hcju/isabelleps huggingface dataset:
sbelow_sctns_def: fixes X :: "'c set"   and sctn :: "'c sctn set" shows "sbelow_sctns X sctn = SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> sbelow_halfspaces sctn)"
true_cls_mset_true_cls: fixes I :: "'b set"   and CC :: "'b literal multiset multiset"   and C :: "'b literal multiset" assumes "I \<Turnstile>m CC"   and "C \<in># CC" shows "I \<Turnstile> C"
step_2_main_similar: fixes A :: "'b mat"   and n :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat (step_2_main n j A) A"
A: shows "bcomp (b, False, size (ccomp c) + 1) @ ccomp c @ [JMP (- size (bcomp (b, False, size (ccomp c) + 1)) - size (ccomp c) - 1)] \<Turnstile> cfs\<box>"
asIMOD_wlsFSwSb: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "iwlsFSwSb (asIMOD SEM)"
some_ground_head_arity: fixes x :: "'v" shows "arity_var x \<le> arity_sym (SOME f. f \<in> local.ground_heads (Var x))"
le_Real: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "Real.cauchy Y" shows "(Real.Real X \<le> Real.Real Y) = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
nat_sum_has_integral_ceiling: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "m < n" shows "((\<lambda>x. f (nat \<lceil>x\<rceil>)) has_integral sum f {m<..n}) {real m..real n}"
nat_mult_eq_1_iff: fixes m :: "nat"   and n :: "nat" shows "(m * n = 1) = (m = 1 \<and> n = 1)"
monad_state_optionT': assumes "monad_state Monad_Overloading.return Monad_Overloading.bind get put" shows "monad_state Monad_Overloading.return Monad_Overloading.bind get put"
dvd_power: fixes n :: "nat"   and x :: "'a" assumes "0 < n \<or> x = (1::'a)" shows "x dvd x ^ n"
filterACS: shows "set (filter_compliant_stateful_ACS G M (rev edgesList)) = edges G - backflows (edges G)"
int_pow_diff: fixes x :: "'a"   and n :: "int"   and m :: "int" assumes "x \<in> carrier G" shows "x [^] (n - m) = x [^] n \<otimes> inv (x [^] m)"
delta_price_eq: fixes n :: "nat"   and T :: "nat"   and N :: "bool stream measure"   and cash_flow :: "bool stream \<Rightarrow> real"   and w :: "bool stream" assumes "Suc n \<le> T" shows "delta_price N cash_flow T n w = (rn_price N cash_flow T (Suc n) (spick w n True) - rn_price N cash_flow T (Suc n) (spick w n False)) / (geom_proc n w * (u - d))"
simps_1: fixes t :: "'c" shows "dbm_entry_bound (Le t) = t"
assms_3: shows "Prj\<^sub>1 \<cdot> h = Prj\<^sub>1 \<cdot> h'"
aux: fixes a :: "'v list"   and k :: "nat"   and j :: "nat" assumes "a = tl (dropWhile ((\<noteq>) u) (tj_stack s))"   and "k < length a"   and "j < k"   and "u \<in> set (tj_stack s)" shows "\<delta> s (a ! k) < \<delta> s (a ! j)"
strict_implies_order: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term" assumes "b \<rhd> a" shows "b \<unrhd> a"
nat_set_cset_bij: obtains f :: "nat set cset \<Rightarrow> nat set" where "bij f"
uI_AE: fixes n :: "nat"   and m :: "nat" shows "AE x in MI. uI (n + m) x \<le> uI n x + uI m ((TI ^^ n) x)"
insert_Diff_single: fixes a :: "'c"   and A :: "'c set" shows "insert a (A - {a}) = insert a A"
size_ipv4addr: fixes x :: "32 word" shows "size x = 32"
atLeastLessThan_iff: fixes i :: "'b word"   and l :: "'b word"   and u :: "'b word" shows "(i \<in> signed.atLeastLessThan l u) = (l \<le>s i \<and> i <s u)"
distinct_39: fixes x4 :: "('a, 'b, 'c) expr"   and x61 :: "('a, 'b, 'c) expr"   and x62 :: "('a, 'b, 'c) expr" shows "Ref x4 \<noteq> Assign x61 x62"
map_bind_pmf: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'e pmf"   and g :: "'e \<Rightarrow> 'd pmf" shows "map_pmf f (M \<bind> g) = M \<bind> (\<lambda>x. map_pmf f (g x))"
product_def: fixes A :: "'e set"   and B :: "'f set" shows "Product_Type.product A B = A \<times> B"
atm_of_eq_atm_of: fixes L :: "'b literal"   and L' :: "'b literal" shows "(atm_of L = atm_of L') = (L = L' \<or> L = - L')"
concurrent_eff: fixes pm :: "'a extended"   and l :: "nat"   and sm :: "'a extended"   and u :: "nat" assumes "idx s pm = Inr l"   and "idx s sm = Inr u" obtains d :: "'a extended list list" where "mapM (concurrent s l u) (substr s l u) = Inr d \<and> set (concat d) = InString ` I ` {x. is_concurrent pm sm x}"
