hcju/isabelleps huggingface dataset:
both_mono1: fixes t :: "'b ttree"   and t' :: "'b ttree"   and t'' :: "'b ttree" assumes "paths t \<subseteq> paths t'" shows "paths (t \<otimes>\<otimes> t'') \<subseteq> paths (t' \<otimes>\<otimes> t'')"
set_disconnected_nodes_locs_impl: shows "set_disconnected_nodes_locs = CD.a_set_disconnected_nodes_locs"
IH: fixes t :: "'a trm"   and u :: "'a trm"   and v :: "'a trm" assumes "subterm t q u" shows "\<exists>a. replace_subterm t q v a"
semiring_normalization_rules_34: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "times x (plus y z) = plus (times x y) (times x z)"
C: fixes x :: "real" shows "cmod (sum_upto \<chi> x) \<le> C"
all_normalized_and_ex_tcsigs_imp_exe_all_normalized_and_ex_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss"   and "all_normalized_and_ex_tcsigs (set cs) (translate_ars arss)" shows "exe_all_normalized_and_ex_tcsigs cs arss"
sup_continuous_Continuous: fixes F :: "'b upred \<Rightarrow> 'b upred" assumes "Continuous F" shows "sup_continuous F"
lunit_char_3: fixes f :: "'a" assumes "ide f" shows "trg f \<star> PB.lunit f = (\<i>' (trg f) \<star> f) \<cdot> local.inv \<a>[trg f, trg f, f]"
assms_1: shows "min_maxsimpchain (x # xs @ [y])"
Sup_upper: fixes x :: "V"   and A :: "V set" assumes "x \<in> A"   and "small A" shows "x \<le> \<Squnion> A"
ex_closure: fixes p :: "('a, 'b) form" shows "\<exists>m. closed 0 (put_unis m p)"
is_basis: fixes p :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "countable_basis p B" shows "topological_space.topological_basis p B"
not_High_Low: 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 Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and H :: "'c set"   and L :: "'c set"   and High :: "'b"   and Low :: "'b"   and V :: "'c" assumes "NonInterferenceInterGraph sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses H L High Low"   and "V \<notin> H" shows "V \<in> L"
mult2_alt_ns_ns_add: fixes X1 :: "'a multiset"   and Y1 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X2 :: "'a multiset"   and Y2 :: "'a multiset" assumes "(X1, Y1) \<in> mult2_alt_ns ns s"   and "(X2, Y2) \<in> mult2_alt_ns ns s" shows "(X1 + X2, Y1 + Y2) \<in> mult2_alt_ns ns s"
sum_eq_1_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite A" shows "(sum f A = 1) = (\<exists>a\<in>A. f a = 1 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
RECOVER_PURE_def: fixes P :: "assn"   and Q :: "assn" shows "RECOVER_PURE P Q \<equiv> P \<Longrightarrow>\<^sub>t Q"
c: shows "??.Stirling_Formula.c = ln (2 * pi) / 2"
mult_zero_1: fixes f :: "nat \<Rightarrow> int" assumes "f \<in> carrier Zp" shows "f \<otimes> \<zero> = \<zero>"
hfunction_restr: fixes r :: "hf"   and x :: "hf" assumes "hfunction r" shows "hfunction (hrestrict r x)"
linear: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "linear f"
cotrace_trace': fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event llist"   and i :: "nat" assumes "cotrace c t" shows "\<exists>c'. trace c (Co_Snapshot.ltake i t) c'"
assms: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> S"   and "y \<in>\<^sub>\<circ> S"   and "x \<noteq> y" shows "R x y"
group_def: fixes G :: "('a, 'b) monoid_scheme" shows "Group.group G \<equiv> Group.monoid G \<and> Group.group_axioms G"
inv_is_inverse: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "category.iso C f" shows "category.inverse_arrows C f (category.inv C f)"
distinct_21: fixes x12 :: "word5" shows "PSR \<noteq> ASR x12"
order_char_poly_smult: fixes A :: "complex mat"   and n :: "nat"   and k :: "complex"   and x :: "complex" assumes "A \<in> carrier_mat n n"   and "k \<noteq> 0" shows "Polynomial.order x (char_poly (k \<cdot>\<^sub>m A)) = Polynomial.order (x / k) (char_poly A)"
fmpred_mono_strong: fixes m :: "('a, 'b) fmap"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>fmlookup m x = Some y; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "fmpred P m" shows "fmpred Q m"
