hcju/isabelleps huggingface dataset:
n_mult_comm: fixes x :: "'a"   and y :: "'a" shows "n x \<cdot> n y = n y \<cdot> n x"
take_is_prefix: fixes n :: "nat"   and xs :: "'e list" shows "prefix (take n xs) xs"
get_object_ptr_simp2: fixes object_ptr :: "(_) object_ptr"   and object_ptr' :: "(_) object_ptr"   and object :: "(_) Object"   and h :: "(_) heap" assumes "object_ptr \<noteq> object_ptr'" shows "get object_ptr (put object_ptr' object h) = get object_ptr h"
degree_of_sum_diff_degree: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "deg R q < deg R p" shows "deg R (p \<oplus>\<^bsub>UP R\<^esub> q) = deg R p"
c_iop_welldef: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G" shows "c_iop G N (N \<bullet> a) = N \<bullet> \<rho> a"
is_projection_on_reduces_norm: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and h :: "'b" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "norm (\<pi> h) \<le> norm h"
preserves_diagrams: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and J :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and D :: "'d \<Rightarrow> 'a" assumes "functor A B F"   and "diagram J A D" shows "diagram J B (F \<circ> D)"
cong: fixes L :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) set"   and L' :: "'d filter \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'c) set"   and Lr :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "landau_symbol L L' Lr"   and "\<forall>\<^sub>F x in F. f x = g x" shows "L F f = L F g"
pdevs_val_permute: fixes xs :: "'a list"   and ys :: "'a list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "set xs = set ys"   and "distinct xs"   and "distinct ys"   and "e \<in> UNIV \<rightarrow> I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> UNIV \<rightarrow> I" and "pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)"
Digamma_real_three_halves_pos: shows "0 < Digamma (3 / 2)"
loc_reach_def: shows "loc_reach \<equiv> \<lambda>lc. {((xa, x), xb). loc_reachp lc xa x xb}"
fv_incr_bv: fixes inc :: "nat"   and lev :: "nat"   and t :: "Core.term" shows "fv (incr_bv inc lev t) = fv t"
D: shows "(\<forall>x\<in>s. (f has_vector_derivative D x) (at x)) \<and> continuous_on s D"
list_set_append_def: fixes a :: "'b list"   and S :: "'b list set" shows "a \<odot> S = (@) a ` S"
delete_edges_def: fixes G :: "'b graph"   and es :: "('b \<times> 'b) set" shows "delete_edges G es \<equiv> \<lparr>nodes = nodes G, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> (e1, e2) \<notin> es}\<rparr>"
A: shows "(l, l') \<in> \<langle>br ls.\<alpha> ls.invar\<rangle>list_rel"
card_gt_0_iff: fixes A :: "'k set" shows "(0 < card A) = (A \<noteq> {} \<and> finite A)"
inf_sup_distrib2: fixes y :: "'a upred"   and z :: "'a upred"   and x :: "'a upred" shows "((y \<or> z) \<and> x) = (y \<and> x \<or> z \<and> x)"
project_Increasing_I: fixes F :: "'a program"   and G :: "'c program"   and func :: "'a \<Rightarrow> 'e" assumes "extend h F \<squnion> G \<in> Increasing (func \<circ> f)" shows "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Increasing func"
inj: fixes ya :: "real" assumes "root_cond (abs_int_poly (reflect_poly p), inverse r, inverse l) ya" shows "inverse (real_of_1 (p, l, r)) = ya"
typeable: shows "\<turnstile> mds\<^sub>1 { c } mds\<^sub>1'"
pset_deriv: fixes a :: "'b"   and r :: "'b rexp" shows "pset (deriv a r) = pderiv a r"
ivl_integral_has_vderiv_on: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on (closed_segment a b) f" shows "((\<lambda>u. ivl_integral a u f) has_vderiv_on f) (closed_segment a b)"
orthogonal_transformation: fixes f :: "'b \<Rightarrow> 'c" shows "orthogonal_transformation f = (linear f \<and> (\<forall>v. norm (f v) = norm v))"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "gcd (gcd a b) c = gcd a (gcd b c)"
assms_2: shows "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s\<^sub>1\<rangle>"
