hcju/isabelleps huggingface dataset:
less_eq_inf_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<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 x :: "'b"   and y :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup" shows "less_eq x y = (inf y x = x)"
Cd_range_disjoint: fixes d :: "'d"   and d' :: "'d"   and A :: "'x set" assumes "d \<noteq> d'" shows "Cd d A \<inter> Cd d' A = {}"
n_circ_9: fixes k :: "nat" shows "contour_integral (n_circ k) f = of_nat k * contour_integral p_circ f"
supporting_hyperplane_rel_boundary: fixes S :: "'b set"   and x :: "'b" assumes "convex S"   and "x \<in> S"   and "x \<notin> rel_interior S" obtains a :: "'b" where "a \<noteq> (0::'b)" and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y" and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
big_forest_path_bot: fixes a :: "'a"   and d :: "'a"   and H :: "'a" assumes "bijective (a * top) \<and> bijective (a[[top]])"   and "a \<le> d"   and "big_forest H d" shows "(d \<sqinter> - a)[[H * a * top]] \<le> bot"
mbisim_thrD2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x2"   and ln :: "'l \<Rightarrow>f nat" assumes "s1 \<approx>m s2"   and "thr s2 t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>x'. thr s1 t = \<lfloor>(x', ln)\<rfloor> \<and> t \<turnstile> (x', shr s1) \<approx> (x, shr s2) \<and> (wset s2 t = None \<or> x' \<approx>w x)"
assms_2: shows "g \<in> injections ({x} \<union> X) Y"
arctan: fixes y :: "real" shows "- (pi / 2) < arctan y \<and> arctan y < pi / 2 \<and> tan (arctan y) = y"
closure_closed: fixes S :: "'b set" assumes "closed S" shows "closure S = S"
map_mmupd_update_less: fixes K :: "'a set"   and K' :: "'a set"   and m :: "'a \<Rightarrow> 'b option"   and v :: "'b" assumes "K \<subseteq> K'" shows "map_mmupd m (K - dom m) v \<subseteq>\<^sub>m map_mmupd m (K' - dom m) v"
nat_pow_mult: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "[n] \<cdot> x \<oplus> [m] \<cdot> x = [(n + m)] \<cdot> x"
inject_4: fixes x51 :: "expr"   and x52 :: "stmt"   and x53 :: "stmt"   and y51 :: "expr"   and y52 :: "stmt"   and y53 :: "stmt" shows "(Cond x51 x52 x53 = Cond y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
Diff_Int_distrib2: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "(A - B) \<inter> C = A \<inter> C - B \<inter> C"
Comp_is_extensional_2: fixes G :: "'a set"   and F :: "'a set" assumes "Comp G F \<noteq> {}" shows "B.is_iso_class G"
supp_perm_eq_test: fixes x :: "'a"   and p :: "perm" assumes "supp x \<sharp>* p" shows "p \<bullet> x = x"
subst_set_reflects_unifiable: fixes \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" assumes "unifiable (subst_set \<sigma> E)" shows "unifiable E"
set1_F_map: fixes l1 :: "'p \<Rightarrow> 'z"   and l2 :: "'q \<Rightarrow> 'aa"   and l3 :: "'r \<Rightarrow> 'ab" shows "set1_F \<circ> mapl_F l1 l2 l3 = (`) l1 \<circ> set1_F"
take_f_join_if: fixes n :: "nat"   and xs :: "'b list"   and I :: "nat set" shows "xs \<Join>\<^sub> I \<down> n = (if n < card (I \<down>< length xs) then xs \<Join>\<^sub> (I \<down>< (I \<rightarrow> n)) else xs \<Join>\<^sub> I)"
simps: fixes a1 :: "'a list"   and a2 :: "'a list" shows "ord_class.lexordp a1 a2 = ((\<exists>y ys. a1 = [] \<and> a2 = y # ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> x < y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> x < y \<and> \<not> y < x \<and> ord_class.lexordp xs ys))"
silent_tl: fixes s :: "'x1 \<times> 'm1"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s' :: "'x1 \<times> 'm1" assumes "\<tau>move1 s ta s'" shows "ta = \<lbrace>\<rbrace>"
ts_okI: fixes ts :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'f" assumes "\<And>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> P t x m" shows "ts_ok P ts m"
