hcju/isabelleps huggingface dataset:
ring_associated_iff: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "a \<sim>\<^bsub>K [X]\<^esub> b = (\<exists>u\<in>Units (K [X]). a = u \<otimes>\<^bsub>K [X]\<^esub> b)"
stutter_extend_simps_2: fixes G :: "('b, 'd) graph_rec_scheme" shows "g_E (stutter_extend G) = stutter_extend_edges (g_V G) (g_E G)"
to_gbg_ext_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b" shows "b_graph.to_gbg_ext (degeneralize_ext ma) m \<equiv> \<lparr>g_V = g_V (degeneralize_ext ma), g_E = g_E (degeneralize_ext ma), g_V0 = g_V0 (degeneralize_ext ma), gbg_F = if bg_F (degeneralize_ext ma) = UNIV then {} else {bg_F (degeneralize_ext ma)}, \<dots> = m\<rparr>"
dependent_and_consistent_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "dependent_and_consistent A b = (consistent A b \<and> 0 < vec.dim (solution_set A 0))"
DEFL_eq_lifted: shows "DEFL('a\<cdot>lifted) = DEFL('a\<^sub>\<bottom>)"
x_plus_helpers_9: fixes x :: "nat" shows "x + 12 = Suc (x + 11)"
pfsubset_imp_ex_fmem: fixes A :: "'c fset"   and B :: "'c fset" assumes "A |\<subset>| B" shows "\<exists>b. b |\<in>| B |-| A"
propValid_def: fixes APs :: "expr iarray"   and g :: "gState"   and i :: "nat" shows "propValid APs g i = (i < IArray.length APs \<and> exprArith g emptyProc (APs !! i) \<noteq> 0)"
card2: shows "n2 = card {i. i < m2 \<and> sel2 i}"
aux_lemma1: fixes l :: "'a list" assumes "set l \<subseteq> K" shows "set (map poly_of_const l) \<subseteq> carrier (K [X])"
Suc_def: fixes n :: "nat" shows "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
assms_1: shows "Complete_Partial_Order.chain (fun_ord (\<le>)) Y"
weak_respection_of_barbs_and_closures_6: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs ((symcl (Rel\<^sup>=))\<^sup>+) CWB"
csol_3: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "csol t0 x0 t t0 = x0"
cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "f = g"   and "x = y" shows "f x = g y"
power_eq_0_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a n = zero) = (a = zero \<and> 0 < n)"
rsquarefree_gcd_pderiv: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "rsquarefree (p div gcd p (pderiv p))"
prems_1: fixes x :: "nat" assumes "lev < x" shows "\<not> loose_bvar1 (t1 $ t2) x"
rep_eq: fixes x :: "('a \<times> 'b) fset" shows "is_fmap x = is_map (fset x)"
L_def: shows "L \<equiv> lambda_system \<Omega> M f"
isFinalState_def: fixes state :: "State"   and decisionVars :: "nat set" shows "isFinalState state decisionVars \<equiv> \<nexists>state'. transition state state' decisionVars"
t: shows "s to y in n = s to x in ([(y, x)] \<bullet> n)"
sees_field_idemp: 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 F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F:T in D" shows "P \<turnstile> D sees F:T in D"
assms: fixes a :: "'a" assumes "\<And>a1. a1 \<in> local.underS a \<Longrightarrow> P a1" shows "P a"
