hcju/isabelleps huggingface dataset:
comm_neutral: fixes a :: "'a multiset" shows "a \<union># {#} = a"
wfP_trancl: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "wfP r" shows "wfP r\<^sup>+\<^sup>+"
\<tau>red1t_max_vars: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "max_vars e' \<le> max_vars e"
finsum_dim: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a vec" assumes "finite A"   and "f \<in> A \<rightarrow> carrier_vec n" shows "dim_vec (finsum V f A) = n"
cases_scheme: fixes r :: "('c, 'd, 'e) pre_digraph_scheme" obtains verts :: "'c set" and arcs :: "'d set" and tail :: "'d \<Rightarrow> 'c" and head :: "'d \<Rightarrow> 'c" and more :: "'e" where "r = \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr>"
assms: fixes i :: "nat" assumes "i \<in> set xs" shows "i < length as"
hyps_C_b: fixes b :: "contour"   and \<beta>' :: "nat \<Rightarrow> contour option"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and x :: "(nat \<times> (nat \<Rightarrow> contour option)) \<times> d"   and c :: "call" assumes "b \<in> ran \<beta>'"   and "\<forall>b'\<in>ran \<beta>'. b' \<le> b"   and "\<forall>b'\<in>contours_in_ve ve. b' \<le> b"   and "x \<in> evalC\<cdot>(Discr (c, \<beta>', ve, b))" shows "case x of (a, b) \<Rightarrow> (case a of (lab, \<beta>) \<Rightarrow> \<lambda>t. \<exists>b'. b' \<in> ran \<beta> \<and> b \<le> b') b"
attractor_set_VVp: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player"   and w :: "'c"   and W :: "'c set" assumes "ParityGame G"   and "v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "w \<in> ParityGame.attractor G p W" shows "v \<in> ParityGame.attractor G p W"
Fimpl_rule: fixes s :: "'a" assumes "I s" shows "ospec (Fimpl s) (\<lambda>(ni, s'). (ni, Falseif) \<in> R s' \<and> I s' \<and> les s s')"
mds_yields_stable_types_def: fixes mds :: "Mode \<Rightarrow> Eg1.var set" shows "mds_yields_stable_types mds \<equiv> \<forall>x. x \<in> mds AsmNoWrite \<union> mds AsmNoReadOrWrite \<longrightarrow> (\<forall>v\<in>\<C>_vars x. v \<in> mds AsmNoWrite \<union> mds AsmNoReadOrWrite)"
nths_minus_eq: fixes xs :: "'a list"   and A :: "nat set"   and B :: "nat set" assumes "{0..<length xs} \<subseteq> A" shows "nths xs (- B) = nths xs (A - B)"
oseq_nhid_eq_id: fixes i :: "nat"   and ANY :: "state \<Rightarrow> state \<Rightarrow> bool" shows "optoy i \<Turnstile> (act TT, other ANY {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (seql i (\<lambda>(\<xi>, l). l \<in> {PToy-:2..PToy-:8} \<longrightarrow> nhid \<xi> = state.id \<xi>))"
prv_cnj_dsj_distrib1: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (cnj \<phi> (dsj \<chi>1 \<chi>2)) (dsj (cnj \<phi> \<chi>1) (cnj \<phi> \<chi>2)))"
disjoint: shows "{k} \<inter> (vars1 - {k}) = {}"
mustT_def: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('a, 'c) com"   and s :: "'b" shows "PL.mustT tval aval c s \<equiv> \<not> PL.mayDiverge tval aval c s"
f_nonneg: fixes x :: "nat" assumes "x\<^sub>0 \<le> x" shows "0 \<le> f x"
top_neq_numeral: fixes i :: "num" shows "top \<noteq> numeral i"
real_poly_pCons: fixes c :: "'a"   and p :: "'a poly" assumes "c \<in> \<real>"   and "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (pCons c p)) \<subseteq> \<real>"
simps: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list" shows "rule_states q \<rightarrow> l qs = insert q (set qs)"
Union_included_multiset: fixes M :: "'a multiset" shows "\<Union> {supp x |x. x \<in># M} \<subseteq> supp M"
