hcju/isabelleps huggingface dataset:
x_y_in_carrier: fixes x :: "'a pmf"   and y :: "'a pmf" assumes "x \<succeq> y" shows "x \<in> \<P> outcomes \<and> y \<in> \<P> outcomes"
netmask_def: fixes I :: "nat set"   and s :: "(nat \<Rightarrow> 'c) \<times> 'd" shows "netmask I s \<equiv> (\<lambda>i. if i \<in> I then Some (fst s i) else None, snd s)"
mset_ran_mem: fixes r :: "int set"   and i :: "int"   and a :: "int \<Rightarrow> 'b" assumes "finite r"   and "i \<in> r" shows "a i \<in># mset_ran a r"
nat_filtration_comp_measurable: fixes p :: "real"   and M :: "bool stream measure"   and f :: "bool stream \<Rightarrow> 'b"   and N :: "'b measure"   and n :: "nat" assumes "infinite_coin_toss_space p M"   and "f \<in> M \<rightarrow>\<^sub>M N"   and "f \<circ> pseudo_proj_True n = f" shows "f \<in> infinite_coin_toss_space.nat_filtration M n \<rightarrow>\<^sub>M N"
single: fixes k :: "'a"   and v :: "'b" shows "P (Poly_Mapping.single k v)"
simps_8: fixes \<phi> :: "'a ltln"   and \<xi> :: "'a ltln" shows "suspendable (\<phi> M\<^sub>n \<xi>) = (suspendable \<phi> \<and> suspendable \<xi> \<or> pure_universal \<phi> \<and> \<xi> = true\<^sub>n)"
assms: shows "\<R>\<^sub>\<bullet> r \<subseteq>\<^sub>\<circ> B"
rbt_min_eq_rbt_min_opt: fixes t :: "('b, unit) RBT_Impl.rbt" assumes "t \<noteq> rbt.Empty"   and "is_rbt t" shows "rbt_min t = rbt_min_opt t"
distinct_15: fixes x31 :: "POPLmark.type"   and x32 :: "trm"   and x41 :: "trm"   and x42 :: "trm" shows "(\<lambda><:x31. x32) \<noteq> x41 \<bullet> x42"
assms_4: shows "T2 \<in> M2 \<rightarrow>\<^sub>M M2"
g_1_lemma: fixes res :: "int \<times> int" assumes "0 \<le> fst res"   and "fst res < int p"   and "0 \<le> snd res"   and "snd res < int q" shows "P_1 res (g_1 res)"
lm01: fixes a :: "(integer \<times> 'c set) set"   and N :: "integer set"   and \<Omega> :: "'c set" assumes "a \<in> soldAllocations'' N \<Omega>" shows "Domain a \<subseteq> N - {seller} \<and> a \<in> allocationsUniverse"
ncoplanar_perm_9: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet B C D A"
simps_2: fixes h :: "heap" shows "typeof\<^bsub>h\<^esub> Null = \<lfloor>NT\<rfloor>"
deadlockD1: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'a" assumes "multithreaded_base.deadlock final r s"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "\<not> final x"   and "wset s t = None" obtains "multithreaded_base.must_sync final r t x (shr s)" and "\<forall>LT. multithreaded_base.can_sync r t x (shr s) LT \<longrightarrow> (\<exists>lt\<in>LT. final_thread.must_wait final s t lt (dom (thr s)))"
psi_not_divergent_init: fixes t :: "nat"   and j :: "nat"   and b :: "nat" assumes "local.prefixes t j \<down>= b" shows "\<psi> j \<triangleright> (e_length b - 1) = b"
fun_relE1: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'e"   and Rv :: "('d \<times> 'e) set"   and t' :: "'e"   and x :: "'c" assumes "(f, f') \<in> Id \<rightarrow> Rv"   and "t' = f' x" shows "(f x, t') \<in> Rv"
bex_Infinitesimal_iff: fixes x :: "'a star"   and z :: "'a star" shows "(\<exists>y\<in>Infinitesimal. x - z = y) = (x \<approx> z)"
step: fixes xs :: "'a llist" assumes "X xs"   and "\<not> lnull xs" shows "(\<forall>x\<in>lset (ltl xs). lhd xs \<le> x) \<and> (X (ltl xs) \<or> lsorted (ltl xs))"
R: shows "R = (\<le>) \<or> R = (\<lambda>x y. y \<le> x)"