hn_refine_ref: fixes m :: "'c nres"   and m' :: "'c nres"   and P :: "assn"   and c :: "'d Heap"   and Q :: "assn"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "m \<le> m'"   and "hn_refine P c Q R m" shows "hn_refine P c Q R m'"
ss_eq: shows "(\<Sum>i<n. monom (f i) i) = (\<Sum>i<n. monom (g i) i)"
partsp_parts_eq: fixes H :: "msg set" shows "partsp (\<lambda>x. x \<in> H) = (\<lambda>x. x \<in> parts H)"
WTrtNew: fixes 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"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option" assumes "is_class P C" shows "P,E,h \<turnstile> new C : Class C"
assms_2: shows "v \<Turnstile>\<^sub>l\<^sub>e d"
bounded_linear_divide: fixes y :: "'c" shows "bounded_linear (\<lambda>x. x / y)"
restrict_map_disj_dom_empty: fixes h :: "'c \<Rightarrow> 'd option"   and h' :: "'c \<Rightarrow> 'd option" assumes "h \<bottom> h'" shows "h |` dom h' = Map.empty"
ltl_prop_equiv_subst_S: fixes S :: "'b ltln set"   and \<phi> :: "'b ltln"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "S \<Turnstile>\<^sub>P subst \<phi> m = (S - dom m \<union> {uu_. \<exists>\<chi> \<chi>'. uu_ = \<chi> \<and> \<chi> \<in> dom m \<and> m \<chi> = Some \<chi>' \<and> S \<Turnstile>\<^sub>P \<chi>'}) \<Turnstile>\<^sub>P \<phi>"
sorted_sort_key: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f (linorder.sort_key less_eq f xs))"
adjoint_dim_col: fixes A :: "'a mat" shows "dim_col (adjoint A) = dim_row A"
valid_Act: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('b, 'pred, 'act) formula" shows "P \<Turnstile> Act \<alpha> x = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x')"
simps_16: fixes a :: "'b"   and b :: "'b" shows "le_rexp (Atom a) (Atom b) = (a \<le> b)"
TESL_interp_stepwise_composition: fixes \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list"   and n :: "nat" shows "\<lbrakk>\<lbrakk> \<Phi>\<^sub>1 @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = \<lbrakk>\<lbrakk> \<Phi>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>"
Suc: fixes j :: "nat" shows "(ord_of i = ord_of j) = (i = j)"
prod_list_filter: fixes f :: "'a \<Rightarrow> bool" shows "prod_list (filter f xs) * prod_list (filter (\<lambda>x. \<not> f x) xs) = prod_list xs"
dghm_proj_components_1: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>ObjMap\<rparr> = (\<lambda>a\<in>\<^sub>\<circ>dg_prod I \<AA>\<lparr>Obj\<rparr>. a\<lparr>i\<rparr>)"
simps_2: fixes a :: "'a"   and w :: "bool"   and ws :: "bool list"   and x :: "'a" shows "bddinsert (Leaf a) (w # ws) x = (if w then Branch (Leaf a) (bddinsert (Leaf a) ws x) else Branch (bddinsert (Leaf a) ws x) (Leaf a))"
Un_iff: fixes c :: "'h"   and A :: "'h set"   and B :: "'h set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
\<tau>inv_def: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool" shows "\<tau>inv trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2 \<equiv> \<forall>s1 s2 tl1 s1' tl2 s2'. bisim s1 s2 \<longrightarrow> trsys1 s1 tl1 s1' \<longrightarrow> trsys2 s2 tl2 s2' \<longrightarrow> bisim s1' s2' \<longrightarrow> tlsim tl1 tl2 \<longrightarrow> \<tau>move1 s1 tl1 s1' = \<tau>move2 s2 tl2 s2'"
chine_in_hom: shows "\<guillemotleft>\<epsilon>.chine : \<epsilon>.dom.apex \<rightarrow>\<^sub>C \<epsilon>.cod.apex\<guillemotright>"
single_is_union: fixes a :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" shows "(\<LM> a  \<RM> = M + N) = (\<LM> a  \<RM> = M \<and> N = \<Empt> \<or> M = \<Empt> \<and> \<LM> a  \<RM> = N)"
inner_vec_def: fixes x :: "('e, 'f) vec"   and y :: "('e, 'f) vec" shows "x \<bullet> y = (\<Sum>i\<in>UNIV. x $ i \<bullet> y $ i)"
rely_mono: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice"   and i :: "'a::refinement_lattice" assumes "(c::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) // (i::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice) // i"
find_sig_reducer_SomeD_aux_2: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and u :: "'t"   and t :: "'a"   and i :: "nat"   and j :: "nat" assumes "find_sig_reducer bs u t i = Some j" shows "j - i < length bs"
