hcju/isabelleps huggingface dataset:
view_partitioned_ind_def: shows "view_partitioned_ind \<equiv> \<forall>execs1 execs2 s t n u. equivalent_states s t u \<and> purged_relation u execs1 execs2 \<longrightarrow> equivalent_states (run n s execs1) (run n t execs2) u"
even_step: fixes n :: "nat" assumes "n \<in> Even" shows "Suc (Suc n) \<in> Even"
bindU_plusU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c"   and returnU :: "udom \<rightarrow> udom\<cdot>'c"   and bindU :: "udom\<cdot>'c \<rightarrow> (udom \<rightarrow> udom\<cdot>'c) \<rightarrow> udom\<cdot>'c"   and plusU :: "udom\<cdot>'c \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c"   and xs :: "udom\<cdot>'c"   and ys :: "udom\<cdot>'c"   and k :: "udom \<rightarrow> udom\<cdot>'c" assumes "class.monad_plus fmapU returnU bindU plusU" shows "bindU\<cdot>(plusU\<cdot>xs\<cdot>ys)\<cdot>k = plusU\<cdot>(bindU\<cdot>xs\<cdot>k)\<cdot>(bindU\<cdot>ys\<cdot>k)"
P2_inv_g_s_rewrite: fixes s :: "nat"   and r :: "nat"   and \<alpha> :: "nat"   and u1 :: "nat"   and v1 :: "nat" assumes "s < order \<G>" shows "\<^bold>g [^] (r * \<alpha> * u1 + v1 * \<alpha>) \<otimes> inv \<^bold>g [^] (u1 + (order \<G> - s)) = \<^bold>g [^] (r * \<alpha> * u1 + v1 * \<alpha>) \<otimes> \<^bold>g [^] s \<otimes> inv \<^bold>g [^] u1"
relcompp_relcomp_eq: fixes r :: "('b \<times> 'd) set"   and s :: "('d \<times> 'c) set" shows "(\<lambda>x xa. (x, xa) \<in> r) OO (\<lambda>x xa. (x, xa) \<in> s) = (\<lambda>x xa. (x, xa) \<in> r O s)"
necessity_D': assumes "All admits_diagonal_reduction" shows "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'b} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'b})"
echelon_form_echelon_form_upt_row: fixes A :: "(('a, 'b) vec, 'c) vec" shows "echelon_form A = echelon_form_upt_row A (nrows A)"
lcons: fixes x :: "'c"   and xs :: "'c llist" shows "monotone orda ordb (\<lambda>f. lcons f x xs)"
foldr_mult_foldl: fixes f :: "'b \<Rightarrow> 'a"   and as :: "'b list"   and v :: "'a" shows "foldr (\<lambda>x r. r * f x) as v = foldl (\<lambda>r x. r * f x) v as"
assms_1: shows "map_\<I> (inv f) (inv g) \<I>,map_\<I> f' g' \<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2"
vfsequence_vlrestriction: fixes k :: "V" assumes "k \<in>\<^sub>\<circ> \<omega>" shows "vfsequence (xs \<restriction>\<^sup>l\<^sub>\<circ> k)"
ex2: shows "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> P2 x = P2'"
push_bit_double: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n (times a (numeral.numeral one plus (num.Bit0 num.One))) = times (push_bit n a) (numeral.numeral one plus (num.Bit0 num.One))"
preserves_ide: fixes a :: "'a" assumes "C.ide a" shows "C.ide (src\<^sub>C a)"
msetext_dersh_compat_cons: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "msetext_dersh gt ys xs" shows "msetext_dersh gt (x # ys) (x # xs)"
inject_2: fixes x2 :: "real \<times> real \<times> real"   and y2 :: "real \<times> real \<times> real" shows "(EqUni x2 = EqUni y2) = (x2 = y2)"
H: shows "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts"
Invariants_sets: shows "sets Invariants = {A \<in> sets M. T -` A \<inter> space M = A}"
alphabet_splitLeaf: fixes t :: "'a tree"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "alphabet (splitLeaf t w\<^sub>a a w\<^sub>b b) = (if a \<in> alphabet t then alphabet t \<union> {b} else alphabet t)"
