hcju/isabelleps huggingface dataset:
l_set_disconnected_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs \<equiv> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs"
uninst_trivial_simps_3: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and g :: "'d"   and u :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and p :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option" shows "CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs (uses(g := u g)) p g = CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs u p g"
relH_in_rangeI_2: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap" assumes "relH as h h'" shows "in_range (h', as)"
distrib_3: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
OUTfromChCorrect_data1: shows "OUTfromChCorrect data1"
is_std_sig_code: fixes \<Sigma> :: "exesignature" shows "is_std_sig (translate_signature \<Sigma>) = exeis_std_sig \<Sigma>"
difList_equi: fixes ts :: "'b set list"   and s :: "'b set" assumes "disjoint_list_rec ts" shows "difference_list s ts = difference_list_opt s ts"
less_le_not_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
bigtheta_trans2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> \<Theta>[F](h)"
wf\<^sub>s\<^sub>tI': fixes S :: "('d, 'c) strand_step list"   and V :: "'c set" assumes "\<Union> (fv\<^sub>r\<^sub>c\<^sub>v ` set S) \<union> \<Union> (fv_r\<^sub>e\<^sub>q assign ` set S) \<subseteq> V" shows "wf\<^sub>s\<^sub>t V S"
impl_of_distinct: fixes xs :: "('b, 'c) alist" shows "distinct (map fst (alist.impl_of xs))"
CCexp_eq: fixes e :: "exp"   and a :: "Arity" shows "ccExp_syn a e = snd (cCCexp e\<cdot>a)"
nonzero_eq_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(a = divide b c) = (times a c = b)"
extract_viewres_t_def: fixes r :: "('a, 'b) viewres" shows "extract_viewres_t r \<equiv> case Rep_viewres r of None \<Rightarrow> Abs_FingerTree Empty | Some (a, t) \<Rightarrow> Abs_FingerTree t"
nominal_ts_axioms: shows "nominal_ts (\<turnstile>\<^sub>L) (\<rightarrow>\<^sub>L)"
toplevel_summands_lang_UN: fixes n :: "nat"   and s :: "'a atom rexp" shows "lang n s = \<Union> (lang n ` toplevel_summands s)"
safe_reac_direct_def': fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and safe :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool"   and cfg :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" shows "safe_reach (computation.concurrent_step direct_memop_step empty_storebuffer_step program_step (\<lambda>p p' is sb. sb)) safe cfg \<equiv> \<forall>cfg'. (computation.concurrent_step direct_memop_step empty_storebuffer_step program_step (\<lambda>p p' is sb. sb))\<^sup>*\<^sup>* cfg cfg' \<longrightarrow> safe cfg'"
vec_smult: fixes \<alpha> :: "'b"   and A :: "'b tensor" shows "vec (\<alpha> \<cdot> A) = map ((*) \<alpha>) (vec A)"
rel_S_witness1: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and tytok :: "('o \<times> ('o \<times> 'p) \<times> 'p \<times> 'q \<times> ('q \<times> 'r) \<times> 'r \<times> 's) itself"   and L1 :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and x :: "('o, 'q, 'b, 'f, 'i, 'l, 's) S"   and y :: "('p, 'r, 'e, 'h, 'k, 'n, 's) S" assumes "rel_S_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok"   and "rel_S L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_S (\<lambda>x (x', y). x' = x \<and> L1 x y) (\<lambda>x (x', y). x' = x \<and> L2 x y) Co1 Co2 Contra1 Contra2 x (rel_S_witness L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y))"
spec_ops_ref_less_last: fixes xs :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and deps :: "'d \<Rightarrow> 'c set"   and r :: "'c" assumes "spec_ops (xs @ [(oid, oper)]) deps"   and "r \<in> deps oper" shows "r < oid"
