hcju/isabelleps huggingface dataset:
nn_integral_add: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x + g x \<partial>M = integral\<^sup>N M f + integral\<^sup>N M g"
cases: fixes a1 :: "('c \<times> 'd) list"   and a2 :: "'d \<Rightarrow> 'c set" assumes "crdt_ops a1 a2" obtains deps :: "'d \<Rightarrow> 'c set" where "a1 = []" and "a2 = depsa__"   | xs :: "('c \<times> 'd) list" and deps :: "'d \<Rightarrow> 'c set" and oid :: "'c" and oper :: "'d" where "a1 = xs @ [(oid, oper)]" and "a2 = deps" and "crdt_ops xs deps" and "oid \<notin> set (map fst xs)" and "\<forall>ref\<in>deps oper. ref \<in> set (map fst xs) \<and> ref < oid"
fls_delta_subdegree: fixes b :: "'a"   and a :: "int" assumes "b \<noteq> (0::'a)" shows "fls_subdegree (Abs_fls (\<lambda>n. if n = a then b else (0::'a))) = a"
distinct_93: fixes x51 :: "'d set"   and x52 :: "('d, 'e, 'f) com"   and x53 :: "('d, 'e, 'f) com" shows "Cond x51 x52 x53 \<noteq> Throw"
goedel_second_pos: assumes "prv (neg (PP \<langle>fls\<rangle>))" shows "prv fls"
divide_divide_times_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and w :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide x y) (divide z w) = divide (times x w) (times y z)"
Induction_ax_imp_induction_axioms: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> Induction_ax" shows "\<exists>A. x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> A \<in> induction_axioms"
point_diff: fixes xA :: "real"   and yA :: "real"   and xB :: "real"   and yB :: "real" shows "Point xA yA - Point xB yB = Point (xA - xB) (yA - yB)"
B: shows "(x, y) \<notin> B\<^sup>*"
iso_can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.can C f" shows "category.iso (free_monoidal_category.comp C) f"
qo_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "qo_on P A = (reflp_on P A \<and> transp_on P A)"
pne13: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "pid1 \<noteq> pid3"
permute_diff: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "perm"   and q :: "perm"   and x :: "'e" assumes "class.pt permute" shows "permute (p - q) x = permute p (permute (- q) x)"
cmod_eqE: fixes z1 :: "complex"   and z2 :: "complex" assumes "cmod z1 = cmod z2" shows "z1 * cnj z1 = z2 * cnj z2"
finprod_const: fixes a :: "'a"   and A :: "'c set" assumes "a \<in> carrier G" shows "(\<Otimes>x\<in>A. a) = a [^] card A"
prems_3: shows "\<^bold>|s\<^bold>| + \<^bold>|r\<^bold>| - gcd \<^bold>|s\<^bold>| \<^bold>|r\<^bold>| \<le> \<^bold>|w\<^bold>|"
assms_2: shows "\<exists>n>0. x [^] n = \<one>"
assms: fixes i :: "'a"   and j :: "'b" assumes "i \<in> Basis"   and "j \<in> Basis" shows "continuous_on S (\<lambda>s. g s i j)"
osafe_Prod: fixes ODE1 :: "('a, 'b) ODE"   and ODE2 :: "('a, 'b) ODE" assumes "osafe ODE1"   and "osafe ODE2"   and "ODE_dom ODE1 \<inter> ODE_dom ODE2 = {}" shows "osafe (OProd ODE1 ODE2)"
euclid_ext_aux_bezout: fixes r' :: "'a"   and r :: "'a"   and a :: "'a"   and b :: "'a"   and s' :: "'a"   and t' :: "'a"   and s :: "'a"   and t :: "'a"   and x :: "'a"   and y :: "'a"   and c :: "'a" assumes "gcd r' r = gcd a b"   and "s' * a + t' * b = r'"   and "s * a + t * b = r"   and "euclid_ext_aux s' s t' t r' r = ((x, y), c)" shows "x * a + y * b = gcd a b"
domain_additive: fixes x :: "brick list"   and y :: "brick list" shows "domain_block (x \<otimes> y) = domain_block x + domain_block y"
incseq_emeasure: fixes B :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range B \<subseteq> sets M"   and "incseq B" shows "incseq (\<lambda>i. emeasure M (B i))"
assms_2: shows "punit.lt (rep_list p) \<oplus> lt q \<prec>\<^sub>t punit.lt (rep_list q) \<oplus> lt p"
fermat_liar_def: shows "local.fermat_liar = [a ^ (n - 1) = 1] (mod n)"
