hcju/isabelleps huggingface dataset:
abel_semigroup_axioms: shows "abel_semigroup ord_term_lin.max"
keyfree_MPairE: fixes X :: "msg"   and Y :: "msg" assumes "\<lbrace>X, Y\<rbrace> \<in> keyfree" obtains "X \<in> keyfree" and "Y \<in> keyfree"
join_le: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion> y \<sqsubseteq> z"
cnf_semantics_monotonous_in_cnf_subsets_if: fixes \<A> :: "'b \<Rightarrow> bool"   and \<Phi> :: "'b formula"   and \<Phi>' :: "'b formula" assumes "\<A> \<Turnstile> \<Phi>"   and "is_cnf \<Phi>"   and "cnf \<Phi>' \<subseteq> cnf \<Phi>" shows "cnf_semantics \<A> (cnf \<Phi>')"
lexordp_eq_simps_4: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "ord.lexordp_eq less (x \<Colon> xs) (y \<Colon> ys) = (less x y \<or> \<not> less y x \<and> ord.lexordp_eq less xs ys)"
ladder_last_n_def: fixes L :: "(nat \<times> nat \<times> nat) list" shows "ladder_last_n L = ladder_n L (length L - 1)"
integrable2: shows "set_integrable lborel (einterval A B) f"
set_unsat_simps_2: fixes I :: "'a list"   and s :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>u (set_unsat I s) = \<B>\<^sub>i\<^sub>u s"
nths_shift_lemma: fixes A :: "nat set"   and xs :: "'g list"   and i :: "nat" shows "map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [i..<i + length xs])) = map fst (filter (\<lambda>p. snd p + i \<in> A) (zip xs [0..<length xs]))"
wf_tuple_length: fixes n :: "nat"   and V :: "nat set"   and x :: "'a option list" assumes "wf_tuple n V x" shows "length x = n"
gen_llength_cons_code: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and n :: "enat"   and s :: "'c" shows "gen_llength_cons g n s = (case g s of Done \<Rightarrow> n | Skip s' \<Rightarrow> gen_llength_cons g n s' | Yield x s' \<Rightarrow> gen_llength_cons g (eSuc n) s')"
CDMode_CTRL_Action1_def: shows "CDMode_CTRL_Action1 = ({}, CDMode_CTRL_Update1)"
weakPar2: fixes Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs"   and P :: "ccs" assumes "Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> P \<parallel> Q'"
assms_2: shows "\<chi> \<noteq> principal_dchar n"
w_4: shows "dist w x = Gromov_product_at x y z"
P14_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c l\<rightarrow> a \<le> c l\<rightarrow> b"
fin_ch_betw: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[f[a .. b ..  c]X]" shows "[[a b c]]"
quot_of_fract_in_normalized_fracts: fixes x :: "'b fract" shows "quot_of_fract x \<in> normalized_fracts"
strict_sorted_equal: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less"   and "linorder.strict_sorted less xs"   and "linorder.strict_sorted less ys"   and "list.set ys = list.set xs" shows "ys = xs"
dual_less_eqI: fixes y :: "'a dual"   and x :: "'a dual" assumes "undual y \<le> undual x" shows "x \<le> y"
obsne: shows "obs \<sigma>' i \<noteq> obs \<sigma> i"
unitary11_gen_special: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary11_gen M"   and "mat_det M = 1" shows "\<exists>a b. M = (a, b, cnj b, cnj a) \<or> M = (a, b, - cnj b, - cnj a)"
matrix_vector_mult_def: fixes m :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $ i $ j * x $ j"
ld: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < n" shows "A $$ (i, j) = (0::'a)"
conjE1_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p .\<and>. q" shows "A \<turnstile> p"
pickFT_pred_def: fixes cl :: "('d, 'e, 'f) cmd list"   and n :: "nat" shows "pickFT_pred cl n \<equiv> n < length cl \<and> finished (cl ! n)"
prefix_code_1: fixes xs :: "'b list" shows "prefix [] xs = True"
inv_loop_step: fixes x :: "nat"   and cf :: "nat \<times> cell list \<times> cell list" assumes "inv_loop x cf"   and "0 < x" shows "inv_loop x (step0 cf tcopy_loop)"
flowsto_subset: fixes X0 :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and T :: "real set"   and CX :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and Y :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and Z :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and S :: "real set"   and CZ :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and W :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set" assumes "flowsto X0 T CX Y"   and "Z \<subseteq> X0"   and "T \<subseteq> S"   and "CX \<subseteq> CZ"   and "Y \<subseteq> W" shows "flowsto Z S CZ W"
