hcju/isabelleps huggingface dataset:
streamValue7: fixes d :: "nat"   and str :: "nat \<Rightarrow> 'b"   and t :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "'b"   and j :: "nat" assumes "\<forall>j\<le>d. str (t + i + k + d + Suc j) = x"   and "str (t + i + k + d) = x"   and "j \<le> Suc d" shows "str (t + i + k + d + j) = x"
2: shows "\<forall>a\<in>{(i, t, s)} \<union> set D. case a of (j, p) \<Rightarrow> \<forall>a\<in>{(i, t, s)} \<union> set D. case a of (k, q) \<Rightarrow> (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"
base_case_invertible_mat: shows "invertible_mat (M_mat [[1], [- 1]] [[], [0]])"
subst_lt: fixes j :: "nat"   and i :: "nat"   and u :: "db" assumes "j < i" shows "DBVar j<u/i> = DBVar j"
0: shows "simplical_face k ` standard_simplex (p - Suc 0) \<subseteq> standard_simplex p"
A5: shows "Esigma (2 ^ k * p * q) = Esigma (2 ^ k) * Esigma p * Esigma q"
widen_Array2: fixes G :: "prog"   and S :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>T.[]" shows "S = NT \<or> (\<exists>S'. S = S'.[] \<and> G\<turnstile>S'\<preceq>T)"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "L'.C.isomorphic a a'" obtains f :: "'a" where "L'.C.in_hom f a a' \<and> L'.C.iso f"
pos_primfun_list_iff: fixes fs :: "('a \<times> 'b) list" shows "pos_primfun_list fs = pos_list (map snd fs)"
Dom_UP: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "Dom (UP \<mu>) = \<^bold>\<langle>B.dom \<mu>\<^bold>\<rangle>"
non_det_assoc: fixes M :: "'a process"   and P :: "'a process"   and Q :: "'a process" shows "(M |-| P |-| Q) = (M |-| (P |-| Q))"
is_subset_1: fixes TA :: "('a, 'b) tree_automaton_rec"   and c :: "'a" assumes "tree_automaton TA"   and "c \<in> ta_initial TA" shows "c \<in> ta_rstates TA"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "dlex_pp_strict x xa = dlex_pm_strict (pp.mapping_of x) (pp.mapping_of xa)"
is_pseudonatural_equivalence: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<sigma>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>1 :: "'c \<Rightarrow> 'd"   and \<rho>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<rho>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "composite_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<rho>\<^sub>0 \<rho>\<^sub>1" shows "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F H \<Phi>\<^sub>H (composite_pseudonatural_transformation.map\<^sub>0 H\<^sub>D \<sigma>\<^sub>0 \<rho>\<^sub>0) (composite_pseudonatural_transformation.map\<^sub>1 src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D src\<^sub>D trg\<^sub>D F G H \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<rho>\<^sub>0 \<rho>\<^sub>1)"
rel_spmf_spmf_of_pmf: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a pmf"   and q :: "'b pmf" shows "rel_spmf R (spmf_of_pmf p) (spmf_of_pmf q) = rel_pmf R p q"
tm_norm'_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t :: "taylor_model"   and a :: "float interval list"   and I :: "float interval list"   and prec :: "nat"   and ord :: "nat" assumes "x \<in>\<^sub>i range_tm e t"   and "develops_at_within e a I"   and "num_params (tm_poly t) \<le> length I" shows "x \<in>\<^sub>i range_tm e (tm_norm' prec ord I a t)"
eq_game_sym: fixes G :: "game"   and H :: "game" shows "eq_game G H = eq_game H G"
CollectD: fixes a :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"
rt_fresher_trans: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt3"
InvariantWatchesDiffer_def: fixes formula :: "Literal list list"   and watch1 :: "nat \<Rightarrow> Literal option"   and watch2 :: "nat \<Rightarrow> Literal option" shows "InvariantWatchesDiffer formula watch1 watch2 \<equiv> \<forall>clause. 0 \<le> clause \<and> clause < length formula \<longrightarrow> watch1 clause \<noteq> watch2 clause"
isOp_imp_wls: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "wls s X"
conv_abscissa_zeta: shows "conv_abscissa fds_zeta = 1"
