hcju/isabelleps huggingface dataset:
parallel_constructive_security_obsf: fixes real1 :: "('y + 'z, 'aa + 'ab) resource"   and ideal1 :: "('ac + 'z, 'ad + 'ab) resource"   and sim1 :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real1 :: "('y, 'aa) \<I>"   and \<I>_inner1 :: "('ac, 'ad) \<I>"   and \<I>_common1 :: "('z, 'ab) \<I>"   and \<A> :: "(('y + 'ae) + 'z + 'af, ('aa + 'ag) + 'ab + 'ah) distinguisher_obsf"   and sim2 :: "('ae, 'ag, 'ai, 'aj) converter"   and ideal2 :: "('ai + 'af, 'aj + 'ah) resource"   and adv1 :: "real"   and real2 :: "('ae + 'af, 'ag + 'ah) resource"   and \<I>_real2 :: "('ae, 'ag) \<I>"   and \<I>_inner2 :: "('ai, 'aj) \<I>"   and \<I>_common2 :: "('af, 'ah) \<I>"   and adv2 :: "real" assumes "constructive_security_obsf real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 (absorb \<A> (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2))))) adv1"   and "constructive_security_obsf real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 (absorb \<A> (obsf_converter (parallel_wiring \<odot> (converter_of_resource real1 |\<^sub>\<propto> 1\<^sub>C)))) adv2" shows "constructive_security_obsf (parallel_wiring \<rhd> real1 \<parallel> real2) (parallel_wiring \<rhd> ideal1 \<parallel> ideal2) (sim1 |\<^sub>= sim2) (\<I>_real1 \<oplus>\<^sub>\<I> \<I>_real2) (\<I>_inner1 \<oplus>\<^sub>\<I> \<I>_inner2) (\<I>_common1 \<oplus>\<^sub>\<I> \<I>_common2) \<A> (adv1 + adv2)"
obtain_trail_in_fcs: fixes a :: "'b"   and a0 :: "'b"   and an :: "'b" assumes "a \<in> arcs G"   and "a0 \<in> face_cycle_set a"   and "an \<in> face_cycle_set a" obtains p :: "'b list" where "trail (tail G a0) p (head G an)" and "p \<noteq> []" and "hd p = a0" and "last p = an" and "set p \<subseteq> face_cycle_set a"
basic_wt_eq_wt: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and wt_sym :: "'a \<Rightarrow> nat"   and \<epsilon> :: "nat"   and extf :: "'a \<Rightarrow> (('a, 'b) tm \<Rightarrow> ('a, 'b) tm \<Rightarrow> bool) \<Rightarrow> ('a, 'b) tm list \<Rightarrow> ('a, 'b) tm list \<Rightarrow> bool"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and s :: "('a, 'b) tm" assumes "kbo_basic gt_sym wt_sym \<epsilon> extf ground_heads_var" shows "kbo_basic.wt wt_sym ground_heads_var s = Lambda_Free_KBO_Std.kbo_std.wt ground_heads_var 0 (\<lambda>_. \<infinity>) wt_sym s"
max_shift: fixes S :: "nat set"   and f :: "nat \<Rightarrow> real"   and x :: "real" assumes "S \<noteq> {}"   and "finite S" shows "Max (f ` S) + x = (MAX p\<in>S. f p + x)"
InAngle_def: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" shows "P InAngle A B C \<equiv> A \<noteq> B \<and> C \<noteq> B \<and> P \<noteq> B \<and> (\<exists>X. Bet A X C \<and> (X = B \<or> B Out X P))"
arcsin_lt_bounded: fixes y :: "real" assumes "- 1 < y"   and "y < 1" shows "- (pi / 2) < arcsin y \<and> arcsin y < pi / 2"
counit_simps_5: shows "trg \<i>[trg f] = trg (trg f)"
rassocl\<^sub>C_def: shows "rassocl\<^sub>C = map_converter lsumr rsuml id id 1\<^sub>C"
word_less_power_trans2: fixes n :: "'b word"   and m :: "nat"   and k :: "nat" assumes "n < 2 ^ (m - k)"   and "k \<le> m"   and "m < LENGTH('b)" shows "n * 2 ^ k < 2 ^ m"