analytic_on_imp_differentiable_at: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and x :: "complex" assumes "f analytic_on S"   and "x \<in> S" shows "f field_differentiable at x"
chi_def: shows "chi = (\<lambda>n t. if real n \<le> t then 1 else 0)"
x_def: shows "x = Octo 1 0 0 0 1 0 0 0"
adapt_stepsize_fa_def: fixes rtol :: "real"   and m :: "nat"   and e :: "real"   and h' :: "real" shows "adapt_stepsize_fa rtol m e h' = floatarith.Num (float_of h') * floatarith.Powr (floatarith.Num (float_of rtol) / floatarith.Num (float_of e)) (inverse (floatarith.Num (float_of (real m) + 1)))"
base: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "rtrancl_path r x [] x"
wffp_append: fixes sl :: "('St, 'U, 'C) state list"   and sl1 :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "wffp sl1"   and "hd sl1 \<in> \<delta> (last sl)" shows "wffp (sl @ sl1)"
formulas_compl: fixes x :: "'a formula"   and S :: "'a set" assumes "x \<in> formulas S" shows "- x \<in> formulas S"
valid_Tree\<^sub>\<alpha>_def: shows "valid_Tree\<^sub>\<alpha> \<equiv> (id ---> rep_Tree\<^sub>\<alpha> ---> id) valid_Tree"
not_le: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(\<not> x \<le>s y) = (y <s x)"
unlr_sup: fixes x :: "'a admS"   and y :: "'a admS" shows "unlr (sup x y) = unlr x \<union> unlr y"
r_singleton_encode: fixes x :: "nat" shows "eval r_singleton_encode [x] \<down>= singleton_encode x"
omega_closed_add: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<in> elts \<omega>"   and "\<beta> \<in> elts \<omega>" shows "\<alpha> + \<beta> \<in> elts \<omega>"
emsapss_encode_def: fixes M :: "bit list"   and emBits :: "nat" shows "emsapss_encode M emBits = (if 2 ^ 64 \<le> length M \<or> 2 ^ 32 * 160 < emBits then [] else emsapss_encode_help1 (sha1 M) emBits)"
RENAMING_def: fixes s :: "'b"   and d :: "'b" shows "RENAMING s d \<equiv> s = d"
either: fixes a :: "protocol" assumes "a \<noteq> tcp"   and "a \<noteq> udp" shows "False"
non_inf_imp_SN_bound: fixes r :: "('c \<times> 'c) set"   and c :: "'c" assumes "non_inf r" shows "SN {(a, b). (b, c) \<in> r \<and> (a, b) \<in> r}"
Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and R :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 Q,R"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Catch c\<^sub>1 c\<^sub>2 Q,A"
sums_SUP: shows "f sums (SUP n. sum f {..<n})"
monom_Suc_2: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "monom (UP Zp) a (Suc n) = monom (UP Zp) a n \<otimes>\<^bsub>UP Zp\<^esub> monom (UP Zp) \<one> 1"
2: shows "x * (x\<^sup>\<star> \<squnion> y\<^sup>\<star>) \<le> x\<^sup>\<star> \<squnion> y\<^sup>\<star>"
fbox_g_ode_subset: fixes U :: "'a \<Rightarrow> real set"   and G :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>s. s \<in> S \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "|x\<acute>=\<lambda>t. f & G on U S @ 0] Q = (\<lambda>s. s \<in> S \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s)))"
mem_existence_ivl_iv_defined_2: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "ll_on_open_it T f X"   and "t \<in> ll_on_open.existence_ivl T f X t0 x0" shows "x0 \<in> X"
induct: fixes P :: "'b tree \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b list" assumes "\<And>xs. P \<langle>\<rangle> xs"   and "\<And>l x r xs. \<lbrakk>P r xs; \<And>xa. P l (x # xa)\<rbrakk> \<Longrightarrow> P \<langle>l, x, r\<rangle> xs" shows "P a0 a1"
bounded_def: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.metric_space dist uniformity open" shows "metric_space.bounded dist S = (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
frac_lt_1: fixes x :: "'a" shows "frac x < (1::'a)"
simps_3: fixes y :: "'d" shows "flat_1 (deny (allow y)) = deny y"
myss_def: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a" shows "myss x y \<equiv> (vwb_lens x \<and> vwb_lens y) \<and> x \<bowtie> y \<and> y \<bowtie> x"
no_zero_divisors: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "a * b \<noteq> (0::'a)"
inf_sup_distrib1: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<inter># (y \<union># z) = x \<inter># y \<union># (x \<inter># z)"