strict_prefix_def: fixes xs :: "'d list"   and ys :: "'d list" shows "strict_prefix xs ys = (prefix xs ys \<and> xs \<noteq> ys)"
of_int_div_of_int_in_Ints_iff: fixes n :: "int"   and m :: "int" shows "(of_int n / of_int m \<in> \<int>) = (m = 0 \<or> m dvd n)"
r_zero: fixes G :: "('d, 'e) ring_scheme"   and x :: "'d" assumes "abelian_monoid G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<zero>\<^bsub>G\<^esub> = x"
rep_inverse: fixes x :: "('a, 'b) mapping" shows "Mapping (mapping.rep x) = x"
ap_gpv_homo: fixes f :: "'i \<Rightarrow> 'f"   and x :: "'i" shows "Generative_Probabilistic_Value.Done f \<diamondop> Generative_Probabilistic_Value.Done x = Generative_Probabilistic_Value.Done (f x)"
right_neutral: fixes a :: "'a" shows "a \<^bold>\<squnion> \<zero> = a"
valid_sops_stmt_sb: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)"   and s' :: "stmt"   and t' :: "nat" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)"   and "(s', t') \<in> prog_configs sb" shows "valid_sops_stmt t' s'"
Buffer_def: fixes ic :: "state \<Rightarrow> 'a"   and oc :: "state \<Rightarrow> 'a" shows "Buffer ic oc \<equiv> \<exists>\<exists> q. IBuffer ic q oc"
rcos_module: fixes x :: "'a"   and x' :: "'a" assumes "Group.group G"   and "x \<in> carrier G"   and "x' \<in> carrier G" shows "(x' \<in> H #> x) = (x' \<otimes> inv x \<in> H)"
assms_2: shows "k = card (Units (residue_ring (p ^ m)))"
norm_pths_2: fixes x :: "'c"   and y :: "'c" shows "(x \<noteq> y) = (\<not> norm (x - y) \<le> 0)"
singular_subdivision_def: fixes p :: "nat" shows "singular_subdivision p \<equiv> frag_extend (\<lambda>f. chain_map p f (simplicial_subdivision p (frag_of (restrict id (standard_simplex p)))))"
correctness: fixes e :: "('d, 'c) expr"   and env :: "'d \<Rightarrow> 'c" shows "execute (compile e) env = eval e env"
less_eq_decr_length_iff: fixes n :: "nat" shows "(n \<le> LENGTH('b) - Suc 0) = (n < LENGTH('b))"
gen_dfsT_assert_def: fixes gds :: "('b, 'a) gen_dfs_struct"   and V0 :: "'b set" shows "gen_dfs_defs.gen_dfsT_assert gds V0 \<equiv> gds_init gds \<bind> WHILE\<^sub>T (gen_dfs_defs.gen_cond gds V0) (gen_dfs_defs.gen_step_assert gds V0)"
atoms_mono: fixes c :: "atom"   and G :: "chmsg set"   and H :: "chmsg set" assumes "c \<in> atoms G"   and "G \<subseteq> H" shows "c \<in> atoms H"
frequently_at_topE': fixes P :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "nat \<Rightarrow> 'b" assumes "\<forall>n. frequently (P n) at_top"   and "filterlim g at_top sequentially" obtains s :: "nat \<Rightarrow> 'b" where "\<And>i. P i (s i)" and "strict_mono s" and "\<And>n. g n \<le> s n"
simps_2: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and a :: "nat" shows "(P,h,sh \<turnstile>\<^sub>i (C, M, pc, Throwing Cs a)) = (let ins = instrs_of P C M in \<exists>C1. Called_context P C1 (ins ! pc) \<and> (\<exists>obj. h a = \<lfloor>obj\<rfloor>))"
szpilrajn: fixes p :: "'a" shows "[order p \<and> regular p]   t := p; WHILE \<not> linear t INV {order t \<and> regular t \<and> p \<le> t}  VAR {\<lambda>(e, t). card {x. regular x \<and> x \<le> - (t \<squnion> t\<^sup>T)}}  DO e := choose_arc (- (t \<squnion> t\<^sup>T)); t := t \<squnion> t * e * t OD   [linear_orderable_1 t \<and> p \<le> t]"