M_consts_2: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "\<not> w \<Turnstile>\<^sub>n \<phi> M\<^sub>n false\<^sub>n"
mod_mult_right_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a (modulo b c)) c = modulo (times a b) c"
whileI: fixes I' :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and r :: "('b \<times> 'b) set" assumes "\<lbrace>I'\<rbrace> c \<lbrace>I\<rbrace>"   and "\<And>s. I s \<Longrightarrow> if b s then I' s else Q s"   and "wf r"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> r" shows "\<lbrace>I\<rbrace> while b c \<lbrace>Q\<rbrace>"
pbij_typedef_lens: shows "pbij_lens typedef\<^sub>L"
alwaysI: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a" assumes "\<And>i. P (\<sigma> |\<^sub>s i)" shows "(\<box>P) \<sigma>"
that: fixes r :: "V" assumes "x = \<langle>r, \<R>\<^sub>\<circ> r\<rangle>"   and "r \<in>\<^sub>\<circ> A" shows "thesis"
fls_mult_one_1: fixes f :: "'a fls" shows "1 * f = f"
simps: fixes a :: "'b formula" shows "(a \<in> AX0) = ((\<exists>F G. a = F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F) \<or> (\<exists>F G H. a = (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H))"
simps_12: fixes a :: "'a local.term"   and b :: "'a local.term" shows "(a \<^bold>\<otimes> b) \<^bold>\<Down> \<^bold>\<I> = \<^bold>\<r>\<^bold>[a \<^bold>\<otimes> b\<^bold>]"
cop_F_increasing: fixes x :: "'a set"   and ds :: "'b set" shows "x \<subseteq> cop_F ds x"
codomain_make_vert: fixes n :: "nat" shows "codomain_block (make_vert_block n) = int n"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b list" shows "map f (x21 # x22) = f x21 # map f x22"
that: fixes p :: "'a" assumes "prime p"   and "p dvd a" shows "thesis"
singleton_iff: fixes b :: "'c"   and a :: "'c" shows "(b \<in> {a}) = (b = a)"
simps: fixes ps :: "('b, 'c) fm list"   and a :: "'c"   and used :: "'c set" shows "witness (ps, a) used = (witness_list ps used, a)"
adj_map: fixes C :: "'a set"   and D :: "'a set" assumes "folding.chamber C"   and "folding.chamber D"   and "C \<sim> D" shows "f ` C \<sim> f ` D"
pure_I_conv: fixes x :: "'a \<Rightarrow> 'b" shows "pure (\<lambda>x. x) \<diamondop> x = x"
l_inv: fixes x :: "'a"   and n :: "nat" assumes "x \<in> A" shows "l ((T ^^ n) x) = l x"
PrimReplace_iff: fixes A :: "hf"   and R :: "hf \<Rightarrow> hf \<Rightarrow> bool"   and v :: "hf" assumes "\<forall>u v v'. u \<^bold>\<in> A \<longrightarrow> R u v \<longrightarrow> R u v' \<longrightarrow> v' = v" shows "(v \<^bold>\<in> PrimReplace A R) = (\<exists>u. u \<^bold>\<in> A \<and> R u v)"
r_def: shows "r = embed_pmf (\<lambda>x. pmf p x + \<epsilon> * (pmf q x - pmf p x))"
IH: fixes \<omega> :: "'a stream" assumes "((\<lambda>xs. \<not> P xs) until alw P) \<omega>" shows "(enat n < sfirst P \<omega>) = (\<not> P (sdrop n \<omega>))"
l: fixes g :: "real \<Rightarrow> 'c"   and y :: "'c"   and h :: "real \<Rightarrow> 'a" assumes "path g"   and "path_image g \<subseteq> U"   and "pathstart g = z"   and "pathfinish g = y"   and "path h"   and "path_image h \<subseteq> C"   and "pathstart h = a"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (h t) = f (g t)" shows "pathfinish h = l y"
