hcju/isabelleps huggingface dataset:
connected_component_of_def: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "connected_component_of X x y \<equiv> \<exists>T. connectedin X T \<and> x \<in> T \<and> y \<in> T"
class_of_in_stalk: fixes A :: "'a set set"   and z :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "A \<in> top.neighborhoods \<pp>"   and "z \<in> \<O> A" shows "st.class_of A z \<in> st.carrier_stalk"
n_st_def: fixes m :: "'b \<Rightarrow> 'a"   and X :: "char list set"   and st :: "'b Abs_State.st" shows "n_st m X st = (\<Sum>x\<in>X. m (lookup st x))"
converseD: fixes a :: "'g"   and b :: "'h"   and r :: "('h \<times> 'g) set" assumes "(a, b) \<in> r\<inverse>" shows "(b, a) \<in> r"
arsinh_real_at_bot: shows "filterlim arsinh at_bot at_bot"
distinct_19: fixes x5 :: "char list" shows "NT \<noteq> Class x5"
resolventIsEntailed: fixes clause1 :: "Literal list"   and clause2 :: "Literal list"   and literal :: "Literal" shows "formulaEntailsClause [clause1, clause2] (resolve clause1 clause2 literal)"
Ds_mono: fixes A :: "'a set option"   and A' :: "'a set option"   and es :: "'a exp list" assumes "A \<sqsubseteq> A'"   and "\<D>s es A" shows "\<D>s es A'"
span_pr: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "span (pr1 a b) (pr0 a b)"
other_stepsI: fixes U :: "('e \<Rightarrow> 'f) \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and J :: "'e set" assumes "\<And>\<sigma> \<sigma>' j. \<lbrakk>U \<sigma> \<sigma>'; j \<in> J\<rbrakk> \<Longrightarrow> \<sigma>' j = \<sigma> j" shows "other_steps U J"
dyad_rec_43_times2: fixes n :: "nat"   and b :: "nat \<Rightarrow> 'b"   and l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and m :: "nat" assumes "0 < n" shows "dyad_rec b l r (2 * ((4 * real m + 3) / 2 ^ Suc n)) = r (dyad_rec b l r (2 * (2 * real m + 1) / 2 ^ n))"
compute_shift_map_keys: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and f :: "'e \<Rightarrow> 'd"   and xs :: "('a \<times> 'e) list" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.shift_map_keys pair_of_term term_of_pair t f (fmap_of_list xs) = fmap_of_list (map (\<lambda>(k, v). (term_powerprod.splus pair_of_term term_of_pair t k, f v)) xs)"
abst_dbfm_graph_aux_def: shows "abst_dbfm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>abst_dbfm_aux name i t1 t2. x1 = (name, i, DBMem t1 t2) \<and> x2 = DBMem (abst_dbtm name i t1) (abst_dbtm name i t2)) \<or> (\<exists>abst_dbfm_aux name i t1 t2. x1 = (name, i, DBEq t1 t2) \<and> x2 = DBEq (abst_dbtm name i t1) (abst_dbtm name i t2)) \<or> (\<exists>abst_dbfm_aux name i A1 A2. x1 = (name, i, DBDisj A1 A2) \<and> x2 = DBDisj (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i, A1)) (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i, A2)) \<and> p (name, i, A1) (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i, A1)) \<and> p (name, i, A2) (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i, A2))) \<or> (\<exists>abst_dbfm_aux name i A. x1 = (name, i, DBNeg A) \<and> x2 = DBNeg (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i, A)) \<and> p (name, i, A) (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i, A))) \<or> (\<exists>abst_dbfm_aux name i A. x1 = (name, i, DBEx A) \<and> x2 = DBEx (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i + 1, A)) \<and> p (name, i + 1, A) (((\<lambda>uu_. uu_) \<circ> abst_dbfm_aux) (name, i + 1, A))))"