FunctorId2: fixes X :: "'o1" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "F @@ X \<in> obj\<^bsub>CatCod F\<^esub> \<and> F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
sqrt_le: fixes n :: "nat" shows "Discrete.sqrt n \<le> n"
exp_scaleR_has_vector_derivative_right: fixes A :: "'b"   and t :: "real"   and T :: "real set" shows "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative exp (t *\<^sub>R A) * A) (at t within T)"
Mapping_cases: fixes x :: "('a, 'b) mapping" obtains y :: "'a \<Rightarrow> 'b option" where "x = mapping.Mapping y" and "y \<in> UNIV"
max_ipv4_addr_max_word: shows "max_ipv4_addr = - 1"
continuous_on_sequentiallyI: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>u a. \<lbrakk>\<forall>n. u n \<in> s; a \<in> s; u \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a" shows "continuous_on s f"
star_right_induct_mult_equal: fixes y :: "'a"   and x :: "'a" assumes "y * x = y" shows "y * x\<^sup>\<star> \<le> y"
P1'_def: shows "P1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P1"
strong_induct: fixes P :: "'a \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a"   and trm :: "trm" assumes "\<And>var z. P z (Var var)"   and "\<And>trm1 trm2 z. \<lbrakk>\<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (App trm1 trm2)"   and "\<And>var trm z. \<lbrakk>var \<sharp> z; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (Lam [var].trm)"   and "\<And>var trm1 trm2 z. \<lbrakk>var \<sharp> z; var \<sharp> trm2; \<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (Let var be trm2 in trm1)" shows "P z trm"
hseqI: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr"   and c :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>S c\<guillemotright>" shows "S.hseq \<nu> \<mu>"
assms_1: shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"
hom_0: fixes x :: "'a poly poly" assumes "x_y * x = 0" shows "x = 0"
connected_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>* v"   and "v \<rightarrow>\<^sup>* w" shows "u \<rightarrow>\<^sup>* w"
simps: fixes Texp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var ttree"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "TTreeAnalysis.prognosis Texp ae as a (\<Gamma>, e, S) = pathsCard (paths (substitute (ExpAnalysis.AnalBinds Texp \<Gamma>\<cdot>ae) (thunks \<Gamma>) (Texp e\<cdot>a \<otimes>\<otimes> TTreeAnalysis.Fstack Texp as S)))"
irreducible_altdef: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "comm_semiring_1.irreducible times one zero x = (x \<noteq> zero \<and> \<not> dvd.dvd times x one \<and> (\<forall>b. dvd.dvd times b x \<longrightarrow> dvd.dvd times x b \<or> dvd.dvd times b one))"
bisim1CASFail: fixes P :: "'a prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and v' :: "'addr val"   and v :: "'addr val"   and v'' :: "'addr val" shows "P,e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3),h \<turnstile> (Throw ad, xs) \<leftrightarrow> ([v', v, v''], xs, length (compE2 e1) + length (compE2 e2) + length (compE2 e3), \<lfloor>ad\<rfloor>)"
canonical_retraction: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and C :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "A \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "C \<in> A" shows "ChamberComplexRetraction X (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A C)"