more_temp_simps3_7: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior" shows "\<not> (sigma \<Turnstile> \<diamond>F) \<equiv> sigma \<Turnstile> \<box>\<not> F"
dist_add_cancel: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "dist (plus a b) (plus a c) = dist b c"
assms_3: shows "path_image (part_circlepath C r a c) \<subseteq> A"
power4_eq_xxxx: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.monoid_mult one times" shows "power.power one times x 4 = times (times (times x x) x) x"
l_assm: fixes i :: "nat" shows "y \<le> ereal (a + real i * c)"
lincomb_list_Nil: fixes c :: "nat \<Rightarrow> 'a" shows "lincomb_list c [] = 0\<^sub>v n"
cBexI: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and A :: "'f cset" assumes "P x"   and "cin x A" shows "cBex A P"
toplevel_summands_finite: fixes r :: "'b rexp" shows "finite (toplevel_summands r)"
equiv_eq1: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "((P \<longleftrightarrow> Q) = true) = (P = Q)"
snd_swap: fixes x :: "'d \<times> 'e" shows "snd (prod.swap x) = fst x"
g: shows "g \<in> N \<rightarrow>\<^sub>M subprob_algebra B"
multpw_refl': fixes ns :: "('a \<times> 'a) set"   and A :: "'a multiset" assumes "locally_refl ns A" shows "(A, A) \<in> multpw ns"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(\<lbrakk>x2\<rbrakk> = \<lbrakk>y2\<rbrakk>) = (x2 = y2)"
assms_9: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> \<le> r\<^sup>T * t\<^sup>\<star>"
simps_5: fixes a :: "'b"   and as :: "'b list"   and f :: "('b, 'c) Kform" shows "subjective a \<^bold>C\<^bsub>as\<^esub> f = (a \<in> set as)"
SemNew: fixes l :: "Location"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var"   and C :: "Class" assumes "l \<notin> Dom (snd s)"   and "t = (update (fst s) x (RVal (Loc l)), (l, C, []) # snd s)" shows " s , New x C \<rightarrow>\<^sub>1  t "
simps_26: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a"   and x51 :: "pdf_type"   and x52 :: "pdf_type" shows "(case PRODUCT x51 x52 of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f5 x51 x52"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and f :: "'fsym"   and Tl :: "'fsym trm list" shows "int \<xi> (Fn f Tl) = intF f (map (int \<xi>) Tl)"
inject_1: fixes x11 :: "'a"   and x12 :: "bool"   and y11 :: "'a"   and y12 :: "bool" shows "(<x11, inM x12> = <y11, inM y12>) = (x11 = y11 \<and> x12 = y12)"
dispatch_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "supported_instruction (fst i) = True \<and> fst i \<noteq> ctrl_type RETT" shows "snd (dispatch_instruction i s) = False"
cod_closed: fixes f :: "'a" assumes "Arr f" shows "Arr (C.cod f)"
rightreg_scalar_multD2: fixes x :: "'r"   and r :: "'r"   and f :: "'r \<Rightarrow> 'b" assumes "x \<notin> R" shows "(r \<currency> f) x = (0::'b)"
map_id: fixes p :: "'d spmf" shows "map_spmf id p = p"
lc_eq_zero_iff: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "(i.lcf p = (0::'b)) = (p = 0)"
leftderivation_implies_derivation: fixes u :: "symbol list" assumes "is_leftderivation u" shows "is_derivation u"
set_cases_1: fixes e :: "'g"   and a :: "('g, 'h, 'i) generat" assumes "e \<in> generat_pures a" obtains "a = Pure e"
empty_def: shows "AA_Set.empty = \<langle>\<rangle>"
i_set0_i_set0_ind_eq: shows "i_set0 = i_set0_ind"
binary_endofunctor_T: shows "binary_endofunctor (\<cdot>) T"
P: fixes s :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "P (s, n)" shows "wp\<^sub>3\<^sub>' c (\<lambda>(ps, n). (Q \<and>* (\<lambda>s. True)) (ps, n div k)) (s, k * n)"
simps_1: fixes \<mu> :: "'c" shows "Cod \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
ARR_comp: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "ARR f"   and "ARR g"   and "DOM f = COD g" shows "ARR (f \<cdot> g)"
vector_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i j k. f (i, j) = f (i, k)" shows "matrix_bounded_idempotent_semiring.vector f"
