.arr \<mu> \<and> src\<^sub>C \<mu> = a \<and> trg\<^sub>C \<mu> = b"
rot_diamond_gen_div_diamond: shows "gen_division (cubeImage diamond_cube) (cubeImage ` {rot_diamond_cube})"
path_image_linepath: fixes a :: "'a"   and b :: "'a" shows "path_image (linepath a b) = closed_segment a b"
distinct_38: fixes x7 :: "CTypeId" shows "CClassT x7 \<noteq> ShortT"
equiv_funI: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "\<exists>c\<^sub>1>0. \<exists>c\<^sub>2>0. \<exists>n. \<forall>m>n. f m \<le> c\<^sub>1 * g m \<and> g m \<le> c\<^sub>2 * f m" shows "f \<cong> g"
K_record_comp: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'c" shows "(\<lambda>x. c) \<circ> f = (\<lambda>x. c)"
ref_simps_4: fixes T :: "Arraytype"   and a :: "nat"   and i :: "nat" shows "ref (arrLoc T a i) = arrV T a"
simps_1: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c" shows "update_by_fun_pair k f [] = (let v = f (0::'c) in if v = (0::'c) then [] else [(k, v)])"
microstep_rtranscl_equal: fixes p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and q :: "('a, 'b, 'c, 'd) seqp" assumes "not_call p"   and "not_choice p"   and "p \<leadsto>\<^bsub>\<Gamma>\<^esub>\<^sup>* q" shows "q = p"
bisim1Sync6: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and v :: "'addr val"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Val v, xs) \<leftrightarrow> ([v], xs, 5 + length (compE2 e1) + length (compE2 e2), None)"
d2_def: shows "ps_P.d2 = prod_list ps_P.dims2"
conj_lin: fixes p :: "fm"   and q :: "fm" assumes "islin p"   and "islin q" shows "islin (Parametric_Ferrante_Rackoff.conj p q)"
delta': fixes S :: "'b set"   and a :: "'b"   and b :: "'b \<Rightarrow> 'a" assumes "finite S" shows "(\<Sum>k\<in>S. if a = k then b k else (0::'a)) = (if a \<in> S then b a else (0::'a))"
ffield_def: fixes r :: "V" shows "\<F>\<^sub>\<bullet> r = \<D>\<^sub>\<bullet> r \<union>\<^sub>\<circ> \<R>\<^sub>\<bullet> r"
bigo_fix: fixes f :: "nat \<Rightarrow> 'c"   and h :: "nat \<Rightarrow> 'c" assumes "(\<lambda>x. f (x + 1)) \<in> O(\<lambda>x. h (x + 1))"   and "f 0 = (0::'c)" shows "f \<in> O(h)"
elims: fixes x :: "'b Trace"   and y :: "nat" assumes "tLength x = y" obtains s :: "'b" where "x = tInit s__" and "y = 0"   | t :: "'b Trace" and s :: "'b" where "x = t \<leadsto> s" and "y = 1 + tLength t"
preserves_inv: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "T (CC.inv f) = local.inv (T f)"
space_closed: fixes M :: "'b measure" shows "sets M \<subseteq> Pow (space M)"
i1: shows "local.adverse 1 i j \<triangleright> Suc xmin = e_snoc (local.adverse z i j \<triangleright> xmin) 1"
1: fixes n :: "nat" assumes "odd n" shows "\<exists>i. odd i \<and> real n = real_of_int i"
get_shadow_root_new_element: fixes ptr' :: "(_) element_ptr"   and new_element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "r \<in> get_shadow_root_locs ptr'" shows "r h h'"
union_single_eq_member: fixes x :: "'d"   and M :: "'d multiset"   and N :: "'d multiset" assumes "add_mset x M = N" shows "x \<in># N"
cHeap_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "cHeap \<Gamma> e = (\<Lambda> a. ae2ce (Aheap \<Gamma> e\<cdot>a))"
x: shows "x \<in> {(ram1, hd vs), (last vs, ram2), (ram2, ram1)} \<union> Edges vs"
pointermap_p_valid_updateI: fixes m :: "('a, 'b) pointermap_scheme"   and a :: "'a"   and u :: "'a pointermap"   and p :: "nat" assumes "pointermap_sane m"   and "getentry m a = None"   and "u = pointermap_insert a m"   and "p = the (getentry u a)" shows "pointermap_p_valid p u"
dom_def: fixes f :: "'a" shows "local.dom f = (if domains f \<noteq> {} then SOME a. a \<in> domains f else null)"
inter_eqvt: fixes p :: "perm"   and A :: "'a set"   and B :: "'a set" shows "p \<bullet> (A \<inter> B) = p \<bullet> A \<inter> p \<bullet> B"
