hcju/isabelleps huggingface dataset:
cong: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "F g A = F h B"
heap_is_wellformed_impl: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel" shows "heap_is_wellformed = l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes get_disconnected_nodes"
liftAll2_def: fixes phi :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and inp1 :: "'d \<Rightarrow> 'b option"   and inp2 :: "'d \<Rightarrow> 'c option" shows "liftAll2 phi inp1 inp2 \<equiv> \<forall>i v1 v2. inp1 i = Some v1 \<and> inp2 i = Some v2 \<longrightarrow> phi v1 v2"
downset_iso_iff: fixes x :: "'a"   and y :: "'a" shows "(\<down> x \<subseteq> \<down> y) = (x \<le> y)"
MapsTo_def: fixes CC :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and X :: "'a"   and Y :: "'a" shows "f maps\<^bsub>CC\<^esub> X to Y \<equiv> f \<in> mor\<^bsub>CC\<^esub> \<and> dom\<^bsub>CC\<^esub> f = X \<and> cod\<^bsub>CC\<^esub> f = Y"
assms_3: shows "Mono\<^bsub>L\<^esub> f"
poly_mod_type_def: fixes m :: "int" shows "poly_mod_type TYPE('b) m \<equiv> m = int CARD('b)"
remdups_id_iff_distinct: fixes xs :: "'c list" shows "(remdups xs = xs) = distinct xs"
emeasure_mono: fixes a :: "'a set"   and b :: "'a set"   and M :: "'a measure" assumes "a \<subseteq> b"   and "b \<in> sets M" shows "emeasure M a \<le> emeasure M b"
abs_model_check_def: fixes uu :: "'i itself"   and uv :: "'j itself"   and uw :: "'k itself"   and ux :: "'l itself"   and sys :: "('h, 'm set, 'n) sa_rec_scheme"   and \<phi> :: "'m ltlc" shows "abs_model_check uu uv uw ux sys \<phi> \<equiv> ltl_to_gba_spec (not\<^sub>c \<phi>) \<bind> (\<lambda>gba. ASSERT (igba gba) \<bind> (\<lambda>_. ASSERT (sa sys) \<bind> (\<lambda>_. inter_spec sys gba \<bind> (\<lambda>(Gprod, map_state). ASSERT (igb_graph Gprod) \<bind> (\<lambda>_. find_ce_spec Gprod \<bind> (\<lambda>ce. case ce of None \<Rightarrow> RETURN None | Some None \<Rightarrow> RETURN (Some None) | Some (Some r) \<Rightarrow> RETURN (Some (Some (map_state \<circ> r)))))))))"
submonoid_is_comm_monoid: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.comm_monoid G"   and "submonoid H G" shows "Group.comm_monoid (G\<lparr>carrier := H\<rparr>)"
rbt_lookup_rbt_bulkload: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "('c \<times> 'd) list" assumes "class.linorder less_eq less" shows "ord.rbt_lookup less (ord.rbt_bulkload less xs) = map_of xs"
B_5: fixes i :: "'a" assumes "i \<in> K" shows "F i \<in> sets (M i)"
faithful_functor_axioms: shows "faithful_functor (\<cdot>\<^sub>B) (\<cdot>) UP"
typing_alt_simps_36: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and x :: "'a collection_literal_part_expr"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>C [x] : \<tau>) = (\<Gamma> \<turnstile>\<^sub>P x : \<tau>)"
size_union: fixes M :: "'b multiset"   and N :: "'b multiset" shows "size (M + N) = size M + size N"
monad_prob_optionT: fixes return :: "'b option \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b option \<Rightarrow> 'c) \<Rightarrow> 'c"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_prob return bind sample" shows "monad_prob (return_option return) (bind_option return bind) (sample_option sample)"
hseq_char: fixes \<nu> :: "'d"   and \<mu> :: "'d" shows "D'.hseq \<nu> \<mu> = (D'.arr \<mu> \<and> D'.arr \<nu> \<and> D'.src \<nu> = D'.trg \<mu>)"