closed_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "closed A f = (\<forall>x\<in>A. \<forall>y\<in>A. x \<squnion>\<^bsub>f\<^esub> y \<in> A)"
similar_mat_witD_5: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "B \<in> carrier_mat n n"
domA_from_set: fixes x :: "'a"   and e :: "'b"   and h :: "('a \<times> 'b) list" assumes "(x, e) \<in> set h" shows "x \<in> domA h"
H: fixes a :: "String.literal list"   and b :: "char list list"   and a' :: "String.literal list" assumes "(a, b) \<in> monom_rel"   and "(a', b) \<in> monom_rel" shows "a = a'"
is_quantum_predicate_def: fixes P :: "complex mat" shows "ps_P.is_quantum_predicate P = (P \<in> carrier_mat ps_P.d ps_P.d \<and> positive P \<and> P \<le>\<^sub>L 1\<^sub>m ps_P.d)"
sum_trading_strat: fixes pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and pf2 :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy pf1"   and "trading_strategy pf2" shows "trading_strategy (qty_sum pf1 pf2)"
has_fps_expansion_shift: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps"   and n :: "nat"   and c :: "'a" assumes "f has_fps_expansion F"   and "n \<le> subdegree F"   and "c = fps_nth F n" shows "(\<lambda>x. if x = (0::'a) then c else f x / x ^ n) has_fps_expansion fps_shift n F"
assms_1: shows "rell_FGl L1 L2 L3 x y"
intros_2: fixes F :: "'b formula"   and G :: "'b formula" shows "F \<^bold>\<rightarrow> F \<^bold>\<or> G \<in> AX10"
rev_map: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "rev (map f xs) = map f (rev xs)"
partition_on_def: fixes A :: "'b set"   and P :: "'b set set" shows "partition_on A P = (\<Union> P = A \<and> disjoint P \<and> {} \<notin> P)"
down_part: shows "down dm lm d (updown' dm lm d \<alpha>) p = (\<Sum>p''\<in>(\<Union>p'\<in>parents d (base {d} p) p. lgrid (base {0..<d} p') {0..<d} lm). \<alpha> p'' * (\<Prod>d' = 0..<Suc d. l2_\<phi> (p'' ! d') (p ! d')))"
rel_gpv''_mono: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R' :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "A \<le> A'"   and "C \<le> C'"   and "R' \<le> R" shows "rel_gpv'' A C R \<le> rel_gpv'' A' C' R'"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> f). f\<lparr>a\<rparr> = b"
z: shows "z \<notin> \<gamma> ` {a..b}"
poly_pinfty_gt_lc: fixes p :: "real poly" assumes "0 < Polynomial.lead_coeff p" shows "\<exists>n. \<forall>x\<ge>n. Polynomial.lead_coeff p \<le> poly p x"
prob_space_return: fixes x :: "'b"   and M :: "'b measure" assumes "x \<in> space M" shows "prob_space (return M x)"
master_integral': fixes p' :: "real"   and a :: "real"   and p :: "real" assumes "p' \<noteq> 0"   and "1 < a" obtains c :: "real" and d :: "real" where "p' < 0 \<longrightarrow> c \<noteq> 0" and "d \<noteq> 0" and "(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p * ln u powr (p' - 1) / u powr (p + 1)))) \<in> \<Theta>(\<lambda>x. c * real x powr p + d * real x powr p * ln (real x) powr p')"
e: shows "e = (\<Union>i\<in>s. e \<inter> f i)"
fold_Option_bind_at_some_point_None_eq_None: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a option"   and l1 :: "('b \<times> 'c) list"   and start :: "'a option"   and l2 :: "('b \<times> 'c) list" assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) l1 start = None" shows "fold (\<lambda>(x, y) option. Option.bind option (f x y)) (l1 @ l2) start = None"
stake_all: fixes xs :: "[:'a:]"   and i :: "Integer" assumes "le\<cdot>(slength\<cdot>xs)\<cdot>i = TT" shows "stake\<cdot>i\<cdot>xs = xs"
range_to_nat: shows "range mod_type_class.to_nat = {0..<CARD('d)}"
numadd_nz: fixes a :: "Ferrack.num"   and b :: "Ferrack.num" assumes "nozerocoeff a"   and "nozerocoeff b" shows "nozerocoeff (numadd a b)"
simps: fixes x :: "'b"   and xs :: "'b List.list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
cblinfun_apply: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>V) x \<in> {f. bounded_clinear f}"