final_insert: fixes xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option"   and ys :: "('a \<times> 'a option) list" assumes "set (xs @ [x]) = set (ys @ [x])"   and "rga_ops (xs @ [x])"   and "insert_ops (ys @ [x])"   and "interp_rga xs = interp_ins ys" shows "interp_rga (xs @ [x]) = interp_ins (ys @ [x])"
partial_equivalence_axioms_def: fixes A :: "'b set"   and equiv :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "partial_equivalence_axioms A equiv \<equiv> transitive A equiv"
card_eq_2_doubleton: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "card A = 2"   and "x \<noteq> y"   and "x \<in> A"   and "y \<in> A" shows "A = {x, y}"
char_eq_0_iff: fixes a :: "nat" shows "(m.dc2c (principal_dchar m) a = 0) = (a \<notin> carrier H)"
netglobal_weakenE: fixes p :: "(((state \<times> (state, msg, pseqp, pseqp label) seqp) \<times> msg list \<times> (msg list, msg, unit, unit label) seqp) net_state, 'a) automaton"   and P :: "(nat \<Rightarrow> state) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> state) \<Rightarrow> bool" assumes "p \<TTurnstile> netglobal P"   and "\<And>\<sigma>. P \<sigma> \<Longrightarrow> Q \<sigma>" shows "p \<TTurnstile> netglobal Q"
subset_cball: fixes d :: "real"   and e :: "real"   and x :: "'a" assumes "d \<le> e" shows "cball x d \<subseteq> cball x e"
eq: shows "real_of_int x - real_of_int y * sqrt (real D) = 1 / (real_of_int x + real_of_int y * sqrt (real D))"
simps_1: fixes u :: "'b"   and d :: "'b"   and v :: "'b" shows "lrev_geom u d v [] = v"
lrf_def: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec"   and ur :: "('a \<times> 'a) set" assumes "Unwinding SES \<V>" shows "Unwinding.lrf SES \<V> ur \<equiv> \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>s'\<in>S\<^bsub>SES\<^esub>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. reachable SES s \<and> s c\<longrightarrow>\<^bsub>SES\<^esub> s' \<longrightarrow> (s', s) \<in> ur"
image_mset_union: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset"   and N :: "'b multiset" shows "image_mset f (M + N) = image_mset f M + image_mset f N"
rk_AA'C'a: fixes A :: "Points"   and A' :: "Points"   and C' :: "Points"   and B :: "Points"   and P :: "Points"   and a :: "Points"   and C :: "Points"   and B' :: "Points" assumes "3 \<le> rk {A, A', C'}"   and "4 \<le> rk {A, B, P, a}"   and "rk {A, B, C, A', B', C', P} = 3" shows "4 \<le> rk {A, A', C', a}"
sum_list_zeroI: fixes xs :: "'b list" assumes "set xs \<subseteq> {0::'b}" shows "sum_list xs = (0::'b)"
inorder_delete: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (WBT.delete \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 x t) = del_list x (Tree2.inorder t)"
solve_factored_ratfps'_def: shows "solve_factored_ratfps' = (\<lambda>p (a, cs). solve_factored_ratfps (smult (inverse a) p) cs)"
acute_per__lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Acute A B C"   and "D \<noteq> E"   and "E \<noteq> F"   and "Per D E F" shows "A B C LtA D E F"
interaction_bounded_by_IO: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and "consider" :: "'e \<Rightarrow> bool"   and n :: "enat"   and input :: "'f" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "interaction_bounded_by consider gpv n"   and "consider out" shows "n \<noteq> 0 \<and> interaction_bounded_by consider (c input) (n - 1)"
of_int_leD: 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 n :: "int"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq (abs (ring_1.of_int one plus zero minus n)) x" shows "n = 0 \<or> less_eq one x"
comp_null_2: fixes f :: "'b" shows "f \<cdot>\<^sub>B B.null = B.null"