scaleC_nonpos_nonpos: fixes a :: "complex"   and b :: "'b" assumes "a \<le> 0"   and "b \<le> (0::'b)" shows "(0::'b) \<le> a *\<^sub>C b"
f10: shows "s'' \<otimes> s \<otimes> (s'' \<otimes> s') \<otimes> (r'' \<otimes> s' \<otimes> r) = s'' \<otimes> s \<otimes> s' \<otimes> (s'' \<otimes> s' \<otimes> (r'' \<otimes> r))"
proj_rep_set_finite: fixes n :: "nat" shows "finite (proj_rep_set n)"
linorder_less_linear: fixes x :: "'b"   and y :: "'b" shows "x < y \<or> x = y \<or> y < x"
assms_2: fixes \<chi> :: "'a ltl" assumes "\<chi> \<in> dom \<pi>" shows "the (\<pi> \<chi>) < max_rank_of \<Sigma> \<chi>"
simps: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf" shows "epo.gt_same ground_heads_var gt_sym extf a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> epo.chkchop_same (epo.gt ground_heads_var gt_sym extf) t s \<and> (\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (epo.gt ground_heads_var gt_sym extf) (args t) (args s)))"
Ext_declares_foo: shows "tprg\<turnstile> mdecl Ext_foo declared_in Ext"
sigma_sets_eq: shows "sigma_sets S {{}, X, S - X, S} = {{}, X, S - X, S}"
basis_indep_card: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set"   and B :: "'b set" assumes "matroid carrier indep"   and "indep X"   and "indep_system.basis carrier indep B" shows "card X \<le> card B"
degree_mult: fixes p :: "'a mpoly"   and q :: "'a mpoly"   and v :: "nat" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "MPoly_Type.degree (p * q) v = MPoly_Type.degree p v + MPoly_Type.degree q v"
mult_closed: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e"   and b :: "'e" assumes "Group G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<cdot>\<^bsub>G\<^esub> b \<in> carrier G"
simps_1: fixes G :: "('c, 'd) pre_digraph"   and c :: "'d list" assumes "wf_digraph G" shows "wf_digraph.mk_cycles_path 0 c = []"
ring_hom_cring_axioms: shows "ring_hom_cring R S \<phi>"
complex_clinearD: fixes f :: "complex \<Rightarrow> complex" assumes "clinear f" obtains c :: "complex" where "f = (*) c"
lexordp_irreflexive': fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" assumes "class.order less_eq less" shows "\<not> ord.lexordp less xs xs"
cs_less_split: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<prec> ys" obtains a :: "'a" and as :: "'a list" where "ys = xs @ a # as"
finsum_empty: fixes G :: "('e, 'f) ring_scheme"   and f :: "'g \<Rightarrow> 'e" assumes "abelian_monoid G" shows "finsum G f {} = \<zero>\<^bsub>G\<^esub>"
g_def: shows "g \<equiv> linepath a (a + complex_of_real s)"
card_Union_disjoint: fixes C :: "'a set set" assumes "pairwise disjnt C"   and "\<And>A. A \<in> C \<Longrightarrow> finite A" shows "card (\<Union> C) = sum card C"
assms: shows "\<bar>f x0\<bar> \<noteq> \<infinity>"
wf_jvm_prog_def: fixes P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog" shows "wf_jvm_prog P \<equiv> \<exists>\<Phi>. wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
lexext_hd_or_tl: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "lexext gt (y # ys) (x # xs)" shows "gt y x \<or> lexext gt ys xs"
dom_res_addition_in: fixes m\<^sub>1 :: "('d, 'c) fmap"   and k :: "'d"   and m\<^sub>2 :: "('d, 'e) fmap"   and v' :: "'e"   and v :: "'c" assumes "m\<^sub>1 $$ k = None"   and "m\<^sub>2 $$ k = Some v'" shows "fmdom' (m\<^sub>1(k $$:= v)) \<lhd> m\<^sub>2 = fmdom' m\<^sub>1 \<lhd> m\<^sub>2 ++\<^sub>f {k $$:= v'}"