countable_subset_image: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(countable B \<and> B \<subseteq> f ` A) = (\<exists>A'. countable A' \<and> A' \<subseteq> A \<and> B = f ` A')"
measurable_on_const_UNIV: fixes k :: "'d" shows "(\<lambda>x. k) measurable_on UNIV"
below_less_iff: fixes i :: "nat"   and k :: "nat" shows "(i \<in> below k) = (i < k)"
one_le_zcount_after_summary: fixes s :: "'sum"   and S :: "'sum antichain"   and t :: "'t" assumes "s \<in>\<^sub>A S" shows "1 \<le> zcount (after_summary {#t#}\<^sub>z S) (results_in t s)"
meet_assoc: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" shows "x \<inter> y \<inter> z = x \<inter> (y \<inter> z)"
Int_lower1: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> B \<subseteq> A"
le_RealI: fixes Y :: "nat \<Rightarrow> rat"   and x :: "real" assumes "Real.cauchy Y"   and "\<forall>n. x \<le> real_of_rat (Y n)" shows "x \<le> Real.Real Y"
simps_1: fixes x :: "Preliminaries.var" shows "nvT (Var x) = {x}"
subset_bound_invariant: fixes r :: "Vars.var list" shows "invariant (\<Rightarrow>) (r_ok r)"
ASSN_ANNOT_def: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a" shows "ASSN_ANNOT A x \<equiv> x"
diff_union_swap2: fixes y :: "'b"   and M :: "'b multiset"   and x :: "'b" assumes "y \<in># M" shows "remove1_mset y (add_mset x M) = add_mset x (remove1_mset y M)"
span_subspace_induct: fixes x :: "'b"   and S :: "'b set"   and P :: "'b set" assumes "x \<in> span S"   and "subspace P"   and "\<And>x. x \<in> S \<Longrightarrow> x \<in> P" shows "x \<in> P"
no_type_error: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "\<Phi> \<turnstile> t:\<sigma> \<surd>" shows "exec_d P t \<sigma> \<noteq> TypeError"
K'_3: fixes n :: "nat" shows "K' n \<subseteq> fm n ` B n"
measure_mono_eq_prod: fixes f :: "complex poly"   and g :: "complex poly" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "mahler_measure_monic (f * g) = mahler_measure_monic f * mahler_measure_monic g"
cring_simprules_20: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = \<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
g: shows "(g \<longlongrightarrow> (0::'b)) F"
order_trans_rules_25: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
has_bochner_integral_mult_right: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. c * f x) (c * x)"
rreq_rrep_fresh_def: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and m :: "msg" shows "rreq_rrep_fresh crt m \<equiv> case m of Rreq hopsc x xa xb xc oipc osnc ipcc xd \<Rightarrow> ipcc \<noteq> oipc \<longrightarrow> oipc \<in> kD crt \<and> (osnc < sqn crt oipc \<or> sqn crt oipc = osnc \<and> the (dhops crt oipc) \<le> hopsc \<and> the (flag crt oipc) = val) | Rrep hopsc dipc dsnc x ipcc \<Rightarrow> ipcc \<noteq> dipc \<longrightarrow> dipc \<in> kD crt \<and> sqn crt dipc = dsnc \<and> the (dhops crt dipc) = hopsc \<and> the (flag crt dipc) = val | _ \<Rightarrow> True"
NEST_def: fixes S :: "'b set" shows "NEST S \<equiv> \<exists>i. \<I> i \<and> (S = BEGIN i \<or> S = END i)"
lhs: shows "(\<Sum>x\<in>#add_mset y A. x) = (\<Sum>x\<in>#A. x) + y"
ag_r_zero: fixes x :: "'a" assumes "x \<in> carrier M" shows "x \<plusminus> \<zero> = x"
not_MTimpnotMT: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and p :: "'b list" assumes "not_MT C p" shows "p \<noteq> []"