i_cut_commute_disj: fixes cut_op1 :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set"   and cut_op2 :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set"   and I :: "'a set"   and t1 :: "'a"   and t2 :: "'a" assumes "cut_op1 = (\<down><) \<or> cut_op1 = (\<down>\<le>) \<or> cut_op1 = (\<down>>) \<or> cut_op1 = (\<down>\<ge>)"   and "cut_op2 = (\<down><) \<or> cut_op2 = (\<down>\<le>) \<or> cut_op2 = (\<down>>) \<or> cut_op2 = (\<down>\<ge>)" shows "cut_op2 (cut_op1 I t1) t2 = cut_op1 (cut_op2 I t2) t1"
x_img: shows "\<pi> ` ps \<subseteq> \<V>'"
carrier_subgroup_generated_subgroup: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G" shows "carrier (subgroup_generated G H) = H"
fset_simps_2: fixes x :: "'b"   and xa :: "'b fset" shows "fset (finsert x xa) = insert x (fset xa)"
fmrestrict_set_fmmap: fixes A :: "'d set"   and f :: "'f \<Rightarrow> 'e"   and m :: "('d, 'f) fmap" shows "fmrestrict_set A (fmmap f m) = fmmap f (fmrestrict_set A m)"
c_cons_aux1: fixes x :: "nat" shows "c_cons x 0 = c_pair 0 x + 1"
omap_option_conv_3: fixes f :: "'d \<Rightarrow> 'c option"   and xx :: "'d option" shows "(omap_option f xx = Some None) = (xx = None)"
Node_in_tree_sigma: fixes X :: "('d \<times> 'd tree \<times> 'd tree) set"   and M :: "'d measure" assumes "X \<in> sets (M \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M tree_sigma M)" shows "{\<langle>l, v, r\<rangle> |l v r. (v, l, r) \<in> X} \<in> sets (tree_sigma M)"
assms_3: shows "\<forall>n. \<exists>m>n. P (s m)"
take_bit_decr_eq: fixes n :: "nat"   and k :: "int" assumes "take_bit n k \<noteq> 0" shows "take_bit n (k - 1) = take_bit n k - 1"
closed_segment_subset: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> S"   and "y \<in> S"   and "convex S" shows "closed_segment x y \<subseteq> S"
hyps: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P c c' Q" shows "ir_hoare P c c' Q"
rel_mset_Zero: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_mset R {#} {#}"
3: shows "sqrt ((\<sigma> i)\<^sup>2 + (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2))\<^sup>2) = sqrt (\<Sum>i\<in>insert i I. (\<sigma> i)\<^sup>2)"
simps: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option"   and \<tau> :: "'d \<Rightarrow> ('e, 'd, 'f) val option"   and e :: "('e, 'd, 'f) expr" shows "(\<sigma>, \<tau>, e)\<^sub>\<tau> = \<tau>"
dis1: shows "deg_fun t \<noteq> deg_fun s \<or> \<not> ord t s"
unat_ucast_up_simp: fixes x :: "'b word" assumes "LENGTH('b) \<le> LENGTH('c)" shows "unat (UCAST('b \<rightarrow> 'c) x) = unat x"
assms: fixes x :: "'a" assumes "x \<in> S" shows "f x \<subseteq> T"
subst_typ_simulates_tsubstT: fixes T :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubstT T \<rho> = subst_typ (map (\<lambda>(x, y). ((x, y), \<rho> x y)) (SOME l. distinct l \<and> set l = tvsT T)) T"
tanh_real_bounds: fixes x :: "real" shows "tanh x \<in> {- 1<..<1}"
list_all2_trans: fixes P1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and P2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P3 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and as :: "'b list"   and bs :: "'c list"   and cs :: "'d list" assumes "\<And>a b c. \<lbrakk>P1 a b; P2 b c\<rbrakk> \<Longrightarrow> P3 a c"   and "list_all2 P1 as bs"   and "list_all2 P2 bs cs" shows "list_all2 P3 as cs"
assms: shows "f \<in> borel_measurable (nat_filtration 0)"
multiplicity_times_unit_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and c :: "'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times c one" shows "factorial_semiring.multiplicity times one p (times c x) = factorial_semiring.multiplicity times one p x"
C1_def: shows "C1 = Max {B (2 * n) / B n |n. n \<le> N}"
set_custom_empty_axioms: fixes N :: "nat" shows "set_custom_empty (PR_CONST (op_ias_empty_sz N))"