poly_eval_power: fixes a :: "'d \<Rightarrow> 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and m :: "nat" shows "poly_eval a (p ^ m) = poly_eval a p ^ m"
divide_pos_pos: fixes x :: "'b"   and y :: "'b" assumes "(0::'b) < x"   and "(0::'b) < y" shows "(0::'b) < x / y"
simps: fixes x :: "nat"   and n :: "nat" shows "\<UU>.token_run x n = run \<up>af\<^sub>G\<^sub>\<UU> (Abs (Unf\<^sub>G \<phi>)) (suffix x w) (n - x)"
lcm_factorial_gcd_factorial: fixes a :: "'a"   and b :: "'a" shows "lcm_factorial a b = normalize (a * b div gcd_factorial a b)"
assms_1: shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
no_matching_Goto: fixes ti :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], s\<rangle> \<Rightarrow> ti"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, ti\<rangle> \<Rightarrow> t"   and "no_matching_Goto \<gamma> p [r]" shows "thesis"
cmult: fixes c :: "'b"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "O[F](\<lambda>x. c * f x) = O[F](f)"
mult_float_interval_mono: fixes A :: "float interval"   and X :: "float interval"   and B :: "float interval"   and Y :: "float interval"   and prec :: "nat" assumes "set_of A \<subseteq> set_of X"   and "set_of B \<subseteq> set_of Y" shows "set_of (mult_float_interval prec A B) \<subseteq> set_of (mult_float_interval prec X Y)"
merge_on_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "merge_on B h bo m"
arr_cod_iff_arr: fixes f :: "'c \<times> 'c" shows "CopxC.arr (CopxC.cod f) = CopxC.arr f"
assms: fixes b :: "'a" assumes "(a, b) \<in> r" shows "SN_on r {b}"
name_swap_bij: fixes a :: "name"   and b :: "name"   and x :: "'a" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
lem: fixes p :: "nat" assumes "prime p"   and "p dvd m"   and "p dvd n" shows "p = 1"
cp0: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'res"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "E X \<tau> = E (\<lambda>_. X \<tau>) \<tau>"
curry_case_prod: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'j" shows "curry (\<lambda>(x, y). f x y) = f"
fun_relD: fixes f :: "'g \<Rightarrow> 'h"   and f' :: "'i \<Rightarrow> 'j"   and A :: "('g \<times> 'i) set"   and B :: "('h \<times> 'j) set"   and x :: "'g"   and x' :: "'i" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
neval: shows "lc \<leadsto>\<^bsup>n\<^esup> lc'"
verticesFrom_empty: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "(verticesFrom f v = []) = (vertices f = [])"
1: shows "termFSwMorph (rec MOD) (recAbs MOD) MOD"
eq_implies_ltl_equiv: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<psi>"
le_extend_mult: fixes c :: "nat"   and a :: "nat"   and b :: "nat" assumes "0 < c"   and "a \<le> b" shows "a \<le> b * c"
assms_3: assumes "\<not> b"   and "\<not> b'" shows "S2 \<le> \<Down>\<^sub>E E R S2'"
assms: shows "ok (e\<lparr>x:A\<rparr>\<lparr>y:B\<rparr>)"
Rep_bit_cases: fixes y :: "bool" assumes "y \<in> UNIV" obtains x :: "bit" where "y = Rep_bit x"
exhaust_sel: fixes ast_effect :: "'a ast_effect" obtains "ast_effect = Effect (adds ast_effect) (dels ast_effect)"
r_distr: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y"
isomorphism_sym: fixes f :: "'a \<Rightarrow> 'b"   and G1 :: "('a, 'c) monoid_scheme"   and G2 :: "('b, 'd) monoid_scheme"   and r' :: "'b \<Rightarrow> 'b"   and r :: "'a \<Rightarrow> 'a" assumes "f \<in> Group.iso G1 G2"   and "\<And>x. x \<in> carrier G1 \<Longrightarrow> r' (f x) = f (r x)"   and "\<And>x. x \<in> carrier G1 \<Longrightarrow> r x \<in> carrier G1"   and "Group.group G1"   and "Group.group G2" shows "\<exists>f\<in>Group.iso G2 G1. \<forall>x\<in>carrier G2. r (f x) = f (r' x)"
