hcju/isabelleps huggingface dataset:
coeffs_poly_of_vec: fixes v :: "'b vec" shows "coeffs (poly_of_vec v) = rev (dropWhile ((=) (0::'b)) (list_of_vec v))"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b" assumes "semigroup f"   and "group_axioms f z inverse" shows "group f z inverse"
card_distinct: fixes xs :: "'c list" assumes "card (set xs) = length xs" shows "cycle xs"
one_le_exp_iff: fixes x :: "real" shows "(1 \<le> exp x) = (0 \<le> x)"
that: fixes a :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A" shows "thesis"
setl_def: shows "setl \<equiv> \<lambda>s. {x. setlp s x}"
ring_l_one: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c" assumes "Ring R"   and "x \<in> carrier R" shows "1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>r\<^bsub>R\<^esub> x = x"
tendsto_in\<^sub>N_def: fixes N :: "'b quasinorm"   and u :: "nat \<Rightarrow> 'b"   and x :: "'b" shows "tendsto_in\<^sub>N N u x = (\<lambda>n. Norm N (u n - x)) \<longlonglongrightarrow> 0"
simps_11: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and x2 :: "'c" shows "(case Tag x2 of Oldf x \<Rightarrow> f1 x | Tag x \<Rightarrow> f2 x | Wit x \<Rightarrow> f3 x) = f2 x2"
hyps: fixes i :: "nat" assumes "mem_idx x as = Some i" shows "i < length as \<and> as ! i = x"
contour_integral_nearby_ends: fixes S :: "complex set"   and p :: "real \<Rightarrow> complex" assumes "open S"   and "path p"   and "path_image p \<subseteq> S" shows "\<exists>d>0. \<forall>g h. valid_path g \<and> valid_path h \<and> (\<forall>t\<in>{0..1}. cmod (g t - p t) < d \<and> cmod (h t - p t) < d) \<and> pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<longrightarrow> path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and> (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f)"
aezfun_setspan_proj_def: fixes A :: "'b set"   and x :: "('a, 'b) aezfun" shows "aezfun_setspan_proj A x \<equiv> Abs_aezfun (aezfun x \<down> A)"
eq_parts_imp_eq: fixes a :: "'e \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and b :: "'e \<Rightarrow> 'f" assumes "a \<in> carrier (DirProds G I)"   and "b \<in> carrier (DirProds G I)"   and "\<And>i. i \<in> I \<Longrightarrow> a i = b i" shows "a = b"
spectrum_imp_pr: fixes \<pp> :: "'a set" assumes "\<pp> \<in> Spec" shows "pr_ideal R \<pp> (+) (\<cdot>) \<zero> \<one>"
treap_of_set_tree_unique: fixes A :: "('a \<times> 'b) set" assumes "finite A"   and "inj_on fst A"   and "inj_on snd A" shows "set_tree (treap_of A) = A"
obs_singleton: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and n :: "'b"   and S :: "'b set" assumes "StandardControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "(\<exists>m. CFG.obs sourcenode targetnode valid_edge n (StandardControlDependencePDG.PDG_BS_s sourcenode targetnode valid_edge Def Use Exit S) = {m}) \<or> CFG.obs sourcenode targetnode valid_edge n (StandardControlDependencePDG.PDG_BS_s sourcenode targetnode valid_edge Def Use Exit S) = {}"
callee_invariant: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and y :: "'c"   and s' :: "'a"   and s :: "'a"   and x :: "'b" assumes "callee_invariant_on callee I \<I>"   and "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "x \<in> outs_\<I> \<I>" shows "I s'"
borel_measurable_continuous_onI: fixes f :: "'c \<Rightarrow> 'd" assumes "continuous_on UNIV f" shows "f \<in> borel_measurable borel"
zero_less_double_add_iff_zero_less_single_add: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (plus a a) = less zero a"
PR: fixes X :: "('a \<times> 'b) topology"   and X' :: "'b topology" assumes "X homeomorphic_space X'" shows "P X = R X'"
row_add_iterate_preserves_pivot_row: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b"   and b :: "'b" assumes "n < nrows A"   and "mod_type_class.to_nat i \<le> n" shows "row_add_iterate A n i j $ i $ b = A $ i $ b"
g'g: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> T" shows "g' x = g x"