considers_def: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and nA :: "nat \<times> formula"   and n :: "nat" shows "considers f nA n = (case snd (f n) of [] \<Rightarrow> False | x # xs \<Rightarrow> x = nA)"
CESem_simps_no_tick_2: fixes e :: "exp"   and x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and r :: "C" shows "(\<N>\<lbrakk> App e x \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<sqsubseteq> ((\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<down>CFn \<rho> x|\<^bsub>r\<^esub>)\<cdot>r"
bot_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "UML_Types.bot_class.bot \<equiv> Abs_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<bottom>"
fcompress_eq: fixes M :: "'d fset" shows "fcompress M = M"
sig_trd_aux_red_rtrancl: fixes args0 :: "'a \<times> ('t \<Rightarrow>\<^sub>0 'b)"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "fst args0 \<in> keys (rep_list (snd args0))" shows "(sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs))\<^sup>*\<^sup>* (snd args0) (sig_trd_aux bs args0)"
inf_par_distrib: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "par_distrib (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) ((c\<^sub>0::'b::refinement_lattice) \<sqinter> (c\<^sub>1::'b::refinement_lattice)) (d::'b::refinement_lattice) = par c\<^sub>0 d \<sqinter> par c\<^sub>1 d"
comp: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and Z :: "'d set set"   and g :: "'c \<Rightarrow> 'd" assumes "ChamberComplexMorphism X Y f"   and "ChamberComplexMorphism Y Z g" shows "ChamberComplexMorphism X Z (g \<circ> f)"
ats: shows "ats \<Turnstile>\<^sub>i \<B>\<I> s"
real_cond_exp_nested_subalg: fixes G :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "subalgebra M G"   and "subalgebra G Invariants"   and "integrable M f" shows "AE x in M. real_cond_exp M Invariants (real_cond_exp M G f) x = real_cond_exp M Invariants f x"
istopology_topology\<^sub>N: fixes N :: "'c quasinorm" shows "istopology (\<lambda>U. \<forall>x\<in>U. \<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U)"
csi: shows "weak_reduction_correspondence_simulation (indRelRTPO TRel) (STCal Source Target)"
idom_divide_hom_axioms: shows "idom_divide_hom hom"
simps_4: fixes \<phi> :: "atomUni fmUni"   and \<psi> :: "atomUni fmUni"   and x :: "real" shows "evalUni (AndUni \<phi> \<psi>) x = (evalUni \<phi> x \<and> evalUni \<psi> x)"
horizontal_composition_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" shows "horizontal_composition V H src trg \<equiv> horizontal_homs V src trg \<and> functor (subcategory.comp (product_category.comp V V) (\<lambda>\<mu>\<nu>. partial_magma.arr V (fst \<mu>\<nu>) \<and> partial_magma.arr V (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))) V (\<lambda>\<mu>\<nu>. H (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<and> horizontal_composition_axioms V H src trg"
neval_SucE: fixes x :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and n :: "nat"   and y :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "x \<leadsto>\<^bsup>Suc n\<^esup> y" obtains y :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "x \<leadsto> y" and "y \<leadsto>\<^bsup>n\<^esup> ya__"
free_gen_gen: fixes P :: "('g \<times> 'h, 'a, 'i) Module_scheme"   and M :: "('g, 'a, 'i) Module_scheme"   and N :: "('h, 'a, 'i) Module_scheme" assumes "FM\<^bsub>R\<^esub> P M N" shows "generator R P (M \<times>\<^sub>c N)"
Var: fixes x :: "Preliminaries.var" shows "\<phi> (Var x)"
cong: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "f = g"   and "x = y" shows "f x = g y"
assms: shows "m \<le>\<^sub>n SPEC (\<lambda>x. I x \<and> \<Phi> (\<alpha> x))"
flush_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i = load_store_type FLUSH" shows "snd (flush_instr i s) = False"
hyps_for'_def: shows "hyps_for' \<equiv> \<lambda>n p. {x. hyps_for'p n p x}"
neval_ZeroE: fixes x :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and y :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)" assumes "conc.neval x 0 y" obtains "y = x"