poincare_between_sum_distances_x_axis_u0v: fixes u' :: "complex"   and v' :: "complex" assumes "of_complex u' \<in> unit_disc"   and "of_complex v' \<in> unit_disc"   and "is_real u'"   and "u' \<noteq> 0"   and "v' \<noteq> 0" shows "(poincare_distance (of_complex u') 0\<^sub>h + poincare_distance 0\<^sub>h (of_complex v') = poincare_distance (of_complex u') (of_complex v')) = (is_real v' \<and> Re u' * Re v' < 0)"
left_id: fixes x :: "int poly fract" shows "rat_poly_times 1 x = x"
simps: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'b" shows "star r a1 a2 = ((\<exists>x. a1 = x \<and> a2 = x) \<or> (\<exists>x y z. a1 = x \<and> a2 = z \<and> r x y \<and> star r y z))"
has_vector_derivative: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "'b" assumes "bounded_bilinear prod"   and "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at x within s)" shows "((\<lambda>x. prod (f x) (g x)) has_vector_derivative prod (f x) g' + prod f' (g x)) (at x within s)"
is_run_def: fixes G :: "('b, 'c) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'b" shows "graph_defs.is_run G r \<equiv> r 0 \<in> g_V0 G \<and> ipath (g_E G) r"
image_Union: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set set" shows "f ` \<Union> S = \<Union> ((`) f ` S)"
alg_multE_1: fixes x :: "'a"   and p :: "'a list" assumes "x \<in> carrier R"   and "p \<in> carrier (poly_ring R)"   and "p \<noteq> []" shows "[\<one>, \<ominus> x] [^]\<^bsub>poly_ring R\<^esub> alg_mult p x pdivides p"
get_min: fixes ts :: "'b tree list" assumes "mset_heap ts \<noteq> {#}"   and "invar ts" shows "get_min ts = Min_mset (mset_heap ts)"
unit: fixes y :: "'a" assumes "y \<in> fst ` set ys" shows "is_unit (lift y)"
normalize_prime: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "normalize p = p"
mset_subset_eq_exists_conv: fixes A :: "'b multiset"   and B :: "'b multiset" shows "(A \<subseteq># B) = (\<exists>C. B = A + C)"
star_inductr_var: fixes y :: "'a"   and x :: "'a" assumes "\<lparr>y\<rparr>x\<lparr>y\<rparr>" shows "\<lparr>y\<rparr>x\<^sup>\<star>\<lparr>y\<rparr>"
polytope_Int_polyhedron: fixes S :: "'b set"   and T :: "'b set" assumes "polytope S"   and "polyhedron T" shows "polytope (S \<inter> T)"
vec_to_iarray_morph: fixes A :: "('b, 'c) vec"   and B :: "('b, 'c) vec" shows "(A = B) = (vec_to_iarray A = vec_to_iarray B)"
subset_singletonD: fixes A :: "'d set"   and x :: "'d" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
node_relator_def: fixes x :: "'b"   and y :: "'b set" shows "node_relator x y = (x \<in> y)"
Sup_upper: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
binop_wf_dfa: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and B :: "nat bdd list \<times> bool list"   and f :: "bool \<Rightarrow> bool \<Rightarrow> bool" assumes "wf_dfa A n"   and "wf_dfa B n" shows "wf_dfa (binop_dfa f A B) n"
prems: shows "Lfilled (k + 1) lholed es les'"
nonpos_Ints_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0"   and "y \<in> \<int>\<^sub>\<le>\<^sub>0" shows "x + y \<in> \<int>\<^sub>\<le>\<^sub>0"
bit_iff_odd_drop_bit: fixes a :: "'a"   and n :: "nat" shows "bit a n = odd (drop_bit n a)"
le_measureD3: fixes A :: "'b measure"   and B :: "'b measure"   and X :: "'b set" assumes "A \<le> B"   and "sets A = sets B" shows "emeasure A X \<le> emeasure B X"
Ball_def: fixes A :: "'f set"   and P :: "'f \<Rightarrow> bool" shows "Ball A P = (\<forall>x. x \<in> A \<longrightarrow> P x)"
\<Psi>_simp: fixes yx :: "'d \<times> 'c" assumes "\<Phi>\<Psi>.DopxC.ide yx" shows "\<Psi> yx = \<Psi>o yx"
ennreal_prod_eq_top: fixes f :: "'c \<Rightarrow> ennreal"   and I :: "'c set" shows "(prod f I = top) = (finite I \<and> (\<forall>i\<in>I. f i \<noteq> 0) \<and> (\<exists>i\<in>I. f i = top))"
dsj_E: fixes A :: "fmla"   and H :: "fmla set"   and C :: "fmla"   and B :: "fmla" assumes "insert A H \<turnstile> C"   and "insert B H \<turnstile> C" shows "insert (A OR B) H \<turnstile> C"
simps_1: fixes r :: "proc \<times> val" shows "\<gamma> \<bottom> r = snd r"
