hcju/isabelleps huggingface dataset:
last_messageI2: fixes i :: "nat"   and xs :: "'b message_af list" assumes "i < length xs"   and "xs ! i \<noteq> NoMsg"   and "\<And>j. \<lbrakk>i < j; j < length xs\<rbrakk> \<Longrightarrow> xs ! j = NoMsg" shows "last_message xs = xs ! i"
ideal_ideal_generated: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "class.ring plus zero minus uminus times" shows "ring.ideal plus zero uminus times (ring.ideal_generated plus zero uminus times S)"
ComplD: fixes c :: "'g"   and A :: "'g set" assumes "c \<in> - A" shows "c \<notin> A"
assms_2: shows "s' \<in> carrier R\<^bsup>\<omega>\<^esup>"
sat_precond_as_pfx: fixes s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and as' :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "sat_precond_as s (as @ as')" shows "sat_precond_as s as"
r: shows "\<exists>m. m \<in> divisor_set n \<and> m \<noteq> 1 \<and> m \<noteq> n"
replace_basis_completely: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and vs :: "'b list"   and us :: "'b list" assumes "VectorSpace smult V"   and "set vs \<subseteq> V"   and "R_scalar_mult.R_lin_independent UNIV smult vs"   and "length us = length vs"   and "set us \<subseteq> R_scalar_mult.RSpan UNIV smult vs"   and "R_scalar_mult.R_lin_independent UNIV smult us" shows "set us \<subseteq> R_scalar_mult.RSpan UNIV smult vs \<and> R_scalar_mult.RSpan UNIV smult vs = R_scalar_mult.RSpan UNIV smult us \<and> R_scalar_mult.R_lin_independent UNIV smult us"
rep_uminus: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "'d \<Rightarrow>\<^sub>0 'c" assumes "module scale" shows "module.rep scale (- r) = - module.rep scale r"
simps_1: fixes \<phi> :: "Formula.formula" shows "remove_neg (formula.Neg \<phi>) = \<phi>"
subst_Lbd: fixes T :: "Syntax.type"   and t :: "trm"   and s :: "trm"   and k :: "nat" shows "(\<lambda>T:t)[s/k]\<^sup>T = \<lambda>T:t[liftL_trm s 0/k + 1]\<^sup>T"
degree_Poly': fixes xs :: "'b list" assumes "xs \<noteq> []" shows "degree (Poly xs) < length xs"
simps_11: fixes ve :: "H1" shows "CX4h ve {X4.Xd1, X4.Xd1', X4.Xd2} = {}"
1: shows "(T, S) \<in> \<langle>A\<rangle>Rs"
contD: fixes f :: "'g \<Rightarrow> 'h"   and C :: "'g set" assumes "cont f"   and "C \<noteq> {}" shows "f (Sup C) = Sup (f ` C)"
step_2_def: fixes A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_2 A = ??.Jordan_Normal_Form_Existence.step_2_main (dim_row A) 0 A"
mult_1_left: fixes a :: "'a" shows "(1::'a) \<cdot> a = a"
openin_open: fixes U :: "'a set"   and S :: "'a set" shows "openin (top_of_set U) S = (\<exists>T. open T \<and> S = U \<inter> T)"
rel_FGl_neg_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co3 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co3' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co4 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co4' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra3 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra3' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra4 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Contra4' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and tytok_F :: "(('ah, 'ai, 'p, 'aj, 'y, 'ak, 'al) G \<times> ('am, 'an, 'q, 'ao, 'z, 'ap, 'al) G \<times> ('aq, 'ar, 'r, 'as, 'aa, 'at, 'al) G \<times> 'ah \<times> 'am \<times> 'aq \<times> 'au \<times> 'av \<times> 'aw \<times> 'ax) itself"   and Co2 :: "'aj \<Rightarrow> 'ao \<Rightarrow> bool"   and Co2' :: "'ao \<Rightarrow> 'as \<Rightarrow> bool"   and Contra2 :: "'ak \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra2' :: "'ap \<Rightarrow> 'at \<Rightarrow> bool"   and tytok_G :: "('ah \<times> 'am \<times> 'aq \<times> 'ai \<times> 'an \<times> 'ar \<times> 'al) itself"   and tytok_FGl :: "('ah \<times> 'am \<times> 'aq \<times> 'ai \<times> 'an \<times> 'ar \<times> 'au \<times> 'av \<times> 'aw \<times> 'al \<times> 'ax) itself" assumes "rel_F_neg_distr_cond Co1 Co1' Co3 Co3' Co4 Co4' Contra1 Contra1' Contra3 Contra3' Contra4 Contra4' tytok_F"   and "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_FGl_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' tytok_FGl"
eq_zero: fixes a :: "nat" assumes "\<not> coprime a d" shows "\<Phi> a = 0"
Disjoint_Eats_E: fixes x :: "tm"   and z :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (Disjoint x z) (insert (Neg (y IN z)) H) \<turnstile> A" shows "insert (Disjoint (Eats x y) z) H \<turnstile> A"
points_index_0_iff: fixes B :: "'b set multiset"   and ps :: "'b set" shows "(B index ps = 0) = (\<forall>b. b \<in># B \<longrightarrow> \<not> ps \<subseteq> b)"
rbtreeify_def: fixes kvs :: "('a \<times> 'b) list" shows "rbtreeify kvs = fst (rbtreeify_g (Suc (length kvs)) kvs)"
phiUses_finite: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "finite (phiUses g n)"
integrable_on_refl: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" shows "f integrable_on cbox a a"
sgnx_poly_nz_1: fixes p :: "real poly"   and x :: "real" assumes "poly p x \<noteq> 0" shows "sgnx (poly p) (at_left x) = sgn (poly p x)"
pcr_cr_eq: shows "pcr_stream (=) = cr_stream"
fun_ordI: fixes ord :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>x. ord (f x) (g x)" shows "fun_ord ord f g"
med0sk1iai_def: fixes t :: "skl1_state" shows "med0sk1iai t \<equiv> \<lparr>signals = signalsInit t\<rparr>"
distinct_read_tmps_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "distinct_read_tmps (xs @ ys) = (distinct_read_tmps xs \<and> distinct_read_tmps ys \<and> read_tmps xs \<inter> read_tmps ys = {})"
ide_compE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.ide (B.VV.comp g f)" obtains "B.VV.seq g f" and "B.VV.seq f g" and "B.VV.comp g f = B.VV.dom f" and "B.VV.comp g f = B.VV.cod g"
widen_Class_Class: fixes G :: "prog"   and C :: "qtname"   and cm :: "qtname" assumes "G\<turnstile>Class C\<preceq>Class cm" shows "G\<turnstile>C\<preceq>\<^sub>C cm"
resolvingBidAlg_def: fixes N :: "'d set"   and \<Omega> :: "'e list"   and bids :: "'d \<times> 'e set \<Rightarrow> 'f"   and random :: "integer" shows "resolvingBidAlg N \<Omega> bids random \<equiv> tiebidsAlg (chosenAllocationAlg N \<Omega> bids random) N (set \<Omega>)"
get_shadow_root_ptr_simp2: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root_ptr' :: "(_) shadow_root_ptr"   and shadow_root :: "('n, 'o, 'p, 'q option) RShadowRoot_scheme"   and h :: "(_) heap" assumes "shadow_root_ptr \<noteq> shadow_root_ptr'" shows "get shadow_root_ptr (put shadow_root_ptr' shadow_root h) = get shadow_root_ptr h"
subgraph_def: fixes H :: "('a, 'b) pre_digraph"   and G :: "('a, 'b) pre_digraph" shows "subgraph H G \<equiv> verts H \<subseteq> verts G \<and> arcs H \<subseteq> arcs G \<and> wf_digraph G \<and> wf_digraph H \<and> compatible G H"
sorted_runs: fixes xs :: "'b list" shows "\<forall>x\<in>set (runs xs). sorted x"
select_convs_1: fixes verts :: "'c set"   and arcs :: "'d set"   and tail :: "'d \<Rightarrow> 'c"   and head :: "'d \<Rightarrow> 'c"   and more :: "'e" shows "verts \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = verts"