basis_list_basis_3: fixes n :: "nat"   and fs :: "'a vec list" assumes "cof_vec_space.basis_list n fs" shows "vectorspace.basis class_ring (module_vec TYPE('a) n) (set fs)"
assms: shows "(w', w'') \<in> (\<Union> (relations (gen_model M w) ` as))\<^sup>+"
p_residue_def: fixes m :: "nat"   and n :: "int" shows "p_residue m n \<equiv> residue (p ^ m) n"
phis_finite: fixes g :: "'a" shows "finite (dom (Mapping.lookup (phis g)))"
nodes_of_uses_eq_NoneD: fixes v :: "'val"   and n :: "'node"   and args :: "'val list" assumes "Mapping.lookup nodes_of_uses v = None"   and "Mapping.lookup (phis g) (n, v) = Some args" obtains "\<forall>n\<in>set (\<alpha>n g). \<forall>vs. Mapping.lookup (uses g) n = Some vs \<longrightarrow> v \<notin> vs"
compat_pair_def: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" shows "compat_pair S NS \<equiv> NS O S \<subseteq> S \<and> S O NS \<subseteq> S"
hom_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
Conseq1: fixes A :: "triple set"   and P' :: "assn"   and c :: "stmt"   and Q :: "assn"   and P :: "assn" assumes "A \<turnstile> {P'} c {Q}"   and "\<forall>s. P s \<longrightarrow> P' s" shows "A \<turnstile> {P} c {Q}"
in_own_span: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n" shows "S \<subseteq> local.span S"
mtf_ub: fixes n :: "nat" shows "t_mtf n + \<Phi> (n + 1) - \<Phi> n \<le> 2 * c_A n - 1 + p_A n - f_A n"
arr_dom: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.arr (Left_a.dom f)"
runit_simps_2: fixes f :: "'a" assumes "ide f" shows "src \<r>[f] = src f"
EE_2: shows "(([u], {#}), ee, sh, ch) \<in> ntrs fg"
sym_preproc_aux_term1_wf_on: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "wfp_on (\<lambda>x y. (x, y) \<in> sym_preproc_aux_term1 d) {x. set (fst (snd (snd x))) \<subseteq> pp_of_term -` dgrad_set d m}"
next_commutes_1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<circle>(F \<and> G) \<equiv> \<circle>F \<and> \<circle>G"
mult_eq: fixes x :: "nat"   and y :: "nat" shows "x \<otimes>\<^bsub>G\<^esub> y = x * y mod n"
in_fst_imageE: fixes x :: "'a"   and S :: "('a \<times> 'b) set" assumes "x \<in> fst ` S" obtains y :: "'b" where "(x, y) \<in> S"
prf_oracle_def: fixes key :: "bool list"   and \<sigma> :: "unit"   and x :: "bool list" shows "local.prf.prf_oracle key \<sigma> x = return_spmf (f key x, ())"
powr_Reals_eq: fixes x :: "complex"   and y :: "complex" assumes "x \<in> \<real>"   and "y \<in> \<real>"   and "0 \<le> Re x" shows "x powr y = complex_of_real (Re x powr Re y)"
wellformed: fixes b :: "'a set" assumes "b \<in># \<B>'" shows "b \<subseteq> \<V>'"
comm_neutral: fixes a :: "'b" shows "a AND - (1::'b) = a"
Confidentiality_Kas_lemma: fixes authK :: "nat"   and A :: "agent"   and evs :: "event list"   and Ta :: "nat" assumes "authK \<in> symKeys"   and "A \<notin> bad"   and "evs \<in> kerbIV"   and "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"   and "Key authK \<in> analz (knows Spy evs)" shows "expiredAK Ta evs"
div_mult1_eq: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (times a b) c = plus (times a (divide b c)) (divide (times a (modulo b c)) c)"
insertCI: fixes a :: "'c"   and B :: "'c set"   and b :: "'c" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> {b} \<union> B"
deriv_c1: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>1 c\<^sub>1 Q,A"
synthetic_div_eq_0_iff: fixes p :: "'b poly"   and c :: "'b" shows "(synthetic_div p c = 0) = (degree p = 0)"
min_y_yz: shows "maximum j = maximum (i \<squnion> j)"
InitState_notmem_InitConf: fixes SA :: "('a, 'b, 'c) seqauto"   and A :: "('a, 'b, 'c) hierauto"   and S :: "'a"   and T :: "'a" assumes "SA \<in> the (CompFun A S)"   and "S \<in> InitConf A"   and "T \<in> States SA"   and "T \<noteq> InitState SA" shows "T \<notin> InitConf A"