poly_eval_minus: fixes a :: "'d \<Rightarrow> 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a (p - q) = poly_eval a p - poly_eval a q"
unit_factor_normalize: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero" shows "unit_factor (normalize a) = one"
STAR_exp_Infinitesimal: fixes x :: "real star" assumes "x \<in> Infinitesimal" shows "(*f* exp) x \<approx> 1"
SBHWriteNonVolatile: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'c memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Write False a (D, f) A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, sb @ [Write\<^sub>s\<^sub>b False a (D, f) (f \<theta>) A L R W], m, \<D>, \<O>, \<R>, \<S>)"
proj_def: fixes v :: "'b"   and u :: "'b" shows "proj v u = (v \<bullet> u / (u \<bullet> u)) *\<^sub>R u"
acquire: fixes x :: "'x"   and ln :: "'l \<Rightarrow>f nat"   and n :: "'l" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "ta = (K$ [], [], [], [], [], convert_RA ln)"   and "\<not> waiting (wset s t)"   and "may_acquire_all (locks s) t ln"   and "0 < ln $ n"   and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)" shows "thesis"
cClose2: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "!P \<longmapsto> a<x> \<prec> P''"   and "x \<sharp> P"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a<x> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>x>(P' \<parallel> P''))"
cltn2_abs_rep_abs_mult: fixes M :: "((real, 3) vec, 3) vec"   and N :: "((real, 3) vec, 3) vec" assumes "invertible M"   and "invertible N" shows "cltn2_abs (cltn2_rep (cltn2_abs M) ** N) = cltn2_abs (M ** N)"
add_pos_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less_eq zero b" shows "less zero (plus a b)"
inv\<^sub>1E_4: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "B\<^sub>2 \<notin> P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2"
exhaust: fixes y :: "'a ref" obtains x :: "nat" where "y = Ref x"
tendsto_at_topI_sequentially: fixes f :: "real \<Rightarrow> 'c"   and y :: "'c" assumes "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) at_top"
no_grey_refs_locs_subseteq_hs_in_sync_locs: shows "no_grey_refs_locs \<subseteq> hs_in_sync_locs"
a_consistent_edom_subsetD: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, e, S)" shows "edom ae \<subseteq> domA \<Gamma> \<union> upds S"
simps_3: fixes x2 :: "'d" shows "\<lfloor>x2\<rfloor> \<noteq> None"
observable_io_target_is_singleton: fixes M :: "('d, 'e, 'f) FSM"   and p :: "'f"   and q :: "'f"   and io :: "('d \<times> 'e) list" assumes "observable M"   and "p \<in> io_targets M q io" shows "io_targets M q io = {p}"
degree_One_pdevs: shows "degree One_pdevs = length Basis_list"
e: fixes z :: "'a" assumes "dist z x < e" shows "z \<in> A"
ell2_norm_def: fixes x :: "'c \<Rightarrow> complex" shows "ell2_norm x = sqrt (\<Squnion> (sum (\<lambda>i. (cmod (x i))\<^sup>2) ` {F. finite F}))"
finite_image_set: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "finite {x. P x}" shows "finite {f x |x. P x}"
map_dual_anti: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<le> g" shows "\<partial>\<^sub>F g \<le> \<partial>\<^sub>F f"
zip_append_0_upt: fixes xs :: "'a list"   and ys :: "'a list" shows "zip (xs @ ys) [0..<length xs + length ys] = zip xs [0..<length xs] @ zip ys [length xs..<length xs + length ys]"