IH: assumes "valid_prefixes rtbl"   and "is_longest_prefix_routing rtbl"   and "has_default_route rtbl"   and "unambiguous_routing rtbl" shows "(routing_table_semantics rtbl addr = act) = (\<exists>rr\<in>set rtbl. prefix_match_semantics (routing_match rr) addr \<and> routing_action rr = act \<and> (\<forall>ra\<in>set rtbl. routing_rule_sort_key ra < routing_rule_sort_key rr \<longrightarrow> \<not> prefix_match_semantics (routing_match ra) addr))"
succs_def: fixes G :: "nat graph_rec"   and v :: "nat" assumes "cfg_doms G" shows "cfg_doms.succs G v = g_E G `` {v}"
prime_elem_one_plus_i_gauss_int: shows "prime_elem (1 + \<i>\<^sub>\<int>)"
nmaps_fresh: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and x :: "name"   and c :: "coname"   and P :: "trm"   and a :: "'a" assumes "\<theta>_n nmaps x to Some (c, P)"   and "a \<sharp> \<theta>_n" shows "a \<sharp> (c, P)"
monoD: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "f y < f x" shows "x < y"
better_AndR_substc: fixes c :: "coname"   and a :: "coname"   and M :: "trm"   and b :: "coname"   and N :: "trm"   and z :: "name"   and P :: "trm" assumes "c \<sharp> ([a].M, [b].N)" shows "AndR <a>.M <b>.N c{c:=z.P} = fresh_fun (\<lambda>a'. Cut <a'>.AndR <a>.M <b>.N a' z.P)"
subalgebra_Span_incl: fixes K :: "'a set"   and V :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "subalgebra K V R"   and "set Us \<subseteq> V" shows "R.Span K Us \<subseteq> V"
is_canon_rewriterD1: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_canon_rewriter rword A u p" shows "p \<in> A"
simps_1: fixes P :: "'a \<Rightarrow> bool" shows "takeWhile P [] = []"
1: shows "X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs"
intros_19: fixes t :: "t"   and v :: "v"   and s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and k :: "i32"   and off :: "nat"   and tp :: "tp"   and vs :: "v list"   and a :: "nat" assumes "types_agree t v"   and "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "store_packed m (nat_of_int k) off (bits v) (tp_length tp) = None" shows "\<lparr>s;vs;[$C ConstInt32 k, $C v, $Store t (Some tp) a off]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
integ_exis_vert: fixes \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "\<exists>(k', \<gamma>'')\<in>two_chain_vertical_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>'' = \<gamma>" shows "line_integral_exists F {i} \<gamma>"
hom_grading_fun_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and d :: "'c \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'c" assumes "class.monoid_add plus zero" shows "monoid_add.hom_grading_fun plus d f = (\<forall>n. (\<forall>s t. f n (plus s t) = plus (f n s) (f n t)) \<and> (\<forall>t. d (f n t) \<le> n \<and> (d t \<le> n \<longrightarrow> f n t = t)))"
simps_2: fixes I :: "('b, 'c, 'a) interp"   and f :: "'b"   and args :: "'a \<Rightarrow> ('b, 'a) trm"   and v :: "(real, 'a) vec" shows "frechet I ($f f args) v = (\<lambda>v'. FunctionFrechet I f (\<chi>i. sterm_sem I (args i) v) (\<chi>i. frechet I (args i) v v'))"
enumerate_append_eq: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'a list" shows "enumerate n (xs @ ys) = enumerate n xs @ enumerate (n + length xs) ys"
leadsTo_imp_LeadsTo: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto> B" shows "F \<in> A \<longmapsto>w B"
elims: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list"   and y :: "'a list" assumes "mergesort_by_rel x xa = y" obtains R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xs :: "'a list" where "x = R" and "xa = xs" and "y = (if length xs < 2 then xs else mergesort_by_rel_merge R (mergesort_by_rel R (fst (mergesort_by_rel_split ([], []) xs))) (mergesort_by_rel R (snd (mergesort_by_rel_split ([], []) xs))))"
Act\<^sub>\<alpha>_eq_iff: fixes \<alpha>1 :: "'d"   and t1 :: "('b, 'c, 'd) Tree\<^sub>\<alpha>"   and \<alpha>2 :: "'d"   and t2 :: "('b, 'c, 'd) Tree\<^sub>\<alpha>" shows "(Act\<^sub>\<alpha> \<alpha>1 t1 = Act\<^sub>\<alpha> \<alpha>2 t2) = (tAct \<alpha>1 (rep_Tree\<^sub>\<alpha> t1) =\<^sub>\<alpha> tAct \<alpha>2 (rep_Tree\<^sub>\<alpha> t2))"