hypreal_sqrt_hrabs2: fixes x :: "real star" shows "(*f* sqrt) (x * x) = \<bar>x\<bar>"
mon_rule_def: shows "mon_rule \<equiv> ([pusall \<rightarrow> \<langle> Game hgidd \<rangle> TT], \<langle> Game hgid1 \<rangle> pusall \<rightarrow> \<langle> Game hgid1 \<rangle> \<langle> Game hgidd \<rangle> TT)"
differentiable_at_withinI: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" assumes "f differentiable at x" shows "f differentiable at x within s"
UP_pre_univ_prop_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "UP_pre_univ_prop R S h \<equiv> ring_hom_cring R S h \<and> UP_cring R"
in_prime_factors_imp_prime: fixes p :: "'a"   and x :: "'a" assumes "p \<in># prime_factorization x" shows "prime p"
regular_complement_top: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_p_algebra.regular x" shows "x \<oplus> \<ominus> x = mtop"
csm': shows "cs\<^bsup>\<pi>'\<^esup> m = cs\<^bsup>\<pi>'\<^esup> k'"
assms: fixes c' :: "real" assumes "ereal c' < c" shows "\<exists>s. s \<bullet> (1::'a) = c' \<and> \<not> fds_converges f s"
filter_distinct_at4: fixes xs :: "'a list"   and as :: "'a list"   and u :: "'a"   and bs :: "'a list"   and us :: "'a list"   and zs :: "'a list" assumes "distinct xs"   and "xs = as @ u # bs"   and "filter (\<lambda>v. v = u \<or> v \<in> set us) xs = u # us"   and "set zs \<inter> set us \<subseteq> {u} \<union> set as" shows "filter (\<lambda>v. v \<in> set us) (zs @ bs) = us"
assms_3: shows "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (mat_leqb_eqc A b c)"
one_local_ring_at_def: shows "one_local_ring_at \<equiv> local.one_rel"
intro: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and num :: "'e set"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and enc :: "'f \<Rightarrow> 'e"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and "proof" :: "'g set"   and prfOf :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and encPf :: "'g \<Rightarrow> 'e"   and Pf :: "'f"   and isTrue :: "'f \<Rightarrow> bool" assumes "CleanRepr_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc fls dsj proof prfOf encPf Pf"   and "Minimal_Truth_Soundness var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num bprv isTrue" shows "Minimal_Truth_Soundness_Proof_Repr var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv isTrue enc proof prfOf encPf Pf"
rA': shows "local.sub2 (dim_row A) l (r, A) = (r', A')"
trm_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<t>[a] : a \<rightarrow> \<one>\<guillemotright>"
same_on_vars_lit: fixes L :: "('b, 'a) Term.term literal"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<forall>v\<in>vars_lit L. \<sigma> v = \<tau> v" shows "subst_lit L \<sigma> = subst_lit L \<tau>"
contrapos_pn: fixes Q :: "bool"   and P :: "bool" assumes "Q"   and "P \<Longrightarrow> \<not> Q" shows "\<not> P"
isPrefixPrefixToLevel_aux: fixes t :: "('b \<times> bool) list"   and l :: "nat"   and i :: "nat" shows "\<exists>s. prefixToLevel_aux t l i @ s = t"
assms_2: fixes c :: "'a" assumes "\<And>a. a \<in> A \<Longrightarrow> c dvd a" shows "c dvd b"
qfree_list_conj: fixes fs :: "'a fm list" assumes "\<forall>\<phi>\<in>set fs. qfree \<phi>" shows "qfree (list_conj fs)"
key_sets_crypts_hash: fixes X :: "msg"   and H :: "msg set" shows "key_sets (Hash X) (crypts H) \<subseteq> key_sets X (crypts H)"