mem_Sigma_mset_iff: fixes a :: "'a"   and b :: "'b"   and A :: "'a multiset"   and B :: "'a \<Rightarrow> 'b multiset" shows "((a, b) \<in># Sigma_mset A B) = (a \<in># A \<and> b \<in># B a)"
finlsts_rec_def: fixes c :: "'b"   and d :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "'c llist" shows "finlsts_rec c d r \<equiv> if r \<in> UNIV\<^sup>\<star> then wfrec finlsts_pred (\<lambda>f. case_llist c (\<lambda>a r. d a r (f r))) r else undefined"
assms_1: shows "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"
has_dom_def: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" shows "has_dom f S \<equiv> \<forall>s. s \<notin> S \<longrightarrow> f s = s"
ext_wf_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_wf ext \<equiv> ext ext \<and> ext_wf_axioms ext"
le_less_trans: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less y z" shows "less x z"
depth_app_1: fixes A :: "'b trm"   and B :: "'b trm" shows "depth A < depth (App A B)"
iso_lunit: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.iso (S.lunit f)"
len\<^sub>A: shows "length cms\<^sub>A = length cms"
relcompp_bot2: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "R OO \<bottom> = \<bottom>"
game_sem_union: fixes I :: "interp"   and \<alpha> :: "game"   and X :: "(variable \<Rightarrow> real) set"   and Y :: "(variable \<Rightarrow> real) set" shows "game_sem I \<alpha> X \<union> game_sem I \<alpha> Y \<subseteq> game_sem I \<alpha> (X \<union> Y)"
finsertE: fixes a :: "'a"   and b :: "'a"   and A :: "'a fset" assumes "a |\<in>| finsert b A" obtains "a = b"   | "a |\<in>| A"
assms_2: fixes s :: "'a" assumes "s \<in> basis" shows "F *\<^sub>V s = G *\<^sub>V s"
cases: fixes a1 :: "'a option"   and a2 :: "('a \<times> 'a option) list" assumes "insert_seq a1 a2" obtains start :: "'a option" and oid :: "'a" where "a1 = start__" and "a2 = [(oid__, start__)]"   | start :: "'a option" and list :: "('a \<times> 'a option) list" and prev :: "'a" and ref :: "'a option" and oid :: "'a" where "a1 = start" and "a2 = list @ [(prev, ref), (oid, Some prev)]" and "insert_seq start (list @ [(prev, ref)])"
exhaust: fixes y :: "('a, 'b) xstate" obtains x1 :: "'a" where "y = Normal x1"   | x2 :: "'a" where "y = Abrupt x2"   | x3 :: "'b" where "y = Fault x3"   | "y = Stuck"
fun_upd_twist: fixes a :: "'b"   and c :: "'b"   and m :: "'b \<Rightarrow> 'c"   and b :: "'c"   and d :: "'c" assumes "a \<noteq> c" shows "m(a := b, c := d) = m(c := d, a := b)"
le_iff_sup: fixes x :: "'a upred"   and y :: "'a upred" shows "(y \<sqsubseteq> x) = ((x \<or> y) = y)"
conforms_XcptLocD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option"   and a :: "loc" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "x = Some (Xcpt (Loc a))" shows "G,s\<turnstile>Addr a\<Colon>\<preceq>Class (SXcpt Throwable)"
inf_sup_ord_2: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type)) y"
col_fst_is_col: fixes M :: "'a mat" shows "col_fst M = col M 0"
ParamDefs_return_target_length: fixes a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and ins :: "'c list"   and outs :: "'c list" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "(p, ins, outs) \<in> set procs" shows "length (ParamDefs (targetnode a)) = length outs"
sel'_someD: fixes s :: "'s"   and P :: "'x \<Rightarrow> bool"   and x :: "'x" assumes "invar s"   and "sel' s P = Some x" shows "x \<in> \<alpha> s \<and> P x"
prime: fixes p :: "nat" assumes "prime p" shows "moebius_mu p = - (1::'a)"
simps: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and xs :: "'varSort"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" shows "qAbs xs x X $[[rho]] = (let x' = pickQFreshEnv xs {x} {X} {rho} in qAbs xs x' X #[[x' \<and> x]]_xs #[[rho]])"
HaddP_commute: fixes x :: "tm"   and y :: "tm"   and z :: "tm" shows "{HaddP x y z, OrdP x} \<turnstile> HaddP y x z"