lfinite_less_induct: fixes xs :: "'b llist"   and P :: "'b llist \<Rightarrow> bool" assumes "lfinite xs"   and "\<And>xs. \<lbrakk>lfinite xs; \<And>zs. llength zs < llength xs \<Longrightarrow> P zs\<rbrakk> \<Longrightarrow> P xs" shows "P xs"
validCI_act: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b" assumes "\<exists>i\<ge>n. active cid (t i)"   and "\<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) (the_enat (dynamic_component.nAct tCMP active cid (enat n) (inf_llist t)))" shows "dynamic_component.eval tCMP active cid t t' n \<gamma>"
times_mat_def: fixes A :: "'a mat"   and B :: "'a mat" shows "A * B \<equiv> mat (dim_row A) (dim_col B) (\<lambda>(i, j). row A i \<bullet> col B j)"
inverse_roots: fixes x :: "'a"   and p :: "int poly" assumes "x \<noteq> (0::'a)" shows "(ipoly (reflect_poly p) x = (0::'a)) = (ipoly p (inverse x) = (0::'a))"
P2: shows "A A' Perp C'' A' \<and> A A' OS C C''"
abs_conv_abscissa_power_leI: fixes f :: "'a fds"   and d :: "ereal"   and n :: "nat" assumes "abs_conv_abscissa f \<le> d" shows "abs_conv_abscissa (f ^ n) \<le> d"
union_is_single: fixes M :: "'a multiset"   and N :: "'a multiset"   and a :: "'a" shows "(M + N = \<LM> a  \<RM>) = (M = \<LM> a  \<RM> \<and> N = \<Empt> \<or> M = \<Empt> \<and> N = \<LM> a  \<RM>)"
fix_const: fixes c :: "'b" shows "(\<mu> x. c) = c"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "pre (C\<^sub>1;; C\<^sub>2) Q = pre C\<^sub>1 (pre C\<^sub>2 Q)"
HMA_M_AA': shows "Mod_Type_Connect.HMA_M A A'"
complete_measure_axioms: fixes M :: "'b measure" shows "complete_measure (completion M)"
gen_lucas_lehmer_sequence'_correct: fixes a :: "int"   and m :: "int"   and n :: "nat" assumes "a \<in> {0..<m}" shows "gen_lucas_lehmer_sequence' m a n = gen_lucas_lehmer_sequence a n mod m"
trees_def: shows "trees \<equiv> \<lambda>S. {x. treesp (\<lambda>x. x \<in> S) x}"
results'_gpv_cases: fixes x :: "'d"   and gpv :: "('d, 'e, 'f) gpv" assumes "x \<in> results'_gpv gpv" obtains generat :: "('d, 'e, ('d, 'e, 'f) rpv) generat" where "generat__ \<in> set_spmf (the_gpv gpv)" and "x \<in> generat_pures generat__"   | generat :: "('d, 'e, ('d, 'e, 'f) rpv) generat" and c :: "('d, 'e, 'f) rpv" and input :: "'f" where "generat \<in> set_spmf (the_gpv gpv)" and "c \<in> generat_conts generat" and "x \<in> results'_gpv (c input)"
select_convs_2: fixes arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'b" shows "refs \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = refs"
image_ideal_subset: fixes h :: "'b \<Rightarrow> 'c"   and F :: "'b set" assumes "\<And>x y. h (x + y) = h x + h y"   and "\<And>x y. h (x * y) = h x * h y" shows "h ` ideal F \<subseteq> ideal (h ` F)"
fold_argmin: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and x :: "'a" shows "fold (\<lambda>x (m, fm). let fx = f x in if fx \<le> fm then (x, fx) else (m, fm)) xs (x, f x) = (argmin f (x # xs), f (argmin f (x # xs)))"
simulationI_axioms: shows "simulationI as\<^sub>0 alstep cs\<^sub>0 clstep R"
xs: shows "filter (\<lambda>(i, Ail). Ail \<noteq> (0::'a)) (map (\<lambda>i. (i, A $$ (i, l))) [Suc l..<dim_row A]) = []"
bounded_linear_imp_has_derivative: fixes f :: "'c \<Rightarrow> 'd"   and net :: "'c filter" assumes "bounded_linear f" shows "(f has_derivative f) net"
path_conforms_with_strategy_prefix: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and P :: "'c llist"   and \<sigma> :: "'c \<Rightarrow> 'c"   and P' :: "'c llist" assumes "ParityGame G"   and "ParityGame.path_conforms_with_strategy G p P \<sigma>"   and "lprefix P' P" shows "ParityGame.path_conforms_with_strategy G p P' \<sigma>"