rep_eq: fixes x :: "('a, 'b) fmap" shows "fmdom' x = dom (fmlookup x)"
VS_com_Deriv: fixes t :: "TP"   and c :: "OBJ"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "(t, c) \<in> VS_com" shows "(t = high \<longrightarrow> (G, c, HighSec) \<in> Deriv) \<and> (t = low \<longrightarrow> (\<exists>\<Phi>. (G, c, Sec \<Phi>) \<in> Deriv))"
PX4d_range: fixes d :: "D2" shows "Field (PX4d d) \<subseteq> {x. X4d x = d}"
StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y \<equiv> \<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then if x \<tau> = null \<or> y \<tau> = null then \<lfloor>\<lfloor>x \<tau> = null \<and> y \<tau> = null\<rfloor>\<rfloor> else \<lfloor>\<lfloor>oid_of (x \<tau>) = oid_of (y \<tau>)\<rfloor>\<rfloor> else invalid \<tau>"
conjunction_in: fixes U :: "nat set"   and p :: "fm"   and q :: "fm" shows "valid_in U (Con' p q) = (valid_in U p \<and> valid_in U q)"
simps_3: fixes x :: "'b trm"   and y :: "'b trm" shows "is_compound (x \<cdot> y) = True"
mds_consistent_def: fixes mds :: "Mode \<Rightarrow> Eg1.var set"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" shows "A.mds_consistent mds \<Gamma> \<S> P \<equiv> \<S> = (mds AsmNoWrite, mds AsmNoReadOrWrite) \<and> dom \<Gamma> = {x. x \<notin> \<C> \<and> A.stable \<S> x} \<and> A.type_stable \<S> P"
step: shows "SRabbr (V' ! i) m1 \<alpha> p m2"
closest_supchamberD_1: fixes F :: "'b set"   and D :: "'b set" assumes "F \<in> Y"   and "codomain.chamber D" shows "codomain.chamber (codomain.closest_supchamber F D)"
take_0: shows "\<psi>\<^sup>D\<^bsub>0\<^esub> = \<bottom>"
intros_8: fixes F :: "'b formula" shows "(F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<^bold>\<not> F \<in> AX10"
jmpl_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (jmpl_instr instr s1)) \<and> t2 = snd (fst (jmpl_instr instr s2))" shows "low_equal t1 t2"
inter_add_left2: fixes x :: "'k"   and N :: "'k multiset"   and M :: "'k multiset" assumes "x \<in># N" shows "add_mset x M \<inter># N = add_mset x (M \<inter># (N - {#x#}))"
rotation_minus_pi_2_y_axis: shows "moebius_circline (moebius_rotation (- pi / 2)) y_axis = x_axis"
associated_iff_same_ideal: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "cring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<sim>\<^bsub>R\<^esub> b = (PIdl\<^bsub>R\<^esub> a = PIdl\<^bsub>R\<^esub> b)"
triangle: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "Turn_into_distance f" shows "turn_into_distance f x z \<le> turn_into_distance f x y + turn_into_distance f y z"
add_closed: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "on_curve a b (ell_field_class.add a p\<^sub>1 p\<^sub>2)"
redT_updT_nta_bisim_inv: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta1 :: "('a, 'b, 'c) new_thread_action"   and ta2 :: "('a, 'd, 'e) new_thread_action"   and ts1 :: "'a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option"   and T :: "'a"   and ts2 :: "'a \<Rightarrow> ('d \<times> 'g \<Rightarrow>f nat) option" assumes "nta_bisim bisim ta1 ta2"   and "(ts1 T = None) = (ts2 T = None)" shows "(redT_updT ts1 ta1 T = None) = (redT_updT ts2 ta2 T = None)"
cl_augment: fixes X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "X \<subseteq> \<E>"   and "x \<in> \<E>"   and "y \<in> \<E>.cl (insert x X) - \<E>.cl X" shows "x \<in> \<E>.cl (insert y X)"