choose_reduce_nat: fixes n :: "nat"   and k :: "nat" assumes "0 < n"   and "0 < k" shows "n choose k = n - 1 choose (k - 1) + (n - 1 choose k)"
type_definition_signed: shows "type_definition Rep_signed Abs_signed UNIV"
norm_triangle_ineq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (plus x y) \<le> norm x + norm y"
tr: shows "((\<xi>, p'), a, \<xi>', q) \<in> automaton.trans A"
bit_flip_bit_iff: fixes m :: "nat"   and a :: "'a"   and n :: "nat" shows "bit (flip_bit m a) n = ((m = n) = (\<not> bit a n) \<and> (2::'a) ^ n \<noteq> (0::'a))"
wpQ_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "wp\<^sub>Q c Q = (\<lambda>s. if \<exists>t p. (c, s) \<Rightarrow> p \<Down> t \<and> Q t < \<infinity> then enat (\<down>\<^sub>t (c, s)) + Q (\<down>\<^sub>s (c, s)) else \<infinity>)"
sum_diff_distrib: fixes Q :: "'a \<Rightarrow> nat"   and P :: "'a \<Rightarrow> nat"   and n :: "'a" assumes "\<forall>x. Q x \<le> P x" shows "sum P {..<n} - sum Q {..<n} = (\<Sum>x<n. P x - Q x)"
eq: shows "lt (sig_trd bs (poly_of_pair p)) = lt (poly_of_pair p)"
moebius_reciprocal: shows "moebius_pt moebius_reciprocal = reciprocal"
cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject: fixes x :: "(_) document_ptr"   and y :: "(_) document_ptr" shows "(cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y) = (x = y)"
weakSimWeakenSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<leadsto><Rel> Q"   and "\<And>\<Psi>' R S \<Psi>''. \<lbrakk>(\<Psi>', R, S) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', R, S) \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>\<^sub>w<Rel> Q"
weight_return_pmf_None: shows "weight_spmf (return_pmf None) = 0"
IH: fixes M :: "(lit, lit list) mapping" shows "E_of_succ (\<lambda>v. set (\<alpha>sl (fold ins_edges cnfi M) v)) = imp_graph (cnf_\<alpha> cnfi) \<union> E_of_succ ((set \<circ>\<circ> \<alpha>sl) M)"
conv_inf_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(bot = id x \<sqinter> y) = (bot = x \<sqinter> id y)"
mD': shows "P \<turnstile> D' sees M', Static :  Ts\<rightarrow>T = (mxs', mxl', ins', xt') in D'"
funion_assoc: fixes A :: "'d fset"   and B :: "'d fset"   and C :: "'d fset" shows "A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)"
bind_eq_Ok_conv: fixes x :: "('a, 'c) result"   and f :: "'c \<Rightarrow> ('a, 'b) result"   and z :: "'b" shows "(x \<bind> f = Ok z) = (\<exists>y. x = Ok y \<and> f y = Ok z)"
connected_graph_def: fixes G :: "('a, 'b) graph" shows "connected_graph G \<equiv> valid_graph G \<and> connected_graph_axioms G"
associated_iff_dvd: 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 a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(normalize a = normalize b) = (dvd.dvd times a b \<and> dvd.dvd times b a)"
execute_parallel_operator_negative_effect_if_i: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and op :: "'a strips_operator"   and v :: "'a" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "op \<in> set ops"   and "v \<in> set (delete_effects_of op)" shows "map_of (effect_to_assignments op) v = Some False"
matches_def: fixes t :: "('d, 'e) Term.term"   and p :: "('d, 'f) Term.term" shows "matches t p = (case match_list (\<lambda>_. t) [(p, t)] of None \<Rightarrow> False | Some x \<Rightarrow> True)"
SentP_sf: fixes x :: "tm" shows "Sigma_fm (SentP x)"
nexts_SPEC: fixes prog :: "program"   and g :: "gState" assumes "gState_inv prog g"   and "program_inv prog" shows "nexts prog g \<le> SPEC (\<lambda>gs. \<forall>g'\<in>ls.\<alpha> gs. (g, g') \<in> gState_progress_rel prog)"
distinct_23: fixes x31 :: "nat"   and x32 :: "nat"   and x61 :: "nat"   and x62 :: "recf" shows "recf.Id x31 x32 \<noteq> Mn x61 x62"
MSB_mono: fixes I1 :: "nat fset list"   and I2 :: "nat fset list" assumes "set I1 \<subseteq> set I2" shows "MSB I1 \<le> MSB I2"