simps_2: fixes div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and i :: "nat"   and xs :: "nat list"   and H :: "'a mat"   and P :: "'a mat"   and K :: "'a mat" shows "local.reduce_column_aux div_op (i # xs) H (P, K) = (let k = div_op (H $$ (i, 0)) (H $$ (0, 0)); P' = addrow_mat (dim_row H) (- k) i 0; K' = addrow (- k) i 0 K in local.reduce_column_aux div_op xs H (P' * P, K'))"
permutes_iff: fixes p :: "'a \<Rightarrow> 'a"   and n :: "'a"   and i :: "'a"   and j :: "'a" assumes "p permutes {..<n}"   and "i < n"   and "j < n" shows "(p i = p j) = (i = j)"
l_def: shows "l \<equiv> limsup (\<lambda>n. ereal (root n (norm (f n))))"
fv_list\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) stateful_strand_step list" shows "fv_list\<^sub>s\<^sub>s\<^sub>t S \<equiv> remdups (concat (map fv_list\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
lActive_equality: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<And>x. \<parallel>c\<parallel>\<^bsub>t x\<^esub> \<Longrightarrow> x \<le> i" shows "\<langle>c \<and> t\<rangle> = i"
assms: shows "y \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}"
divergent_p: shows "\<not> summable (\<lambda>n. inverse (p n))"
ModPon_def: fixes x :: "hf"   and y :: "hf"   and z :: "hf" shows "ModPon x y z \<equiv> y = q_Imp x z"
addfunsetD_supp: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and M :: "'b set" assumes "f \<in> addfunset A M" shows "supp f \<subseteq> A"
addr_loc_type_def: shows "h.addr_loc_type \<equiv> \<lambda>P h a. lfp (\<lambda>p x1 x2. (\<exists>U F T fm D. x1 = CField D F \<and> x2 = T \<and> typeof_addr a = \<lfloor>U\<rfloor> \<and> P \<turnstile> class_type_of U has F:T (fm) in D) \<or> (\<exists>T n' n. x1 = ACell n \<and> x2 = T \<and> typeof_addr a = \<lfloor>Array_type T n'\<rfloor> \<and> n < n'))"
assms_1: shows "P \<longmapsto> \<alpha> \<prec> P'"
no_write_to_read_only_memory_append: fixes \<S> :: "nat \<Rightarrow> bool option"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "no_write_to_read_only_memory \<S> (xs @ ys) = (no_write_to_read_only_memory \<S> xs \<and> no_write_to_read_only_memory (share xs \<S>) ys)"
in_alist_imp_in_map_of: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and name :: "String.literal"   and ars :: "(String.literal \<times> String.literal set list) list" assumes "alist_conds arss"   and "(name, ars) \<in> set arss" shows "translate_ars arss name = Some (map_of ars)"
mult_cancel_right2: fixes a :: "'b"   and c :: "'b" shows "(a * c = c) = (c = (0::'b) \<or> a = (1::'b))"
diamond_edges_are_valid_4: shows "valid_path (snd diamond_bot_right_edge)"
normalize_poly_old_def: fixes f :: "'b poly" shows "normalize f = Polynomial.smult (inverse (unit_factor (lead_coeff f))) f"
concurrentD1: fixes s1 :: "'a"   and s2 :: "'a" assumes "s1 \<parallel> s2" shows "\<not> s1 \<prec> s2"
inject_3: fixes x4 :: "'a OCL_Types.type"   and y4 :: "'a OCL_Types.type" shows "(Collection x4 = Collection y4) = (x4 = y4)"
assms_5: shows "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>h h'"
f5: shows "t \<otimes> (snd x \<otimes> r \<ominus> s \<otimes> fst x) = \<zero>"
path_entry_is_trancl: fixes G :: "nat graph_rec"   and E :: "(nat \<times> nat) set"   and l :: "nat list"   and v :: "nat" assumes "cfg_doms G"   and "path_entry E l v"   and "l \<noteq> []" shows "(0, v) \<in> E\<^sup>+"
sterms_maximal: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and q :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "q \<in> sterms \<Gamma> p" shows "sterms \<Gamma> q = {q}"
assm: shows "v=v1\<parallel>v2 \<and> v2=v3\<parallel>v4"
swaprows_mat_carrier: fixes n :: "nat"   and k :: "nat"   and l :: "nat" shows "swaprows_mat n k l \<in> carrier_mat n n"