subsumes_accessible_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and r\<^sub>1 :: "'a"   and r\<^sub>2 :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst" shows "substitution.subsumes_accessible r\<^sub>1 r\<^sub>2 s = (r\<^sub>2 \<in> RID\<^sub>L (the (s r\<^sub>1)) \<longrightarrow> LID\<^sub>S (the (s r\<^sub>2)\<^sub>\<sigma>) \<subseteq> doms (the (s r\<^sub>1)))"
ordered_model_resolvent_def: fixes I :: "'at set"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" shows "ordered_model_resolvent I P1 P2 C = (\<exists>L. C = P1 - {L} \<union> (P2 - {complement L}) \<and> (validated_part I P1 = {} \<and> strictly_maximal_literal P1 L) \<and> strictly_maximal_literal (validated_part I P2) (complement L))"
select_convs_2: fixes iinput :: "Proc \<Rightarrow> InputsOrNi"   and ioutput :: "Proc \<Rightarrow> InputsOrNi"   and ichosen :: "InputsOrNi"   and iallInput :: "InputsOrNi set"   and more :: "'a" shows "ioutput \<lparr>iinput = iinput, ioutput = ioutput, ichosen = ichosen, iallInput = iallInput, \<dots> = more\<rparr> = ioutput"
singleton_group: fixes a :: "'e" shows "Group.group (singleton_group a)"
alive_trivial_simps_1: fixes b :: "bool"   and s :: "Store" shows "alive (boolV b) s"
semilattice_neutr_set_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "semilattice_neutr_set f z \<equiv> semilattice_neutr f z"
pair_no_loops: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs H" shows "fst e \<noteq> snd e"
ext_inject: fixes nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p"   and nothinga' :: "unit"   and doctype' :: "char list"   and document_element' :: "(_) element_ptr option"   and disconnected_nodes' :: "(_) node_ptr list"   and more' :: "'p" shows "(\<lparr>RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RDocument.nothing = nothinga', doctype = doctype', document_element = document_element', disconnected_nodes = disconnected_nodes', \<dots> = more'\<rparr>) = (nothinga = nothinga' \<and> doctype = doctype' \<and> document_element = document_element' \<and> disconnected_nodes = disconnected_nodes' \<and> more = more')"
not_eq_order_implies_strict: fixes a :: "'a word"   and b :: "'a word" assumes "a \<noteq> b"   and "a \<le>s b" shows "a <s b"
redT_updTs_invariant: fixes I :: "'t \<Rightarrow> 'i option"   and ts :: "'t \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option"   and m :: "'m"   and t :: "'t"   and x :: "'x"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and ln :: "'a \<Rightarrow>f nat"   and ln' :: "'a \<Rightarrow>f nat" assumes "ts_inv P I ts m"   and "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "thread_oks ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "ts_inv P (upd_invs I P \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>) (redT_updTs ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', ln'))) m'"
cn_weak: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d" assumes "Regions' X v n not_in_X" shows "\<forall>c. 0 < v c"
initd_def: fixes C :: "qtname" shows "initd C = inited C \<circ> globs \<circ> snd"
bullet_App: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" shows "(App s\<^sup>\<bullet> t\<^sup>\<bullet>, (App s t)\<^sup>\<bullet>) \<in> {\<rightarrow>\<^sub>\<beta>}\<^sup>="
order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
has_contour_integral_trivial_iff: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "complex" shows "(f has_contour_integral i) (linepath a a) = (i = 0)"
assms_5: shows "path_conforms_with_strategy p P' \<sigma>"
app_simps_3: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c"   and u :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_app unapp (app t u)"
cong_scale: fixes k :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "k \<noteq> (0::'a)" shows "[a = b] (mod c) = [k * a = k * b] (mod k * c)"
Sum1: fixes P :: "pi"   and V :: "residual"   and Q :: "pi" assumes "P \<longmapsto> V" shows "P \<oplus> Q \<longmapsto> V"
rxa_in_Rxa: fixes a :: "'a"   and r :: "'a" assumes "a \<in> carrier R"   and "r \<in> carrier R" shows "r \<cdot>\<^sub>r a \<in> R \<diamondsuit>\<^sub>p a"
ladder_shift_n_drop: fixes N :: "nat"   and n :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_shift_n N (drop n L) = drop n (ladder_shift_n N L)"