induct: fixes x :: "('a, 'b) trm"   and P :: "('a, 'b) trm \<Rightarrow> bool" assumes "dsafe x"   and "\<And>i. P (Var i)"   and "\<And>r. P (Const r)"   and "\<And>args i. \<lbrakk>\<And>i. dsafe (args i); \<And>i. P (args i)\<rbrakk> \<Longrightarrow> P ($f i args)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dsafe \<theta>\<^sub>1; P \<theta>\<^sub>1; dsafe \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Plus \<theta>\<^sub>1 \<theta>\<^sub>2)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dsafe \<theta>\<^sub>1; P \<theta>\<^sub>1; dsafe \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Times \<theta>\<^sub>1 \<theta>\<^sub>2)"   and "\<And>\<theta>. dfree \<theta> \<Longrightarrow> P (Differential \<theta>)"   and "\<And>i. P ($' i)" shows "P x"
assms_2: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "P n (m mod n)" shows "P m n"
remove1_tl: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "remove1 (hd xs) xs = tl xs"
rel_bsingleton: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c"   and x2 :: "'d" shows "rel_bset R (bsingleton x1) (bsingleton x2) = R x1 x2"
sfis_add: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> sfis f M"   and "b \<in> sfis g M" shows "a + b \<in> sfis (\<lambda>w. f w + g w) M"
part2_def: shows "part2 = snd \<circ> part12"
fst_snd_flip: fixes xy :: "'a \<times> 'b" shows "fst xy = (snd \<circ> (\<lambda>(x, y). (y, x))) xy"
A: shows "x * \<top> \<sqinter> x ^ o = x"
simps_2: fixes v :: "nat \<Rightarrow> 'a"   and x :: "nat"   and c :: "'a" shows "v \<Turnstile>\<^sub>a Geq x c = (c \<le> v x)"
dvd_div_mult_self: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times (divide b a) a = b"
mcont: fixes x :: "'a" shows "mcont local.tSup local.tllist_ord lub ord (\<lambda>xs. f x xs (TCons x xs))"
steps_z_alpha_complete: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle>"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "Z \<subseteq> V"   and "Z' \<noteq> {}" shows "\<exists>Z''. A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z''\<rangle> \<and> Z' \<subseteq> Z''"
ide_leg0: shows "ide fg\<^sub>0h\<^sub>1.p\<^sub>0"
set_integral_cong_set: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "set_borel_measurable M A f"   and "set_borel_measurable M B f"   and "AE x in M. (x \<in> A) = (x \<in> B)" shows "set_lebesgue_integral M B f = set_lebesgue_integral M A f"
fnmember_elim: fixes x :: "'b"   and xs :: "'b fset" assumes "x |\<notin>| xs" obtains "x \<notin> \<langle>xs\<rangle>\<^sub>f"
has_vderiv_on_compose: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> real"   and T :: "real set"   and g' :: "real \<Rightarrow> real" assumes "(f has_vderiv_on f') (g ` T)"   and "(g has_vderiv_on g') T" shows "(f \<circ> g has_vderiv_on (\<lambda>x. g' x *\<^sub>R f' (g x))) T"
pt_list_append: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('b \<times> 'b) list"   and xs :: "'a list" assumes "pt TYPE('a) TYPE('b)" shows "(pi1 @ pi2) \<bullet> xs = pi1 \<bullet> pi2 \<bullet> xs"
while_false_symeval: fixes \<Gamma> :: "'a \<Rightarrow> 'a"   and b :: "'a upred"   and P :: "'a hrel" assumes "\<Gamma> \<dagger> b = false" shows "\<Gamma> \<Turnstile> while\<^sup>\<top> b do P od = \<Gamma> \<Turnstile> II"
ta_seq_consist_into_non_speculative: fixes P :: "'a prog"   and vs :: "'b \<times> addr_loc \<Rightarrow> ('b val \<times> bool) option"   and obs :: "('b, 'c) obs_event action llist"   and vs' :: "'b \<times> addr_loc \<Rightarrow> 'b val set" assumes "ta_seq_consist P vs obs"   and "\<forall>adal. set_option (vs adal) \<subseteq> vs' adal \<times> UNIV" shows "non_speculative P vs' obs"