servK_not_AKcryptSK: fixes B :: "agent"   and A :: "agent"   and SK :: "nat"   and Ts :: "nat"   and evs :: "event list"   and K :: "nat" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key SK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key SK \<notin> analz (knows Spy evs)"   and "SK \<in> symKeys"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "\<not> AKcryptSK SK K evs"
maxsimpchain_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "folding_g.gallery (x # xs)" shows "folding_g.gallery xs"
front_tickFree_implies_tickFree: fixes t :: "'c event list"   and a :: "'c event" assumes "front_tickFree (t @ [a])" shows "tickFree t"
Max_eq_iff: fixes A :: "'a word set"   and m :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(signed.Max A = m) = (m \<in> A \<and> (\<forall>a\<in>A. a \<le>s m))"
sum_list_listset: fixes as :: "'b list"   and As :: "'b set list" assumes "as \<in> listset As" shows "sum_list as \<in> (\<Sum>A\<leftarrow>As. A)"
ts_bool_False: fixes x :: "nat \<Rightarrow> bool list"   and i :: "nat" assumes "ts x"   and "x i \<noteq> [True]" shows "x i = [False]"
realizable_ics_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "realizable n ns n'" obtains ns' :: "'node SDG_node list" where "n ics-ns'\<rightarrow>\<^sub>d* n'"
simple_imp_good_ruleset: fixes rs :: "'a rule list" assumes "simple_ruleset (rs::'a rule list)" shows "good_ruleset (rs::'a rule list)"
c_pair_strict_mono2: fixes y1 :: "nat"   and y2 :: "nat"   and x :: "nat" assumes "y1 < y2" shows "c_pair x y1 < c_pair x y2"
Rep_inverse: fixes x :: "'b" shows "Abs (Rep x) = x"
winding_number_rectpath_outside: fixes a1 :: "complex"   and a3 :: "complex"   and z :: "complex" assumes "Re a1 \<le> Re a3"   and "Im a1 \<le> Im a3"   and "z \<notin> cbox a1 a3" shows "winding_number (rectpath a1 a3) z = 0"
minus_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "x - y = (fst x - fst y, snd x - snd y)"
graph_of_mat_def: fixes A :: "'a mat" shows "graph_of_mat A = (let n = dim_row A in Let {..<n} (Restr {ij. A $$ ij \<noteq> (0::'a)}))"
not_iIN_iMODb_subset: fixes d' :: "nat"   and m :: "nat"   and n' :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < d'"   and "m \<noteq> Suc 0" shows "\<not> [n'\<dots>,d'] \<subseteq> [ r, mod m, c ]"
keys_def: shows "RBT_Set2.keys \<equiv> map_fun mapping_rbt.impl_of id RBT_Impl.keys"
a_comm_group: fixes G :: "('c, 'd) ring_scheme" assumes "abelian_group G" shows "comm_group (add_monoid G)"
HF_hfset: fixes a :: "hf" shows "HF (hfset a) = a"
semIntAll_termFSwSbImorph: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "termFSwSbImorph (semInt SEM) (semIntAbs SEM) (asIMOD SEM)"
loop_stepB: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"   and "B\<^sub>1 = {}"   and "u \<in> L" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 {u} B\<^sub>2 S (L - {u})"
trans_greater_equiv: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b = Greater"   and "cmp b c = Equiv" shows "cmp a c = Greater"
fbrelation_vunion: fixes r :: "V"   and s :: "V" shows "fbrelation (r \<union>\<^sub>\<circ> s) = (fbrelation r \<and> fbrelation s)"
map_vector_def: fixes f :: "'g \<Rightarrow> 'e"   and v :: "('g, 'f) Finite_Cartesian_Product.vec" shows "map_vector f v \<equiv> \<chi>i. f (v $h i)"
ds_ord_iso_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y" shows "x \<cdot> z \<sqsubseteq> y \<cdot> z"