assms_1: shows "constructive_security_obsf real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 (absorb \<A> (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2))))) adv1"
edges_subsets: shows "edges_of_walk xs \<subseteq> edges_of_walk ys"
pq_eq2: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "int p * int q = 2 * ((int p * int q - 1) div 2) + 1"
rem_initial_append2: fixes b :: "bit"   and xs :: "bit list"   and ys :: "bit list" assumes "rem_initial b xs = []" shows "rem_initial b (xs @ ys) = rem_initial b ys"
no_id: fixes v :: "'v"   and w :: "'w" shows "(v, w, v) \<notin> E"
action_needed_vars_def: fixes a :: "('a, 'b) fmap \<times> 'c"   and s :: "('a, 'b) fmap" shows "action_needed_vars a s \<equiv> {v \<in> fmdom' s. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s v}"
UN_atMost_UNIV: shows "\<Union> (range atMost) = UNIV"
POR_arg2_rel_def: shows "POR_arg2_rel \<equiv> \<lambda>i. case i of One \<Rightarrow> \<bottom> | Two \<Rightarrow> ValTT | Three \<Rightarrow> ValFF"
alternate_length: fixes xs :: "'b list" shows "length (alternate xs) = (1 + length xs) div 2"
insert_subset: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "((x::'b) \<triangleright> (A::'b set) \<subseteq> (B::'b set)) = (x \<in> B \<and> A \<subseteq> B)"
diameter_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "bounded T" shows "diameter S \<le> diameter T"
less_parity_def: fixes x :: "parity"   and y :: "parity" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
split_relation_def: fixes xs :: "'d list" shows "split_relation xs \<equiv> \<lambda>(as, bs) i. i \<le> length xs \<and> as = take i xs \<and> bs = drop i xs"
one_plus_numeral_commute: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "num" assumes "class.numeral plus" shows "plus one (numeral.numeral one plus x) = plus (numeral.numeral one plus x) one"
integrable_fst': fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) f" shows "integrable lborel (\<lambda>x. LBINT y. f (x, y))"
fv_set_mono: fixes M :: "('c, 'd) Term.term set"   and N :: "('c, 'd) Term.term set" assumes "M \<subseteq> N" shows "fv\<^sub>s\<^sub>e\<^sub>t M \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t N"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'c"   and x :: "('c, 'd) phi \<Rightarrow> 'c" assumes "eq_onp (\<lambda>xf. True) xa xa"   and "eq_onp (\<lambda>xf. True) x x" shows "Abs_lifted_boolean_algebra xa \<sqinter> Abs_lifted_boolean_algebra x = Abs_lifted_boolean_algebra (\<lambda>f. xa f \<sqinter> x f)"
Some: shows "lookup (\<lambda>x. x = k) pairs = Some v"
stat_ref_G: shows "statically_complete_calculus Bot_G Inf_G (\<Turnstile>G) Red_I_G Red_F_G"
simps_1: fixes x :: "('a, 'b) Combinators"   and y :: "('a, 'b) Combinators" shows "first_destNet (x \<oplus> y) = first_destNet x"
inject: fixes x21 :: "'b"   and x22 :: "'b heap list"   and y21 :: "'b"   and y22 :: "'b heap list" shows "(Hp x21 x22 = Hp y21 y22) = (x21 = y21 \<and> x22 = y22)"
matches_ipassmt_iface_replace_srcip_mexpr_case_Iface: fixes ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and p :: "('c, 'd) tagged_packet_scheme"   and p_ips :: "('c word \<times> nat) list"   and \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ifce :: "iface"   and a :: "action" assumes "ipassmt_sanity_nowildcards ipassmt"   and "ipassmt_sanity_disjoint ipassmt"   and "ipassmt (Iface (p_iiface p)) = Some p_ips \<and> p_src p \<in> ipcidr_union_set (set p_ips)" shows "matches (common_matcher, \<alpha>) (ipassmt_iface_replace_srcip_mexpr ipassmt ifce) a p = matches (common_matcher, \<alpha>) (Match (IIface ifce)) a p"