THE_b: shows "(THE y. \<langle>a, y\<rangle> \<in>\<^sub>\<circ> r) = b"
ev_short_count_le: shows "\<forall>\<^sup>\<infinity>n. pf_short_count n < 1 / 2"
set_return_bind: fixes x :: "'d"   and A :: "'d \<Rightarrow> 'c set" shows "{x} \<bind> A = A x"
min_enat_simps_2: fixes q :: "enat" shows "min q 0 = 0"
powr_fast_growth_tendsto: fixes g :: "'a \<Rightarrow> real"   and F :: "'a filter"   and f :: "'a \<Rightarrow> real"   and n :: "real"   and k :: "real" assumes "g \<in> O[F](f)"   and "0 \<le> n"   and "1 < k"   and "filterlim f at_top F"   and "\<forall>\<^sub>F x in F. 0 \<le> g x" shows "(\<lambda>x. g x powr n) \<in> o[F](\<lambda>x. k powr f x)"
strict_from_inj_image: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and f :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n"   and "f \<in> Cauchy_Binet.F_inj n m" shows "strict_from_inj n f ` {0..<n} = f ` {0..<n}"
signs_are_cols_R: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" shows "\<forall>i<length signs. col (alt_matrix_A_R signs subsets) i = vec (length subsets) (\<lambda>j. z_R (subsets ! j) (signs ! i))"
dropreject: fixes m :: "'a match_expr"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Drop], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Reject], s\<rangle> \<Rightarrow> t"
Gp_inherited: fixes G :: "('e, 'f) Group_scheme"   and K :: "'e set"   and L :: "'e set" assumes "Group G"   and "G \<guillemotright> K "   and "G \<guillemotright> L "   and "K \<subseteq> L" shows "\<natural>\<^bsub>\<natural>\<^bsub>G\<^esub>L\<^esub>K = \<natural>\<^bsub>G\<^esub>K"
square_free_factor: fixes a :: "'c poly"   and p :: "'c poly" assumes "a dvd p"   and "square_free p" shows "square_free a"
linord_of_list_total_on: fixes xs :: "'b list" shows "total_on (set xs) (linord_of_list xs)"
quotient_rep_ex: fixes H :: "'a set"   and x :: "'b" assumes "H \<in> carrier (G LMod stabiliser x)" shows "rep H \<in> H"
trms_proj\<^sub>l\<^sub>s\<^sub>t_subset_1: fixes l :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) strand_step) list"   and B :: "('c strand_label \<times> ('a, 'b) strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>t (proj l A) \<subseteq> trms\<^sub>l\<^sub>s\<^sub>t (proj l (A @ B))"
fresh_list_append: fixes a :: "'a"   and xs :: "'b list"   and ys :: "'b list" shows "a \<sharp> (xs @ ys) = (a \<sharp> xs \<and> a \<sharp> ys)"
mem_append_def: shows "mem_append \<equiv> map_fun Rep_mem (map_fun Rep_bytes Abs_mem) (@)"
set_pivot_positions_main_gen: fixes ze :: "'b"   and A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" shows "set (pivot_positions_main_gen ze A nr nc i j) \<subseteq> {0..<nr} \<times> {0..<nc}"
cases: fixes x :: "nat \<times> nat \<times> 'b mat" obtains i :: "nat" and j :: "nat" and A :: "'b mat" where "x = (i, j, A)"
maximal_wrt_subst: fixes A :: "'a"   and \<sigma> :: "'s"   and C :: "'a literal multiset" assumes "maximal_wrt (A \<cdot>a \<sigma>) (C \<cdot> \<sigma>)" shows "maximal_wrt A C"
card_of_def: fixes A :: "'c set" shows "|A| = (SOME r. card_order_on A r)"
aform_form_def: fixes prec :: "nat"   and form :: "form"   and xs :: "(real \<times> real pdevs) list" shows "aform_form prec form xs = dRETURN (approx_form_aform prec form xs)"
\<pi>_def_on_edges_to_S: fixes v :: "'v"   and u :: "'v" assumes "v \<in> S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u))"   and "u \<noteq> r"   and "(u, v) \<in> edges rg" shows "\<pi>' Q \<pi> u u \<noteq> None"