insert_before_is_strongly_dom_component_safe: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node :: "(_) node_ptr"   and child :: "(_) node_ptr option"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> insert_before ptr node child \<rightarrow>\<^sub>h h'" shows "is_strongly_scdom_component_safe ({ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node} \<union> (case child of None \<Rightarrow> {} | Some ref \<Rightarrow> {cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 ref})) {} h h'"
gen_binomial_real'': fixes y :: "real"   and x :: "real"   and a :: "real" assumes "\<bar>y\<bar> < x" shows "(\<lambda>n. (a gchoose n) * x powr (a - real n) * y ^ n) sums (x + y) powr a"
hrcis_Ex: fixes z :: "complex star" shows "\<exists>r a. z = hrcis r a"
C_def: shows "C \<equiv> \<Union>s\<in>A\<^sup>\<star>. {infsuff A s}"
w2_def: shows "w2 = drop (n - length (u @ v)) w"
blinding_of_tree_hash: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "bo \<le> vimage2p h h (=)" shows "blinding_of_tree h bo \<le> vimage2p (hash_tree h) (hash_tree h) (=)"
dp_heap_axioms: shows "dp_heap P Q lookup lookup_st update update_st"
dvd_unit_factor_div: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times b a" shows "unit_factor (divide a b) = divide (unit_factor a) (unit_factor b)"
Proj_orthog_ccspan_union: fixes X :: "'b set"   and Y :: "'b set" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> Y\<rbrakk> \<Longrightarrow> is_orthogonal x y" shows "Proj (ccspan (X \<union> Y)) = Proj (ccspan X) + Proj (ccspan Y)"
fdia_def: fixes R :: "('e \<times> 'f) set"   and f :: "'f \<Rightarrow> 'd"   and x :: "'e" shows " |R\<rangle> f x = \<Squnion>{f y |y. (x, y) \<in> R}"
Taylor: shows "((\<lambda>xa. (1 - xa)\<^sup>2 *\<^sub>R (h ^ 3 / 2) *\<^sub>R (f'' (h * xa + t, x (h * xa + t)) $ (1, f (h * xa + t, x (h * xa + t))) $ (1, f (h * xa + t, x (h * xa + t))) + f' (h * xa + t, x (h * xa + t)) $ (0, f' (h * xa + t, x (h * xa + t)) $ (1, f (h * xa + t, x (h * xa + t)))))) has_integral x (t + h) - x t - h *\<^sub>R f (t, x t) - (h\<^sup>2 / 2) *\<^sub>R f' (t, x t) $ (1, f (t, x t))) {0..1}"
sel_19: fixes x71 :: "'d list"   and x72 :: "(('e, 'd) Term.term \<times> ('e, 'd) Term.term) list"   and x73 :: "(('e, 'd) Term.term \<times> ('e, 'd) Term.term) list" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle> = x71"
MClkReply_change: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "\<turnstile> MClkReply send rcv cst p \<longrightarrow> <MClkReply send rcv cst p>_(cst!p, rtrner send!p, caller rcv!p)"
assms_3: shows "primitive_extractor (disc, sel) m = (as, ms)"
elims: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and xa :: "'c"   and xb :: "nat \<Rightarrow> 'd"   and xc :: "nat"   and y :: "'c \<times> 'd \<times> 'c" assumes "run\<^sub>t x xa xb xc = y" obtains \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c" and q\<^sub>0 :: "'c" and w :: "nat \<Rightarrow> 'd" and i :: "nat" where "x = \<delta>" and "xa = q\<^sub>0" and "xb = w" and "xc = i" and "y = (run \<delta> q\<^sub>0 w i, w i, run \<delta> q\<^sub>0 w (Suc i))"
snth_sset: fixes s :: "'b stream"   and n :: "nat" shows "s !! n \<in> sset s"
Skolem_list_nth: fixes k :: "nat"   and P :: "nat \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<forall>i<k. \<exists>x. P i x) = (\<exists>xs. length xs = k \<and> (\<forall>i<k. P i (xs ! i)))"
K33_not_comb_planar: fixes G :: "('c, 'd) pre_digraph" assumes "K\<^bsub>3,3\<^esub> G" shows "\<not> comb_planar G"