betw_b_in_path: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and ac :: "'a set" assumes "[[a b c]]"   and "path ac a c" shows "b \<in> ac"
cball_trivial: fixes x :: "'b" shows "cball x 0 = {x}"
ode_e_impl_def: shows "ode_e_impl = (\<lambda>(ode_ops, uu_). ode_expression ode_ops)"
collapse_3: fixes unboxed :: "('a, 'b, 'c) unboxed" assumes "is_OpUbx2 unboxed" shows "OpUbx2 (un_OpUbx2 unboxed) = unboxed"
word_exp_length_eq_0: shows "2 ^ LENGTH('b) = 0"
dis_log3_def: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" shows "dis_log3 \<A> = TRY sample_uniform (order \<G>) \<bind> (\<lambda>w. let (h, w) = ((\<^bold>g [^] w, g' [^] w), w) in \<A> h \<bind> (\<lambda>w'. return_spmf [w = w'] (mod order \<G>))) ELSE return_spmf False"
simps_4: fixes I :: "('b, 'c, 'a) interp"   and \<phi> :: "('b, 'c, 'a) formula"   and \<psi> :: "('b, 'c, 'a) formula" shows "fml_sem I (\<phi> && \<psi>) = fml_sem I \<phi> \<inter> fml_sem I \<psi>"
leg_simps_1: shows "C.arr doms.\<nu>.leg0"
member_le_sum: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "i \<in> A"   and "\<And>x. x \<in> A - {i} \<Longrightarrow> (0::'c) \<le> f x"   and "finite A" shows "f i \<le> sum f A"
o_def: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "f \<circ> g = (\<lambda>x. f (g x))"
simps_8: fixes x31 :: "expr"   and x32 :: "expr"   and x1 :: "lname" shows "x31.[x32] \<noteq> LVar x1"
head_in_verts: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "snd e \<in> pverts G"
Image_Id_on: fixes A :: "'e set"   and B :: "'e set" shows "Id_on A `` B = A \<inter> B"
cp_thm: fixes \<phi> :: "atom fm"   and d :: "int"   and xs :: "int list" assumes "nqfree \<phi>"   and "\<forall>a\<in>set (atoms\<^sub>0 \<phi>). hd_coeff_is1 a"   and "\<forall>a\<in>set (atoms\<^sub>0 \<phi>). divisor a dvd d"   and "0 < d" shows "(\<exists>x. Z.I \<phi> (x # xs)) = (\<exists>j\<in>{0..d - 1}. Z.I (inf\<^sub>- \<phi>) (j # xs) \<or> (\<exists>(i, ks)\<in>set (lbounds (atoms\<^sub>0 \<phi>)). Z.I \<phi> ((i - \<langle>ks,xs\<rangle> + j) # xs)))"
simps: fixes I :: "('sf, 'sc, 'sz) interp"   and R :: "((real, 'sz) vec \<times> (real, 'sz) vec) set" shows "extendc I R = \<lparr>Functions = Functions I, Predicates = Predicates I, Contexts = case_sum (Contexts I) (\<lambda>() _. R), Programs = Programs I, ODEs = ODEs I, ODEBV = ODEBV I\<rparr>"
ball_empty: fixes P :: "'a \<Rightarrow> bool" shows "Ball {} P = True"
smaller_l_child: fixes l :: "int"   and x :: "int" shows "l_child l x < r_child l x"
P2_real_model_end_def: fixes M :: "'a \<times> 'a"   and bs :: "('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'e"   and \<A>3 :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'e \<Rightarrow> 'd spmf" shows "P2_real_model_end M bs \<A>3 = (let (x0, x1) = M; ((h0, h1, a, b0, b1), s) = bs in sample_uniform (order \<G>) \<bind> (\<lambda>u0. sample_uniform (order \<G>) \<bind> (\<lambda>u1. sample_uniform (order \<G>) \<bind> (\<lambda>v0. sample_uniform (order \<G>) \<bind> (\<lambda>v1. let z0 = b0 [^] u0 \<otimes> h0 [^] v0 \<otimes> x0; w0 = a [^] u0 \<otimes> \<^bold>g [^] v0; e0 = (w0, z0); z1 = (b1 \<otimes> inv \<^bold>g) [^] u1 \<otimes> h1 [^] v1 \<otimes> x1; w1 = a [^] u1 \<otimes> \<^bold>g [^] v1; e1 = (w1, z1) in \<A>3 e0 e1 s \<bind> (\<lambda>out. return_spmf ((), out)))))))"