internal_case_prod_conv: fixes c :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and a :: "'c"   and b :: "'d" shows "??.Product_Type.internal_case_prod c (a, b) = c a b"
antimono_def: fixes f :: "'c \<Rightarrow> 'd" shows "antimono f = (\<forall>x y. x \<le> y \<longrightarrow> f y \<le> f x)"
vectorize_poly_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "vectorize_poly p = Abs_poly_mapping (\<lambda>k. proj_poly k p)"
image_poly_eval_focus_ideal: fixes a :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "poly_eval (\<lambda>x. monomial (a x) 0) ` focus X ` ideal F = ideal (poly_eval (\<lambda>x. monomial (a x) 0) ` focus X ` F) \<inter> P[- X]"
lebesgue_integral_count_space_finite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A" shows "integral\<^sup>L (count_space A) f = sum f A"
pre_ranks_range: fixes n :: "nat"   and \<nu> :: "'a"   and q :: "'b" shows "pre_ranks (\<lambda>q. state_rank q n) \<nu> q \<subseteq> {0..max_rank}"
general_compatibility: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and q :: "(symbol \<times> character list) list"   and n :: "nat" assumes "p \<in> \<P> k u"   and "q \<in> \<P> k u"   and "charslength p = charslength (take n q)"   and "charslength p \<le> k"   and "admissible (p @ drop n q)" shows "p @ drop n q \<in> \<P> k u"
bij_2: shows "G\<^sub>2' = map_ugraph f G\<^sub>2"
post_choose_pending_def: fixes u :: "'v"   and vo :: "'v option"   and s0 :: "'s"   and s :: "'s" shows "post_choose_pending u vo s0 s \<equiv> gen_rwof' s0 \<and> gen_cond s0 \<and> stack s0 \<noteq> [] \<and> u = hd (stack s0) \<and> inres (choose_pending u vo s0) s \<and> stack s = stack s0 \<and> (\<forall>x. gds_is_discovered gds x s = gds_is_discovered gds x s0) \<and> (case vo of None \<Rightarrow> pending s0 `` {u} = {} \<and> pending s = pending s0 | Some v \<Rightarrow> v \<in> pending s0 `` {u} \<and> pending s = pending s0 - {(u, v)})"
deflation_cast: fixes d :: "'a defl" shows "deflation (cast\<cdot>d)"
smult_l_distr: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and a :: "'e"   and b :: "'e"   and x :: "'g" assumes "module R M"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M" shows "(a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x"
nAct_not_same_active: fixes c :: "'id"   and n :: "nat"   and t :: "cnf llist"   and n' :: "enat" assumes "\<langle>c #\<^bsub>enat n\<^esub>t\<rangle> < \<langle>c #\<^bsub>n'\<^esub>t\<rangle>"   and "\<not> lfinite t \<or> n' - 1 < llength t" shows "\<exists>i\<ge>n. enat i < n' \<and> enat i < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
assms_1: shows "fv t \<subseteq> subst_domain s"
vsubset_vsingletonD: fixes A :: "V"   and x :: "V" assumes "A \<subseteq>\<^sub>\<circ> set {x}" shows "A = 0 \<or> A = set {x}"
hseqE: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "hseq \<nu> \<mu>" obtains "arr \<mu>" and "arr \<nu>" and "src \<nu> = trg \<mu>"
inj_moebius_circline: fixes M :: "moebius" shows "inj (moebius_circline M)"
snd_hd: fixes xs :: "('b, 'c) oalist_ntm" assumes "fst (list_of_oalist_ntm xs) \<noteq> []" shows "snd (OAlist_hd_ntm xs) = OAlist_lookup_ntm xs (fst (OAlist_hd_ntm xs))"
Rep_Abs_mod: fixes m :: "int" shows "Rep_bit1 (Abs_bit1 (m mod int CARD('b bit1))) = m mod int CARD('b bit1)"
A3: shows "weak_reduction_simulation (Rel\<inverse>) (STCal Source Target)"
CAST: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ" assumes "G \<rhd> c : HighSec" shows "G \<rhd> c : Sec (\<lambda>(s, t, \<beta>). s \<equiv>\<^sub>\<beta> t)"
3: shows "\<forall>xa. xa \<in> X \<longrightarrow> x \<in> D xa"