hpar: shows "src\<^sub>D \<mu> = src\<^sub>D \<mu>' \<and> trg\<^sub>D \<mu> = trg\<^sub>D \<mu>'"
major: shows "\<langle>while (b) cx,s\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>Skip,s'\<rangle>"
ih: shows "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> pderivs_lang (PSuf s) r2"
find_index_eqI: fixes i :: "nat"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "i \<le> length xs"   and "\<forall>j<i. \<not> P (xs ! j)"   and "i < length xs \<Longrightarrow> P (xs ! i)" shows "find_index P xs = i"
widen_null_array: fixes P :: "'b prog"   and A :: "ty" shows "P \<turnstile> NT \<le> A\<lfloor>\<rceil>"
paths: shows "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
case_eq_if: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and sum :: "'d + 'e" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (if isl sum then f1 (projl sum) else f2 (projr sum))"
simps_1: fixes f :: "'i \<Rightarrow> 'g +\<^sub>\<bottom> 'h" shows "Bottom \<bind> f = Bottom"
inverse_unique: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "category C"   and "category.inverse_arrows C f g" shows "category.inv C f = g"
fv_subterms: fixes t :: "('e, 'd) Term.term" shows "fv\<^sub>s\<^sub>e\<^sub>t (subterms t) = fv t"
exhaust: fixes y :: "'e\<cdot>'f writer" obtains "y = \<bottom>"   | w :: "'f" and a :: "'e" where "y = Writer\<cdot>w\<cdot>a"
empty_open: shows "{} open\<^bsub>S\<^esub>"
i_expand_i_take_mod: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" assumes "n mod k = 0" shows "f \<odot> k \<Down> n = f \<Down> (n div k) \<odot> k"
Nil_eq_unfoldr_iff: fixes g :: "('b, 'c) generator"   and s :: "'c" shows "([] = list.unfoldr g s) = (\<not> list.has_next g s)"
fmlookup_dom_iff: fixes x :: "'c"   and m :: "('c, 'd) fmap" shows "(x |\<in>| fmdom m) = (\<exists>a. fmlookup m x = Some a)"
stoch_proc_filt_gen: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc F X" shows "stoch_proc_filt M X borel n = fct_gen_subalgebra M (stream_space borel) (proj_stoch_proc X n)"
ApproximationTr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> carrier K"   and "0 \<le> v x" shows "v x \<le> v (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> x)^\<^bsup>K Suc n\<^esup>)"
trace_callee_eqD: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf"   and xs :: "('e \<times> 'f) list"   and x :: "'e" assumes "trace_callee_eq callee1 callee2 A p q"   and "set xs \<subseteq> A \<times> UNIV"   and "x \<in> A" shows "trace_callee callee1 p xs x = trace_callee callee2 q xs x"
less_cSUP_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (f ` A)" shows "less a (Sup (f ` A)) = (\<exists>x\<in>A. less a (f x))"
upt_Cons_conv: fixes j :: "nat"   and n :: "nat" assumes "j < n" shows "[j..<n] = j # [j + 1..<n]"
distinct_2: fixes x2 :: "nat"   and x1 :: "nat" shows "Neg x2 \<noteq> Pos x1"
image_subsetI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
resolution_class_size1: fixes S :: "'a set multiset" assumes "S \<in># \<P>" shows "\<v> = \<k> * int (size S)"
right_idem: fixes a :: "'c"   and b :: "'c" shows "min (min a b) b = min a b"
T'_def: shows "T' = (\<lambda>n x. - real (j + n) * c n * inverse x ^ Suc (j + n))"
ideal_cont_Rxa: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> I" shows "R \<diamondsuit>\<^sub>p a \<subseteq> I"
strongly_preferred_def: fixes a :: "'c"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c" shows "a \<prec>[R] b \<equiv> R a b \<and> \<not> R b a"
dist_les_12: fixes nat :: "nat" shows "ValN\<cdot>nat \<notsqsubseteq> ValFF"