simps_3: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cTimeF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cTimeF"
simps_6: fixes n :: "nat"   and m :: "nat"   and F :: "'b formula"   and G :: "'b formula" shows "cut_bound n m (F \<^bold>\<rightarrow> G) = cut_bound (cut_bound m n F) m G"
negligible_translation_rev: fixes c :: "'c"   and S :: "'c set" assumes "negligible ((+) c ` S)" shows "negligible S"
assms_3: shows "strategy_attracts_via p \<sigma> v A W"
jkbpC_jkbpCSn_jAction_eq: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> Clock.jkbpCn n" shows "jAction Clock.MC t = jAction (Clock.MCSn n) (clock_sim t)"
cauchy_inverse: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "\<not> Real.vanishes X" shows "Real.cauchy (\<lambda>n. inverse (X n))"
maximalAllocationBridgingTheorem: fixes N :: "integer set"   and \<Omega> :: "'c list"   and b :: "integer \<times> 'c set \<Rightarrow> 'd" assumes "finite N"   and "distinct \<Omega>" shows "maximalStrictAllocations N (set \<Omega>) b = maximalStrictAllocationsAlg N \<Omega> b"
bddfg: shows "bdd_above ((\<lambda>x. f x + g x) ` R)"
weight_return_spmf: fixes x :: "'c" shows "weight_spmf (return_spmf x) = 1"
cSup_eq_Sup_fin: fixes X :: "'a multiset set" assumes "finite X"   and "X \<noteq> {}" shows "Sup X = subset_mset.Sup_fin X"
simps_4: fixes x :: "'b"   and y :: "'b" shows "extB.Val x + extB.Val y = extB.Val (x + y)"
reduce_semantics: fixes M :: "('a, 'b) kripke"   and w :: "'b"   and p :: "'a pfm" shows "(M, w \<Turnstile>\<^sub>! p) = (M, w \<Turnstile>\<^sub>! reduce p)"
dy_fake_msg_def: fixes b :: "agent set"   and i :: "msg set"   and c :: "chan set" shows "dy_fake_msg b i c = synth (analz (extr b i c))"
monom_eq_iff': fixes c :: "'c"   and n :: "nat"   and d :: "'c"   and m :: "nat" shows "(monom c n = monom d m) = (c = d \<and> (c = (0::'c) \<or> n = m))"
injectivity_union: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and D :: "'a set"   and P :: "'a set \<Rightarrow> bool"   and Q :: "'a set \<Rightarrow> bool" assumes "A \<union> B = C \<union> D"   and "P A"   and "P C"   and "Q B"   and "Q D"   and "\<And>S T. \<lbrakk>P S; Q T\<rbrakk> \<Longrightarrow> S \<inter> T = {}" shows "A = C \<and> B = D"
mat_default_def: fixes d :: "'b"   and n :: "nat" shows "default\<^sub>m d n = mat n n (\<lambda>(i, j). if i = j then d else (0::'b))"
mf_def: shows "mf = (\<lambda>g. compose J g f) \<circ> proj"
of_nat_eq_1_iff: fixes n :: "nat" shows "(of_nat n = (1::'b)) = (n = 1)"
butlast_rev: fixes xs :: "'d list" shows "butlast (rev xs) = rev (tl xs)"
regularCard_def: fixes r :: "('b \<times> 'b) set" shows "regularCard r \<equiv> \<forall>K. K \<subseteq> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
