hcju/isabelleps huggingface dataset:
power_Suc2: fixes a :: "'a"   and n :: "nat" shows "a\<^bsup>Suc n\<^esup> = a\<^bsup>n\<^esup> \<cdot> a"
filter_for_biflows_def: fixes E :: "('a \<times> 'a) list" shows "filter_for_biflows E \<equiv> filter (\<lambda>e. (snd e, fst e) \<in> set E) E"
adds_univariate_linear: fixes X :: "'c set"   and s :: "'c \<Rightarrow>\<^sub>0 nat"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "card X \<le> 1"   and "s \<in> .[X]"   and "t \<in> .[X]" obtains "s adds t"   | "t adds s"
distinctClosed: fixes p :: "(name \<times> name) list"   and xs :: "'d list" shows "distinct (p \<bullet> xs) = distinct xs"
card_of_Times_commute: fixes A :: "'a set"   and B :: "'b set" shows "|A \<times> B| =o |B \<times> A|"
rat_roots2_def: fixes p :: "rat poly" shows "rat_roots2 p = (let a = coeff p 2; b = coeff p 1; c = coeff p 0; b2a = b / (2 * a); bac = b2a\<^sup>2 - c / a in map ((+) (- b2a)) (sqrt_rat bac))"
entry_valid: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and p :: "'b" assumes "flowgraph fg" shows "proc_of fg (entry fg p) = p"
id: shows "(a = (0::'a) \<or> f = 0) = False"
inv_right: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "bijection f" shows "f (inv f a) = a"
lt2_per2__lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Per A B C"   and "Per A' B' C'"   and "A B Lt A' B'"   and "B C Lt B' C'" shows "A C Lt A' C'"
uminus_octo_def: shows "uminus \<equiv> corec_octo (\<lambda>x. - Ree x) (\<lambda>x. - Im1 x) (\<lambda>x. - Im2 x) (\<lambda>x. - Im3 x) (\<lambda>x. - Im4 x) (\<lambda>x. - Im5 x) (\<lambda>x. - Im6 x) (\<lambda>x. - Im7 x)"
dbm_not_lt_impl: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "\<not> a \<prec> b" shows "b \<prec> a \<or> a = b"
no_overlaps_append: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "('d, 'f) flow_entry_match list"   and y :: "('d, 'f) flow_entry_match list" assumes "no_overlaps \<gamma> (x @ y)" shows "no_overlaps \<gamma> y"
lfinite_ltake: fixes n :: "enat"   and xs :: "'b llist" shows "lfinite (ltake n xs) = (lfinite xs \<or> n < \<infinity>)"
card_inj_subs: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj_on f A"   and "B \<subseteq> A" shows "card (f ` B) = card B"
substEnv_updEnv_fresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and xs :: "'varSort" assumes "good X"   and "good Y"   and "fresh ys y X" shows "rho [x \<leftarrow> X]_xs &[Y / y]_ys = rho &[Y / y]_ys [x \<leftarrow> X]_xs"
weakBisimWeakCoinduct: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<X> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<approx> Q"
neg_numeral_le_neg_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus (numeral.numeral one plus m)) (uminus one)"
path_split_snoc: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" obtains a' :: "'a" and as' :: "'a list" where "as = as' @ [a']" and "CFG.path sourcenode targetnode valid_edge n as' (sourcenode a')" and "valid_edge a'" and "n' = targetnode a'"
space_eq_univ: fixes M :: "real measure" assumes "real_distribution M" shows "space M = UNIV"
pull_back_def: fixes p :: "'a" shows "pull_back p = dual_map (src.tangent_space p) push_forward"
bilinear_bounded: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bilinear h" shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
strict_iff_order: fixes a :: "'a"   and b :: "'a" shows "lt a b = (le a b \<and> a \<noteq> b)"
homeomorphism_image2: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphism S T f g" shows "g ` T = S"
nt_mult_assoc0: fixes a :: "'a"   and m :: "int"   and n :: "int" assumes "a \<in> carrier A" shows "m\<triangleright>n\<triangleright>a\<^bsub>A\<^esub>\<^bsub>A\<^esub> = (m * n)\<triangleright>a\<^bsub>A\<^esub>"
"3_6": fixes c :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (\<langle>c: t \<in> s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
correctCompositionSecrets_subcomp2: fixes C :: "specID"   and xb :: "specID"   and xc :: "Secrets" assumes "correctCompositionSecrets C"   and "xb \<in> subcomponents C"   and "xc \<in> specSecrets xb" shows "xc \<in> specSecrets C"