hyps: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'f option" assumes "thread_oks ts TAS"   and "I t = \<lfloor>i\<rfloor>"   and "ts t = \<lfloor>x\<rfloor>" shows "upd_invs I Q TAS t = \<lfloor>i\<rfloor>"
\<Gamma>_wf_1: fixes f :: "'fun"   and T :: "('fun, 'atom) Term.term list"   and t :: "('fun, 'var) Term.term" assumes "Fun f T \<sqsubseteq> \<Gamma> t" shows "0 < arity f"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d"   and I :: "'c set" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"   and "ideal I R addition multiplication zero unit"   and "ideal_in_kernel_axioms \<eta> R zero' I" shows "ideal_in_kernel \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit' I"
\<Phi>_suc: shows "\<Phi> C M ! (pc + 1) = \<lfloor>(ST', LT')\<rfloor>"
is_expansion_exp_neg: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "is_expansion (MS (MSLCons (C, e) xs) f) basis"   and "basis_wf basis"   and "e < 0" shows "is_expansion (MS (powser_ms_aux' exp_series_stream (MSLCons (C, e) xs)) (\<lambda>x. exp (f x))) basis"
elims_2: fixes x :: "nat"   and xa :: "nat" assumes "balanced1_arith x xa" obtains a :: "nat" and b :: "nat" where "x = a" and "xa = b" and "10 * (b + 1) \<le> 25 * (a + 1)"
three_ins_3: fixes e :: "'a"   and i :: "'a"   and t :: "'a" shows "e \<in> set [i, t, e]"
cong_eval_sym: fixes l :: "bdt"   and r :: "bdt" shows "l \<sim> r = r \<sim> l"
Diagonalize_Tensor_Arr_Unity: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t \<^bold>\<otimes> \<^bold>\<I>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
U_consts_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "\<not> w \<Turnstile>\<^sub>n \<phi> U\<^sub>n false\<^sub>n"
tseytin_tran1: fixes G :: "'a formula"   and F :: "'a formula"   and H :: "'a formula"   and S :: "'a formula \<Rightarrow> 'a formula"   and \<A> :: "'a \<Rightarrow> bool" assumes "G \<in> set (subformulae F)"   and "H \<in> set (tseytin_tran1 S G)"   and "\<forall>J\<in>set (subformulae F). tseytin_update \<A> F \<Turnstile> J = tseytin_update \<A> F \<Turnstile> S J" shows "tseytin_update \<A> F \<Turnstile> H"
map_14: fixes f :: "'c \<Rightarrow> 'b"   and x141 :: "'c ltlc"   and x142 :: "'c ltlc" shows "map_ltlc f (x141 M\<^sub>c x142) = map_ltlc f x141 M\<^sub>c map_ltlc f x142"
distinct_57: fixes x41 :: "'a fm"   and x42 :: "'a fm"   and x91 :: "nat"   and x92 :: "'a fm" shows "And x41 x42 \<noteq> ExN x91 x92"
assms_1: fixes n :: "nat" assumes "m \<le> n" shows "(\<Sum>k\<le>N. c k * f (n + k)) = (0::'a)"
wf_semiJoin_2: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and J :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set"   and QQ :: "(nat set \<times> 'b option list set) set"   and st :: "nat set"   and t :: "'b option list" assumes "getIJ getIJ"   and "1 \<le> card J"   and "wf_query n J Q Qn"   and "non_empty_query Q"   and "covering J Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"   and "QQ = (\<lambda>tab. New_max.semiJoin tab (st, t)) ` Q" shows "non_empty_query QQ"
sets_eq_bot2: fixes M :: "'c measure" shows "({{}} = sets M) = (M = bot)"
dom_restrict: fixes m :: "'a \<Rightarrow> 'b option"   and A :: "'a set" shows "dom (m |` A) = dom m \<inter> A"
Pbij_extends_transitive: fixes \<beta> :: "(Location \<times> Location) set"   and \<gamma> :: "(Location \<times> Location) set"   and \<delta> :: "(Location \<times> Location) set" assumes "Pbij_extends \<beta> \<gamma>"   and "Pbij_extends \<gamma> \<delta>" shows "Pbij_extends \<beta> \<delta>"
right_totalE: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e" assumes "right_total A" obtains x :: "'d" where "A x y"
gen_gba_impl_rel_eext_def: fixes Rm :: "('k \<times> 'n) set"   and Rl :: "('j \<times> ('l \<Rightarrow> 'm \<Rightarrow> bool)) set" shows "\<langle>Rm, Rl\<rangle>gen_gba_impl_rel_eext = {(\<lparr>gbai_L = Li, \<dots> = mi\<rparr>, \<lparr>gba_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
essentially_surjective: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "essentially_surjective_functor A B F"   and "partial_magma.ide B b" shows "\<exists>a. partial_magma.ide A a \<and> category.isomorphic B (F a) b"