labeled_entailment_lifting: fixes NL1 :: "('f \<times> 'l) set"   and NL2 :: "('f \<times> 'l) set" shows "entails_\<G> NL1 NL2 = (fst ` NL1 \<Turnstile>\<inter>\<G> fst ` NL2)"
ball_UN: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>z\<in>\<Union> (B ` A). P z) = (\<forall>x\<in>A. \<forall>z\<in>B x. P z)"
typing_ctxt_correct1: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and E :: "ctxt"   and r :: "trm"   and T :: "Syntax.type" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T ctxt_subst E r : T" shows "\<exists>U. \<Gamma> , \<Delta> \<turnstile>\<^sub>T r : U \<and> \<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T \<Leftarrow> U"
abs_eq: fixes x :: "'e" shows "blinfun_inner_left x = Blinfun (\<lambda>y. y \<bullet> x)"
Abs_eq_apps_conv: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list" shows "(Abs r = s \<degree>\<degree> ss) = (Abs r = s \<and> ss = [])"
init: fixes \<sigma> :: "'a"   and s :: "'b" assumes "(\<sigma>, s) \<in> init A" shows "P (\<sigma>, s)"
RG_proj_mult_rightdelta': fixes G :: "'a set"   and g' :: "'a"   and r :: "'b"   and g :: "'a"   and s :: "'b" assumes "Group G"   and "g' \<in> G" shows "aezfun_setspan_proj G (r \<delta>\<delta> g * (s \<delta>\<delta> g')) = aezfun_setspan_proj G (r \<delta>\<delta> g) * (s \<delta>\<delta> g')"
bst_partition: fixes p :: "'b"   and t :: "'b tree"   and l' :: "'b tree"   and r' :: "'b tree" assumes "Splay_Heap.partition p t = (l', r')"   and "bst_wrt (\<le>) t" shows "bst_wrt (\<le>) \<langle>l', p, r'\<rangle>"
functor_axioms: shows "functor C.VV.comp D.VV.comp FF"
dims_tensor_deep_model: fixes rs :: "nat list"   and y :: "nat"   and m :: "real mat convnet" assumes "deep_model_correct_params_y rs y"   and "remove_weights m = deep_model_l rs" shows "dims (tensors_from_net m $ y) = replicate (2 * deep_model_correct_params.N_half rs) (last rs)"
inext_append_eq1: fixes A :: "nat set"   and B :: "nat set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "Max A < iMin B"   and "n \<in> A"   and "n \<noteq> Max A" shows "inext n (A \<union> B) = inext n A"
Lambert_W'_eqI: fixes w :: "real"   and x :: "real" assumes "w \<le> - 1"   and "w * exp w = x" shows "Lambert_W' x = w"
triangle_right': shows "\<r>[g] \<cdot> (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> \<l>\<^sup>-\<^sup>1[g] = g"
bind_return_pmf': fixes N :: "'h pmf" shows "N \<bind> return_pmf = N"
distinct_19: fixes term1 :: "'d"   and term2 :: "'d"   and psi :: "('d, 'e, 'f) psi"   and psi1' :: "('d, 'e, 'f) psi"   and psi2' :: "('d, 'e, 'f) psi" shows "term1\<langle>term2\<rangle>.psi \<noteq> psi1' \<parallel> psi2'"
list_slice_append_mod: fixes xs :: "'b list"   and k :: "nat"   and ys :: "'b list" assumes "length xs mod k = 0" shows "list_slice (xs @ ys) k = list_slice xs k @ list_slice ys k"
\<tau>move1TryThrow: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr"   and C :: "String.literal"   and V :: "'b"   and e :: "('b, 'c, 'addr) exp" shows "\<tau>move1 P h (try Throw a catch(C V) e)"
of_int_1: shows "\<guillemotleft>1\<guillemotright> = \<one>"
quasi_isometry_betweenD_7: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_between X Y f" shows "0 \<le> C"