bij_betw_id: fixes A :: "'d set" shows "bij_betw id A A"
replace_0: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and j :: "nat"   and a :: "nat \<Rightarrow> nat"   and x :: "nat \<Rightarrow> nat" assumes "kuhn_simplex p n base upd s"   and "j < n"   and "a \<in> s"   and "\<forall>x\<in>s - {a}. x j = 0"   and "x \<in> s" shows "x \<le> a"
fresh_6: fixes a :: "name"   and x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "a \<sharp> [x3\<noteq>x2]x1 = (a \<sharp> x3 \<and> a \<sharp> x2 \<and> a \<sharp> x1)"
smooth_on_cong: fixes k :: "enat"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S g"   and "open S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "k-smooth_on S f"
orbit_perm_rem: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "bij f"   and "x \<noteq> y" shows "orbit (perm_rem y f) x = orbit f x - {y}"
less_SucE: fixes m :: "nat"   and n :: "nat" assumes "order.greater (Suc n) m" obtains "order.greater n m"   | "m = n"
HBL1_inp: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv \<phi>" shows "prv (instInp Pinp \<langle>\<phi>\<rangle>)"
inject: fixes x1 :: "fid_t"   and x2 :: "nat"   and y1 :: "fid_t"   and y2 :: "nat" shows "(x1 $ x2 = y1 $ y2) = (x1 = y1 \<and> x2 = y2)"
span_eq_iff: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b set" assumes "Modules.module scale" shows "(module.span scale s = s) = module.subspace scale s"
euler_trail_def: fixes a :: "'a \<times> nat \<times> 'a"   and u :: "'a"   and p :: "('a \<times> nat \<times> 'a) list"   and v :: "'a" shows "H.euler_trail a u p v \<equiv> H.trail a u p v \<and> set p = arcs (add_arc a) \<and> set (H.awalk_verts u p) = verts (add_arc a)"
Extra_single: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and str :: "char list"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (Match (Extra str)) a p = \<alpha> a p"
linf_norm_vec_def: fixes v :: "'b vec" shows "\<parallel>v\<parallel>\<^sub>\<infinity> \<equiv> Missing_Lemmas.max_list (map abs (list_of_vec v) @ [0::'b])"
i: shows "fst (prod_dfs A B x) ! i1 ! i2 = Some k"
j_2: shows "\<forall>x\<in>f. x j = p"
matrix_to_iarray_snd_Gauss_Jordan_upt_k_PA: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'd) vec" assumes "k < ncols A" shows "matrix_to_iarray (snd (Gauss_Jordan_upt_k_PA A k)) = snd (Gauss_Jordan_upt_k_iarrays_PA (matrix_to_iarray A) k)"
step_5: shows "sorted_wrt (\<lambda>x y. y \<le> x) \<alpha>s"
h_le_lim_h_plus: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "flow_network.h \<Delta> g e \<le> (\<Squnion>n. flow_network.h_plus \<Delta> g n e)"
foldM_rev: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a + 'b"   and s :: "'b"   and li :: "'c list"   and ll :: "'c" shows "foldM f s (li @ [ll]) = foldM f s li \<bind> (\<lambda>t. f t ll)"
T_in: shows "Triple p a (prod_list ys) \<in> set (build_fpc p a (r div y) ys @ xs)"
separation_normal_compact: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "closed T"   and "S \<inter> T = {}" obtains U :: "'a set" and V :: "'a set" where "open U" and "compact (closure U)" and "open V" and "S \<subseteq> U" and "T \<subseteq> V" and "U \<inter> V = {}"
hyps_2: shows "\<Gamma>\<turnstile> cfg'' \<rightarrow>\<^sup>* cfg\<^sub>2"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "Adj.C.seq g f" shows "G (g \<cdot>\<^sub>C f) = G g \<cdot>\<^sub>D G f"
wf_inorm: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "WS1S_Equivalence_Checking.wf \<Sigma> n r" shows "WS1S_Equivalence_Checking.wf \<Sigma> n (inorm r)"