assms_3: shows "a * b0 + b = c * b0 + d"
Seq_MtransC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and c1' :: "('a, 'c) com"   and s' :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.MtransC tval aval (c1, s) (c1', s')" shows "PL.MtransC tval aval (c1 ;; c2, s) (c1' ;; c2, s')"
proper_interval_set_less_eqI: fixes ao :: "'a option"   and x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "proper_interval ao (Some x)"   and "\<forall>a\<in>A. x \<le> a"   and "\<forall>b\<in>B. x \<le> b" shows "A \<sqsubseteq>' - B \<inter> above ao"
Stirling_less: fixes n :: "nat"   and k :: "nat" assumes "n < k" shows "Stirling n k = 0"
elims: fixes x :: "nat list"   and y :: "nat rexp" assumes "w2rexp x = y" obtains "x = []" and "y = rexp.One"   | a :: "nat" and as :: "nat list" where "x = a # as" and "y = Times (Atom a) (w2rexp as)"
perm_simps_7: fixes p :: "perm" shows "p \<bullet> ANil = ANil"
sup_bool_def: fixes P :: "bool"   and Q :: "bool" shows "P \<squnion> Q = (P \<or> Q)"
epi_implies_arr: fixes f :: "'a" assumes "A.epi f" shows "A.arr f"
lsc_uminus: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 (\<lambda>x. - f x) = usc_at x0 f"
some_geodesic_is_geodesic_segment_1: fixes x :: "'a"   and y :: "'a" shows "geodesic_segment_between {x--y} x y"
finprod_reindex: fixes f :: "'c \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "f \<in> h ` A \<rightarrow> carrier G"   and "inj_on h A" shows "finprod G f (h ` A) = (\<Otimes>x\<in>A. f (h x))"
case_optionE: fixes P :: "bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b option" assumes "case x of None \<Rightarrow> P | \<lfloor>y\<rfloor> \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'b" where "x = \<lfloor>y\<rfloor>" and "Q y"
l_count_cons: fixes x :: "'b"   and y :: "'b"   and ys :: "'b list" shows "l_count x (y # ys) = l_count x [y] + l_count x ys"
bsingleton_def: fixes x :: "'a" shows "bsingleton x = binsert x bempty"
simps_1: fixes x1 :: "char list"   and y1 :: "char list" shows "(Forward x1 = Forward y1) = (x1 = y1)"
IH: fixes K :: "'a trm" assumes "depth K < depth M" shows "P S K"
Abs_proj2_inverse: fixes y :: "(real, 3) vec set" assumes "y \<in> non_zero_vectors // proportionality" shows "Rep_proj2 (Abs_proj2 y) = y"
hyps_1: fixes gpv :: "('a, 'b, 'c) gpv" shows "interaction_bound' gpv \<le> interaction_any_bound gpv"
compute_nr_of_jordan_blocks_def: fixes A :: "'a mat"   and ev :: "'a"   and k :: "nat" shows "local.compute_nr_of_jordan_blocks A ev k = 2 * local.dim_gen_eigenspace A ev k - local.dim_gen_eigenspace A ev (k - 1) - local.dim_gen_eigenspace A ev (Suc k)"
assms_3: shows "integrable M (\<lambda>x. f x * g x)"
3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
mult_carrier_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * B \<in> carrier_mat nr nc"
distinct_5: fixes x2 :: "'c" shows "\<lbrakk>x2\<rbrakk> \<noteq> \<stileturn>"
comp_fun_idem: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'a" assumes "comp_fun_idem f" shows "f x \<circ> f x = f x"
corthogonal_mat_def: fixes A :: "'b mat" shows "corthogonal_mat A \<equiv> let B = mat_adjoint A * A in diagonal_mat B \<and> (\<forall>i<dim_col A. B $$ (i, i) \<noteq> (0::'b))"
q_def: shows "q = snd (rat_to_normalized_int_poly p)"
scale_left_commute: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and x :: "'d" assumes "module scale" shows "scale a (scale b x) = scale b (scale a x)"
mod_1_coprime_nat: fixes n :: "nat"   and a :: "nat"   and b :: "nat" assumes "0 < n"   and "[a ^ n = 1] (mod b)" shows "coprime a b"
MMD_plurality_rule_def: fixes A :: "'c set"   and Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set" shows "MMD_plurality_rule A Is P \<equiv> choiceSet A (MMD Is P)"
conjugate_zero_iff_vec: fixes v :: "'c vec"   and n :: "nat" shows "(conjugate v = 0\<^sub>v n) = (v = 0\<^sub>v n)"
sqrt_2_not_rat: shows "sqrt 2 \<notin> \<rat>"
tauPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel" shows "\<tau>.P \<leadsto>\<guillemotleft>Rel\<guillemotright> \<tau>.Q"
map_upd_nonempty: fixes t :: "'a \<Rightarrow> 'b option"   and k :: "'a"   and x :: "'b" shows "t(k \<mapsto> x) \<noteq> \<epsilon>"