awalk_conv: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "awalk u p v = (set (pawalk_verts u p) \<subseteq> pverts G \<and> set p \<subseteq> parcs G \<and> hd (pawalk_verts u p) = u \<and> last (pawalk_verts u p) = v \<and> pcas u p v)"
finite_UNIV_card_ge_0: assumes "finite UNIV" shows "0 < CARD('b)"
i_take_nth: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "i < n" shows "f \<Down> n ! i = f i"
insf_def: fixes s :: "hf"   and k :: "hf"   and y :: "hf" shows "insf s k y \<equiv> nonrestrict s \<lbrace>k\<rbrace> \<triangleleft> \<langle>k, y\<rangle>"
purgeIdle_simps_2: fixes st :: "'a"   and sl :: "('a, 'b, 'c) state list" shows "GM_sec_model.purgeIdle (Idle st # sl) = GM_sec_model.purgeIdle sl"
eq_sym_Unity_conv: fixes x :: "bool" shows "(x = (() = ())) = x"
reduced_GB_is_reduced_GB_finite: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite F" shows "is_reduced_GB (reduced_GB F)"
Quotient_unit: shows "Quotient (eq_onp (\<lambda>x. x \<in> {0..1})) Linear_Aggregation_Algebras.unit.Abs_unit Linear_Aggregation_Algebras.unit.Rep_unit cr_unit"
at_right_eq: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "at_right x = (INF a\<in>{x<..}. principal {x<..<a})"
hyps_2: fixes xs' :: "'a list" assumes "mset xs' = mset xs" shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
ccsqrt_mult: fixes a :: "complex"   and b :: "complex" shows "ccsqrt (a * b) = ccsqrt a * ccsqrt b \<or> ccsqrt (a * b) = - ccsqrt a * ccsqrt b"
Busemann_function_isometry: fixes f :: "'b \<Rightarrow> 'c"   and xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b" assumes "isometry f" shows "Busemann_function_at (Gromov_extension f xi) (f x) (f y) = Busemann_function_at xi x y"
scale_one_on: fixes x :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" assumes "x \<in> manifold_eucl.dest.tangent_space p" shows "1 *\<^sub>R x = x"
sum_clauses_1: fixes f :: "'d \<Rightarrow> 'c" shows "sum f {} = (0::'c)"
carrier_stalk_def: shows "carrier_stalk \<equiv> dlim \<FF> \<rho> (neighborhoods x)"
ide_compE: fixes g :: "'c"   and f :: "'c" assumes "C.ide (g \<cdot> f)" obtains "C.seq g f" and "C.seq f g" and "g \<cdot> f = C.dom f" and "g \<cdot> f = C.cod g"
hyps_2: shows "d' \<Rightarrow>\<^sub>G d''"
less_le_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<prec> b"   and "b \<preceq> c" shows "a \<prec> c"
perm_3: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list"   and x3 :: "char list"   and x1 :: "NominalSequents.var"   and x2 :: "form" shows "pi \<bullet> x3 \<nabla> [x1].x2 = (pi \<bullet> x3) \<nabla> [(pi \<bullet> x1)].(pi \<bullet> x2)"
sup_aci_1: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
ex_finite_adds_term: fixes d :: "'a \<Rightarrow> nat"   and S :: "'t set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` S)"   and "pp_of_term ` S \<subseteq> dgrad_set d m" obtains T :: "'t set" where "finite T" and "T \<subseteq> S" and "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. t adds\<^sub>t s"
r_def: shows "r = rec_nat S2 (\<lambda>_ S'. {s \<in> S\<^sub>r. \<exists>t\<in>S'. (s, t) \<in> E})"
inf_prefix_Cons: fixes a :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(a # xs) \<sqinter> (a # ys) = a # xs \<sqinter> ys"
Im_sin_nonneg: fixes z :: "complex" assumes "Re z = 0"   and "0 \<le> Im z" shows "0 \<le> Im (sin z)"