ln_ge_zero_iff: fixes x :: "real" assumes "0 < x" shows "(0 \<le> ln x) = (1 \<le> x)"
A_5: fixes i :: "'a" assumes "i \<in> J" shows "E i \<in> sets (M i)"
simps: fixes a1 :: "fm set"   and a2 :: "fm" shows "a1 \<turnstile> a2 = ((\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> H) \<or> (\<exists>H. a1 = H \<and> a2 = extra_axiom) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> boolean_axioms) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> equality_axioms) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> special_axioms) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> HF_axioms) \<or> (\<exists>A H. a1 = H \<and> a2 = A \<and> A \<in> induction_axioms) \<or> (\<exists>H A B H'. a1 = H \<union> H' \<and> a2 = B \<and> H \<turnstile> A IMP B \<and> H' \<turnstile> A) \<or> (\<exists>H A B i. a1 = H \<and> a2 = SyntaxN.Ex i A IMP B \<and> H \<turnstile> A IMP B \<and> atom i \<sharp> B \<and> (\<forall>C\<in>H. atom i \<sharp> C)))"
invert: shows "\<forall>X\<in>P. (X \<union> {x \<in> B - B'. f x = X}) \<inter> C = X \<and> (X \<union> {x \<in> B - B'. g x = X}) \<inter> C = X"
cancels_to_self: fixes l :: "(bool \<times> 'd) list" shows "cancels_to l l"
own_dist: shows "(\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {}"
simps_3: fixes \<Sigma> :: "'b list"   and n :: "nat"   and m :: "nat"   and M :: "nat" assumes "formula \<Sigma>" shows "formula.rexp_of n (FIn m M) = rexp.Inter (TIMES [Full, Atom (Arbitrary_Except2 m M), Full]) (formula.ENC n {m})"
subdegree_decompose': fixes n :: "nat"   and f :: "'a fps" assumes "n \<le> subdegree f" shows "f = fps_shift n f * fps_X ^ n"
Diff_Un: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A - (B \<union> C) = (A - B) \<inter> (A - C)"
im_d_gchainsTr1_5: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat"   and j :: "nat" assumes "0 < n"   and "d_gchain G n f"   and "i \<le> n"   and "f i \<in> f ` {i. i \<le> n} - {f 0}"   and "(LEAST j. f j \<in> f ` {i. i \<le> n} - {f 0}) = j" shows "f ` {i. i \<le> j - Suc 0} = {f 0}"
is_class_Object: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "is_class P Object"
ln_less_self: fixes x :: "real" assumes "0 < x" shows "real_ln x < x"
listProd_def: fixes as :: "'c list"   and bs :: "'d list" shows "as \<times> bs \<equiv> concat (map (\<lambda>a. listProd1 a bs) as)"
distinct_find: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and x :: "'b" assumes "distinct (map f xs)"   and "x \<in> set xs" shows "find (\<lambda>x'. f x' = f x) xs = Some x"
se_preserves_finiteness1: fixes l :: "('c, 'd) label"   and c :: "('c, 'd) conf"   and c' :: "('c, 'd) conf" assumes "finite_label l"   and "se c l c'"   and "\<forall>e\<in>pred c. finite (Bexp.vars e)" shows "\<forall>e\<in>pred c'. finite (Bexp.vars e)"
rref: shows "reduced_row_echelon_form_upt_k A k"
red_diff_rtrancl: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "(red F)\<^sup>*\<^sup>* (p - q) 0" obtains s :: "'t \<Rightarrow>\<^sub>0 'c" where "(red F)\<^sup>*\<^sup>* p s" and "(red F)\<^sup>*\<^sup>* q s"
curryE: fixes f :: "'e \<times> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f" assumes "curry f a b" obtains "f (a, b)"
n_distrib1_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(n x \<oplus> n y) \<cdot> n z = n x \<cdot> n z \<oplus> n y \<cdot> n z"