borel_measurableI_less: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "\<And>y. {x \<in> space M. f x < y} \<in> sets M" shows "f \<in> borel_measurable M"
rfb_sup_dual: shows "sup_dual fb\<^sub>\<R>"
INF_le_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 f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'b" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq (Inf (f ` A)) x = (\<forall>y. less x y \<longrightarrow> (\<exists>i\<in>A. less (f i) y))"
next_change_multiplicity'_def: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" shows "next_change_multiplicity' c0 c1 loc t n \<equiv> n \<noteq> 0 \<and> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t) \<and> c1 = c0\<lparr>c_pts := (c_pts c0)(loc := update_zmultiset (c_pts c0 loc) t n), c_work := (c_work c0)(loc := c_work c0 loc + frontier_changes (update_zmultiset (c_pts c0 loc) t n) (c_pts c0 loc))\<rparr>"
simps_2: fixes b :: "bexp"   and res :: "bool"   and S :: "'av Abs_State.st option" shows "bfilter (bexp.Not b) res S = bfilter b (\<not> res) S"
current_set_error_code: shows "\<forall>s a. current (set_error_code s a) = current s"
pair_sym_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs (subdivide G e w)"   and "w \<notin> pverts (subdivide G e w)" shows "pair_sym_digraph (subdivide (subdivide G e w) e w)"
analz_NI_I_K_analz_NI_EI: fixes NI :: "msg set"   and K :: "msg set"   and I :: "msg set" assumes "NI \<subseteq> payload"   and "K \<subseteq> range LtK"   and "I \<subseteq> valid" shows "analz (NI \<union> I \<union> K \<union> Tags) \<subseteq> synth (analz (NI \<union> extractable K I \<union> K \<union> Tags)) \<union> - payload"
insort_insert_insort_key: fixes f :: "'c \<Rightarrow> 't"   and x :: "'c"   and xs :: "'c list" assumes "f x \<notin> f ` set xs" shows "ord_term_lin.insort_insert_key f x xs = ord_term_lin.insort_key f x xs"
Trans: shows "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"
chordal_circle_cvec_cmat_def: fixes a :: "complex \<times> complex"   and r :: "real" shows "chordal_circle_cvec_cmat a r = (let (a1, a2) = a in (4 * a2 * cnj a2 - (cor r)\<^sup>2 * (a1 * cnj a1 + a2 * cnj a2), - 4 * a1 * cnj a2, - 4 * cnj a1 * a2, 4 * a1 * cnj a1 - (cor r)\<^sup>2 * (a1 * cnj a1 + a2 * cnj a2)))"
l: shows "llength (llist_of p) \<le> 2 * N"
continuous_imp_bigo_1: fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous (at x within A) f" shows "f \<in> O[at x within A](\<lambda>_. 1::'b)"
ffun_minus_plus: fixes f :: "('a, 'b) ffun"   and g :: "('a, 'b) ffun" assumes "fdom f \<inter> fdom g = {}" shows "f + g - g = f"
less_oSuc_eq_le: fixes x :: "ordinal"   and y :: "ordinal" shows "(x < oSuc y) = (x \<le> y)"
set_integrable_cong_AE: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x\<in>A in M. f x = g x"   and "A \<in> sets M" shows "set_integrable M A f = set_integrable M A g"
double_fold_D0: fixes v :: "'a" assumes "v \<in> D0 - C0" shows "g (f v) = v"
sasp_op_to_strips_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op :: "('a, 'b) sas_plus_operator" shows "\<phi>\<^sub>O \<Psi> op \<equiv> let pre = sas_plus_operator.precondition_of op; add = effect_of op in Let (concat (map (\<lambda>(v, a). map (Pair v) (filter ((\<noteq>) a) (the (range_of \<Psi> v)))) (effect_of op))) (operator_for pre add)"
eq_from_inner_prod: fixes v :: "complex Matrix.vec"   and w :: "complex Matrix.vec" assumes "dim_vec v = dim_vec w"   and "\<langle>v|w\<rangle> = 1"   and "\<langle>v|v\<rangle> = 1"   and "\<langle>w|w\<rangle> = 1" shows "v = w"
mkM_simps_1: fixes T :: "'s Trace set" shows "worlds (Clock.mkM T) = T"
P1: shows "Bet A C E \<and> Cong C E A C"
productdivides: fixes x :: "nat"   and a :: "nat"   and b :: "nat" assumes "x mod a = 0"   and "x mod b = 0"   and "prime a"   and "prime b"   and "a \<noteq> b" shows "x mod (a * b) = 0"