Abs_oalist_inject: fixes x :: "('a \<times> 'b) list \<times> 'a key_order"   and y :: "('a \<times> 'b) list \<times> 'a key_order" assumes "x \<in> Collect ko.oalist_inv"   and "y \<in> Collect ko.oalist_inv" shows "(Abs_oalist x = Abs_oalist y) = (x = y)"
a_set_mode_locs_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_mode_locs shadow_root_ptr \<equiv> all_args (put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode_update)"
sup_commute: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<union># y = y \<union># x"
assms_2: fixes \<rho> :: "'a OCL_Types.type" assumes "\<tau> = Sequence \<rho>"   and "\<rho> < \<sigma>" shows "P"
vec_smult_def: fixes \<alpha> :: "'b"   and \<beta> :: "'b list" shows "vec_smult \<alpha> \<beta> = map ((*) \<alpha>) \<beta>"
rel_induct_subset: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and I :: "('b \<times> 'b) set"   and u :: "'b" assumes "view_partition P D R"   and "weakly_step_consistent P D R"   and "locally_respects P I D R" shows "rel_induct P I D u \<subseteq> R u"
maintainedI: fixes A :: "('g, 'e) labeled_graph"   and G :: "('g, 'f) labeled_graph"   and B :: "('g, 'e) labeled_graph" assumes "\<And>f. graph_homomorphism A G f \<Longrightarrow> extensible (A, B) G f" shows "maintained (A, B) G"
coboundedI: fixes A :: "'a word set"   and a :: "'a word" assumes "finite A"   and "a \<in> A" shows "a \<le>s signed.Max A"
seqr_mono: fixes P\<^sub>2 :: "('b, 'c) urel"   and P\<^sub>1 :: "('b, 'c) urel"   and Q\<^sub>2 :: "('c, 'd) urel"   and Q\<^sub>1 :: "('c, 'd) urel" assumes "P\<^sub>1 \<sqsubseteq> P\<^sub>2"   and "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2" shows "P\<^sub>1 ;; Q\<^sub>1 \<sqsubseteq> P\<^sub>2 ;; Q\<^sub>2"
discounted_mult: fixes r :: "real"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and Y :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "\<forall>n w. discounted_value r (\<lambda>m x. X m x * Y m x) n w = X n w * discounted_value r Y n w"
hyp2_axiom8: shows "\<exists>a b c. \<not> B\<^sub>K a b c \<and> \<not> B\<^sub>K b c a \<and> \<not> B\<^sub>K c a b"
mig_componentwise_def: fixes a :: "'b"   and b :: "'b" shows "mig_componentwise a b = (\<Sum>i\<in>Basis. mig_real (a \<bullet> i) (b \<bullet> i) *\<^sub>R i)"
enc_respects_pred_iff_indRelR_respects_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_respects_pred Pred = rel_respects_pred indRelR Pred"
power: fixes n :: "nat"   and m :: "nat" shows "\<chi> (n ^ m) = \<chi> n ^ m"
Class_is_Left_Coset: fixes g :: "'a" assumes "g \<in> R" shows "contained.additive.Class g = additive.kernel.Left_Coset g I"
assms: shows "lookup (row_to_poly ts r) t \<noteq> (0::'b)"
NSTD_1: fixes a :: "nat"   and b :: "cell list"   and c :: "cell list" assumes "\<not> TSTD (a, b, c)" shows "rec_exec rec_NSTD [trpl_code (a, b, c)] = Suc 0"
P2_real_model_end_def: fixes M :: "'grp \<times> 'grp"   and bs :: "('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'e"   and \<A>3 :: "'grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<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)))))))"
con_rews_2: fixes a :: "'b"   and list :: "['b]" shows "a : list \<noteq> \<bottom>"
ivl_disj_un_one_2: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{..<l} \<union> {l..<u} = {..<u}"
C: shows "C = {(I, S, E)} WHILE Less (V a) (N b) DO a ::= Plus (V a) (N 1)"
degree_reflect_poly_le: fixes p :: "'a poly" shows "degree (reflect_poly p) \<le> degree p"
all_final_exceptI: fixes s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('x \<times> 'b \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'd wait_set_status option) \<times> 'a set"   and Ts :: "'a set" assumes "\<And>t. not_final_thread s t \<Longrightarrow> t \<in> Ts" shows "all_final_except s Ts"
img_is_subgroup: shows "subgroup ((\<lambda>f. sum' f I) ` carrier (sum_group I (\<lambda>i. free_Abelian_group (S i)))) (free_Abelian_group (\<Union> (S ` I)))"