card_Vi_le_NVLi: fixes i :: "nat"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" assumes "1 < i \<and> matching V E M" shows "card (V_i i V E M L) \<le> N V L i"
P_f: fixes c :: "'a acom" assumes "P c" shows "P (f c)"
lemma5_2: fixes r :: "dB"   and r' :: "dB" assumes "r \<rightarrow>\<^sub>l r'" shows "NF r'"
\<theta>\<^sub>f': shows "\<guillemotleft>\<theta>\<^sub>f' : tab\<^sub>0 f \<star> w\<^sub>f' \<rightarrow> u\<^sub>f\<guillemotright>"
functionOnFirstEqualsSecond: fixes R :: "('a \<times> 'b) set"   and z :: "'a \<times> 'b" assumes "runiq R"   and "z \<in> R" shows "R ,, fst z = snd z"
square_free_heuristic_def: fixes f :: "int poly" shows "square_free_heuristic f = (let lc = lead_coeff f in find (\<lambda>p. comm_monoid_mult_class.coprime lc p \<and> separable_impl p f) [2, 3, 5, 7, 11, 13, 17, 19, 23])"
uniqueD: fixes l :: "('c \<times> 'd) list"   and x :: "'c"   and y :: "'d"   and x' :: "'c"   and y' :: "'d" assumes "unique l"   and "(x, y) \<in> set l"   and "(x', y') \<in> set l"   and "x = x'" shows "y = y'"
Dom_f: shows "\<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"
\<E>: shows "E' \<in> h.mthr.if.\<E> (init_fin_lift_state status (h.start_state f P C M vs))"
module_hom_axioms_def: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" shows "module_hom_axioms s1 s2 f \<equiv> (\<forall>b1 b2. f (b1 + b2) = f b1 + f b2) \<and> (\<forall>r b. f (s1 r b) = s2 r (f b))"
elem_xs: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> vcard xs" shows "xs\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i"
subgroupE_1: fixes H :: "int set" assumes "subgroup H (Z n)" shows "H \<subseteq> carrier (Z n)"
Oops: fixes evso :: "event list"   and A :: "agent"   and B :: "agent"   and K :: "nat"   and NA :: "nat"   and NB :: "nat"   and X :: "msg" assumes "evso \<in> yahalom"   and "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, X\<rbrace> \<in> set evso" shows "Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso \<in> yahalom"
S5S6LeadstoS6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and sigma :: "behavior" assumes "sigma \<Turnstile> S5 rmhist p \<leadsto> S6 rmhist p" shows "sigma \<Turnstile> S5 rmhist p \<or> S6 rmhist p \<leadsto> S6 rmhist p"
formulaUnsatIffImpliesEmptyClause: fixes formula :: "Literal list list" shows "formulaEntailsClause formula [] = (\<not> satisfiable formula)"
fls_def: fixes \<pi>l :: "('state \<times> 'aprop set) stream list" shows "fls \<pi>l \<equiv> False"
Lim_def: fixes A :: "'f filter"   and f :: "'f \<Rightarrow> 'e" shows "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
lsc_at_def: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "lsc_at x0 f = (\<forall>X l. X \<longlonglongrightarrow> x0 \<and> (f \<circ> X) \<longlonglongrightarrow> l \<longrightarrow> f x0 \<le> l)"
b0: shows "b $ 0 \<noteq> (0::'a)"
prod_add_distrib: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite I" shows "(\<Prod>i\<in>I. f i + g i) = (\<Sum>J\<in>Pow I. prod f J * prod g (I - J))"
baIANow: fixes \<phi> :: "'b \<Rightarrow> bool"   and tCMP :: "'c \<Rightarrow> cnf \<Rightarrow> 'b"   and c :: "'c"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and active :: "'c \<Rightarrow> cnf \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'b" assumes "\<phi> (tCMP c (t n))"   and "active c (t n)" shows "dynamic_component.eval tCMP active c t t' n (dynamic_component.ba \<phi>)"