match_list_True: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and ms :: "'c match_expr list"   and a :: "action"   and p :: "'d" assumes "match_list \<gamma> ms a p" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) ms) Undecided = (case a of Accept \<Rightarrow> Decision FinalAllow | Drop \<Rightarrow> Decision FinalDeny | Log \<Rightarrow> Undecided | Reject \<Rightarrow> Decision FinalDeny | Empty \<Rightarrow> Undecided)"
least_bounded_gr: fixes upper :: "nat"   and P :: "nat \<Rightarrow> bool"   and m :: "nat" assumes "\<exists>x<upper. P x" shows "(LEAST x. x < upper \<and> P x) = (LEAST x. x < upper + m \<and> P x)"
sum_vs: fixes U :: "'b set"   and E :: "'b set"   and V :: "'b set" assumes "Subspace.subspace U E"   and "Subspace.subspace V E"   and "vectorspace E" shows "vectorspace (U + V)"
NCSaux1: fixes p :: "('b, 'c) Combinators list"   and x :: "'b"   and y :: "'b"   and l :: "'b set list"   and u :: "'b"   and v :: "'b" assumes "noDenyAll p"   and "{x, y} \<in> set l"   and "all_in_list p l"   and "singleCombinators p"   and "FWNormalisationCore.sorted (DenyAllFromTo x y # p) l"   and "{x, y} \<noteq> firstList p"   and "DenyAllFromTo u v \<in> set p" shows "{x, y} \<noteq> {u, v}"
abs_dif_fiX_bound_C_aux2: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and x :: "real"   and y :: "real"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "\<forall>l\<in>C. \<bar>f l - g l\<bar> \<le> x"   and "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> y"   and "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>g l - g m\<bar> \<le> y"   and "p \<in> C"   and "q \<in> C"   and "r \<in> C" shows "y \<le> \<Delta> \<longrightarrow> \<bar>fiX f p r - fiX g q r\<bar> \<le> x"
hom_zero: shows "hom (0::'a) = (0::'b)"
are_endpoints_in_S_cross_ratio_correct: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" assumes "are_endpoints_in_S p q a b" shows "cross_ratio_correct p q a b"
step_thread_eq_Some_activeD: fixes update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and t' :: "'t"   and taxm\<sigma>' :: "(('l, 't, 'x, 'm, 'w, 'o) thread_action \<times> 'x \<times> 'm) option \<times> 'a" assumes "step_thread update_state s t = \<lfloor>(t', taxm\<sigma>')\<rfloor>" shows "t' = t \<and> t \<in> active_threads s"
simps_2: fixes t :: "'a rose_tree"   and i :: "nat"   and "is" :: "nat list" shows "tree_at t (i # is) = tree_at (children t ! i) is"
l11_30: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.CongA Bet Cong D E F D' E' F'" shows "Tarski_neutral_dimensionless.LeA Bet Cong A' B' C' D' E' F'"
int_pow_mult: fixes x :: "'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier \<G>" shows "x [^] (i + j) = x [^] i \<otimes> x [^] j"
valid_pub_OR_def: shows "valid_pub_OR = {(x0, x1). x0 \<in> valid_pub0 \<and> x1 \<in> valid_pub1}"
lcs_adds: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lcs :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and u :: "'b"   and t :: "'b" assumes "class.lcs_powerprod plus minus zero lcs"   and "plus.adds plus s u"   and "plus.adds plus t u" shows "plus.adds plus (lcs s t) u"
of_rat_0: shows "of_rat 0 = (0::'a)"
internal_iface_name_subset: fixes i1 :: "char list"   and i2 :: "char list" shows "internal_iface_name_subset i1 i2 = ({i. internal_iface_name_match i1 i} \<subseteq> {i. internal_iface_name_match i2 i})"
le: shows "\<forall>n. real (f n) \<le> C\<^sub>3 * real n + C\<^sub>4"
denote_while_n_sum_mat_seq: fixes \<rho> :: "complex mat"   and dims :: "nat list"   and x1 :: "nat \<Rightarrow> complex mat"   and x2 :: "com" assumes "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "x1 0 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "x1 1 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "partial_density_operator \<rho>"   and "state_sig.well_com dims x2"   and "measurement (state_sig.d dims) 2 x1" shows "matrix_seq (state_sig.d dims) (matrix_sum (state_sig.d dims) (\<lambda>n. denote_while_n (x1 0) (x1 1) (state_sig.denote dims x2) n \<rho>))"