lam_eval_prod: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and h :: "'a" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>h : a \<rightarrow> exp b c\<guillemotright>" shows "\<Lambda> a b c (C (eval b c) (local.prod h b)) = h"
alphas_1: fixes bs :: "atom set"   and x :: "'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'b" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
word_and_def: fixes a :: "'b word"   and b :: "'b word" shows "a AND b = word_of_int (uint a AND uint b)"
diff_single_eq_union: fixes x :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" assumes "x \<in># M" shows "(M - {#x#} = N) = (M = add_mset x N)"
new_list_gt_gmean: fixes xs :: "real list" assumes "pos xs"   and "0 < het xs" shows "\<exists>xs'. gmean xs < gmean xs' \<and> \<gamma>_eq (xs', xs) \<and> het xs' < het xs \<and> pos xs'"
count_terminals_of_word: fixes w :: "symbol list" assumes "is_word w" shows "count_terminals w = length w"
ln_exp_cf7_lower_bound_neg: fixes x :: "real" assumes "x \<le> 0"   and "0 < numer_cf7 x" shows "ln (exp_cf7 x) \<le> x"
UN_subset_iff: fixes A :: "'e \<Rightarrow> 'd set"   and I :: "'e set"   and B :: "'d set" shows "(\<Union> (A ` I) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
pterm_induct: fixes P :: "pterm \<Rightarrow> bool"   and t :: "pterm" assumes "\<And>x. P (Pconst x)"   and "\<And>x. P (Pvar x)"   and "\<And>cs. (\<And>pat t. (pat, t) |\<in>| cs \<Longrightarrow> P t) \<Longrightarrow> P (Pabs cs)"   and "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (t $\<^sub>p u)" shows "P t"
Rep_pdata_tuple: fixes D :: "'b pdata" shows "Rep_pdata D = (PDataValue D, PDataSpace D)"
pth_d: fixes x :: "'b" shows "x + (0::'b) \<equiv> x"
tdec2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and d :: "dec"   and s2 :: "'b state"   and err :: "v error_result" assumes "evaluate_dec ck [] env s1 d (s2, Rerr err)" shows "evaluate_top ck env s1 (Tdec d) (s2, Rerr err)"
sep_conj_sep_true: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "P h" shows "sep_algebra.sep_conj plus sep_disj P (\<lambda>s. True) h"
inverse_finite_2_def: shows "inverse = (\<lambda>x. x)"
igSubstIGAbsSTR_imp_igSubstIGAbs: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igSubstIGAbsSTR MOD" shows "igSubstIGAbs MOD"
sublist_length: fixes i :: "nat"   and j :: "nat"   and xs :: "'c list" assumes "i \<le> j"   and "j < length xs" shows "length (WB_Sort.sublist xs i j) = 1 + j - i"
induct: fixes P :: "'c event list \<times> 'c event set \<times> 'c event list \<Rightarrow> bool"   and a0 :: "'c event list \<times> 'c event set \<times> 'c event list" assumes "\<And>X. P ([], X, [])"   and "\<And>X y t. (\<And>x xa. y \<notin> X \<Longrightarrow> P ([], X, t)) \<Longrightarrow> P ([], X, y # t)"   and "\<And>x s X. (\<And>xa xb. x \<notin> X \<Longrightarrow> P (s, X, [])) \<Longrightarrow> P (x # s, X, [])"   and "\<And>x s X y t. \<lbrakk>\<And>xa xb. \<lbrakk>x \<in> X; y \<in> X; x = y\<rbrakk> \<Longrightarrow> P (s, X, t); \<And>xa xb. \<lbrakk>x \<in> X; y \<notin> X\<rbrakk> \<Longrightarrow> P (x # s, X, t); \<And>xa xb. \<lbrakk>x \<notin> X; y \<notin> X\<rbrakk> \<Longrightarrow> P (s, X, y # t); \<And>xa xb. \<lbrakk>x \<notin> X; y \<notin> X\<rbrakk> \<Longrightarrow> P (x # s, X, t); \<And>xa xb. \<lbrakk>x \<notin> X; \<not> y \<notin> X\<rbrakk> \<Longrightarrow> P (s, X, y # t)\<rbrakk> \<Longrightarrow> P (x # s, X, y # t)" shows "P a0"