trans_F: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq>\<^sub>F Q"   and "Q \<sqsubseteq>\<^sub>F S" shows "P \<sqsubseteq>\<^sub>F S"
euclidean_space_basis_vector_def: shows "euclidean_space_basis_vector \<equiv> map_fun id Abs_euclidean_space (\<lambda>x. indicat_real {x})"
step_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Seq c1 c2, s) \<rightarrow> u" obtains c\<^sub>1' :: "('e, 'd, 'f) com" and s' :: "('e, 'f) xstate" where "u = (Seq c\<^sub>1' c2, s')" and "\<Gamma>\<turnstile> (c1, s) \<rightarrow> (c\<^sub>1', s')"   | "c1 = Skip" and "u = (c2, s)"   | sa :: "'e" where "u = (Throw, Normal sa)" and "c1 = Throw" and "s = Normal sa"   | f :: "'f" where "s = Fault f" and "u = (Skip, Fault f)" and "redex c1 = Seq c1 c2"   | "s = Stuck" and "u = (Skip, Stuck)" and "redex c1 = Seq c1 c2"
rep_eq: fixes x :: "('a, 'b) alist" shows "DAList.lookup x = map_of (alist.impl_of x)"
get_disconnected_nodes_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr" assumes "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "type_wf h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_disconnected_nodes document_ptr"
intros_4: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and b :: "'d"   and c\<^sub>1 :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and c\<^sub>2 :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "eval\<^sub>B mem b"   and "sifum_lang_no_dma.no_await c\<^sub>1"   and "(((c\<^sub>1, mds), mem), (c\<^sub>2, mds'), mem') \<in> (sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B)\<^sup>+"   and "sifum_lang_no_dma.is_final c\<^sub>2" shows "(((Await b c\<^sub>1, mds), mem), (c\<^sub>2, mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
invf_inv: fixes K :: "('e, 'f) Ring_scheme"   and x :: "'e" assumes "Corps K"   and "x \<in> carrier K - {\<zero>\<^bsub>K\<^esub>}" shows "x\<^bsup>\<hyphen> K\<^esup> \<in> carrier K \<and> x\<^bsup>\<hyphen> K\<^esup> \<cdot>\<^sub>r\<^bsub>K\<^esub> x = 1\<^sub>r\<^bsub>K\<^esub>"
conf_sys_xcpt: fixes h :: "heap"   and C :: "char list"   and P :: "'a prog" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "P,h \<turnstile> Addr (addr_of_sys_xcpt C) :\<le> Class C"
continuous_on_cases_le: fixes s :: "'b set"   and h :: "'b \<Rightarrow> real"   and a :: "real"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on {t \<in> s. h t \<le> a} f"   and "continuous_on {t \<in> s. a \<le> h t} g"   and "continuous_on s h"   and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t" shows "continuous_on s (\<lambda>t. if h t \<le> a then f t else g t)"
tendsto_Im2: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im2 (g x)) \<longlongrightarrow> Im2 a) F"
closure_of_minimal_eq: fixes S :: "'c set"   and X :: "'c topology"   and T :: "'c set" assumes "S \<subseteq> topspace X"   and "closedin X T" shows "(X closure_of S \<subseteq> T) = (S \<subseteq> T)"
le_infI2: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and x :: "'c"   and a :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq b x" shows "less_eq (inf a b) x"
tdghm_is_tdghm_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<beta>\<^esub> \<BB>"
neval_S_n_3: assumes "y \<in> loc_reach lc" shows "z \<in> loc_reach lc"
preserves_ide: fixes a :: "'a" assumes "Adj.C.ide a" shows "Adj.D.ide (G' a)"
tc_tail: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.tail p \<noteq> 0" shows "punit.tc (punit.tail p) = punit.tc p"