in_out_arcs_blocks: fixes v :: "'a" shows "arc_to_block ` in_arcs G v = arc_to_block ` out_arcs G v"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceHom smult V smult T"   and "VectorSpaceEnd_axioms V T" shows "VectorSpaceEnd smult V T"
assms_3: fixes t :: "'a ttree"   and t' :: "'a ttree"   and x :: "'a" assumes "P t t'"   and "possible t x"   and "possible t' x" shows "P (nxt t x) (nxt t' x)"
semiring_simprules_3: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
prod_zero_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.semidom plus minus zero times one"   and "finite A" shows "(comm_monoid_mult.prod times one f A = zero) = (\<exists>a\<in>A. f a = zero)"
XY: shows "simple_distributed M (\<lambda>x. (X x, Y x)) Pxy"
commute: fixes a :: "'a"   and b :: "'a" shows "ord.min (\<sqsubseteq>) a b = ord.min (\<sqsubseteq>) b a"
filters_upsets: fixes X :: "'a set" assumes "filterp X" shows "X \<in> upsets"
case_split_left: fixes w :: "'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "reflexive (w \<squnion> z)"   and "w * x \<le> y"   and "z * x \<le> y" shows "x \<le> y"
simps_2: fixes uv :: "'a tree23"   and uw :: "'a"   and ts :: "'a tree23s" shows "len (TTs uv uw ts) = len ts + 1"
lookup_except_when: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" shows "lookup (except p S) = (\<lambda>t. lookup p t when t \<notin> S)"
ffb_g_ode: fixes f :: "'a \<Rightarrow> 'a"   and T :: "real set"   and S :: "'a set"   and \<phi> :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> bool"   and Q :: "'a set" assumes "local_flow f T S \<phi>" shows "fb\<^sub>\<F> x\<acute>=\<lambda>t. f & G on \<lambda>s. T S @ 0 Q = {s. s2p S s \<longrightarrow> (\<forall>t\<in>T. (\<forall>\<tau>\<in>down T t. G (\<phi> \<tau> s)) \<longrightarrow> s2p Q (\<phi> t s))}"
Sup_less_least: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and w :: "'d"   and P :: "'d \<Rightarrow> 'c"   and Q :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v < w \<Longrightarrow> less_eq (P v) Q" shows "less_eq (complete_lattice.Sup_less Sup P w) Q"
lift_f_ker_Q: shows "order.greater_eq (ker lift_f) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>"
element_ptr_kinds_eq_h3: shows "element_ptr_kinds h3 = element_ptr_kinds h'"
StrongEq_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e"   and Y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "X \<triangleq> Y \<equiv> \<lambda>\<tau>. \<lfloor>\<lfloor>X \<tau> = Y \<tau>\<rfloor>\<rfloor>"
psubst_closed: fixes i :: "nat"   and f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form" shows "closed i (psubst f p) = closed i p"
chinese_remainder_nat: fixes A :: "'b set"   and m :: "'b \<Rightarrow> nat"   and u :: "'b \<Rightarrow> nat" assumes "finite A"   and "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)" shows "\<exists>x. \<forall>i\<in>A. [x = u i] (mod m i)"
lambda_predicates_3_0: fixes \<phi> :: "\<o>" shows "[[\<^bold>\<lambda>\<^sup>0 \<phi> \<^bold>= \<phi>]]"
86: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<le> x \<squnion> (z \<squnion> y)"
is_tm_dghm_axioms_def: fixes \<alpha> :: "V"   and \<FF> :: "V" shows "is_tm_dghm_axioms \<alpha> \<FF> \<equiv> \<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha> \<and> \<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
cases_scheme: fixes r :: "('b, 'c) sparc_state_scheme" obtains cpu_reg :: "CPU_register \<Rightarrow> word32" and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32" and sys_reg :: "sys_reg \<Rightarrow> word32" and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option" and mmu :: "MMU_state" and cache :: "cpu_cache" and dwrite :: "(int \<times> word32 \<times> CPU_register) list" and state_var :: "sparc_state_var" and traps :: "Trap set" and undef :: "bool" and more :: "'c" where "r = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