symvars_def: fixes s :: "'c \<Rightarrow> nat" shows "Store.symvars s = range (\<lambda>v. symvar v s)"
mHom_ex_zero: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme" assumes "R module M"   and "R module N" shows "mzeromap M N \<in> mHom R M N"
pt_bij2: fixes x :: "'a"   and pi :: "('b \<times> 'b) list"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "x = rev pi \<bullet> y" shows "pi \<bullet> x = y"
lm107: fixes X :: "'c set"   and Z :: "'c set" shows "sum (chi X) (Z - X) = (0::'b)"
Equiv_Exec_sym2: fixes equiv_states_sym :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and equiv_states :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'f"   and localState1 :: "'g \<Rightarrow> 'e"   and localState2 :: "'h \<Rightarrow> 'd"   and input_fun1 :: "'f \<Rightarrow> 'i message_af"   and input_fun2 :: "'f \<Rightarrow> 'j message_af"   and output_fun1 :: "'g \<Rightarrow> 'k message_af"   and output_fun2 :: "'h \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and trans_fun2 :: "'j message_af \<Rightarrow> 'h \<Rightarrow> 'h"   and k1 :: "nat"   and k2 :: "nat"   and c1 :: "'g"   and c2 :: "'h" assumes "equiv_states_sym = (\<lambda>s1 s2. equiv_states s2 s1)" shows "Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 = Equiv_Exec m equiv_states_sym localState2 localState1 input_fun2 input_fun1 output_fun2 output_fun1 trans_fun2 trans_fun1 k2 k1 c2 c1"
rev_concat: fixes xs :: "'b list list" shows "rev (concat xs) = concat (map rev (rev xs))"
discounted_rfr: fixes n :: "nat"   and w :: "bool stream" shows "discounted_value r (prices Mkt risk_free_asset) n w = 1"
intensional_simps_3: shows "\<not> #False \<equiv> #True"
pred_alt_def_1: fixes name :: "name"   and vs :: "value list" shows "vno_abs (Vconstr name vs) = (True \<and> list_all vno_abs vs)"
fsmult_def: fixes a :: "'f"   and v :: "('f, 'g) aezfun \<Rightarrow> 'v" shows "a \<currency>\<currency> v \<equiv> a \<delta>\<delta> (0::'g) \<currency> v"
healthy_monoD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "mono_trans t"
is_unit_iff_degree: fixes p :: "'c poly" assumes "p \<noteq> 0" shows "is_unit p = (degree p = 0)"
min_less_iff_conj: fixes z :: "'a Word.word"   and x :: "'a Word.word"   and y :: "'a Word.word" shows "(z <s signed.min x y) = (z <s x \<and> z <s y)"
Trans: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<equiv>\<^sub>s Q"   and "Q \<equiv>\<^sub>s R" shows "P \<equiv>\<^sub>s R"
as_empty_def: fixes uu :: "unit" shows "as_empty uu \<equiv> (array_of_list [], 0::'d)"
case_1: fixes f1 :: "i32 \<Rightarrow> 'a"   and f2 :: "i64 \<Rightarrow> 'a"   and f3 :: "f32 \<Rightarrow> 'a"   and f4 :: "f64 \<Rightarrow> 'a"   and x1 :: "i32" shows "(case ConstInt32 x1 of ConstInt32 x \<Rightarrow> f1 x | ConstInt64 x \<Rightarrow> f2 x | ConstFloat32 x \<Rightarrow> f3 x | ConstFloat64 x \<Rightarrow> f4 x) = f1 x1"
subst_apply_pairs_singleton: fixes t :: "('c, 'e) Term.term"   and s :: "('c, 'e) Term.term"   and \<delta> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "[(t, s)] \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta> = [(t \<cdot> \<delta>, s \<cdot> \<delta>)]"
plus'_def: fixes n :: "int'"   and m :: "int'" shows "plus' n m = Abs_int' (Rep_int' n + Rep_int' m)"
multiset_induct: fixes P :: "'b multiset \<Rightarrow> bool"   and M :: "'b multiset" assumes "P {#}"   and "\<And>x M. P M \<Longrightarrow> P (x, M)" shows "P M"
mdqd: shows "(\<exists>x. Ifm bbs (x # bs) p) = (Ifm bbs (i # bs) (evaldjf (\<lambda>j. simpfm (subst0 (C j) (simpfm (minusinf (fst (unit p)))))) [1..snd (snd (unit p))]) \<or> Ifm bbs (i # bs) (evaldjf (\<lambda>(b, j). simpfm (subst0 (Add b (C j)) (fst (unit p)))) (concat (map (\<lambda>b. map (Pair b) [1..snd (snd (unit p))]) (fst (snd (unit p)))))))"
cont: shows "continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