simps_4: shows "IN sA4 = {data6, data7, data13}"
simps_20: fixes v :: "char list"   and va :: "'a exp"   and vb :: "char list list"   and vc :: "'a exp" shows "throw_of (RI (v,va) ; vb \<leftarrow> vc) = None"
simps_5: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "Set \<tau> \<squnion> \<sigma> = (case \<sigma> of Collection \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Set \<rho> \<Rightarrow> Set (\<tau> \<squnion> \<rho>) | OrderedSet \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Bag \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Sequence \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | _ \<Rightarrow> OclSuper)"
m: fixes i :: "nat" assumes "m < i" shows "a i = 0"
PPsI: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set" assumes "keys t \<subseteq> X" shows "t \<in> .[X]"
ord_le_eq_trans: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b"   and "b = c" shows "a \<subseteq># c"
while_vrt_def: fixes b :: "'a upred"   and p :: "'a upred"   and v :: "(nat, 'a) uexpr"   and S :: "'a hrel" shows "while b invr p vrt v do S od = while\<^sub>\<bottom> b do S od"
simps_15: shows "OUTfromV data15 = {stA6}"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and llist :: "'c llist" shows "P (case llist of LNil \<Rightarrow> f1 | x ## xa \<Rightarrow> f2 x xa) = ((llist = LNil \<longrightarrow> P f1) \<and> (\<forall>x21 x22. llist = x21 ## x22 \<longrightarrow> P (f2 x21 x22)))"
1: shows "\<turnstile> DBSpec \<longrightarrow> FullInit"
orecvmsg_recv_weaken: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and a :: "'b seq_action"   and Q :: "'b \<Rightarrow> bool" assumes "orecvmsg P \<sigma> a"   and "\<And>\<sigma> a. P \<sigma> a \<Longrightarrow> Q a" shows "recvmsg Q a"
pre_adj_if_postL: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree"   and l' :: "('b \<times> nat) tree"   and b :: "'b" assumes "invar \<langle>l, (a, lv), r\<rangle>"   and "post_del l l'" shows "pre_adjust \<langle>l', (b, lv), r\<rangle>"
thm_cont_e_3: fixes v :: "i" shows "[Contingent (E!\<^sup>-) in v]"
exists_maximal_extension: shows "\<exists>M\<in>\<S>. \<forall>L\<in>\<S>. M \<lesssim> L \<longrightarrow> L = M"
reflection_of_barbs_and_closures_2: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_reflects_binary_pred Rel (\<lambda>b. hasBarb b CWB)" shows "rel_reflects_binary_pred (Rel\<^sup>+) (\<lambda>b. hasBarb b CWB)"
simps_1: fixes q :: "('a, 'b) fm" shows "([] \<^bold>\<leadsto> q) = q"
simps_4: fixes a :: "('a, 'b) rbt"   and k :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and c :: "('a, 'b) rbt"   and s :: "'a"   and y :: "'b"   and d :: "('a, 'b) rbt" shows "combine (Branch B a k x b) (Branch B c s y d) = (case combine b c of Empty \<Rightarrow> balance_left a k x (Branch B Empty s y d) | Branch R b2 t z c2 \<Rightarrow> Branch R (Branch B a k x b2) t z (Branch B c2 s y d) | Branch B b2 t z c2 \<Rightarrow> balance_left a k x (Branch B (Branch B b2 t z c2) s y d))"
gallery_double_cross_not_minimal_Cons1: fixes B :: "'a set"   and C :: "'a set"   and D :: "'a set"   and Cs :: "'a set list" assumes "B \<in> f \<turnstile> folding.\<C>"   and "C \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "D \<in> f \<turnstile> folding.\<C>"   and "folding.gallery (B # C # Cs @ [D])" shows "\<not> min_gallery (B # C # Cs @ [D])"
coprime: shows "coprime e ((P - 1) * (Q - 1))"
hyps_4: fixes P :: "'state" shows "P \<Turnstile> x = L_transform.valid (EF (L (\<alpha>, F, f), P)) (L_transform x)"
non_infer_chain: shows "chain R (ldrop (enat k) Sts)"
has_vector_derivative_const: fixes c :: "'a"   and net :: "real filter" shows "((\<lambda>x. c) has_vector_derivative (0::'a)) net"
elims_3: fixes x :: "'b extended"   and xa :: "'b extended" assumes "\<not> x \<le> xa" obtains xb :: "'b" and y :: "'b" where "x = Fin xb" and "xa = Fin y" and "\<not> xb \<le> y"   | v :: "'b" where "x = \<infinity>" and "xa = Fin v__"   | "x = \<infinity>" and "xa = -\<infinity>"   | v :: "'b" where "x = Fin v" and "xa = -\<infinity>"