openin_diff: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin U S"   and "closedin U T" shows "openin U (S - T)"
simps_11: fixes a :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (a\<bullet>length) A = \<D> a A"
bind_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b set" shows "Set.bind A f = {x. \<exists>B\<in>f ` A. x \<in> B}"
pl_dvdm_imp_p_dvdm: fixes l :: "nat"   and a :: "int poly"   and b :: "int poly" assumes "l \<noteq> 0"   and "poly_mod.dvdm (p ^ l) a b" shows "a dvdm b"
real_polynomial_function_sum: fixes I :: "'c set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> real" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> real_polynomial_function (\<lambda>x. f x i)" shows "real_polynomial_function (\<lambda>x. sum (f x) I)"
encoders_are_abstract: fixes v :: "i"   and x :: "\<nu>" shows "[(\<^bold>\<exists>F. \<lbrace>x\<^sup>P,F\<rbrace>) \<^bold>\<rightarrow> \<lparr>A!,x\<^sup>P\<rparr> in v]"
simps: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf"   and a1 :: "'d spmf"   and a2 :: "'g spmf" shows "trace_callee_closure callee1 callee2 A p q a1 a2 = (\<exists>xs. a1 = trace_callee_aux callee1 p xs \<and> a2 = trace_callee_aux callee2 q xs \<and> set xs \<subseteq> A \<times> UNIV)"
sign_changes_poly_times_root_minus_x: fixes g :: "'a poly"   and a :: "'a" assumes "g \<noteq> 0"   and "(0::'a) < a" shows "0 < sign_changes (coeffs ([:a, - (1::'a):] * g)) - sign_changes (coeffs g) \<and> odd (sign_changes (coeffs ([:a, - (1::'a):] * g)) - sign_changes (coeffs g))"
lens_comp_indep_cong_left: fixes Z :: "'e \<Longrightarrow> 'f"   and X :: "'g \<Longrightarrow> 'e"   and Y :: "'h \<Longrightarrow> 'e" assumes "mwb_lens Z"   and "X ;\<^sub>L Z \<bowtie> Y ;\<^sub>L Z" shows "X \<bowtie> Y"
collapse: fixes option :: "'a option" assumes "option \<noteq> None" shows "pure (the option) = option"
assms_1: shows "(\<lambda>n. f n - real_of_int (g n)) \<longlonglongrightarrow> 0"
square_eq_iff: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.idom times one plus zero minus uminus" shows "(times a a = times b b) = (a = b \<or> a = uminus b)"
bot_least: fixes a :: "'a" shows "(0::'a) \<le> a"
neg_lt: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare" shows "neg\<cdot>(Ord.lt compare\<cdot>x\<cdot>y) = Ord.le compare\<cdot>y\<cdot>x"
iffs: fixes getM :: "(Literal \<times> bool) list"   and more :: "'a"   and getM' :: "(Literal \<times> bool) list"   and more' :: "'a" shows "(\<lparr>getM = getM, \<dots> = more\<rparr> = \<lparr>getM = getM', \<dots> = more'\<rparr>) = (getM = getM' \<and> more = more')"
nat_term_compare_inv_conv: fixes to :: "'b nat_term_order" shows "nat_term_compare_inv to = key_compare (key_order_of_nat_term_order_inv to)"
coplanar_4_ter: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and \<gamma> :: "Points" assumes "rk {A, B, C} = 3"   and "rk {A, B, \<gamma>} = 2" shows "rk {A, B, C, \<gamma>} = 3"
bad_sticky1: fixes s2 :: "'s2" assumes "bad2 s2" shows "callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"
distinct_12: fixes x4 :: "nat"   and x3 :: "key" shows "aNum x4 \<noteq> aKey x3"
hol_d: fixes y :: "complex" assumes "y \<in> U" shows "d y holomorphic_on U"
cons_genPrefixE: fixes x :: "'a"   and xs :: "'a list"   and zs :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(x # xs, zs) \<in> genPrefix r" obtains y :: "'a" and ys :: "'a list" where "zs = y # ys" and "(x, y) \<in> r" and "(xs, ys) \<in> genPrefix r"
Int_Collect: fixes x :: "'c"   and A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
Gauss_Jordan: fixes A :: "(('b, 'd) vec, 'c) vec" shows "\<exists>P. invertible P \<and> Gauss_Jordan A = P ** A \<and> reduced_row_echelon_form (Gauss_Jordan A)"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "Ferrack.num" shows "dvdnumcoeff (CN n c t) = (\<lambda>g. g dvd c \<and> dvdnumcoeff t g)"
finite_Collapse_state: fixes Q :: "hf set" assumes "Q \<in> states Collapse_dfa // Collapse.eq_right_lang" shows "finite Q"