inv_def: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one" shows "Dmonoid.inv prod one x = (THE y. prod x y = one \<and> prod y x = one)"
rel_FGl_def: fixes L1 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and L3 :: "'x \<Rightarrow> 'aj \<Rightarrow> bool"   and Co1 :: "'s \<Rightarrow> 'af \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'ag \<Rightarrow> bool"   and Co3 :: "'y \<Rightarrow> 'ak \<Rightarrow> bool"   and Co4 :: "'z \<Rightarrow> 'al \<Rightarrow> bool"   and Contra1 :: "'u \<Rightarrow> 'ah \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra3 :: "'aa \<Rightarrow> 'am \<Rightarrow> bool"   and Contra4 :: "'ab \<Rightarrow> 'an \<Rightarrow> bool" shows "rel_FGl L1 L2 L3 Co1 Co2 Co3 Co4 Contra1 Contra2 Contra3 Contra4 = rel_F (rel_G L1 L2 Co1 Co2 Contra1 Contra2) L1 L3 Co1 Co3 Co4 Contra1 Contra3 Contra4"
commute: fixes a :: "'a::type"   and b :: "'a::type" shows "(a::'a::type) \<sqinter> (b::'a::type) = b \<sqinter> a"
acyclic_6_implies_4a: fixes x :: "'a" assumes "acyclic_6 x" shows "acyclic_4a x"
disjoint_iff: fixes A :: "'e set"   and B :: "'e set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
simps_5: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "formula_of_constraint (t1, False, MLess, t2) = formula.Neg (formula.Less t1 t2)"
r'_def: shows "r' \<equiv> {(x, y). (x, y) \<in> r \<and> x \<notin> M \<and> y \<notin> M}"
tiebreaker_lifting_axioms_def: fixes Prec_F_g :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "tiebreaker_lifting_axioms Prec_F_g \<equiv> \<forall>g. minimal_element (Prec_F_g g) UNIV"
scale_sum_right: fixes a :: "'a"   and f :: "'c \<Rightarrow> ('a, 'b) vec"   and A :: "'c set" shows "a *s sum f A = (\<Sum>x\<in>A. a *s f x)"
OUTfromVCorrect1_data9: shows "OUTfromVCorrect1 data9"
distinct_concat_iff: fixes xs :: "'b list list" shows "distinct (concat xs) = (distinct (removeAll [] xs) \<and> (\<forall>ys. ys \<in> set xs \<longrightarrow> distinct ys) \<and> (\<forall>ys zs. ys \<in> set xs \<and> zs \<in> set xs \<and> ys \<noteq> zs \<longrightarrow> set ys \<inter> set zs = {}))"
below_def: fixes n :: "nat" shows "below n = {i. i < n}"
not_Cons_self2: fixes x :: "'e"   and xs :: "'e list" shows "x # xs \<noteq> xs"
path_conforms_VVp: fixes v :: "'a"   and p :: "Player"   and w :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a"   and Ps :: "'a llist" assumes "v \<in> VV p"   and "w = \<sigma> v"   and "path_conforms_with_strategy p (LCons w Ps) \<sigma>" shows "path_conforms_with_strategy p (LCons v (LCons w Ps)) \<sigma>"
emeasure_mono: fixes a :: "'c set"   and b :: "'c set"   and M :: "'c measure" assumes "a \<subseteq> b"   and "b \<in> sets M" shows "emeasure M a \<le> emeasure M b"
functional_converse_relation: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "left_unique R"   and "right_total R" shows "\<forall>y. \<exists>!x. R x y"
\<R>\<^sub>1_closed_glob_consistent: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "closed_glob_consistent (\<R>\<^sub>1 \<Gamma>' \<S>' P')"
simps_1: fixes x :: "brick list" shows "is_tangle_diagram (basic x) = True"
map_poly_simps: fixes f :: "'d \<Rightarrow> 'c"   and c :: "'d"   and p :: "'d poly" shows "map_poly f (pCons c p) = (if c = (0::'d) \<and> p = 0 then 0 else pCons (f c) (map_poly f p))"