lossless1: fixes pk :: "'a" shows "lossless_gpv \<I>_full (\<A>1 pk)"
rwof_eq_DFS_invar: fixes G :: "('a, 'b) graph_rec_scheme"   and param :: "('a, ('a, 'c) state_scheme, 'c) gen_parameterization" assumes "param_DFS G" shows "rwof (param_DFS_defs.init param) (param_DFS_defs.cond G param) (param_DFS_defs.step G param) = DFS_invar G param"
dirichlet_prod'_cong: fixes x :: "real"   and f :: "nat \<Rightarrow> 'a"   and f' :: "nat \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a"   and x' :: "real" assumes "\<And>n. \<lbrakk>0 < n; real n \<le> x\<rbrakk> \<Longrightarrow> f n = f' n"   and "\<And>y. \<lbrakk>1 \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> g y = g' y"   and "x = x'" shows "dirichlet_prod' f g x = dirichlet_prod' f' g' x'"
orderE: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" obtains "a = sup a b"
order_le_imp_less_or_eq: fixes x :: "'d"   and y :: "'d" assumes "x \<le> y" shows "x < y \<or> x = y"
lt_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.lt p = (if p = 0 then punit.min_term else ordered_powerprod_lin.Max (keys p))"
der: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(f has_real_derivative f' x) (at x)"
start_reachable: shows "G.reachable s\<^sub>0"
right_invertible_iff_invertible: fixes M :: "((real, 'a) vec, 'a) vec" shows "(\<exists>N. M ** N = mat 1) = invertible M"
A: shows "ipurge_tr g_I L High xs = filter (\<lambda>x'. L x' = Low) xs"
assoc_mult_mat_vec: fixes A :: "'a mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'a mat"   and n\<^sub>3 :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "v \<in> carrier_vec n\<^sub>3" shows "A * B *\<^sub>v v = A *\<^sub>v (B *\<^sub>v v)"
totient_prime_power_Suc: fixes p :: "nat"   and n :: "nat" assumes "prime p" shows "totient (p ^ Suc n) = p ^ n * (p - 1)"
partitions_zero: fixes p :: "nat \<Rightarrow> nat" shows "(p partitions 0) = (p = (\<lambda>i. 0))"
2: shows "index_to_residue s k \<in> UNIV \<rightarrow> carrier (residue_ring (p ^ k))"
map_entry_code: fixes k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and m :: "('c, 'd) mapping" shows "Mapping.map_entry k f m = (case Mapping.lookup m k of None \<Rightarrow> m | Some v \<Rightarrow> Mapping.update k (f v) m)"
balI: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "c \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> c'"   and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'"   and "stack c' = stack c" shows "c \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^esub> c'"
igWlsBinp_def: fixes MOD :: "('o, 'p, 'varSort, 'sort, 'q, 'r, 's, 't, 'u) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 't option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
eq_nextl_def: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.eq_nextl M \<equiv> {(u, v). dfa.nextl M (dfa.init M) u = dfa.nextl M (dfa.init M) v}"
equivalent_objects_reflexive: fixes a :: "'a" assumes "B.obj a" shows "B.equivalent_objects a a"
cut0: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "{} \<turnstile> A"   and "insert A H \<turnstile> B" shows "H \<turnstile> B"
frag_cmul_def: fixes c :: "int"   and a :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul c a = Abs_poly_mapping (\<lambda>x. c * lookup a x)"
All2_Eats_E: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "atom i \<sharp> (t, u)"   and "insert (All2 i t A) (insert (A(i::=u)) H) \<turnstile> B" shows "insert (All2 i (Eats t u) A) H \<turnstile> B"
eval_to: fixes t :: "lam"   and t' :: "lam" assumes "t \<Down> t'" shows "val t'"
h_diff_le_g: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_diff i e \<le> g e"
l_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "inv x \<otimes> x = \<one>"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "l_set_tag_name type_wf set_tag_name set_tag_name_locs"   and "l_set_tag_name_get_tag_name_axioms get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs" shows "l_set_tag_name_get_tag_name type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"