holomorphic_on_mult: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f holomorphic_on s"   and "g holomorphic_on s" shows "(\<lambda>z. f z * g z) holomorphic_on s"
nofail_SPEC: fixes m :: "'a nres" assumes "nofail m" shows "m \<le> SPEC (\<lambda>_. True)"
aqgrp_carrier: fixes H :: "'a set" assumes "A +> H" shows "set_rcs (b_ag A) H = set_ar_cos A H"
zero_uexpr_def: shows "0 = \<guillemotleft>0::'c\<guillemotright>"
edom_def: fixes m :: "'a \<Rightarrow> 'b" shows "edom m = {x. m x \<noteq> \<bottom>}"
retract_ab: fixes x :: "'Fab"   and y :: "'Fb" assumes "R\<^sup>*\<^sup>* (\<pi>_Fabb x) y" shows "\<exists>z. Eab x z \<and> y = \<pi>_Fabb z"
has_partial_vector_derivative_def: fixes F :: "'a \<Rightarrow> 'b"   and base_vec :: "'a"   and F' :: "'b"   and a :: "'a" shows "has_partial_vector_derivative F base_vec F' a \<equiv> ((\<lambda>x. F (a - (a \<bullet> base_vec) *\<^sub>R base_vec + x *\<^sub>R base_vec)) has_vector_derivative F') (at (a \<bullet> base_vec))"
subseq_to_zero: fixes w :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "w [i \<rightarrow> 0] = []"
ruler_suf: fixes w :: "'b list"   and z :: "'b list"   and v :: "'b list" assumes "w \<le>s z \<cdot> v" shows "w \<bowtie>\<^sub>s v"
rel: shows "(xs, ys) \<in> ns_mul_ext ns s"
ladder_cut_prefix_i: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat" assumes "index < length L"   and "0 < length L" shows "ladder_i (ladder_cut_prefix d L) index = ladder_i L index - d"
irreducible_monic_coprime: fixes f :: "rat poly"   and g :: "rat poly" assumes "monic f"   and "irreducible f"   and "monic g"   and "irreducible g"   and "f \<noteq> g" shows "comm_monoid_mult_class.coprime f g"
empty_\<X>: fixes k :: "nat" assumes "length Doc < k" shows "\<X> k = {}"
GDERIV_DERIV_compose: fixes f :: "'a \<Rightarrow> real"   and x :: "'a"   and df :: "'a"   and g :: "real \<Rightarrow> real"   and dg :: "real" assumes "GDERIV f x :> df"   and "(g has_real_derivative dg) (at (f x))" shows "GDERIV (\<lambda>x. g (f x)) x :> dg *\<^sub>R df"
Map_UP: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.arr V\<^sub>B \<mu>" shows "concrete_category.Map (strictified_bicategory.UP V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<mu>) = \<mu>"
assms_3: fixes F :: "'a set" assumes "F \<in> {x \<in> A. P x} // R" shows "card F = k"
WTrtCons: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and es :: "char list exp list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> e : T"   and "P,E,h \<turnstile> es [:] Ts" shows "P,E,h \<turnstile> e # es [:] T # Ts"
psub: fixes F' :: "'a set set" assumes "F' \<subset> F" shows "S \<subset> affine hull S \<inter> \<Inter> F'"
broadcasts_unique: fixes i :: "nat"   and j :: "nat"   and m :: "'a \<times> ('a, 'b) operation" assumes "i \<noteq> j"   and "Broadcast m \<in> set (history i)" shows "Broadcast m \<notin> set (history j)"
tensor_from_lookup_def: fixes ds :: "nat list"   and e :: "nat list \<Rightarrow> 'b" shows "tensor_from_lookup ds e = tensor_from_vec ds (tensor_vec_from_lookup ds e)"
lemma2_6_2_b: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset" assumes "trans r"   and "(M, N) \<in> mul_eq r" shows "r \<down>m M \<subseteq> r \<down>m N"