elims_3: fixes x :: "'a Line"   and xa :: "'a Line" assumes "\<not> x \<parallel>  xa" obtains lineA :: "'a Line" and lineB :: "'a Line" where "x = lineA" and "xa = lineB" and "\<not> (direction lineA = 0 \<or> direction lineB = 0 \<or> (\<exists>k. k \<noteq> (0::'a) \<and> direction lineB = k ** direction lineA))"
le_supI2: fixes x :: "'g multiset"   and b :: "'g multiset"   and a :: "'g multiset" assumes "x \<subseteq># b" shows "x \<subseteq># a \<union># b"
eq_fin_execute: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and p :: "'b"   and w\<^sub>2 :: "'a list" assumes "transition_system_traces ex en ind"   and "transition_system.path ex en w\<^sub>1 p"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2" shows "fold ex w\<^sub>1 p = fold ex w\<^sub>2 p"
dyn_def: shows "\<forall>c x. dyn c x = 0"
that: fixes x :: "'b" assumes "x \<in> A" shows "[f x = g x] (mod a)"
inj_compose: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "inj f"   and "inj g" shows "inj (f \<circ> g)"
assms: shows "\<Psi> \<turnstile> M \<leftrightarrow> N"
select_convs_2: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b" shows "flows_fix \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_fix"
\<Phi>_pc: shows "\<Phi> C M ! pc = \<lfloor>(ST, LT)\<rfloor>"
prolongation_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.prolongation \<P> betw a b \<equiv> {x. \<exists>ab. betw a b x \<and> x \<in> ab \<and> ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b}"
sin_periodic_pi: fixes x :: "real" shows "sin (x + pi) = - sin x"
cong_mk_mat: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and f' :: "nat \<times> nat \<Rightarrow> 'a" assumes "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> f (i, j) = f' (i, j)" shows "mk_mat nr nc f = mk_mat nr nc f'"
incrIndexList_help9: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and x :: "nat" assumes "incrIndexList ls m nmax"   and "x \<in> set (butlast ls)" shows "x \<le> nmax - 2"
edge_pred_def: fixes M :: "'c pre_map" shows "edge_pred M = inv (edge_succ M)"
tl_take: fixes n :: "nat"   and xs :: "'d list" shows "tl (take n xs) = take (n - 1) (tl xs)"
deg_fun_eq_0_iff: fixes s :: "'a \<Rightarrow> 'b" assumes "finite (supp_fun s)" shows "(deg_fun s = (0::'b)) = (s = 0)"
row_space_mult_unit: fixes P :: "'c mat"   and A :: "'c mat"   and b :: "'d" assumes "P \<in> Units (ring_mat TYPE('c) (dim_row A) b)" shows "row_space (P * A) = row_space A"
closureE: fixes x :: "'a llist"   and a :: "'a llist set"   and c :: "'a llist set" assumes "x \<in> closure a" obtains "\<not> c closed"   | "\<not> a \<subseteq> c"   | "x \<in> c"
tmsub_def: fixes s :: "tm"   and t :: "tm" shows "tmsub s t \<equiv> if s = t then CP 0\<^sub>p else tmadd s (tmneg t)"
zorder_eqI: fixes s :: "complex set"   and z :: "complex"   and g :: "complex \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex"   and n :: "int" assumes "open s"   and "z \<in> s"   and "g holomorphic_on s"   and "g z \<noteq> 0"   and "\<And>w. \<lbrakk>w \<in> s; w \<noteq> z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powr of_int n" shows "zorder f z = n"
res_c3: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "residuated f" shows "residual f (f (residual f (f x))) = residual f (f x)"
lookup_proj_poly: fixes k :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" shows "lookup (proj_poly k p) t = lookup p (term_of_pair (t, k))"
freshInp_def: fixes xs :: "'c"   and x :: "'d"   and inp :: "'e \<Rightarrow> ('f, 'g, 'c, 'd, 'h) qTerm set option" shows "freshInp xs x inp \<equiv> liftAll (fresh xs x) inp"
bird_simps_3: shows "right bird = pure succ \<diamondop> (pure recip \<diamondop> bird)"