subst_same_Var: fixes \<phi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var" shows "subst \<phi> (Var x) x = \<phi>"
infsetsum_uminus: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Sum>\<^sub>ax\<in>A. - f x) = - infsetsum f A"
log4: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e1 :: "exp0"   and s' :: "'b state"   and err :: "v error_result"   and op0 :: "lop"   and e2 :: "exp0" assumes "evaluate ck env s e1 (s', Rerr err)" shows "evaluate ck env s (Log op0 e1 e2) (s', Rerr err)"
swapBinp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and binp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs set option" shows "binp %%[x \<and> y]_xs \<equiv> lift (swapAbs xs x y) binp"
sets_eq: shows "set (zip xs XS) = set (zip ys YS)"
sterms_cterms: fixes p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and q :: "('a, 'b, 'c, 'd) seqp" assumes "p \<in> cterms \<Gamma>"   and "q \<in> sterms \<Gamma> p"   and "wellformed \<Gamma>" shows "q \<in> cterms \<Gamma>"
unx_some_lemmas_2: fixes u :: "'a"   and x :: "'a \<times> 'a" assumes "unapp u = Some x" shows "unfree u = None"
ring_simprules_19: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
max2_def: fixes r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "wo_rel r" shows "max2 r a b \<equiv> if (a, b) \<in> r then b else a"
assms_3: fixes i :: "'agent" assumes "i \<in> agents" shows "SD (R i) (sds R) q"
mcont_lzip: shows "mcont (prod_lub lSup lSup) (rel_prod (\<sqsubseteq>) (\<sqsubseteq>)) lSup (\<sqsubseteq>) (\<lambda>(x, y). lzip x y)"
sba2_export: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- x \<squnion> y) = - x \<squnion> - y"
star_slide_var: fixes x :: "'a ppath set" shows "pp_prod (pp_star x) x = pp_prod x (pp_star x)"
sup_conv_inf: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> y = - (- x \<sqinter> - y)"
hd_last_sorted: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "ccw'.sortedP x xs"   and "1 < length xs" shows "ccw' x (hd xs) (last xs)"
bij_2: fixes x :: "'b" assumes "P x" shows "Q (g x)"
ht_distinct_ls_rehash: fixes l :: "('a \<times> 'b) list list" shows "ht_distinct (ls_rehash l)"
mat_of_row_carrier_1: fixes y :: "'b vec"   and n :: "nat" assumes "y \<in> carrier_vec n" shows "mat_of_row y \<in> carrier_mat 1 n"
ereal_infinity_cases: fixes a :: "ereal" assumes "a \<noteq> \<infinity>"   and "a \<noteq> - \<infinity>" shows "\<bar>a\<bar> \<noteq> \<infinity>"
Diag_Diagonalize_3: fixes t :: "'c local.term" assumes "Arr t" shows "Cod \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
longest_Lyndon_suf_longest: fixes w :: "'a list"   and v' :: "'a list" assumes "w \<noteq> \<epsilon>"   and "v' \<le>s w"   and "Lyndon v'" shows "\<^bold>|v'\<^bold>| \<le> \<^bold>|LynSuf w\<^bold>|"
refl_sred: fixes t :: "dB" shows "t \<rightarrow>\<^sub>s t"
x: fixes a :: "real"   and b :: "real" assumes "0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b" shows "a \<noteq> b"
pair_sym_digraph_subdivide: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b"   and w :: "'b" assumes "pair_sym_digraph G"   and "e \<in> parcs G"   and "w \<notin> pverts G" shows "pair_sym_digraph (subdivide G e w)"
zero_rev_flow_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> E" shows "f (v, u) = (0::'capacity)"
split_sel_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> option = Some (the option) \<and> \<not> P (f2 (the option))))"
point_rep_number_split: fixes A :: "'b set multiset"   and B :: "'b set multiset"   and x :: "'b" shows "(A + B) rep x = A rep x + B rep x"
field_inverse: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "inverse a * a = (1::'a)"
gt_antisym: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wary s"   and "wary t"   and "local.kbo_std.gt t s" shows "\<not> local.kbo_std.gt s t"
invar_found: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and d :: "nat" assumes "f_invar c src dst PRED d" shows "Graph.bfs_spec c src dst (Some (d, PRED))"
hensel_hypothesis_weakened: shows "val_Zp (f' \<bullet> a) < val_Zp (f \<bullet> a)"