get_add_neq: fixes x :: "'fun"   and y :: "'fun"   and e :: "'fenv_ubx"   and v :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "x \<noteq> y" shows "Fubx_get (Fubx_add e x v) y = Fubx_get e y"
a_rcosI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H"   and "H \<subseteq> carrier R"   and "x \<in> carrier R" shows "h \<oplus> x \<in> H +> x"
injectivity: fixes x :: "'a"   and y :: "'a" assumes "f x = f y" shows "x = y"
errMOD_igSubstIGVar1STR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igVarIPresIGWls MOD"   and "igSubstIGVar1 MOD" shows "igSubstIGVar1STR (errMOD MOD)"
map_simp: fixes f :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f" shows "\<F>\<^sub>SC.map f = f"
orderI: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" assumes "a = a \<oplus> b" shows "b \<preceq> a"
basevars: fixes vs :: "state \<Rightarrow> 'b"   and c :: "'b" assumes "basevars vs" shows "\<exists>u. vs u = c"
simps_1: fixes c :: "'c" shows "get_const (Le c) = c"
abs_summable_on_altdef': fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "(f abs_summable_on A) = set_integrable (count_space B) A f"
subgroup_inverse_iff: fixes x :: "'a" assumes "B.additive.invertible x"   and "x \<in> B" shows "(B.additive.inverse x \<in> I) = (x \<in> I)"
fps_const_compose: fixes a :: "'a"   and b :: "'a fps" shows "fps_const a oo b = fps_const a"
betw4_sym: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "[[a b c d]] = [[d c b a]]"
coeff_smult: fixes a :: "'b"   and p :: "'b poly"   and n :: "nat" shows "coeff (smult a p) n = a * coeff p n"
norm_12: shows "\<box>(\<lambda>s. False) = (\<lambda>s. False)"
emeasure_buffon_set'_short: shows "emeasure lborel (buffon_set' l d) = ennreal l"
Bonk_Schramm_extension_hyperbolic: shows "Gromov_hyperbolic_subset (deltaG TYPE('b)) UNIV"
size_3: shows "\<^bold>|\<epsilon>\<^bold>| = 0"
f: fixes ij :: "'n \<times> 'n" shows "ij \<in> G ^^ f ij"
wt: shows "fv body \<subseteq> {this} \<union> set pns"
logic_actual_nec_4: fixes \<phi> :: "\<o>" shows "[[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<A>\<^bold>\<A>\<phi>]]"
plus_p_mod_def: fixes x :: "int"   and y :: "int" assumes "x \<in> {0..<p}"   and "y \<in> {0..<p}" shows "plus_p p x y = (x + y) mod p"
ring_hom_memI: fixes R :: "('e, 'f) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and S :: "('g, 'h) ring_scheme" assumes "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"   and "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>" shows "h \<in> ring_hom R S"
singleton: fixes y :: "'a"   and x :: "'a"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "y \<noteq> x" shows "ext gt [y] [x] = gt y x"
cases: fixes a1 :: "Arity list"   and a2 :: "stack_elem list" assumes "stack_consistent a1 a2" obtains "a1 = []" and "a2 = []"   | S :: "stack_elem list" and a :: "Arity" and as :: "Arity list" and e1 :: "exp" and e2 :: "exp" where "a1 = a # asa__" and "a2 = Alts e1 e2 # Sa__" and "Astack Sa__ \<sqsubseteq> a" and "stack_consistent asa__ Sa__"   | as :: "Arity list" and S :: "stack_elem list" and x :: "Vars.var" where "a1 = asb__" and "a2 = Upd x__ # Sb__" and "stack_consistent asb__ Sb__"   | as :: "Arity list" and S :: "stack_elem list" and x :: "Vars.var" where "a1 = as" and "a2 = Arg x # S" and "stack_consistent as S"
extended_Gromov_product_tendsto_PInf_a_b: fixes a :: "'c"   and u :: "'b \<Rightarrow> 'c Gromov_completion"   and v :: "'b \<Rightarrow> 'c Gromov_completion"   and F :: "'b filter"   and b :: "'c" assumes "((\<lambda>n. extended_Gromov_product_at a (u n) (v n)) \<longlongrightarrow> \<infinity>) F" shows "((\<lambda>n. extended_Gromov_product_at b (u n) (v n)) \<longlongrightarrow> \<infinity>) F"
gVarInj_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) Recursion.model_scheme" shows "gVarInj MOD \<equiv> \<forall>xs x y. gVar MOD xs x = gVar MOD xs y \<longrightarrow> x = y"