assms_2: shows "[\<^bold>\<A>\<^bold>\<not>\<^bold>\<not>\<lparr>P,x\<^sup>P\<rparr> in v]"
induct: fixes P :: "'c rexp \<Rightarrow> bool"   and a0 :: "'c rexp" assumes "P Zero"   and "P Full"   and "\<And>r. P (rexp.Not r)"   and "\<And>r s. \<lbrakk>P r; P s\<rbrakk> \<Longrightarrow> P (Plus r s)"   and "\<And>r s. \<lbrakk>P r; P s\<rbrakk> \<Longrightarrow> P (rexp.Inter r s)"   and "P One"   and "\<And>v. P (Atom v)"   and "\<And>v va. P (Times v va)"   and "\<And>v. P (Star v)"   and "\<And>v. P (Pr v)" shows "P a0"
induct: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> (('c \<times> 'd) \<times> color) tree \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "'d"   and a2 :: "(('c \<times> 'd) \<times> color) tree" assumes "\<And>x y. P x y \<langle>\<rangle>"   and "\<And>x y l a b r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x y l; cmp x a = GT \<Longrightarrow> P x y r\<rbrakk> \<Longrightarrow> P x y (B l (a, b) r)"   and "\<And>x y l a b r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x y l; cmp x a = GT \<Longrightarrow> P x y r\<rbrakk> \<Longrightarrow> P x y (R l (a, b) r)" shows "P a0 a1 a2"
to_simple_rule_list_def: shows "to_simple_rule_list \<equiv> map (apsnd (\<lambda>x. case x of (Accept, b) \<Rightarrow> b | (Drop, b) \<Rightarrow> Drop))"
advantage3_def: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" shows "advantage3 \<A> \<equiv> spmf (dis_log3 \<A>) True"
map_eq_conv: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "'g list"   and g :: "'g \<Rightarrow> 'f" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
inj_on_insert: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and A :: "'b set" shows "inj_on f (insert a A) = (inj_on f A \<and> f a \<notin> f ` (A - {a}))"
simps_2: fixes c1 :: "('a, 'b) com"   and c2 :: "('a, 'b) com" shows "noWhile (c1 ;; c2) = (noWhile c1 \<and> noWhile c2)"
assms_4: shows "A *\<^sub>v v = 0\<^sub>v n"
Nonce_synth_eq: fixes N :: "fresh_t"   and H :: "msg set" shows "(NonceF N \<in> synth H) = (NonceF N \<in> H)"
preca_ex_def: fixes x :: "'a \<Rightarrow> bool" shows "preca_ex x = (x (1::'a) = (\<not> x (0::'a)))"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and order :: "order" shows "P (case order of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = ((order = Eq \<longrightarrow> P f1) \<and> (order = Lt \<longrightarrow> P f2) \<and> (order = Gt \<longrightarrow> P f3))"
bisim: shows "(cms\<^sub>1, mem\<^sub>1) \<approx> (cms\<^sub>2, mem\<^sub>2)"
Un_UNIV_right: fixes A :: "'e set" shows "A \<union> UNIV = UNIV"
take_bit_word_ctz_eq: fixes w :: "'b word" shows "take_bit LENGTH('b) (word_ctz w) = word_ctz w"
insert_Diff: fixes a :: "'e"   and A :: "'e set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
homotopic_paths_reversepath_D: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q" shows "homotopic_paths s (reversepath p) (reversepath q)"
if_loc_reach: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and e :: "'d"   and c\<^sub>1 :: "('a, 'c, 'd) Stmt"   and c\<^sub>2 :: "('a, 'c, 'd) Stmt"   and annos :: "'a ModeUpd list"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b" assumes "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars"   and "((c', mds'), mem') \<in> sifum_security_init.loc_reach dma (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((sifum_modes.annotate (Stmt.If e c\<^sub>1 c\<^sub>2) annos, mds), mem)" shows "c' = sifum_modes.annotate (Stmt.If e c\<^sub>1 c\<^sub>2) annos \<and> mds' = mds \<or> (\<exists>mem''. ((c', mds'), mem') \<in> sifum_security_init.loc_reach dma (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((c\<^sub>1, sifum_modes.update_annos mds annos), mem'')) \<or> (\<exists>mem''. ((c', mds'), mem') \<in> sifum_security_init.loc_reach dma (sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((c\<^sub>2, sifum_modes.update_annos mds annos), mem''))"