restrict_shared_out: fixes a :: "nat"   and L :: "nat set"   and m :: "nat \<Rightarrow> bool option"   and A :: "nat set" assumes "a \<notin> L" shows "(m \<ominus>\<^bsub>A\<^esub> L) a = map_option ((\<or>) (a \<in> A)) (m a)"
absorb_iff1: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (inf a b = a)"
g': shows "g' = snd (snd (splitFace g u v f vs))"
a: shows "(\<Squnion>n. set_nn_integral M (\<Union>n. (T ^^ n) --` A) (F n)) = \<integral>\<^sup>+ x. (\<Squnion>n. F n x * indicator (\<Union>n. (T ^^ n) --` A) x) \<partial>M"
distinct_3: fixes x11 :: "('a, 'c) trm"   and x12 :: "('a, 'c) trm"   and x3 :: "('a, 'b, 'c) formula" shows "Geq x11 x12 \<noteq> ! x3"
b_e_type_cvtop_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and t1 :: "t"   and cvtop :: "cvtop"   and t :: "t"   and sx :: "sx option" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Cvtop t1 cvtop t sx" shows "\<exists>ts''. ts = ts'' @ [t] \<and> ts' = ts'' @ [arity_1_result e]"
int_pow_eq_id: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier G" shows "(x [^] i = \<one>) = (int (ord x) dvd i)"
rel_generat_reflI: fixes x :: "('e, 'f, 'g) generat"   and A :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>y. y \<in> generat_pures x \<Longrightarrow> A y y"   and "\<And>out. out \<in> generat_outs x \<Longrightarrow> B out out"   and "\<And>cont. cont \<in> generat_conts x \<Longrightarrow> C cont cont" shows "rel_generat A B C x x"
preserves_reflects_arr: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "natural_transformation A B F G \<tau>" shows "partial_magma.arr B (\<tau> f) = partial_magma.arr A f"
callee_invariant: fixes y :: "'d"   and s' :: "'s"   and s :: "'s"   and x :: "'c" assumes "(y, s') \<in> set_spmf (right s x)"   and "I s"   and "x \<in> outs_\<I> \<I>r" shows "I s'"
ucast_up_mono_le: fixes x :: "'b word"   and y :: "'b word" assumes "LENGTH('b) \<le> LENGTH('c)"   and "x \<le> y" shows "UCAST('b \<rightarrow> 'c) x \<le> UCAST('b \<rightarrow> 'c) y"
simps_2: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "lhd (iterates f x) = x"
idem_T: fixes P :: "'a process" shows "P \<sqsubseteq>\<^sub>T P"
hfref_weaken_pre': fixes P :: "'a \<Rightarrow> bool"   and R :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'b \<Rightarrow> assn)"   and P' :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c Heap"   and h :: "'a \<Rightarrow> 'd nres"   and S :: "'d \<Rightarrow> 'c \<Rightarrow> assn" assumes "\<And>x. \<lbrakk>P x; rdomp (fst R) x\<rbrakk> \<Longrightarrow> P' x"   and "(f, h) \<in> [P']\<^sub>a R \<rightarrow> S" shows "(f, h) \<in> [P]\<^sub>a R \<rightarrow> S"
diff_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "diff_list xs ys \<equiv> fold removeAll ys xs"
subring_is_ring: fixes R :: "('e, 'f) ring_scheme"   and H :: "'e set" assumes "ring R"   and "subring H R" shows "ring (R\<lparr>carrier := H\<rparr>)"
fr_eq_msubst2: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list"   and x :: "'b" assumes "islin p" shows "(\<exists>x. Ifm vs (x # bs) p) = (Ifm vs (x # bs) (minusinf p) \<or> Ifm vs (x # bs) (plusinf p) \<or> Ifm vs (x # bs) (subst0 (CP 0\<^sub>p) p) \<or> (\<exists>(n, t)\<in>set (uset p). Ifm vs (x # bs) (msubst2 p (n *\<^sub>p (- 2)\<^sub>p) t)) \<or> (\<exists>(c, t)\<in>set (uset p). \<exists>(d, s)\<in>set (uset p). Ifm vs (x # bs) (msubst2 p ((- 2)\<^sub>p *\<^sub>p c *\<^sub>p d) (tm.Add (tm.Mul d t) (tm.Mul c s)))))"
simps_2: fixes P :: "char list"   and ts :: "'a list" shows "Neg P ts\<^sup>c = Pos P ts"
get_root_node_not_node_same: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "ptr |\<in>| object_ptr_kinds h"   and "\<not> is_node_ptr_kind ptr" shows "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r ptr"
fv_t2_bound: shows "fv (t \<cdot> \<theta>2) \<subseteq> fv t \<union> range_vars \<theta>2"
order_iff_strict: fixes a :: "'b"   and b :: "'b" shows "codomain.greater_eq b a = (codomain.greater b a \<or> a = b)"