l_zero: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> x = x"
lead_coeff_numeral: fixes n :: "num" shows "lead_coeff (numeral n) = numeral n"
fplus_maybe_simps_1: fixes ys :: "'c\<cdot>maybe" shows "fplus\<cdot>\<bottom>\<cdot>ys = \<bottom>"
naturality: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "\<Phi> (C.VV.cod f) \<cdot>\<^sub>D H\<^sub>DoFF.map f = FoH\<^sub>C.map f \<cdot>\<^sub>D \<Phi> (C.VV.dom f)"
nb: shows "set (List_Supplement.embed \<psi>) = {[op] |op. op \<in> set \<psi>}"
f_def: fixes x :: "'a"   and y :: "'a" shows "f x y = (if (x, y) \<in> ord \<and> (y, x) \<in> ord then Eq else if (x, y) \<in> ord then Lt else Gt)"
gen_sqrt_upper: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "sqrt x \<le> sqrtu x n"
empty_not_insert_fset_2: fixes x :: "'b"   and S :: "'b fset" shows "insert_fset x S \<noteq> {||}"
liftAll_True: fixes inp :: "'b \<Rightarrow> 'a option" shows "liftAll (\<lambda>v. True) inp"
add_rel_well_defined: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b"   and z :: "'a set"   and z' :: "'a set" assumes "x \<in> Sigma I \<FF>1"   and "y \<in> Sigma I \<FF>1"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y"   and "z' \<in> I"   and "z' \<subseteq> fst x"   and "z' \<subseteq> fst y" shows "stk1.class_of z (add_str1 z (\<rho>1 (fst x) z (snd x)) (\<rho>1 (fst y) z (snd y))) = stk1.class_of z' (add_str1 z' (\<rho>1 (fst x) z' (snd x)) (\<rho>1 (fst y) z' (snd y)))"
pell_valuation_fund_sol_ge_2: shows "2 \<le> pell_valuation (case fund_sol of (x, y) \<Rightarrow> (int x, int y))"
Gt_lt_conv: fixes x :: "'b"   and y :: "'b" shows "(c1 x y = Gt) = c1.lt y x"
5: shows "c' = \<lparr>store = (store c)(v := snd sv), pred = insert (\<lambda>\<sigma>. \<sigma> sv = adapt_aexp e (store c) \<sigma>) (pred c)\<rparr>"
float_of_fp64_inverse: fixes a :: "64 word" shows "fp64_of_float (float_of_fp64 a) = a"
ss_exps_Vals_NoneI: fixes es :: "'c exp list" assumes "ss_exps es = None" shows "\<exists>vs. es = map Val vs"
splitAt_distinct_ram_fst: fixes vs :: "'a list"   and ram :: "'a" assumes "distinct vs" shows "ram \<notin> set (fst (splitAt ram vs))"
lens_comp_anhil: fixes X :: "'d \<Longrightarrow> 'e" assumes "wb_lens X" shows "0\<^sub>L ;\<^sub>L X = 0\<^sub>L"
pfinite_inline: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and \<I>' :: "('k, 'l) \<I>"   and s :: "('h, 'i, 'k, 'l) converter" assumes "pfinite_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; pfinite_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>\<rbrakk> \<Longrightarrow> pfinite_gpv \<I>' (run_converter s x)"   and "pfinite_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "pfinite_gpv \<I>' (inline run_converter gpv s)"
UN_disjointed_eq: fixes A :: "nat \<Rightarrow> 'b set" shows "\<Union> (range (disjointed A)) = \<Union> (range A)"
additive_principal_omega_exp: fixes x :: "ordinal" shows "additive_principal (\<omega> ** x)"
lookup_pair_Cons: fixes k :: "'a"   and v :: "'e"   and xs :: "('a \<times> 'e) list"   and k0 :: "'a" assumes "oalist_inv_raw ((k, v) # xs)" shows "lookup_pair ((k, v) # xs) k0 = (if k = k0 then v else lookup_pair xs k0)"
ins: shows "NewThread t x m'' \<in> set (concat (map (thr_a \<circ> snd) (TTAS @ [(T, TA)])))"
intro: fixes V\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>B :: "'r \<Rightarrow> 'r"   and src\<^sub>B :: "'r \<Rightarrow> 'r"   and trg\<^sub>B :: "'r \<Rightarrow> 'r"   and V\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>C :: "'s \<Rightarrow> 's"   and src\<^sub>C :: "'s \<Rightarrow> 's"   and trg\<^sub>C :: "'s \<Rightarrow> 's"   and V\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and H\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and \<a>\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't \<Rightarrow> 't"   and \<i>\<^sub>D :: "'t \<Rightarrow> 't"   and src\<^sub>D :: "'t \<Rightarrow> 't"   and trg\<^sub>D :: "'t \<Rightarrow> 't"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'s \<Rightarrow> 't"   and \<Phi>\<^sub>G :: "'s \<times> 's \<Rightarrow> 't" assumes "bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"   and "pseudofunctor 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 G \<Phi>\<^sub>G" shows "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B 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"