E: fixes N :: "'a quasinorm" shows "N \<le> N"
conj_disj_not_abs: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(P \<and> (\<not> P \<or> Q)) = (P \<and> Q)"
integral_nonneg_AE: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "AE x in M. 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
assms_6: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
\<sigma>\<tau>_def: shows "\<sigma>\<tau> \<equiv> instInpP \<sigma> 0 \<tau>"
supp_Act: fixes \<alpha> :: "'a" shows "supp (L_action.Act \<alpha>) = supp \<alpha>"
split_inf_Seq: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, s) \<rightarrow> \<dots>(\<infinity>)" shows "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> \<dots>(\<infinity>) \<or> (\<exists>s'. \<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow>\<^sup>* (Skip, s') \<and> \<Gamma>\<turnstile> (c\<^sub>2, s') \<rightarrow> \<dots>(\<infinity>))"
o_id: fixes f :: "'d \<Rightarrow> 'e" shows "f \<circ> id = f"
atms_less_eq_imp_lit_less_eq_neg: fixes C :: "'b literal multiset"   and A :: "'b"   and L :: "'b literal" assumes "\<And>B. B \<in> atms_of C \<Longrightarrow> B \<le> A"   and "L \<in># C" shows "L \<le> Neg A"
product_simps_1: fixes AA :: "('c, 'd) dca list" shows "dca.alphabet (intersect_list AA) = \<Inter> (dca.alphabet ` set AA)"
a_minus_def: fixes x :: "'a"   and y :: "'a" shows "- x - - y = - (- - x \<squnion> - y)"
preserves_dom: fixes f :: "'b" assumes "D.arr f" shows "D.dom (trg\<^sub>D f) = trg\<^sub>D (D.dom f)"
lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "prod g {..<Suc n} = g 0 * (\<Prod>i<n. g (Suc i))"
intros: fixes i :: "nat"   and j :: "nat"   and xs :: "'a list" assumes "i < j"   and "j < length xs"   and "xs ! j < xs ! i" shows "(i, j) \<in> inversions xs"
assms_3: fixes a' :: "'a" assumes "a' \<in> local.Above B" shows "(a, a') \<in> r"
no_loops_in_iapath: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and a :: "'a \<times> 'a" assumes "gen_iapath V u p v"   and "a \<in> set p" shows "fst a \<noteq> snd a"
poly_map_poly_code: fixes h :: "'b \<Rightarrow> 'a"   and p :: "'b poly"   and x :: "'a" shows "poly (map_poly h p) x = fold_coeffs (\<lambda>a b. h a + x * b) p (0::'a)"
poly_add_minus_zero_iff: fixes p :: "'a list"   and q :: "'a list" shows "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
nsum_suc: fixes R :: "('c, 'd) aGroup_scheme"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "\<Sigma>\<^sub>e R f Suc n = \<Sigma>\<^sub>e R f n \<plusminus>\<^bsub>R\<^esub> f (Suc n)"
out_lea__out: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and E :: "'a"   and D :: "'a"   and F :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet E D F"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.Out Bet B A C"
proots_within_uminus: fixes p :: "'b poly"   and s :: "'b set" shows "proots_within (- p) s = proots_within p s"
menger_trivial: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and n :: "nat"   and S :: "'c set" assumes "v0_v1_Digraph G v0 v1"   and "DisjointPaths G v0 v1 paths"   and "card paths = n"   and "Separation G v0 v1 S" shows "n \<le> card S"
surj_def: fixes f :: "'d \<Rightarrow> 'c" shows "top_pres (\<P> f) = (\<forall>y. \<exists>x. y = f x)"
to_poly_mult_simp_1: fixes b :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "b \<in> carrier R"   and "f \<in> carrier (UP R)" shows "to_polynomial R b \<otimes>\<^bsub>UP R\<^esub> f = b \<odot>\<^bsub>UP R\<^esub> f"
dickson: fixes P1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A1 :: "'a set"   and P2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A2 :: "'b set" assumes "wqo_on P1 A1"   and "wqo_on P2 A2" shows "wqo_on (prod_le P1 P2) (A1 \<times> A2)"