circ_one: shows "(1::'a)\<^sup>\<star> = (1::'a)\<^sup>\<star>\<^sup>\<star>"
c: shows "steps0 (Suc 0, l, r) (adjust0 ap) stpa = (sa, la, ra)"
assms_1: shows "Case_Labeling.CTXT (Suc OC1) CT OC (ovalidNF R g Q)"
map_eq_append_conv: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and ys :: "'g list"   and zs :: "'g list" shows "(map f xs = ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
primes_omega_power: fixes n :: "nat"   and x :: "nat" assumes "0 < n" shows "primes_omega (x ^ n) = primes_omega x"
cases: fixes a1 :: "'d set"   and a2 :: "'d \<Rightarrow> nat"   and a3 :: "'d \<Rightarrow> intv"   and a4 :: "('d \<times> 'd) set" assumes "valid_region a1 a2 a3 a4" obtains X\<^sub>0 :: "'d set" and X :: "'d set" and I :: "'d \<Rightarrow> intv" and r :: "('d \<times> 'd) set" and k :: "'d \<Rightarrow> nat" where "a1 = X" and "a2 = k" and "a3 = I" and "a4 = r" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}" and "refl_on X\<^sub>0 r" and "trans r" and "total_on X\<^sub>0 r" and "\<forall>x\<in>X. valid_intv (k x) (I x)"
comm_semiring_1_ow_def: fixes U :: "'c set"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" shows "comm_semiring_1_ow U times one plus zero \<equiv> zero_neq_one_ow U one zero \<and> comm_semiring_0_ow U plus zero times \<and> comm_monoid_mult_ow U times one"
invar_set: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "set_kdt (Node k v l r) = set_kdt l \<union> set_kdt r"
finite_ReZ_segments_shiftpah: fixes g :: "real \<Rightarrow> complex"   and z :: "complex"   and s :: "real" assumes "finite_ReZ_segments g z"   and "s \<in> {0..1}"   and "path g"   and "pathfinish g = pathstart g" shows "finite_ReZ_segments (shiftpath s g) z"
flow_in_domain: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "flow t0 x0 t \<in> X"
L2_set_le_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" shows "L2_set f A \<le> sum f A"
P: shows "\<forall>r\<in>set rs. P (get_match r)"
handshake_invL_def: shows "handshake_invL = (\<lambda>s. (atS gc hs_noop_locs s \<longrightarrow> sys_hs_type s\<down> = ht_NOOP) \<and> (atS gc hs_get_roots_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetRoots) \<and> (atS gc hs_get_work_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetWork) \<and> (atS gc hs_mut_in_muts_locs s \<longrightarrow> gc_mut s\<down> \<in> gc_muts s\<down>) \<and> (atS gc hs_init_loop_locs s \<longrightarrow> (\<forall>x. x \<notin> gc_muts s\<down> \<longrightarrow> sys_hs_pending x s\<down> \<or> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc hs_init_loop_not_done_locs s \<longrightarrow> (\<forall>x. x \<in> gc_muts s\<down> \<longrightarrow> \<not> sys_hs_pending x s\<down> \<and> \<not> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc hs_init_loop_done_locs s \<longrightarrow> (sys_hs_pending (gc_mut s\<down>) s\<down> \<or> sys_ghost_hs_in_sync (gc_mut s\<down>) s\<down>) \<and> (\<forall>x. x \<in> gc_muts s\<down> \<and> x \<noteq> gc_mut s\<down> \<longrightarrow> \<not> sys_hs_pending x s\<down> \<and> \<not> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc hs_done_locs s \<longrightarrow> (\<forall>x. sys_hs_pending x s\<down> \<or> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc hs_done_loop_locs s \<longrightarrow> (\<forall>x. x \<notin> gc_muts s\<down> \<longrightarrow> \<not> sys_hs_pending x s\<down>)) \<and> (atS gc hs_none_pending_locs s \<longrightarrow> (\<forall>x. \<not> sys_hs_pending x s\<down>)) \<and> (atS gc hs_in_sync_locs s \<longrightarrow> (\<forall>x. sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc hs_out_of_sync_locs s \<longrightarrow> (\<forall>x. \<not> sys_hs_pending x s\<down> \<and> \<not> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc hp_Idle_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_Idle) \<and> (atS gc hp_IdleInit_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_IdleInit) \<and> (atS gc hp_InitMark_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_InitMark) \<and> (atS gc hp_IdleMarkSweep_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_IdleMarkSweep) \<and> (atS gc hp_Mark_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_Mark))"