well_formed_def: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list" shows "well_formed procs \<equiv> distinct_fst procs \<and> (\<forall>(p, ins, outs, c)\<in>set procs. wf_proc (p, ins, outs, c))"
Image_Domain: fixes R :: "('e \<times> 'f) set"   and A :: "'e set" assumes "Domain R = A" shows "R `` A = Range R"
assms: shows "\<CC>\<lparr>Obj\<rparr> = []\<^sub>\<circ>"
distinct_258: fixes x15 :: "Syntax.term"   and x11 :: "Syntax.term" shows "Auth x15 \<noteq> Prj1 x11"
split_asm: fixes P :: "'g \<Rightarrow> bool"   and f1 :: "'g"   and f2 :: "nat \<Rightarrow> 'g"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (\<not> (nat = 0 \<and> \<not> P f1 \<or> (\<exists>x2. nat = Suc x2 \<and> \<not> P (f2 x2))))"
stable_on_CD_on: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "CD_on ds X = X"
simps_10: fixes f :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "map_decision f (deny x2) = deny (f x2)"
carrier_vs_is_self: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and N :: "'j set" assumes "module R M" shows "carrier (M\<lparr>carrier := N\<rparr>) = N"
hnorm_scaleHR: fixes a :: "real star"   and x :: "'a star" shows "hnorm (scaleHR a x) = \<bar>a\<bar> * hnorm x"
sdescending_stuck: fixes w :: "'a stream" assumes "sdescending w" obtains u :: "'a list" and a :: "'a" where "w = u @- sconst a"
col_zero: fixes j :: "nat"   and nc :: "nat"   and nr :: "nat" assumes "j < nc" shows "col (0\<^sub>m nr nc) j = 0\<^sub>v nr"
prepend_at_ge: fixes vs :: "nat list"   and n :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "length vs \<le> n" shows "(vs \<odot> f) n = f (n - length vs)"
res: shows "reconstruction sl_impl m2 state u (Polynomial.smult lu u) lu d r vs res cands = fs"
tts_AB_C_transfer: fixes \<UU>A :: "'d set"   and \<UU>B :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and \<UU>C :: "'f set"   and A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'i \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> \<UU>A; b \<in> \<UU>B\<rbrakk> \<Longrightarrow> f a b \<in> \<UU>C"   and "Domainp A = (\<lambda>x. x \<in> \<UU>A)"   and "bi_unique A"   and "right_total A"   and "Domainp B = (\<lambda>x. x \<in> \<UU>B)"   and "bi_unique B"   and "right_total B"   and "Domainp C = (\<lambda>x. x \<in> \<UU>C)"   and "bi_unique C"   and "right_total C" shows "\<exists>rcdt. rel_fun A (rel_fun B C) f rcdt"
bij_betw_partition_of: fixes A :: "'a set" shows "bij_betw ((//) A) {R. equiv A R} {P. partition_on A P}"
single_one: shows "monomial (1::'d) (0::'c) = 1"
advantage_nonneg: fixes \<eta> :: "nat"   and \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp set spmf" shows "0 \<le> advantage \<eta> \<A>"
last_t_fusion: fixes x :: "'a \<times> ('b \<times> 'a) list"   and y :: "'a \<times> ('b \<times> 'a) list" assumes "Dioid_Models_New.last x = first y" shows "Dioid_Models_New.last (t_fusion x y) = Dioid_Models_New.last y"
dioph_dfs_bij: fixes n :: "nat"   and ks :: "int list"   and l :: "int"   and i :: "int"   and k :: "nat" shows "(fst (dioph_dfs n ks l) ! int_encode i = Some k \<and> dioph_is_node ks l i) = (k < length (snd (dioph_dfs n ks l)) \<and> snd (dioph_dfs n ks l) ! k = i)"
set: fixes x1 :: "'k"   and x2 :: "'k stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
c1: fixes nhip :: "nat"   and pre :: "nat set" shows "P (0, unk, val, Suc 0, nhip, pre)"
simps_12: fixes v :: "'a" shows "local.le_rexp (Atom v) One = False"
effective_matrix_kauff_mat_3: fixes ws :: "wall" assumes "is_tangle_diagram ws" shows "mat (rat_poly.row_length (kauff_mat ws)) (length (kauff_mat ws)) (kauff_mat ws)"