mult_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<sqinter> z \<le> y \<sqinter> z"
bigo_abs2: fixes f :: "'c \<Rightarrow> 'd" shows "f \<in> O(\<lambda>x. \<bar>f x\<bar>)"
mask_eq_0_eq_x: fixes x :: "'a word"   and w :: "'a word" shows "(x && w = 0) = (x && ~~ w = x)"
ordertype_diff: fixes \<beta> :: "V"   and \<delta> :: "V"   and \<alpha> :: "V" assumes "\<beta> + \<delta> = \<alpha>"   and "\<delta> \<in> elts \<alpha>"   and "Ord \<alpha>" shows "ordertype (elts \<alpha> - elts \<beta>) VWF = \<delta>"
S: fixes H :: "'b pl set"   and p :: "'b pl"   and q :: "'b pl"   and r :: "'b pl" shows "H |- (p -> q -> r) -> (p -> q) -> p -> r"
rep_eq: fixes x :: "'a upred" shows "\<lbrakk>[x]\<^sub>u\<rbrakk>\<^sub>e = (\<lambda>A. \<forall>A'. \<lbrakk>x\<rbrakk>\<^sub>e A')"
one_aezfun_apply_eq: shows "aezfun 1 (0::'b) = (1::'a)"
list_ex_cong: fixes xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> bool"   and g :: "'d \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "list_ex f xs = list_ex g ys"
eball_def: fixes z :: "'a"   and r :: "ereal" shows "eball z r = {z'. ereal (dist z z') < r}"
invar_N_ss_Vis: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat"   and u :: "nat" assumes "nf_invar c src dst PRED C N d"   and "u \<in> N" shows "\<exists>v. PRED u = Some v"
bottom_le: fixes x :: "'a" shows "\<bottom> \<sqsubseteq>\<^bsub>r\<^esub> x"
trans_rel: fixes K :: "'a set"   and A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and D :: "'a set" assumes "K = Cn A"   and "B \<preceq>\<^bsub>K\<^esub> C"   and "C \<preceq>\<^bsub>K\<^esub> D" shows "B \<preceq>\<^bsub>K\<^esub> D"
preorder_rel_prodI: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.preorder orda (mk_less orda)"   and "class.preorder ordb (mk_less ordb)" shows "class.preorder (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
lift_wf: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and agents' :: "'a set"   and alts' :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "sds_lowering agents alts sds agents' alts'"   and "pref_profile_wf agents' alts' R" shows "pref_profile_wf agents alts (lift_pref_profile agents' alts' agents alts R)"
igSubstIGOp_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstGOp MOD" shows "igSubstIGOp (fromMOD MOD)"
pair_measure_countable: fixes S1 :: "'d set"   and S2 :: "'e set" assumes "countable S1"   and "countable S2" shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
eq_onp_mono_iff: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(eq_onp P \<le> eq_onp Q) = (P \<le> Q)"
as': shows "length as = length vs \<and> v = as \<bullet>\<cdot> vs"
Tn_quasi_measure_preserving: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "qmpt M T" shows "T ^^ n \<in> quasi_measure_preserving M M"
preserves_local_guarantee_compliance_def2: fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and \<R> :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of" shows "sifum_refinement.preserves_local_guarantee_compliance dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C \<R> \<equiv> \<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. sifum_security_init.respects_own_guarantees dma\<^sub>A \<C>_vars\<^sub>A eval\<^sub>A (c\<^sub>A, mds\<^sub>A) \<and> (((c\<^sub>A, mds\<^sub>A), mem\<^sub>A), (c\<^sub>C, mds\<^sub>C), mem\<^sub>C) \<in> \<R> \<longrightarrow> sifum_security_init.respects_own_guarantees dma\<^sub>C \<C>_vars\<^sub>C eval\<^sub>C (c\<^sub>C, mds\<^sub>C)"
length_drop: fixes n :: "nat"   and xs :: "'c list" shows "length (drop n xs) = length xs - n"
int_decode_inverse: fixes n :: "nat" shows "int_encode (int_decode n) = n"