base: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "f a \<noteq> b" shows "f a \<in> segment f a b"
spmf_restrict_spmf_inside: fixes x :: "'c"   and A :: "'c set"   and p :: "'c spmf" assumes "x \<in> A" shows "spmf (p \<upharpoonleft> A) x = spmf p x"
poly_order_carrier_axioms: shows "poly_order_carrier default (\<succ>) power_mono discrete"
inputChainAlpha: fixes Xs :: "name set"   and xvec :: "name list"   and N :: "'d"   and P :: "('d, 'e, 'f) psi"   and Ys :: "name set"   and p :: "(name \<times> name) list" assumes "Xs \<sharp>* inputChain xvec N P"   and "Ys \<sharp>* N"   and "Ys \<sharp>* P"   and "set p \<subseteq> Xs \<times> Ys" shows "inputChain xvec N P = inputChain (p \<bullet> xvec) (p \<bullet> N) (p \<bullet> P)"
assign_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and x :: "'a"   and e :: "'c"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((x \<leftarrow> e, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "c' = Stop \<and> mds = mds' \<and> mem' = mem(x := eval\<^sub>A mem e)"
star_right_induct_var: fixes x :: "'a"   and y :: "'a" shows "x * y \<le> x \<longrightarrow> x * y\<^sup>\<star> \<le> x"
c_ipurge_ipurge_tr: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and ys :: "'f list"   and u' :: "'e"   and xs :: "'f list" assumes "refl I"   and "\<not> (\<exists>v\<in>sinks I D u ys. (v, u') \<in> I)" shows "c_ipurge I D u' (xs @ ipurge_tr I D u ys) = c_ipurge I D u' (xs @ ys)"
that: fixes TAC :: "('Q set option, 'L) tree_automaton_rec" assumes "ta_lang TAC = ta_lang TA"   and "complete_tree_automaton TAC A" shows "thesis"
delete_eq: fixes k :: "'d" shows "delete k = filter (\<lambda>(k', uu_). k \<noteq> k')"
unstream_concat_trans_gen: fixes g :: "('e list, 'f) generator"   and xs :: "'e list"   and s :: "'f" shows "unstream (concat_trans g) (xs, s) = xs @ concat (unstream g s)"
lemma3_5_DD_v: fixes r :: "('a \<times> 'a) set"   and ars :: "('b \<times> 'a \<times> 'b) set"   and \<tau> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>' :: "'b \<times> ('a \<times> 'b) list"   and \<tau>' :: "'b \<times> ('a \<times> 'b) list"   and \<upsilon> :: "'b \<times> ('a \<times> 'b) list"   and \<upsilon>' :: "'b \<times> ('a \<times> 'b) list"   and \<tau>'' :: "'b \<times> ('a \<times> 'b) list" assumes "trans r"   and "irrefl r"   and "DD ars r (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "DD ars r (\<tau>', \<upsilon>, \<upsilon>', \<tau>'')" shows "DD ars r (\<tau>, (fst \<sigma>, snd \<sigma> @ snd \<upsilon>), (fst \<sigma>', snd \<sigma>' @ snd \<upsilon>'), \<tau>'')"
rep_eq: fixes x :: "('a, 'b) lifted_distrib_lattice_top"   and xa :: "('a, 'b) lifted_distrib_lattice_top" shows "(x < xa) = ((\<forall>f. Rep_lifted_distrib_lattice_top x f \<le> Rep_lifted_distrib_lattice_top xa f) \<and> \<not> (\<forall>f. Rep_lifted_distrib_lattice_top xa f \<le> Rep_lifted_distrib_lattice_top x f))"
continuous_openin_preimage: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set"   and U :: "'d set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "openin (top_of_set T) U" shows "openin (top_of_set S) (S \<inter> f -` U)"
right_idem: fixes a :: "'d"   and b :: "'d" shows "inf (inf a b) b = inf a b"
privateKey_neq_publicKey: fixes b :: "bool"   and A :: "agent"   and b' :: "bool"   and A' :: "agent" shows "invKey (publicKey b A) \<noteq> publicKey b' A'"
group_commutes_pow: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<otimes>\<^bsub>P\<^esub> y = y \<otimes>\<^bsub>P\<^esub> x"   and "x \<in> carrier P"   and "y \<in> carrier P" shows "x [^]\<^bsub>P\<^esub> n \<otimes>\<^bsub>P\<^esub> y = y \<otimes>\<^bsub>P\<^esub> x [^]\<^bsub>P\<^esub> n"