Crypt_synth: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> synth H" obtains "Crypt K X \<in> H"   | "X \<in> synth H" and "Key K \<in> H"
f: shows "f \<in> \<Omega> \<rightarrow> \<Omega>'"
get_root_node_si_is_l_get_root_node: shows "l_get_root_node get_root_node_si Shadow_DOM.get_parent"
contour_integral_part_circlepath_h: fixes r :: "real"   and s :: "complex" assumes "0 < r" shows "contour_integral (part_circlepath 0 r 0 pi) (f s) = integral {0..pi} (h r s)"
ANDConc: fixes a :: "('c set set, 'd) Combinators"   and p :: "('c set set, 'd) Combinators list" assumes "allNetsDistinct (a # p)" shows "allNetsDistinct p"
Limsup_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" shows "Limsup F f = (INF P\<in>{P. eventually P F}. Sup (f ` {x. P x}))"
hyps: assumes "set (map (\<lambda>a. case a of (a, b) \<Rightarrow> f a b) xys) \<subseteq> G" shows "T (\<Sum>a\<leftarrow>xys. case a of (a, b) \<Rightarrow> f a b) = (\<Sum>a\<leftarrow>xys. case a of (x, y) \<Rightarrow> T (f x y))"
nth_maps: fixes pns :: "'a list"   and Ts :: "'b list"   and i :: "nat"   and E :: "'a \<Rightarrow> 'b option" assumes "length pns = length Ts"   and "distinct pns"   and "i < length Ts" shows "(E(pns [\<mapsto>] Ts)) (pns ! i) = \<lfloor>Ts ! i\<rfloor>"
distinct_34: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "While x61 x62 \<noteq> x41 ;; x42"
norm_eq_sqrt_inner: fixes x :: "'c" shows "norm x = sqrt (x \<bullet> x)"
has_locks_iff: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and n :: "nat" shows "(has_locks l t = n) = (l = None \<and> n = 0 \<or> (\<exists>n'. l = \<lfloor>(t, n')\<rfloor> \<and> Suc n' = n) \<or> (\<exists>t' n'. l = \<lfloor>(t', n')\<rfloor> \<and> t' \<noteq> t \<and> n = 0))"
map_T_id: fixes x :: "('n, 'o, 'p, 'q, 'r, 's) T" shows "map_T id id id id id x = x"
less_element_ptr_def: fixes x :: "(_) element_ptr"   and y :: "(_) element_ptr" shows "x < y \<equiv> x \<le> y \<and> \<not> y \<le> x"
nn_integral_pmf_eq_1: fixes p :: "'h pmf" shows "(\<Sum>\<^sup>+ x. ennreal (pmf p x)) = 1"
abs_def: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and Abs :: "'k set \<Rightarrow> 'l"   and Rep :: "'l \<Rightarrow> 'k set"   and x :: "'k" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
diff_numeral_special_8: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus (numeral.numeral one plus m)) (uminus one) = neg_numeral.sub minus plus one num.One m"
transpose_transpose: fixes A :: "'a mat" shows "A\<^sup>t\<^sup>t = A"
left_inverse: fixes a :: "'d" shows "- a + a = (0::'d)"
monom_uminus: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b" shows "MPoly_Type.monom m (- a) = - MPoly_Type.monom m a"
mkPoint_img_2: fixes x :: "'s"   and a :: "'s" assumes "\<guillemotleft>x : S.unity \<rightarrow> a\<guillemotright>" shows "S.mkPoint a (S.img x) = x"
exhaust: fixes y :: "'a event" obtains x1 :: "'a" where "y = ev x1"   | "y = tick"
prod_list_dvd: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "x dvd prod_list xs"
rep_eq: fixes x :: "'e \<Rightarrow> ('g, 'f) generator" shows "generator (fix_gen x) = fix_gen_raw (generator \<circ> x \<circ> id)"
regular_sup: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "regular z" shows "(x \<le> z \<and> y \<le> z) = (- - (x \<squnion> y) \<le> z)"
cases: fixes x :: "instr list" obtains "x = []"   | r :: "instr" and rs :: "instr list" where "x = r # rs"
finsum_insert: fixes F :: "'e set"   and a :: "'e"   and f :: "'e \<Rightarrow> 'c" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier M"   and "f a \<in> carrier M" shows "finsum M f (insert a F) = f a \<oplus>\<^bsub>M\<^esub> finsum M f F"
card_of_empty3: fixes A :: "'c set" assumes "|A| \<le>o |{}|" shows "A = {}"
wt_u_ge_t: shows "wt u \<ge>\<^sub>p wt t"
causality: fixes xs :: "'a list" assumes "op_history xs" shows "hb_consistent xs"