pred_inject_1: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'a" shows "pred_F P1 P2 (F a) = P1 a"
aezfun: fixes x :: "('b, 'a) aezfun" shows "aezfun x \<in> aezfun_set"
simps_7: fixes \<Psi> :: "'e" shows "guarded (\<lbrace>\<Psi>\<rbrace>) = False"
intersect_num_commute: fixes b1 :: "'b set"   and b2 :: "'b set" shows "b1 |\<inter>| b2 = b2 |\<inter>| b1"
extremum: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and a :: "'b" assumes "ordering_top less_eq less top" shows "less_eq a top"
C'_def: shows "(C\<^sub>0', C\<^sub>1') = (if dist (case p\<^sub>0\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case p\<^sub>0\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R))"
is_zero_cases: fixes x :: "('a, 'b) IEEE.float" assumes "is_zero x" obtains "x = 0"   | "x = - 0"
split_sum_all: fixes P :: "'c + 'd \<Rightarrow> bool" shows "(\<forall>x. P x) = ((\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x)))"
polysub_norm: fixes p :: "poly"   and q :: "poly" assumes "isnpoly p"   and "isnpoly q" shows "isnpoly (p -\<^sub>p q)"
obtain_rep: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "good X"   and "fresh xs x' X" shows "\<exists>X'. (X, X') \<in> swapped \<and> good X' \<and> Abs xs x X = Abs xs x' X'"
diagonalize_def: fixes A :: "(('a, 'e) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "local.diagonalize A bezout = (let (P, S, Q) = diagonalize_JNF (Mod_Type_Connect.from_hma\<^sub>m A) bezout in (Mod_Type_Connect.to_hma\<^sub>m P, Mod_Type_Connect.to_hma\<^sub>m S, Mod_Type_Connect.to_hma\<^sub>m Q))"
eucl_of_list_eucl_nth: fixes xs :: "real list"   and i :: "nat" assumes "length xs = DIM('b)"   and "i < DIM('b)" shows "eucl_of_list xs $\<^sub>e i = xs ! i"
distinct_5: fixes x2 :: "'d" shows "Lt x2 \<noteq> \<infinity>"
induct: fixes P :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"   and a0 :: "'a word"   and a1 :: "'a word" assumes "\<And>a b. (a \<noteq> b \<Longrightarrow> P a (b - 1)) \<Longrightarrow> P a b" shows "P a0 a1"
concurrent_append_delete_independent_technical: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e :: "'b"   and ir :: "'a" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Append i e) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e) \<in> set (node_deliver_messages xs)" shows "hb (i, Append i e) (ir, Delete is e)"
nat_pow_inv: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and i :: "nat" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
allE_id: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and y :: "'a"   and \<phi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "y \<in> var"   and "\<phi> \<in> fmla"   and "prv (all y \<phi>)" shows "prv \<phi>"
simps_3: shows "matcheq_matchNone (MatchNot MatchAny) = True"
filter_sup_absorb_inf: fixes F :: "'a set"   and G :: "'a set" assumes "ord_class.filter F"   and "ord_class.filter G" shows "filter_sup (F \<inter> G) G = G"
in_hhomE: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C b\<guillemotright>" obtains "C.arr \<mu>" and "C.obj a" and "C.obj b" and "src\<^sub>C \<mu> = a" and "trg\<^sub>C \<mu> = b"
szip_sset_fst: fixes u :: "'c stream"   and v :: "'d stream" shows "fst ` sset (u ||| v) = sset u"
urel_integer_power: fixes x :: "integer"   and x' :: "int"   and y :: "integer"   and y' :: "nat" assumes "urel_integer x x'"   and "urel_integer y (int y')" shows "urel_integer (power_p_integer pp x y) (power_p p x' y')"
prems_4: shows "InvariantClCharacterization (getCl state') (getC state') (getM state')"
eq_vecI: fixes w :: "'a vec"   and v :: "'a vec" assumes "\<And>i. i < dim_vec w \<Longrightarrow> v $ i = w $ i"   and "dim_vec v = dim_vec w" shows "v = w"
clop_top_Fix: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "\<top> \<in> Fix f"