catch_put: fixes s :: "'s"   and m :: "'m"   and m' :: "'m" shows "catch (put s m) m' = put s (catch m m')"
choose_smaller_in_sort: fixes cs :: "(String.literal \<times> String.literal) set"   and c :: "String.literal"   and s :: "String.literal set" assumes "wf_subclass_loc cs"   and "c \<in> s"   and "c \<notin> normalize_sort cs s"   and "sort_ex cs s" obtains c' :: "String.literal" where "c' \<in> s" and "class_les cs c' c"
T'_subset: fixes a :: "(int, 'n) vec" shows "T' a \<subseteq> cbox 0 1"
int_A: fixes i :: "nat" shows "integrable M (\<lambda>x. indicat_real (A i) x *\<^sub>R f x)"
add_rng_of_frac_fundamental_lemma: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set"   and r :: "'c"   and s :: "'c"   and r' :: "'c"   and s' :: "'c" assumes "eq_obj_rng_of_frac R S"   and "(r, s) \<in> carrier \<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes>\<^bsub>R\<^esub> (s' \<otimes>\<^bsub>R\<^esub> r \<ominus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r') = \<zero>\<^bsub>R\<^esub>\<rparr>"   and "(r', s') \<in> carrier \<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes>\<^bsub>R\<^esub> (s' \<otimes>\<^bsub>R\<^esub> r \<ominus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r') = \<zero>\<^bsub>R\<^esub>\<rparr>" shows "(r |\<^bsub>\<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes>\<^bsub>R\<^esub> (s' \<otimes>\<^bsub>R\<^esub> r \<ominus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r') = \<zero>\<^bsub>R\<^esub>\<rparr>\<^esub> s) \<oplus>\<^bsub>eq_obj_rng_of_frac.rec_rng_of_frac R S\<^esub> (r' |\<^bsub>\<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes>\<^bsub>R\<^esub> (s' \<otimes>\<^bsub>R\<^esub> r \<ominus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r') = \<zero>\<^bsub>R\<^esub>\<rparr>\<^esub> s') = (s' \<otimes>\<^bsub>R\<^esub> r \<oplus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r' |\<^bsub>\<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes>\<^bsub>R\<^esub> (s' \<otimes>\<^bsub>R\<^esub> r \<ominus>\<^bsub>R\<^esub> s \<otimes>\<^bsub>R\<^esub> r') = \<zero>\<^bsub>R\<^esub>\<rparr>\<^esub> s \<otimes>\<^bsub>R\<^esub> s')"
suf_rev_pref_iff: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>s ys = (rev xs \<le>p rev ys)"
zero_all: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and c :: "'a"   and u :: "'a \<Rightarrow> real" assumes "finite X"   and "Regions.valid_region X k I r"   and "c \<in> X"   and "u \<in> Regions.region X I r"   and "u c = 0" shows "zero c (Regions.region X I r)"
msg_okI: fixes m :: "msg"   and \<sigma> :: "nat \<Rightarrow> state" assumes "\<And>num' sid'. m = Pkt num' sid' \<Longrightarrow> num' \<le> no (\<sigma> sid')" shows "msg_ok \<sigma> m"
fps_of_ratfps_uminus: fixes x :: "'b ratfps" shows "fps_of_ratfps (- x) = - fps_of_ratfps x"
reduc_fair_def: fixes Ns :: "'a set llist" shows "reduc_fair Ns = (Inf_from (Liminf_llist Ns - Sup_llist (lmap Red_F Ns)) \<subseteq> Sup_llist (lmap Red_I Ns))"
split_spec_def: fixes X :: "'a set" shows "split_spec X = SPEC (\<lambda>(A, B). X \<subseteq> A \<union> B)"
cases: fixes x :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" obtains 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" where "x = (qs, B)"
is_starfun_starext: fixes F :: "'c star \<Rightarrow> 'c star"   and f :: "'c \<Rightarrow> 'c" assumes "is_starext F f" shows "F = *f* f"
simps_1: fixes uu :: "'c"   and B :: "'c"   and uv :: "'d" shows "Cod (MkArr uu B uv) = B"