fps_add_nth: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "(f + g) $ n = f $ n + g $ n"
s_max: fixes x :: "real" shows "s x \<le> s_stop"
setdist_sing_in_set: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "setdist {x} S = 0"
scale_sum_left: fixes f :: "'d \<Rightarrow> 'a"   and A :: "'d set"   and x :: "'c" shows "sum f A *b x = (\<Sum>a\<in>A. f a *b x)"
leftmost_prepend_word: fixes i :: "nat"   and v :: "symbol list"   and u :: "symbol list" assumes "leftmost i v"   and "is_word u" shows "leftmost (i + length u) (u @ v)"
assms_6: shows "m2 \<le>\<^sub>L B"
ran_restrictD: fixes y :: "'c"   and m :: "'d \<Rightarrow> 'c option"   and A :: "'d set" assumes "y \<in> ran (m |` A)" shows "\<exists>x\<in>A. m x = Some y"
that: fixes \<rho> :: "'a formula" assumes "\<Turnstile> \<Gamma> \<^bold>\<rightarrow> \<rho>"   and "\<Turnstile> \<rho> \<^bold>\<rightarrow> \<Delta>"   and "atoms \<rho> \<subseteq> atoms \<Gamma>"   and "atoms \<rho> \<subseteq> atoms \<Delta>" shows "thesis"
dghm_proj_snd_HomCod: fixes \<AA> :: "V"   and \<BB> :: "V" shows "\<pi>\<^sub>D\<^sub>G\<^sub>.\<^sub>2 \<AA> \<BB>\<lparr>HomCod\<rparr> = \<BB>"
wa_ref_trans: fixes WC :: "'a while_algo"   and WB :: "'b while_algo"   and \<alpha>1 :: "'a \<Rightarrow> 'b"   and WA :: "'c while_algo"   and \<alpha>2 :: "'b \<Rightarrow> 'c" assumes "wa_refine WC WB \<alpha>1"   and "wa_refine WB WA \<alpha>2" shows "wa_refine WC WA (\<alpha>2 \<circ> \<alpha>1)"
l5_12_a: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "A B Le A C \<and> B C Le A C"
quot: shows "quotient_of (fs.gs.\<mu> i j) = (num, denom)"
nat_to_pr_into_pr: fixes n :: "nat" shows "nat_to_pr n \<in> PrimRec1"
ring_tOp_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r y \<cdot>\<^sub>r z = x \<cdot>\<^sub>r (y \<cdot>\<^sub>r z)"
open_Int_closure_eq_empty: fixes S :: "'a set"   and T :: "'a set" assumes "open S" shows "(S \<inter> closure T = {}) = (S \<inter> T = {})"
leq_antisym: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> leq"   and "(y, x) \<in> leq" shows "x = y"
countable_UN: fixes A :: "'d \<Rightarrow> 'c set"   and X :: "'d set"   and M :: "'c measure" assumes "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
word_ao_absorbs_8: fixes x :: "'a word"   and y :: "'a word" shows "x AND y OR x = x"
sfirst_eSuc: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "\<not> P \<omega>" shows "sfirst P \<omega> = eSuc (sfirst P (stl \<omega>))"
fst_neq: fixes A :: "'a"   and B :: "'a"   and C :: "'b"   and D :: "'b" assumes "A \<noteq> B" shows "(A, C) \<noteq> (B, D)"
PiE_I: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x"   and "\<And>x. x \<notin> A \<Longrightarrow> f x = undefined" shows "f \<in> Pi\<^sub>E A B"
SET_CR6: fixes evs6 :: "event list"   and NonceCCA :: "nat"   and KC2 :: "nat"   and KC3 :: "nat"   and cardSK :: "nat"   and i :: "nat"   and C :: "agent"   and NC3 :: "nat"   and CardSecret :: "nat" assumes "evs6 \<in> set_cr"   and "Nonce NonceCCA \<notin> used evs6"   and "KC2 \<in> symKeys"   and "KC3 \<in> symKeys"   and "cardSK \<notin> symKeys"   and "Notes (CA i) (Key cardSK) \<notin> set evs6"   and "Gets (CA i) \<lbrace>Crypt KC3 \<lbrace>Agent C, Nonce NC3, Key KC2, Key cardSK, Crypt (invKey cardSK) (Hash \<lbrace>Agent C, Nonce NC3, Key KC2, Key cardSK, Pan (pan C), Nonce CardSecret\<rbrace>)\<rbrace>, Crypt (pubEK (CA i)) \<lbrace>Key KC3, Pan (pan C), Nonce CardSecret\<rbrace>\<rbrace> \<in> set evs6" shows "Says (CA i) C (Crypt KC2 \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce NC3, Agent (CA i), Nonce NonceCCA\<rbrace>, certC (pan C) cardSK (XOR (CardSecret, NonceCCA)) onlySig (priSK (CA i)), cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace>) # Notes (CA i) (Key cardSK) # evs6 \<in> set_cr"