congruent_triangle_sss_6: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "dist c1 b1 = dist b2 c2"
P_def: shows "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
hyps: assumes "list_all (\<lambda>i. i < length signs) I"   and "list_all (\<lambda>s. s = 1 \<or> s = 0 \<or> s = - 1) signs" shows "prod_list (map ((!) (map_sgas signs)) I) = 1 \<or> prod_list (map ((!) (map_sgas signs)) I) = 0"
goals: shows "matches \<gamma> m' a p = matches \<gamma> m a p \<and> normalized_nnf_match m'"
d_max'_def: shows "d_max' = truncate_up p (real_divr p 1 a - alpha * a)"
simps: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "a1 \<approx> a2 = (\<exists>\<R> lc\<^sub>1 lc\<^sub>2. a1 = lc\<^sub>1 \<and> a2 = lc\<^sub>2 \<and> strong_low_bisim_mm \<R> \<and> (lc\<^sub>1, lc\<^sub>2) \<in> \<R>)"
IH: fixes ys :: "'a list" shows "cnt x (start_remove xs x ys) \<le> Suc (cnt x ys)"
pat_rule: fixes p :: "'a"   and p' :: "'a"   and t' :: "'a"   and T :: "'b itself" assumes "p \<equiv> p'"   and "ID p' t' T" shows "ID p t' T"
parallel_converter_map_converter: fixes f :: "'h \<Rightarrow> 'n"   and g :: "'o \<Rightarrow> 'j"   and f' :: "'p \<Rightarrow> 'l"   and g' :: "'m \<Rightarrow> 'q"   and conv1 :: "('n, 'o, 'p, 'q) converter"   and f'' :: "'i \<Rightarrow> 'r"   and g'' :: "'s \<Rightarrow> 'k"   and conv2 :: "('r, 's, 'p, 'q) converter" shows "map_converter f g f' g' conv1 |\<^sub>\<propto> map_converter f'' g'' f' g' conv2 = map_converter (map_sum f f'') (map_sum g g'') f' g' (conv1 |\<^sub>\<propto> conv2)"
4: shows "y \<noteq> [] \<longrightarrow> 0 < length y"
holomorphic_on_compose_gen: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex"   and t :: "complex set" assumes "f holomorphic_on s"   and "g holomorphic_on t"   and "f ` s \<subseteq> t" shows "g \<circ> f holomorphic_on s"
intros_6: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com" assumes "\<not> bval b s" shows "(WHILE b DO c, s) \<Rightarrow> Suc 0 \<Down> s"
UP_root_div_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "a \<in> carrier R" shows "UP_cring.UP_root_div R f a \<in> carrier (UP R)"
f_arg_min_list_f: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "f (arg_min_list f xs) = Min (f ` set xs)"
HMA_row: shows "(Mod_Type_Connect.HMA_I ===> Mod_Type_Connect.HMA_M ===> Mod_Type_Connect.HMA_V) (\<lambda>i a. Matrix.row a i) row"
iff_contradiction: fixes A :: "bool" assumes "(\<not> A) = A" shows "False"
is_subseq_closed: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and s' :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "is_subseq_of s s'" shows "s' \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
restrict_SN_def: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "restrict_SN r s = {(a, b) |a b. (a, b) \<in> r \<and> SN_on s {a}}"
prv_eql_subst_trm_rev2: fixes x :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "prv (eql t1 t2)" shows "prv (imp (subst \<phi> t2 x) (subst \<phi> t1 x))"
untime_last_eq_filter_last: fixes s :: "'b message_af list" assumes "untime s \<noteq> []" shows "Msg (last (untime s)) = last (filter (\<lambda>x. x \<noteq> NoMsg) s)"
tiebreaker_lifting_def: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G :: "'b inference set"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F_g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F_g \<equiv> standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I \<and> tiebreaker_lifting_axioms Prec_F_g"
c_manifolds_axioms: shows "c_manifolds k charts charts_eucl"
reachable_project_imp_reachable: fixes C :: "'c set"   and F :: "'a program"   and G :: "'c program"   and x :: "'a" assumes "C \<subseteq> reachable (extend h F \<squnion> G)"   and "x \<in> reachable (F \<squnion> project h C G)" shows "\<exists>y. h (x, y) \<in> reachable (extend h F \<squnion> G)"