leadsto_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "P \<leadsto> Q \<equiv> \<box>(P \<longrightarrow> \<diamond>Q)"
ss: shows "dom_kildall start = fst (iter f step start (unstables r step start))"
hyps_7: shows "(D @ TVarB X Q # \<Gamma>)\<turnstile>T2<:T11"
sums_hurwitz_zeta: assumes "0 < 1"   and "1 < Re s" shows "(\<lambda>n. (of_nat n + complex_of_real 1) powr - s) sums hurwitz_zeta 1 s"
state_mem_mod_1011_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and addr :: "word36"   and val :: "word8"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr> \<and> t2 = s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>" shows "low_equal t1 t2"
ccpo_Sup_singleton: fixes x :: "'a" shows "\<Squnion> {x} = x"
scale_minus_right: fixes a :: "'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" shows "a \<odot> - x = - (a \<odot> x)"
eval_fps_X_power: fixes m :: "nat"   and z :: "'a" shows "eval_fps (fps_X ^ m) z = z ^ m"
comparator_deg_comp: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "comparator cmp" shows "comparator (deg_comp cmp)"
strong_inducts: fixes P :: "'e \<Rightarrow> 'd action \<Rightarrow> bool"   and z :: "'e"   and action :: "'d action" assumes "\<And>a1 a2 z. P z (a1\<lparr>a2\<rparr>)"   and "\<And>a1 list a2 z. P z (a1\<lparr>\<nu>*list\<rparr>\<langle>a2\<rangle>)"   and "\<And>z. P z (\<tau>)" shows "P z action"
oalist_inv_raw_sort_oalist_aux: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "oalist_inv_raw ko (sort_oalist_aux ko xs)"
aux7a: fixes a :: "'a set set"   and b :: "'a set set"   and p :: "('a set set, 'b) Combinators list"   and c :: "'a set set"   and d :: "'a set set"   and po :: "'b" assumes "DenyAllFromTo a b \<in> set p"   and "allNetsDistinct (AllowPortFromTo c d po # p)"   and "a \<noteq> c \<or> b \<noteq> d" shows "twoNetsDistinct a b c d"
fs0_gso0: assumes "0 < length (map of_int_hom.vec_hom fs)" shows "map of_int_hom.vec_hom fs ! 0 = gs1.gso 0"
vec_neg: fixes x :: "complex vec" assumes "x \<in> carrier_vec d" shows "\<ominus>\<^bsub>V\<^esub> x = - x"
ts'_def: shows "ts' \<equiv> map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>). (p, is, \<theta>, dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb, \<D>, acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>, \<R>)) ts"
ps_def: shows "ps = snd (prod_dfs A B (0, 0))"
segment_free: fixes a :: "'a" assumes "a \<notin> carrier D" shows "segment D a = carrier D"
has_contour_integral_const_linepath: fixes c :: "complex"   and b :: "complex"   and a :: "complex" shows "((\<lambda>x. c) has_contour_integral c * (b - a)) (linepath a b)"
infer_sndE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and P :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> (Snd P) = Some \<tau>" shows "\<exists>T S. infer \<Gamma> P = Some (TPair T S) \<and> \<tau> = S"
initial: fixes p :: "'e"   and A :: "('f, 'g, 'e, 'h) FSM_scheme" assumes "p = initial A" shows "p \<in> nodes A"
i_Exec_Stream_take: fixes n :: "nat"   and trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream trans_fun input c \<Down> n = f_Exec_Comp_Stream trans_fun (input \<Down> n) c"
F'rel_def: fixes R1 :: "'m \<Rightarrow> 'p \<Rightarrow> bool"   and R2 :: "'n \<Rightarrow> 'q \<Rightarrow> bool"   and R3 :: "'l \<Rightarrow> 'o \<Rightarrow> bool" shows "F'rel R1 R2 R3 = (BNF_Def.Grp (F'in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y}) (F'map fst fst fst))\<inverse>\<inverse> OO BNF_Def.Grp (F'in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y}) (F'map snd snd snd)"
inv_naturality: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "(G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) \<cdot>\<^sub>D D.inv (\<tau>\<^sub>1 (C.dom \<mu>)) = D.inv (\<tau>\<^sub>1 (C.cod \<mu>)) \<cdot>\<^sub>D (\<tau>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>)"
PR_CONST_def: fixes x :: "'i" shows "PR_CONST x \<equiv> x"
prems_7: shows "InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"