simps_2: fixes n :: "nat"   and t :: "dB"   and k :: "nat" shows "strip_context n (Abs t) k = Abs (strip_context n t (Suc k))"
ras_empty_def: fixes uu :: "unit" shows "ras_empty uu \<equiv> (array_of_list [], 0::'d)"
abs_eq: fixes xa :: "'d"   and x :: "'b" shows "Pred\<^sub>\<alpha> xa x = abs_Tree\<^sub>\<alpha> (tPred xa x)"
rem_effectless_works_4': fixes as :: "('a \<times> ('b, 'c) fmap) list"   and A :: "('a \<times> ('b, 'c) fmap) set" assumes "as \<in> valid_plans A" shows "rem_effectless_act as \<in> valid_plans A"
l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs \<and> l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf"
minus_eq: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_group G" shows "x \<ominus>\<^bsub>G\<^esub> y = x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> y"
fromSingleton_simp: fixes xs :: "'a list"   and x :: "'a" shows "(fromSingleton xs = Inr x) = ([x] = xs)"
none: fixes \<F>' :: "'a set set" assumes "finite \<F>'"   and "\<F>' \<subseteq> \<F>" shows "S \<inter> \<Inter> \<F>' \<noteq> {}"
fin: fixes C :: "'a set set" assumes "C \<subseteq> \<B>"   and "\<Union> C = topspace X" shows "\<exists>C'. finite C' \<and> C' \<subseteq> C \<and> \<Union> C' = topspace X"
sprod_one: fixes m :: "'a" assumes "m \<in> carrier M" shows "1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s m = m"
strict_prefix_simps_3: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "strict_prefix (x # xs) (y # ys) = (x = y \<and> strict_prefix xs ys)"
runit_in_hom_2: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>\<r>\<^sub>C[f] : f \<star>\<^sub>C src\<^sub>C f \<Rightarrow>\<^sub>C f\<guillemotright>"
compress_normalize_primitive_nnf: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and C :: "'c \<Rightarrow> 'b"   and m :: "'b match_expr"   and f :: "'c negation_type list \<Rightarrow> ('c list \<times> 'c list) option"   and m' :: "'b match_expr" assumes "wf_disc_sel disc_sel C"   and "normalized_nnf_match m"   and "compress_normalize_primitive disc_sel C f m = Some m'" shows "normalized_nnf_match m'"
capacity_finite: fixes e :: "'a \<times> 'a" shows "capacity \<Delta> e \<noteq> \<top>"
induct: fixes P :: "nat list \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "nat"   and a2 :: "nat list" assumes "\<And>y i u. (\<lbrakk>i < length y; \<not> u ! i < y ! i\<rbrakk> \<Longrightarrow> P y (Suc i) u) \<Longrightarrow> P y i u" shows "P a0 a1 a2"
convex_translation: fixes S :: "'b set"   and a :: "'b" assumes "convex S" shows "convex ((+) a ` S)"
UN_Un: fixes M :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and B :: "'c set" shows "\<Union> (M ` (A \<union> B)) = \<Union> (M ` A) \<union> \<Union> (M ` B)"
that: fixes \<pi>\<^sub>\<A> :: "'a ltl \<Rightarrow> nat option" assumes "dom \<pi> = dom \<pi>\<^sub>\<A>"   and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<A> \<Longrightarrow> \<pi>\<^sub>\<A> \<chi> = mojmir_def.smallest_accepting_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<chi>)) w {q. dom \<pi>\<^sub>\<A> \<up>\<Turnstile>\<^sub>P q}"   and "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (M_fin \<pi>\<^sub>\<A>, UNIV) w"   and "\<And>\<chi>. \<chi> \<in> dom \<pi>\<^sub>\<A> \<Longrightarrow> accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi>\<^sub>\<A> \<chi>) w" shows "thesis"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom ((\<mu> \<star> \<nu>) \<star> \<pi>))"
left_unit_mult_rng_of_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> carrier rel" shows "\<one>\<^bsub>rec_monoid_rng_of_frac\<^esub> \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) = (r |\<^bsub>rel\<^esub> s)"
roots_mod_prime_bound: fixes p :: "nat"   and n :: "nat"   and c :: "nat" assumes "prime p"   and "0 < n" shows "card {x \<in> {..<p}. [x ^ n = c] (mod p)} \<le> n"