induced_modulus_altdef1: fixes d :: "nat" shows "\<Phi>.induced_modulus d = (d dvd d__ \<and> (\<forall>a b. coprime a d__ \<and> coprime b d__ \<and> [a = b] (mod d) \<longrightarrow> \<Phi> a = \<Phi> b))"
norm_Pair2: fixes x :: "'a" shows "norm (x, 0::'b) = norm x"
fr_rot: fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A * B \<Longrightarrow>\<^sub>A C" shows "B * A \<Longrightarrow>\<^sub>A C"
shd_state_is_none: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and t :: "(String.literal \<times> value list) stream" shows "state_eq None (make_full_observation e None r p t)"
C_eq_if_mr_eq: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators list" assumes "applied_rule_rev C x b = applied_rule_rev C x a"   and "a \<noteq> []"   and "b \<noteq> []" shows "C (list2FWpolicy a) x = C (list2FWpolicy b) x"
letrec2: fixes funs :: "(char list \<times> char list \<times> exp0) list"   and ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0" assumes "\<not> allDistinct (map (\<lambda>x. case x of (x, y, z) \<Rightarrow> x) funs)" shows "evaluate ck env s (Letrec funs e) (s, Rerr (Rabort Rtype_error))"
dependent_def: fixes P :: "'c set" shows "vs2.dependent P = (\<exists>a\<in>P. a \<in> vs2.span (P - {a}))"
valid_W_inv_sys_mem_store_buffersD_3: fixes s :: "'g process_name \<Rightarrow> ('d, 'g, 'e, 'f) Model.local_state"   and p :: "'g process_name"   and fl' :: "bool"   and ws :: "('d, 'e, 'f) mem_store_action list"   and r :: "'f"   and fl :: "bool" assumes "sys_mem_store_buffers p s = mw_fM fl' # ws"   and "mw_Mark r fl \<in> set ws"   and "valid_W_inv s" shows "fl = sys_fM s \<and> r \<in> ghost_honorary_grey (s p) \<and> sys_mem_lock s = Some p \<and> white r s \<and> filter is_mw_Mark ws = [mw_Mark r fl]"
size_rcat_lem: fixes wl :: "'b word list" shows "length (concat (map to_bl wl)) = length wl * LENGTH('b)"
Disjoint_commute: fixes t :: "tm"   and u :: "tm" shows "{Disjoint t u} \<turnstile> Disjoint u t"
chain_index: fixes f :: "nat \<Rightarrow> 'a set"   and F :: "'a set" assumes "is_chain f"   and "finite F"   and "F \<subseteq> \<Union> (range f)" shows "\<exists>n. F \<subseteq> f n"
ChamberComplexApartmentSystemTriangle_def: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" shows "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z \<equiv> ChamberComplexWithApartmentSystem X \<A> \<and> ChamberComplexApartmentSystemTriangle_axioms X \<A> A B B' C D E z"
eqset_imp_iff: fixes A :: "'d set"   and B :: "'d set"   and x :: "'d" assumes "A = B" shows "(x \<in> A) = (x \<in> B)"
choose_pivot_def: fixes uu :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and uv :: "'d \<Rightarrow> 'c"   and uw :: "'d list"   and lo :: "nat"   and hi :: "nat" shows "choose_pivot uu uv uw lo hi = SPEC (\<lambda>k. lo \<le> k \<and> k \<le> hi)"
tab_simps_5: shows "C.cod \<rho> = r \<star>\<^sub>C f"
integral_const_real: fixes a :: "real"   and b :: "real"   and c :: "'a" shows "integral {a..b} (\<lambda>x. c) = content {a..b} *\<^sub>R c"