Nml_Trg: fixes t :: "'a E.term" assumes "E.Nml t" shows "E.Nml (E.Trg t)"
measurement_lowner_le_one1: fixes M0 :: "complex mat"   and M1 :: "complex mat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m d" shows "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m d"
relative_to_subset: fixes S :: "'b set"   and T :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "S \<subseteq> T \<and> P S" shows "(P relative_to T) S"
simps_2: fixes l :: "tree"   and r :: "tree" shows "count_nodes (Node l r) = 1 + count_nodes l + count_nodes r"
simps_2: fixes ao :: "'a option"   and n :: "nat"   and y :: "'a"   and ys :: "'a list" shows "proper_interval_set_Compl_aux ao n [] (y # ys) = (let m = CARD('a) - n; (len_y, y') = length_last (y # ys) in m \<noteq> len_y \<and> (m = len_y + 1 \<longrightarrow> \<not> proper_interval (Some y') None))"
type_less_x_Set_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Set \<rho>"   and "\<rho> < \<sigma>" shows "\<tau> < Set \<sigma>"
StrongG_Body: shows "StrongG \<rhd> body : SSpec Call"
NDSourceExistsDSource: fixes S :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" shows "S \<in> Sources i C \<and> S \<notin> DSources i C \<longrightarrow> (\<exists>Z. S \<in> DSources i Z \<and> Z \<in> Sources i C)"
id_eq_prop_prop_7_b: fixes v :: "i"   and p :: "\<o>" shows "[p \<^bold>= p in v]"
subo: shows "Subobjs\<^sub>R P C' Cs'"
safety_def2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "safety P = (\<forall>\<sigma>. (\<forall>i. \<exists>\<beta>. P (stake i \<sigma> @- \<beta>)) \<longrightarrow> P \<sigma>)"
BB_PROTECT: fixes p :: "'a" shows "p \<equiv> BB_PROTECT p"
prev_maximal: fixes j :: "nat"   and i :: "nat"   and j' :: "nat"   and i' :: "nat"   and ji'' :: "nat \<times> nat"   and k :: "nat" assumes "prev j i = Some (j', i')"   and "(ji'', j, i) \<in> pair_less"   and "ji'' \<in> IJ k" shows "(ji'', j', i') \<in> pair_less \<or> ji'' = (j', i')"
is_processT5_S7: fixes t :: "'b event list"   and P :: "'b process"   and A :: "'b event set" assumes "t \<in> T P"   and "(t, A) \<notin> F P" shows "\<exists>x. x \<in> A \<and> t @ [x] \<in> T P"
Bot_cond: fixes C :: "'f" shows "\<G>_F C \<inter> Bot_G \<noteq> {} \<longrightarrow> C \<in> Bot_F"
a: shows "( \<Gamma> \<Rightarrow>* \<Delta>, n) \<in> derivable (ext R R2 M N)"
rel_Grp: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "list_all2 (Grp A f) = Grp {x. set x \<subseteq> A} (map f)"
axioms_1: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite_total_preorder_on carrier le" shows "total_preorder_on carrier le"
simps: fixes pm :: "nat"   and ipm :: "int"   and ip :: "int"   and x :: "int"   and n :: "int" shows "root_int_main' pm ipm ip x n = (let xpm = x ^ pm; xp = xpm * x in if xp \<le> n then (x, xp = n) else root_int_main' pm ipm ip ((n div xpm + x * ipm) div ip) n)"
semiring_simprules_1: fixes R :: "('d, 'e) ring_scheme"   and x :: "'d"   and y :: "'d" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
nrcons_invlevel: fixes n :: "nat"   and t :: "('a, 'b) FingerTreeStruc"   and nd :: "('a, 'b) Node" assumes "FingerTreeStruc.is_leveln_ftree n t"   and "FingerTreeStruc.is_leveln_node n nd" shows "FingerTreeStruc.is_leveln_ftree n (FingerTreeStruc.nrcons t nd)"
reachable_target: fixes q :: "'c"   and A :: "('d, 'c) ngba"   and p :: "'c"   and r :: "('d \<times> 'c) list" assumes "q \<in> NGBA.reachable A p"   and "NGBA.path A r q" shows "target r q \<in> NGBA.reachable A p"