singletonE: fixes b :: "'d"   and a :: "'d" assumes "b \<in> {a}" obtains "b = a"
harm_expand_2: shows "harm (Suc 0) = (1::'a)"
fun_relD1: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'f \<Rightarrow> 'g"   and Ra :: "('d \<times> 'f) set"   and Rr :: "('e \<times> 'g) set"   and x :: "'d"   and r :: "'e" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f x = r" shows "\<forall>x'. (x, x') \<in> Ra \<longrightarrow> (r, f' x') \<in> Rr"
unit_factor_poly_i_def: fixes ops :: "'b arith_ops_record"   and xs :: "'b list" shows "unit_factor_poly_i ops xs = cCons_i ops (arith_ops_record.unit_factor ops (lead_coeff_i ops xs)) []"
i: shows "i \<equiv> if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1"
to_fun_formula: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "f \<in> carrier P"   and "x \<in> carrier R" shows "to_fun f x = (\<Oplus>i\<in>{..deg R f}. f i \<otimes> x [^] i)"
continuous_map_eq: fixes X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_map X X' f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "continuous_map X X' g"
safe_lpd_fv_regex: fixes r :: "'a regex"   and s :: "'a regex"   and test :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'d" assumes "local.safe_regex Futu Strict r"   and "s \<in> lpd test i r" shows "fv_regex fv s = fv_regex fv r"
trancl_unfold_left: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>+ = r O r\<^sup>*"
right_unique_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "right_unique R" shows "right_unique (list_all2 R)"
bind_fv2_no_occs_unchanged: fixes v :: "variable \<times> typ"   and t :: "Core.term"   and lev :: "nat" assumes "\<not> occs (case v of (x, xa) \<Rightarrow> Fv x xa) t" shows "bind_fv2 v lev t = t"
root_rev_iff: fixes u :: "'b list"   and t :: "'b list" shows "rev u \<in> rev t* = u \<in> t*"
rel_map_2: fixes Sa :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a stream"   and g :: "'d \<Rightarrow> 'b"   and y :: "'d stream" shows "stream_all2 Sa x (smap g y) = stream_all2 (\<lambda>x y. Sa x (g y)) x y"
Exit_Use_empty: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and V :: "'c" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "V \<in> Use Exit" shows "False"
execn_Seq': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and s' :: "('e, 'f) xstate"   and c2 :: "('e, 'd, 'f) com"   and s'' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> =n\<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> =n\<Rightarrow> s''" shows "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> =n\<Rightarrow> s''"
in_sstart: fixes s :: "'c stream"   and S :: "'c set"   and x :: "'c"   and xs :: "'c list" shows "(s \<in> sstart S (x # xs)) = (shd s = x \<and> stl s \<in> sstart S xs)"
X_def: fixes i :: "nat" shows "local.X i = [:- x i, 1::'a:]"
minus_Assertion_def: fixes p :: "'a Assertion"   and q :: "'a Assertion" shows "p - q = p \<sqinter> - q"
order_trans_rules_23: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
atLeastAtMost_diff_ends: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b - {a, b} = ord.greaterThanLessThan less a b"
RHS: shows "integral UNIV (\<lambda>x. integral UNIV (\<lambda>y. f (y, x))) = integral (cbox a b) (\<lambda>x. integral (cbox (g1 x) (g2 x)) (\<lambda>y. g (y, x)))"
ex_assignment: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b" assumes "Hintikka A H" shows "assign A H i \<noteq> {}"
zero: shows "A $ (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> (GREATEST m. \<not> is_zero_row_upt_k m k A) + (1::'rows) \<le> n) $ j = (0::'a)"
boundedE: fixes b :: "'b multiset"   and c :: "'b multiset"   and a :: "'b multiset" assumes "b \<union># c \<subseteq># a" obtains "b \<subseteq># a" and "c \<subseteq># a"