select_object_any_defined\<^sub>S\<^sub>e\<^sub>q: fixes \<tau> :: "'d state \<times> 'd state"   and f :: "'f \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'e"   and s_set :: "'f list" assumes "\<tau> \<Turnstile> \<delta> select_object_any\<^sub>S\<^sub>e\<^sub>q f s_set" shows "s_set \<noteq> []"
subst_o: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm"   and T :: "'c trm" shows "subst (subst \<pi>1 \<circ> \<pi>2) T = subst \<pi>1 (subst \<pi>2 T)"
ub1_sgn: shows "\<forall>x\<ge>ub1. \<forall>p\<in>set ps. sgn (poly p x) = sgn_pos_inf p"
subset_diff: fixes B :: "'c set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "B \<subseteq> A"   and "finite A" shows "F g A = F g (A - B) \<^bold>* F g B"
list_of_oalist_of_list_id: fixes xs :: "('a \<times> 'b) list \<times> 'o" assumes "oalist_inv xs" shows "list_of_oalist (oalist_of_list xs) = xs"
monofunE: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "monofun f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
nonzero_mult_div_cancel_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide (times a b) a = b"
arcos_fixed: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set"   and a :: "'c"   and h :: "'c" assumes "aGroup A"   and "A +> H"   and "a \<in> carrier A"   and "h \<in> H" shows "a \<uplus>\<^bsub>A\<^esub> H = (h \<plusminus>\<^bsub>A\<^esub> a) \<uplus>\<^bsub>A\<^esub> H"
wtl_suc_pc: fixes pc :: "nat" assumes "wtl ins c 0 s\<^sub>0 \<noteq> \<top>"   and "pc + 1 < length ins" shows "wtl (take (pc + 1) ins) c 0 s\<^sub>0 \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! (pc + 1)"
b_e_type_br_if_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Br_if i" shows "i < length (label \<C>)"
T_tree23_of_list_def: fixes as :: "'a list" shows "T_tree23_of_list as = T_leaves as + T_join_all (leaves as) + 1"
assms_2: shows "bij_betw \<phi> S.Univ UNIV"
assms_6: shows "l\<^sub>2' = [pns [\<mapsto>] vs]"
ipresFreshAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'l, 'm, 'var, 'h, 'i, 'n) model_scheme" shows "ipresFreshAll h hA MOD \<equiv> ipresFresh h MOD \<and> ipresFreshAbs hA MOD"
scale_left_imp_eq: fixes a :: "complex"   and x :: "'b"   and y :: "'b" assumes "a \<noteq> 0"   and "a *\<^sub>C x = a *\<^sub>C y" shows "x = y"
Auth_fresh_not_AKcryptSK: fixes authK :: "nat"   and evs :: "event list"   and servK :: "nat" assumes "Key authK \<notin> used evs"   and "evs \<in> kerbIV_gets" shows "\<not> AKcryptSK authK servK evs"
cas_ends: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "cas u p v"   and "cas u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
is_processT9_S_swap: fixes s :: "'c event list"   and P :: "'c process" assumes "s \<notin> D P" shows "s @ [tick] \<notin> D P"
Polygamma_real_even_neg: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "0 < n"   and "even n" shows "Polygamma n x < 0"
p_pow_nonzero_2: fixes n :: "nat" shows "[p] \<cdot> \<one> [^] n \<noteq> \<zero>"
simps_1: fixes c :: "int" shows "sqr (Pc c) = Pc (c * c)"
comp_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "dual_category C" shows "dual_category.comp C g f = C f g"
subdistl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "\<lparr>z\<rparr>x\<lparr>z \<cdot> (x + y)\<rparr>"