nonneg: fixes i :: "'n"   and j :: "'n" shows "0 \<le> A $h i $h j"
monotonicity_L: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set"   and B :: "'b set" assumes "Tarskian_logic Cn"   and "A \<subseteq> B" shows "Cn A \<subseteq> Cn B"
WhileFReds: fixes b :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 c :: "char list exp" assumes "P \<turnstile> \<langle>b,s\<rangle> \<rightarrow>* \<langle>false,s'\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s\<rangle> \<rightarrow>* \<langle>unit,s'\<rangle>"
delta_hedging_trading_strat: fixes N :: "bool stream measure"   and q :: "real"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "der \<in> borel_measurable (G matur)" shows "trading_strategy (delta_hedging N der matur)"
L2_set_def: fixes f :: "'d \<Rightarrow> real"   and A :: "'d set" shows "L2_set f A = sqrt (\<Sum>i\<in>A. (f i)\<^sup>2)"
HomFtorDist': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e"   and g :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "(Hom\<^bsub>C\<^esub>[X,f]) ;;\<^bsub>SET'\<^esub> (Hom\<^bsub>C\<^esub>[X,g]) = Hom\<^bsub>C\<^esub>[X,f ;;\<^bsub>C\<^esub> g]"
l1_defs_2: shows "l1_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0\<rparr>}"
intros: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and H :: "'a"   and \<Sigma> :: "'b list" assumes "Fubx_get F f = Some fd"   and "pc = length (body fd)" shows "Subx.final (State F H [Frame f pc \<Sigma>])"
degree_add_eq_right: fixes p :: "'b poly"   and q :: "'b poly" assumes "degree p < degree q" shows "degree (p + q) = degree q"
substitutes_unitE: fixes y :: "'a"   and M :: "'a trm"   and X :: "'a trm" assumes "substitutes Unit y M X" shows "X = Unit"
suff_finpref_iff: fixes r :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "r \<in> A\<^sup>\<star>"   and "t \<in> A\<^sup>\<infinity>" shows "(r \<in> finpref A t) = (t \<in> suff A r)"
1: shows "x * y\<^sup>\<star> \<le> x\<^sup>\<star> \<squnion> y\<^sup>\<star>"
bind_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'b measure" shows "M \<bind> f = (if space M = {} then count_space {} else join (distr M (subprob_algebra (f (SOME x. x \<in> space M))) f))"
ipresIGSubst_def: fixes h :: "'h \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'sort, 'l, 'm, 'h, 'n, 'o) model_scheme"   and MOD' :: "('p, 'q, 'varSort, 'r, 's, 'm, 'i, 't, 'u) model_scheme" shows "ipresIGSubst h MOD MOD' \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> h (igSubst MOD ys Y y X) = igSubst MOD' ys (h Y) y (h X)"
c_fst_less_arg: fixes x :: "nat" assumes "0 < x" shows "c_fst x < x"
mC_C_eff_cont_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis" shows "PL_Indis.mC_C_eff_cont aval tval cval indis theta c d s t P F \<equiv> \<forall>I i j. I \<in> P \<and> i \<in> I \<and> j \<in> F I \<longrightarrow> (PL.eff aval tval cval c s i, PL.eff aval tval cval d t j) \<in> indis \<and> (PL.cont aval tval cval c s i, PL.cont aval tval cval d t j) \<in> theta"
least_unique: fixes k :: "'d"   and A :: "'d set" assumes "k \<in> A"   and "k \<le> least A" shows "k = least A"
bisimE: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sub>l Q" shows "P \<leadsto>\<^sub>l[Strong_Late_Bisim.bisim] Q"
X_square_integrable: fixes n :: "nat" shows "integrable M (\<lambda>x. (X n x)\<^sup>2)"
simps_1: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and a :: "action"   and p :: "'d" shows "match_list \<gamma> [] a p = False"
U: shows "UNIV = (\<lambda>(a, b, c, d). \<lparr>sprFst = a, sprLst = b, sprCRel = c, \<dots> = d\<rparr>) ` (UNIV \<times> UNIV \<times> UNIV \<times> UNIV)"
openin_open_eq: fixes s :: "'c set"   and t :: "'c set" assumes "open s" shows "openin (top_of_set s) t = (open t \<and> t \<subseteq> s)"