closed_def: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open" shows "topological_space.closed open S = open (- S)"
simps_3: fixes t1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.meldUniq (t1 # bq1) (t2 # bq2) = (if SkewBinomialHeapStruc.rank t1 < SkewBinomialHeapStruc.rank t2 then t1 # SkewBinomialHeapStruc.meldUniq bq1 (t2 # bq2) else if SkewBinomialHeapStruc.rank t2 < SkewBinomialHeapStruc.rank t1 then t2 # SkewBinomialHeapStruc.meldUniq (t1 # bq1) bq2 else SkewBinomialHeapStruc.ins (SkewBinomialHeapStruc.link t1 t2) (SkewBinomialHeapStruc.meldUniq bq1 bq2))"
bal_split_left_2: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ (a, b) # rs) t)" shows "height (Node ls a) = height (Node (ls @ (a, b) # rs) t)"
BB_eq_concat_seqs: fixes K :: "nat set"   and j0 :: "nat"   and j :: "nat" assumes "K \<in> \<K> j0 j"   and "j \<le> j0" shows "BB j0 j K = concat (seqs j0 j K)"
point_path_def: fixes \<gamma> :: "'a \<Rightarrow> 'b" shows "point_path \<gamma> \<equiv> \<exists>c. \<gamma> = (\<lambda>x. c)"
induced_group_homomorphism: shows "group_homomorphism induced kernel.Partition ([\<cdot>]) (kernel.Class \<one>) G' (\<cdot>') \<one>'"
diff_0_eq_0: fixes n :: "nat" shows "0 - n = 0"
normalize_alt_simps_16: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and x :: "'a expr"   and xs :: "'a expr list"   and ys :: "'a expr list" shows "(\<Gamma> \<turnstile>\<^sub>L x # xs \<Rrightarrow> ys) = (\<exists>y ysa. ys = y # ysa \<and> \<Gamma> \<turnstile> x \<Rrightarrow> y \<and> \<Gamma> \<turnstile>\<^sub>L xs \<Rrightarrow> ysa)"
fps_X_compose_fps_exp: fixes a :: "'b" shows "fps_X oo fps_exp a = fps_exp a - 1"
sign_def: fixes p :: "'a \<Rightarrow> 'a" shows "sign p = (if evenperm p then 1 else - 1)"
A: shows "(evsFR4, S, A, U) \<in> protocol"
is_basis_def: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and B :: "'c set" assumes "vector_space scale" shows "vector_space.is_basis scale V B = (B \<subseteq> V \<and> \<not> module.dependent scale B \<and> V \<subseteq> module.span scale B \<and> card B = vector_space.dim scale V)"
hyps: shows "wlp_while_n (M 0) (M 1) WS k P \<le>\<^sub>L wlp_while_n (M 0) (M 1) WS k Q"
foundation10: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x"   and "\<tau> \<Turnstile> \<delta> y" shows "(\<tau> \<Turnstile> x and y) = (\<tau> \<Turnstile> x \<and> \<tau> \<Turnstile> y)"
PP: shows "P \<in> Units (ring_mat TYPE('a) nr undefined)"
C_eq_normalizePr: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "DenyAll \<in> set (policy2list p)"   and "allNetsDistinct (policy2list p)"   and "all_in_list (policy2list p) (Nets_List p)" shows "Cp (list2FWpolicy (normalizePr p)) = Cp p"
loopi_def: fixes F :: "'b \<Rightarrow> 'b set"   and I :: "'b \<Rightarrow> bool" shows "LOOP F INV I  \<equiv> F\<^sup>*"
is_full: fixes a :: "'c"   and a' :: "'c"   and g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "C.ide a"   and "C.ide a'"   and "\<guillemotleft>g : local.map a' \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] local.map a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> local.map f = g"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'b list"   and xs\<^sub>2 :: "'b list"   and ys\<^sub>1 :: "'b list"   and ys\<^sub>2 :: "'b list" shows "(xs\<^sub>1 \<cdot> xs\<^sub>2 = ys\<^sub>1 \<cdot> ys\<^sub>2) = (if \<^bold>|xs\<^sub>1\<^bold>| \<le> \<^bold>|ys\<^sub>1\<^bold>| then xs\<^sub>1 = take \<^bold>|xs\<^sub>1\<^bold>| ys\<^sub>1 \<and> xs\<^sub>2 = drop \<^bold>|xs\<^sub>1\<^bold>| ys\<^sub>1 \<cdot> ys\<^sub>2 else take \<^bold>|ys\<^sub>1\<^bold>| xs\<^sub>1 = ys\<^sub>1 \<and> drop \<^bold>|ys\<^sub>1\<^bold>| xs\<^sub>1 \<cdot> xs\<^sub>2 = ys\<^sub>2)"