A2: shows "A2 \<in> carrier_mat (m - 1) n"
sff: shows "square_free_factorization p (c, bs1 @ (prod_list bs, i) # bs2)"
equalityD1: fixes A :: "'c set"   and B :: "'c set" assumes "A = B" shows "A \<subseteq> B"
prems_6: shows "c2 \<sqsubseteq> c1"
sample_coprime_def: fixes N :: "nat" shows "sample_coprime N = spmf_of_set (coprime_set N)"
transfer_implies_transfer_5: shows "((=) ===> (=) ===> (\<longrightarrow>)) transfer_implies transfer_implies"
eq_is_closed_def: fixes S :: "('c, 'd) eq_object_scheme"   and A :: "'c set" shows "is_closed\<^bsub>S\<^esub> A = (A \<subseteq> carrier S \<and> closure_of\<^bsub>S\<^esub> A = A)"
trg_def: fixes \<mu> :: "'a" shows "trg \<mu> = (if arr \<mu> then trg\<^sub>B \<mu> else null)"
sel_1: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ReZ (x + y) = ReZ x + ReZ y"
arr_to_hfun_in_hexp: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "arr_to_hfun f \<^bold>\<in> hexp (ide_to_hf (local.dom f)) (ide_to_hf (cod f))"
min_maxsimpchainI_betw: fixes X :: "'b set set"   and x :: "'b set"   and y :: "'b set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "x \<noteq> y"   and "SimplicialComplex.maxsimpchain X (x # xs @ [y])"   and "\<And>ys. SimplicialComplex.maxsimpchain X (x # ys @ [y]) \<Longrightarrow> order.greater_eq (length ys) (length xs)" shows "SimplicialComplex.min_maxsimpchain X (x # xs @ [y])"
assms_2: shows "emeasure M (space M) \<noteq> 0"
select_convs_3: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "document_element \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = document_element"
nonneg_bounded: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
stable_Join_Always1: fixes F :: "'b program"   and A :: "'b set"   and G :: "'b program" assumes "F \<in> stable A"   and "G \<in> invariant A" shows "F \<squnion> G \<in> Always A"
assms: shows "obs_intra m \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}"
iso_iff_mono_and_retraction: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "category.iso C f = (category.mono C f \<and> category.retraction C f)"
norm_ge_zero: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "0 \<le> norm x"
nat_term_compD4'': fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and u :: "'b"   and a :: "'b"   and v :: "'b"   and b :: "'b" assumes "comparator cmp"   and "nat_term_comp cmp"   and "fst (rep_nat_term u) = fst (rep_nat_term a)"   and "fst (rep_nat_term v) = fst (rep_nat_term b)"   and "snd (rep_nat_term u) \<le> snd (rep_nat_term v)"   and "snd (rep_nat_term a) = snd (rep_nat_term b)"   and "cmp a b \<noteq> Gt" shows "cmp u v \<noteq> Gt"
word_plus_strict_mono_right: fixes y :: "'b word"   and z :: "'b word"   and x :: "'b word" assumes "y < z"   and "x \<le> x + z" shows "x + y < x + z"
psubsetI: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B"   and "A \<noteq> B" shows "A \<subset> B"
Orbit_Relation_memI: fixes G :: "('c \<Rightarrow> 'c) set"   and S :: "'c set"   and y :: "'c"   and x :: "'c" assumes "transformation_group G S"   and "\<exists>\<alpha>\<in>G. y = \<alpha> x"   and "x \<in> S" shows "(x, y) \<in> transformation_group.Orbit_Relation G S"
less_imp_production_subseteq_interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and D :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "C < D" shows "ground_resolution_with_selection.production S N C \<subseteq> ground_resolution_with_selection.interp S N D"
rcoset_mult_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "[mod I:] (I +> x) \<Otimes> (I +> y) = I +> x \<otimes> y"
compactin_contractive: fixes X' :: "'c topology"   and S :: "'c set"   and X :: "'c topology" assumes "compactin X' S"   and "topspace X' = topspace X"   and "\<And>U. openin X U \<Longrightarrow> openin X' U" shows "compactin X S"
minus_fds_def: fixes f :: "'b fds"   and g :: "'b fds" shows "f - g = fds (\<lambda>n. fds_nth f n - fds_nth g n)"