F: shows "Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>Some ` (H ` ran (heap (fst \<tau>)) - {None})\<rfloor>\<rfloor>) = \<lfloor>\<lfloor>Some ` (H ` ran (heap (fst \<tau>)) - {None})\<rfloor>\<rfloor>"
PreEnvironmentJView_axioms: shows "PreEnvironmentJView jkbp jview"
clop_top: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "top_pres f"
sheaf_spec_morphisms_def: fixes U :: "'a set set"   and V :: "'a set set" shows "sheaf_spec_morphisms U V \<equiv> \<lambda>s\<in>\<O> U. restrict s V"
QAP_App: fixes \<Gamma> :: "(name \<times> ty) list"   and p :: "trm"   and q :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty"   and s :: "trm"   and t :: "trm" assumes "\<Gamma> \<turnstile> p \<leftrightarrow> q : T\<^sub>1\<rightarrow>T\<^sub>2"   and "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1" shows "\<Gamma> \<turnstile> App p s \<leftrightarrow> App q t : T\<^sub>2"
finite_same_card_bij_on_ext_funcset: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "\<exists>f. f \<in> A \<rightarrow>\<^sub>E B \<and> bij_betw f A B"
CanonicalCompWellDefined: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and d :: "(ZF, ZF) Expression"   and B :: "ZF"   and d' :: "(ZF, ZF) Expression"   and C :: "ZF" assumes "ZFAxioms T"   and "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> d : B"   and "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : B \<turnstile> d' : C" shows "CanonicalComp T [A,d,B]\<^bsub>T\<^esub> [B,d',C]\<^bsub>T\<^esub> = [A,sub d in d',C]\<^bsub>T\<^esub>"
rep_def: fixes H :: "'a set"   and x :: "'b" assumes "H \<in> carrier (G LMod stabiliser x)" shows "rep H = (SOME y. y \<in> H)"
acyclic_impl_antisym_rtrancl: fixes r :: "('b \<times> 'b) set" assumes "acyclic r" shows "antisym (r\<^sup>*)"
iso_runit: fixes a :: "'a" assumes "ide a" shows "local.iso \<r>\<^sub>1[a]"
linear_pat_induct: fixes t :: "Term_Class.term"   and P :: "Term_Class.term \<Rightarrow> bool" assumes "linear t"   and "\<And>s. P (Free s)"   and "\<And>name args. \<lbrakk>linears args; \<And>arg. arg \<in> set args \<Longrightarrow> P arg\<rbrakk> \<Longrightarrow> P (name $$ args)" shows "P t"
pdevs_of_real_def: fixes x :: "'a" shows "pdevs_of_real x = (x, zero_pdevs)"
nonneg_cexprD: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and \<sigma> :: "nat \<Rightarrow> val" assumes "nonneg_cexpr V \<Gamma> e"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "0 \<le> extract_real (cexpr_sem \<sigma> e)"
is_le_rel_le: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "is_le_rel rel"   and "rel x y" shows "x \<le> y"
0: shows "A1 \<circ> A2 \<otimes> B1 \<circ> B2 \<noteq> []"
translation_inverse_subset: fixes a :: "'c"   and V :: "'c set"   and S :: "'c set" assumes "(+) (- a) ` V \<subseteq> S" shows "V \<subseteq> (+) a ` S"
not_zero_aB: shows "\<not> is_zero_row_upt_k a (Suc k) B"
l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"
fpxs_nth_cases: fixes y :: "rat \<Rightarrow> 'b" assumes "y \<in> {f. is_fpxs f}" obtains x :: "'b fpxs" where "y = fpxs_nth x"
rawpsubst_subst_fresh_switch: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list"   and s :: "'trm"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s"   and "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t"   and "distinct (map snd txs)"   and "s \<in> trm"   and "y \<in> var"   and "y \<notin> snd ` set txs" shows "rawpsubst (subst \<phi> s y) txs = rawpsubst \<phi> (txs @ [(s, y)])"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "m\<tau>move1 a1 a2 a3" obtains s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x1" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and x' :: "'x1" and ln' :: "'l \<Rightarrow>f nat" and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action" where "a1 = s" and "a2 = (t, ta)" and "a3 = s'" and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "thr s' t = \<lfloor>(x', ln')\<rfloor>" and "\<tau>move1 (x, shr s) ta (x', shr s')"