simps_1: shows "sch_to_pr Base_zero = (\<lambda>x. 0)"
QCongA_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.QCongA Bet Cong a \<equiv> \<exists>A B C. A \<noteq> B \<and> C \<noteq> B \<and> (\<forall>X Y Z. Tarski_neutral_dimensionless.CongA Bet Cong A B C X Y Z = a X Y Z)"
P2_S1_def: fixes \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'd) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'd \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'd) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'd \<Rightarrow> 'f spmf)"   and \<sigma> :: "bool"   and z :: "'e" shows "P2_S1 \<A> \<sigma> z = (let (\<A>1, \<A>2, \<A>3) = \<A> in \<A>1 \<sigma> z \<bind> (\<lambda>((h0, h1, a, b0, b1), s). assert_spmf (h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> a \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>) \<bind> (\<lambda>_. \<A>2 (h0, h1, a, b0, b1) s \<bind> (\<lambda>(((in1, in2, in3), r), s'). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (h, a, b) ((in1, in2, in3), r) \<bind> (\<lambda>(out_zk_funct, uu_). assert_spmf out_zk_funct \<bind> (\<lambda>_. let l = b0 \<otimes> inv (h0 [^] r) in return_spmf (if l = \<one> then False else True, (h0, h1, a, b0, b1), s')))))))"
lt_alt: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "i.lpp p = i.ordered_powerprod_lin.Max (keys p)"
line_convex_combination12: fixes u :: "real"   and a :: "'b"   and b :: "'b"   and i :: "real"   and j :: "real" shows "(1 - u) *\<^sub>R {a -- b}\<^bsub>i\<^esub> + u *\<^sub>R {a -- b}\<^bsub>j\<^esub> = {a -- b}\<^bsub>i + u * (j - i)\<^esub>"
leq_prv_LLq_Num: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "prv (LLq (Num m) (Num n))"
is_unit_normalize: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a one" shows "normalize a = one"
preserves_reflects_arr: fixes f :: "'a1 \<times> 'a2" shows "B.arr (G f) = A1xA2.arr f"
right_idem: fixes a :: "'c"   and b :: "'c" shows "a \<sqinter> b \<sqinter> b = a \<sqinter> b"
elims: fixes x :: "((int \<times> int) set set, int) Combinators"   and y :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent \<Rightarrow> unit decision option" assumes "C x = y" obtains "x = DenyAll" and "y = D\<^sub>U"   | xa :: "(int \<times> int) set set" and ya :: "(int \<times> int) set set" where "x = DenyAllFromTo xa__ ya__" and "y = deny_all_from_to xa__ ya__"   | xa :: "(int \<times> int) set set" and ya :: "(int \<times> int) set set" and p :: "int" where "x = AllowPortFromTo xaa__ yaa__ p" and "y = allow_from_to_port p xaa__ yaa__"   | xa :: "((int \<times> int) set set, int) Combinators" and ya :: "((int \<times> int) set set, int) Combinators" where "x = xa \<oplus> ya" and "y = C ya \<Oplus> C xa"
sourcesD_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "a \<in> weak_composition.sources V H \<mu>" shows "weak_composition.weak_unit V H a"