inserted_item_ident: fixes a :: "'b"   and xs :: "'b list"   and e :: "'b"   and i :: "'b option" assumes "a \<in> set (insert_spec xs (e, i))"   and "a \<notin> set xs" shows "a = e"
between_identity: shows "\<forall>a b. Bet a b a \<longrightarrow> a = b"
Borsukian_def: fixes S :: "'a set" shows "Borsukian S \<equiv> \<forall>f. continuous_on S f \<and> f ` S \<subseteq> - {0} \<longrightarrow> (\<exists>a. homotopic_with_canon (\<lambda>h. True) S (- {0}) f (\<lambda>x. a))"
dom_in_domains: fixes f :: "'a arrow_of_spans_data" assumes "V.domains f \<noteq> {}" shows "V.dom f \<in> V.domains f"
inj_dual_iff: fixes x :: "'a"   and y :: "'a" shows "(\<partial> x = \<partial> y) = (x = y)"
simps_133: fixes x101 :: "stmt"   and x102 :: "stmt"   and x91 :: "stmt"   and x92 :: "qtname"   and x93 :: "vname"   and x94 :: "stmt" shows "x101 Finally x102 \<noteq> Try x91 Catch(x92 x93) x94"
SOSou_iff_strong_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "((\<forall>S T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<longrightarrow> (\<exists>S'. S \<longmapsto>Source S' \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)) \<and> strong_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*) \<and> strong_reduction_simulation (Rel\<inverse>) (STCal Source Target))"
reduce_def: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "subseqs.reduce P s n = (SOME r'. strict_mono r' \<and> P n (s \<circ> r'))"
cases: fixes a :: "'a"   and r :: "('b \<times> 'a) set" assumes "a \<in> Range r" obtains aa :: "'b" and b :: "'a" where "a = b" and "(aa, b) \<in> r"
simps_14: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Ln a) vs = ln (interpret_floatarith a vs)"
sumr_diff_mult_const2: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and r :: "'c" shows "sum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)"
elims: fixes x :: "nat"   and y :: "Value \<Rightarrow> CValue \<Rightarrow> bool" assumes "similar' x = y" obtains "x = 0" and "y = similar'_base"   | n :: "nat" where "x = Suc n" and "y = similar'_step (similar' n)"
intros: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and stack :: "'b \<Rightarrow> 'c list"   and c :: "'b"   and T :: "'b list"   and c' :: "'b" assumes "balance_trace step stack"   and "traces.trace step c T c'"   and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'"   and "stack c' = stack c" shows "balance_trace.bal step stack c T c'"
FactGroup_inj_on: shows "inj_on (\<lambda>X. the_elem (f ` X)) (carrier (G Mod kernel G H f))"
poly_has_sgnx_values_2: fixes p :: "real poly"   and a :: "real" assumes "p \<noteq> 0" shows "(poly p has_sgnx 1) (at_right a) \<or> (poly p has_sgnx - 1) (at_right a)"
totalize_act_def: fixes act :: "('b \<times> 'b) set" shows "totalize_act act \<equiv> act \<union> Id_on (- Domain act)"
relevant_vars_Entry: fixes V :: "'var"   and S :: "'node SDG_node set" assumes "V \<in> rv S (CFG_node (_Entry_))"   and "(_High_) \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "V \<in> L"
maxbidAlg_def: fixes a :: "('c \<times> 'd set) set"   and N :: "'c set"   and \<Omega> :: "'d set" shows "maxbidAlg a N \<Omega> \<equiv> bidMaximizedBy a N \<Omega> Elsee 0"
fun_upd_other: fixes z :: "'a"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "z \<noteq> x" shows "(f(x := y)) z = f z"
inf_commute: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<inter># y = y \<inter># x"