ldrop_lnull: fixes xs :: "'c llist"   and n :: "enat" assumes "lnull xs" shows "ldrop n xs = LNil"
RED_props_2: fixes \<tau> :: "ty" shows "CR2 \<tau>"
DF_subset: fixes A :: "'d set"   and B :: "'d set" assumes "A \<noteq> {}"   and "A \<subseteq> B" shows "DF B \<sqsubseteq>\<^sub>F\<^sub>D DF A"
simps_13: fixes b :: "char list exp"   and e :: "char list exp" shows "fv (while (b) e) = fv b \<union> fv e"
hyp2_axiom11: shows "\<forall>X Y. (\<exists>a. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B\<^sub>K a x y) \<longrightarrow> (\<exists>b. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B\<^sub>K x b y)"
unique_connecting_path_unique: fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and xs :: "'e list"   and w :: "'e" assumes "Tree G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "xs = v \<leadsto>\<^bsub>G\<^esub> w"
subsetD: fixes x :: "'a" assumes "x \<in> F" shows "x \<in> E"
trd_red_rtrancl: fixes fs :: "('a \<Rightarrow>\<^sub>0 'd) list"   and p :: "'a \<Rightarrow>\<^sub>0 'd" shows "(punit.red (set fs))\<^sup>*\<^sup>* p (punit.trd fs p)"
pairwise_iff: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream" shows "PTA_Reachability.pairwise P xs = (\<forall>i. P (xs !! i) (xs !! Suc i))"
inf_cancel_left2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (inf (uminus x) a) (inf x b) = bot"
peval_eval: fixes p :: "'b pfm"   and h :: "char list \<Rightarrow> bool"   and g :: "'b fm \<Rightarrow> bool" assumes "static p" shows "eval h g (lower p) = peval h (g \<circ> lower) p"
\<sigma>_deduct: fixes x :: "'var"   and M :: "('fun, 'var) Term.term set" assumes "x \<in> subst_domain \<sigma>" shows "M \<turnstile>\<^sub>c \<sigma> x"
\<Psi>_simp: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "\<phi>\<psi>.\<Psi> yx = \<phi>\<psi>.\<Psi>o yx"
distr_return: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and x :: "'a" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "x \<in> space M" shows "distr (return M x) N f = return N (f x)"
min_nat_of_len_of: fixes n :: "'b sat" shows "min (nat_of n) LENGTH('b) = nat_of n"
assms_2: shows "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"
id2: shows "ffp (R j) = subresultant j (F 1) (F 2)"
conjunctive_def: shows "Apply.conjunctive = {x. \<forall>y z. x (inf y z) = inf (x y) (x z)}"
no_collision_reactI: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real"   and S :: "real set" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"   and "\<And>t. t \<in> S \<Longrightarrow> safe_distance_normal.u a\<^sub>e v\<^sub>e s\<^sub>e \<delta> t \<noteq> movement.s a\<^sub>o v\<^sub>o s\<^sub>o t" shows "\<not> safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> S"
ordIso_imp_ordLeq: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set" assumes "ordIso2 r r'" shows "ordLeq3 r r'"
MWLfSteps_det_cases_5: fixes b :: "'exp"   and c :: "('exp, 'id) MWLfCom"   and m :: "'id \<Rightarrow> 'val"   and W :: "('exp, 'id) MWLfCom list"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (while b do c od) m W m'" obtains "W = [c;while b do c od]" and "m' = m" and "BMap (E b m)"   | "W = []" and "m' = m" and "\<not> BMap (E b m)"
continuous_on_Arccos_real: shows "continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} Arccos"
less_eq_num1_def: fixes a :: "1"   and b :: "1" shows "(a \<le> b) = (Rep_num1 a \<le> Rep_num1 b)"
is_static_qmdecl_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "is_static (qmdecl sig m) = is_static m"
mkArr_in_hom: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "A \<subseteq> S.Univ"   and "B \<subseteq> S.Univ"   and "F \<in> A \<rightarrow> B" shows "\<guillemotleft>S.mkArr A B F : S.mkIde A \<rightarrow>\<^sub>S S.mkIde B\<guillemotright>"
less_eq_top_Top: fixes x :: "'a top" shows "x \<le> Top"
respection: shows "rel_weakly_respects_barbs (indRelRTPO TRel) (STCalWB SWB TWB)"