steps_decomp: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph_Defs.steps E (xs @ ys)"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "Graph_Defs.steps E xs \<and> Graph_Defs.steps E ys \<and> E (last xs) (hd ys)"
r_constant: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and x :: "'a" assumes "bibd point_set block_collection u_block_size index"   and "x \<in> point_set" shows "block_collection rep x * (u_block_size - 1) = index * (int (card point_set) - 1)"
is_monomial_set_subset: fixes B :: "('d \<Rightarrow>\<^sub>0 'e) set"   and A :: "('d \<Rightarrow>\<^sub>0 'e) set" assumes "is_monomial_set B"   and "A \<subseteq> B" shows "is_monomial_set A"
Abs_lgenerator'_cases: fixes x :: "('d, 'c) lgenerator'" obtains y :: "'c \<Rightarrow> ('d, 'c) step" where "x = Abs_lgenerator' y" and "y \<in> {g. productive g}"
RM_2_b: fixes \<chi> :: "\<o>"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<^bold>\<box>\<chi> in v]" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi> in v]"
assms_4: fixes K :: "'a set" assumes "K \<in> s" shows "\<exists>a b. K = cbox a b"
indRelRTPO_impl_TRel_is_weak_reduction_contrasimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_contrasimulation (indRelRTPO TRel) (STCal Source Target)" shows "weak_reduction_contrasimulation (TRel\<^sup>+) Target"
split_sel_asm: fixes P :: "'d \<Rightarrow> bool"   and f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and envT :: "('e, 'f) envT" shows "P (case envT of EnvT x \<Rightarrow> f x) = (\<not> (envT = EnvT (run_env envT) \<and> \<not> P (f (run_env envT))))"
rel_setD2: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'f set"   and B :: "'g set"   and y :: "'g" assumes "rel_set R A B"   and "y \<in> B" shows "\<exists>x\<in>A. R x y"
fac_def: shows "fac \<equiv> \<mu> fac. \<Lambda> n. If n =\<^sub>B 0 then 1 else n * fac\<cdot>(n - 1)"
runit_char_2: fixes f :: "'a" assumes "Right.ide f" shows "Right.R (Right.runit f) = Right.comp (f \<star> \<i>[src f]) \<a>[f, src f, src f]"
that: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set" assumes "{(h', U'). (h', U') \<in> set (ps\<^sub>+) \<and> poly_deg h' = d} = {(h, U)}"   and "card U = i" shows "thesis"
shift1_inv_preserved: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and x :: "'x" assumes "shift1_inv (qs, B)"   and "1 < card B"   and "(h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m)"   and "x = (SOME y. y \<in> U)" shows "shift1_inv (shift_list (h, U) x qs, B - {(h, U)})"
length_append_repeat: fixes i :: "'a list"   and a :: "nat"   and y :: "'a" shows "length i \<le> length (i @ repeat a y)"
ennreal_SUP_of_nat_eq_top: shows "Sup (range of_nat) = top"
h: shows "2 ^ 4 * 23 * 47 = 17296"
ithI: fixes f :: "'b \<Rightarrow> 'c"   and B :: "('b \<Rightarrow> 'c) set"   and i :: "'b"   and x :: "'c" assumes "f \<in> B"   and "f i = x" shows "x \<in> ith B i"
h2: shows "\<forall>X. \<phi> (\<Inter> X) = \<Inter> (\<P> \<phi> X)"
T_thm_1_lub: shows "lub P cl = lub {x. (x, f x) \<in> r \<and> x \<in> A} cl"
minGraphProps'_def: fixes g :: "graph" shows "minGraphProps' g \<equiv> \<forall>f\<in>\<F> g. 2 < |vertices f| \<and> distinct (vertices f)"
assms_3: fixes x :: "'a"   and y :: "'b" shows "h x (0::'c) y = y"
simps: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'c" shows "(r OO s) a1 a2 = (\<exists>a b c. a1 = a \<and> a2 = c \<and> r a b \<and> s b c)"
n_leaves_nodes: fixes t :: "'c bt" shows "n_leaves t = Suc (n_nodes t)"
WTrtCall: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and U :: "ty"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and meth :: "(String.literal list \<times> 'addr expr) option"   and D :: "String.literal"   and es :: "'addr expr list"   and Ts' :: "ty list" assumes "P,E,h \<turnstile> e : U"   and "class_type_of' U = \<lfloor>C\<rfloor>"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = meth in D"   and "P,E,h \<turnstile> es [:] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts" shows "P,E,h \<turnstile> e\<bullet>M(es) : T"