gf: fixes x :: "nat \<Rightarrow> real"   and i :: "nat" assumes "x \<in> S" shows "g i x = f x i"
min_gallery_rev: fixes xs :: "'a set list" assumes "min_gallery xs" shows "min_gallery (rev xs)"
ads_d_def: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.antidomain_semiringl antidomain_op plus times one zero less_eq less" shows "antidomain_semiringl.ads_d antidomain_op x = antidomain_op (antidomain_op x)"
IH: assumes "ys \<in> valid_lists S G"   and "zs \<in> valid_lists S' G'" shows "xs \<in> valid_lists (S \<union> S') (G \<squnion> (G' \<squnion> S G\<times> S'))"
case_one_wrong_hyp: fixes j :: "nat"   and k :: "nat" assumes "\<nexists>t. local.prefixes t j \<up>" shows "\<exists>n>k. \<phi> (the (s (\<psi> j \<triangleright> n))) \<noteq> \<psi> j"
rev_I'_def: fixes x :: "'b" shows "rev_I' x = (if x = uv then rev_I uv else if x = rev_I uv then uv else if x \<in> arcs I then rev_H' x else x)"
vD_invalidate_vD_not_dests: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> vD (invalidate rt dests)" shows "dip \<in> vD rt \<and> dests dip = None"
ex_power_ivl2: fixes b :: "nat"   and k :: "nat" assumes "2 \<le> b"   and "2 \<le> k" shows "\<exists>n. b ^ n < k \<and> k \<le> b ^ (n + 1)"
matrix_mul_diag_matl: fixes f :: "'f \<Rightarrow> 'd"   and A :: "(('d, 'e) vec, 'f) vec" shows "diag_mat f ** A = (\<chi>i j. f i * A $ i $ j)"
hyps_3: shows "q \<in> dgrad_p_set d m"
drop_eq_nths: fixes n :: "nat"   and xs :: "'c list" shows "drop n xs = nths xs {i. n \<le> i}"
qfree_asubst\<^sub>2: fixes l :: "nat"   and u :: "nat"   and a :: "atom" shows "qfree (asubst\<^sub>2 l u a)"
simps_40: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "('c, 'd) fm \<Rightarrow> 'b"   and f4 :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and f5 :: "('c, 'd) fm \<Rightarrow> 'b"   and f6 :: "'d \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and x41 :: "('c, 'd) fm"   and x42 :: "('c, 'd) fm" shows "(case x41 \<^bold>\<or> x42 of Pro x \<Rightarrow> f1 x | Nom x \<Rightarrow> f2 x | \<^bold>\<not> x \<Rightarrow> f3 x | x \<^bold>\<or> xa \<Rightarrow> f4 x xa | \<^bold>\<diamond> x \<Rightarrow> f5 x | \<^bold>@ x xa \<Rightarrow> f6 x xa) = f4 x41 x42"
simps_2: fixes x :: "'b"   and l :: "nat"   and k :: "nat"   and i_begin :: "nat"   and i_end :: "nat"   and blocks :: "(nat \<times> nat) list"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_3_c x l k ((i_begin, i_end) # blocks) A = Let (if i_end = l then A else ??.Jordan_Normal_Form_Existence.step_3_c_inner_loop (A $$ (i_end, k) / x) l i_end (Suc i_end - i_begin) A) (??.Jordan_Normal_Form_Existence.step_3_c x l k blocks)"
folds_plus_simp: fixes rs :: "'a rexp set" assumes "finite rs" shows "lang (\<Uplus>rs) = \<Union> (lang ` rs)"
c: shows "inconsist (amalg01 f) (f \<triangleright> n) \<down>"
unique_authKeys: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and Ka' :: "nat"   and Ta' :: "msg"   and X' :: "msg" assumes "Says Kas A (Crypt Ka \<lbrace>Key authK, Agent Tgs, Ta, X\<rbrace>) \<in> set evs"   and "Says Kas A' (Crypt Ka' \<lbrace>Key authK, Agent Tgs, Ta', X'\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "A = A' \<and> Ka = Ka' \<and> Ta = Ta' \<and> X = X'"
filter_oseq_def: fixes p :: "'e \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'e option" shows "filter_oseq p s \<equiv> \<lambda>i. case s i of None \<Rightarrow> None | Some x \<Rightarrow> if p x then Some x else None"