Can_and_Nml_implies_Ide: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Can V src trg t"   and "bicategorical_language.Nml V src trg t" shows "bicategorical_language.Ide V src trg t"
standard_sound: fixes P :: "'c \<Rightarrow> bool" shows "sound \<guillemotleft> P \<guillemotright>"
msubst_I: fixes p :: "fm"   and c :: "poly"   and d :: "poly"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and t :: "tm"   and s :: "tm" assumes "islin p"   and "isnpoly c"   and "isnpoly d" shows "Ifm vs (x # bs) (msubst p ((c, t), d, s)) = Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) p"
extremeI: fixes e :: "'b"   and X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "e \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> r x e" shows "extreme X r e"
finite_PDERIVS: fixes R :: "'b rexp set" assumes "finite R" shows "finite (PDERIVS R)"
wp_xs: shows "wellformed_policy2Pr (removeShadowRules2 xs)"
prod_index_def: fixes less_eq0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and idx0 :: "'a bound \<Rightarrow> 'a \<Rightarrow> nat"   and size0 :: "'a bound \<Rightarrow> nat"   and less_eq1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and idx1 :: "'b bound \<Rightarrow> 'b \<Rightarrow> nat"   and size1 :: "'b bound \<Rightarrow> nat" shows "prod_index less_eq0 less0 idx0 size0 less_eq1 less1 idx1 size1 \<equiv> index_locale less_eq0 less0 idx0 size0 \<and> index_locale less_eq1 less1 idx1 size1"
card_mono: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subseteq> B" shows "card A \<le> card B"
absorb2: fixes b :: "'a"   and a :: "'a" assumes "b \<sqsubseteq> a" shows "a \<sqinter> b = b"
is_float_uminus: assumes "is_float (- aa)" shows "is_float (- (- aa))"
neq_iff: fixes x :: "'b"   and y :: "'b" shows "(x \<noteq> y) = (x < y \<or> y < x)"
lift_orient_lit: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and L :: "'b literal"   and t :: "'b trm"   and s :: "'b trm"   and p :: "sign"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.orient_lit_inst trm_ord L t s p \<sigma>" shows "basic_superposition.orient_lit trm_ord (subst_lit L \<sigma>) (t \<lhd> \<sigma>) (s \<lhd> \<sigma>) p"
enum_ex_finite_5_def: fixes P :: "finite_5 \<Rightarrow> bool" shows "enum_ex P = (P a\<^sub>1 \<or> P a\<^sub>2 \<or> P a\<^sub>3 \<or> P a\<^sub>4 \<or> P a\<^sub>5)"
hs_dfs_\<alpha>_def: fixes S :: "('a, unit) HashMap.hashmap \<times> 'a list" shows "hs_dfs_\<alpha> S \<equiv> let (Q, W) = S in (hs.\<alpha> Q, W)"
exec_Seq': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and s' :: "('e, 'f) xstate"   and c2 :: "('e, 'd, 'f) com"   and s'' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow> s''" shows "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow> s''"
reachable_trans: fixes q :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and p :: "'d"   and s :: "'d" assumes "q \<in> reachable A p"   and "s \<in> reachable A q" shows "s \<in> reachable A p"
deg_const: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_ring R"   and "a \<in> carrier R" shows "deg R (monom (UP R) a 0) = 0"
obtains_base: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r' :: "nat \<Rightarrow>f value option" shows "obtains s r e s' r' [] = (s = s' \<and> r = r')"
F_finite: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "f \<in> Field leq" shows "f e \<noteq> \<top>"
wf\<^sub>t\<^sub>r\<^sub>m_def: fixes t :: "('fun, 'b) Term.term" shows "wf\<^sub>t\<^sub>r\<^sub>m t \<equiv> \<forall>f T. Fun f T \<sqsubseteq> t \<longrightarrow> length T = arity f"
add_anno_def: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and upd :: "'Var ModeUpd" shows "\<Gamma> \<oplus> upd = (\<lambda>x. Some (to_total \<Gamma> x)) |` add_anno_dom \<Gamma> upd"
dvd_div_iff_mult: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "c \<noteq> zero"   and "dvd.dvd times c b" shows "dvd.dvd times a (divide b c) = dvd.dvd times (times a c) b"