hcju/isabelleps huggingface dataset:
is_shadow_root_ptr_not_node_ptr: fixes node_ptr :: "(_) node_ptr" shows "\<not> is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)"
Cons_less_iff: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(x # xs < ys) = (\<exists>y ys'. ys = y # ys' \<and> (x < y \<or> x = y \<and> xs < ys'))"
dns3: fixes x :: "'a"   and y :: "'a" shows "ad (ad (x + y)) = ad (ad x) + ad (ad y)"
sum_lower: fixes I :: "'b set"   and i :: "'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "finite I"   and "i \<in> I"   and "\<And>i. i \<in> I \<Longrightarrow> (0::'c) \<le> f i"   and "x < f i" shows "x < sum f I"
linear_inj_imp_surj: fixes f :: "('d, 'e) vec \<Rightarrow> ('d, 'e) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f" shows "surj f"
simps_1: fixes v :: "'d \<Rightarrow> 'g" shows "switch v [] = Skip"
sup_absorb1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x + y = x"
lex_pm_refl: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "lex_pm s s"
set_sort_oalist: fixes xs :: "('a \<times> 'b) list"   and ko :: "'a nat_term_order" assumes "distinct (map fst xs)" shows "set (sort_oalist_ko_ntm' ko xs) = {kv \<in> set xs. snd kv \<noteq> (0::'b)}"
normal_subgroup_set_mult_closed: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and N :: "'c set" assumes "Group.group G"   and "M \<lhd> G"   and "N \<lhd> G" shows "M <#>\<^bsub>G\<^esub> N \<lhd> G"
bcontfun\<^sub>ND_1: fixes f :: "'b \<Rightarrow> 'c" assumes "f \<in> space\<^sub>N bcontfun\<^sub>N" shows "continuous_on UNIV f"
simps_8: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "is_\<mu>LTL (\<phi> U\<^sub>n \<psi>) = (is_\<mu>LTL \<phi> \<and> is_\<mu>LTL \<psi>)"
parts_crypt_sign_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and n :: "nat"   and A :: "nat" assumes "s \<turnstile> s'"   and "Crypt (SesK SK) (Sign n A) \<in> parts (used s')"   and "Crypt (SesK SK) (Sign n A) \<notin> parts (used s)" shows "(Asset n, SesKey SK) \<in> s' \<or> SesKey SK \<in> spied s'"
lub_spmf_subprob: shows "\<integral>\<^sup>+ x. (\<Squnion>p\<in>Y. ennreal (spmf p x)) \<partial>count_space UNIV \<le> 1"
Rep_numeral: fixes w :: "num" shows "Rep (numeral w) = numeral w mod n"
x: shows "LeftDerives1 v (fst d) (snd d) x \<and> LeftDerivation x D w"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('a, 'b) atm" shows "U.satL \<xi> (Neg a) = (\<not> U.satA \<xi> a)"
inner_sum_left: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'a" shows "sum f A \<bullet> y = (\<Sum>x\<in>A. f x \<bullet> y)"
exhaust_sel: fixes pred :: "'a Predicate.pred" obtains "pred = pred.Pred (pred.eval pred)"
homeomorphic_map_closedness: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set" assumes "homeomorphic_map X Y f"   and "U \<subseteq> topspace X" shows "closedin Y (f ` U) = closedin X U"
finite_distinct: fixes X :: "'c set" shows "finite {xs. distinct xs \<and> set xs = X}"
star_1r: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x \<le> x\<^sup>\<star>"
i_Exec_Stream_Init_append: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input1 :: "'e list"   and input2 :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream_Init trans_fun (input1 \<frown> input2) c = f_Exec_Comp_Stream_Init trans_fun input1 c \<frown> i_Exec_Comp_Stream_Init trans_fun input2 (f_Exec_Comp trans_fun input1 c) \<Up> Suc 0"
dom_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ" shows "local.dom (mkIde A) = mkIde A"
HLD_Stream: fixes X :: "'b set"   and x :: "'b"   and \<omega> :: "'b stream" shows "HLD X (x ## \<omega>) = (x \<in> X)"
leg1_in_hom_2: shows "\<guillemotleft>g' : g' \<Rightarrow> g'\<guillemotright>"
Atm_Sbis: fixes atm :: "'atom" assumes "compatAtm atm" shows "Atm atm \<approx>s Atm atm"
inf_add_left: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.top_weight less_eq less top plus zero" shows "plus top a = top"
weak_conv_def: fixes F_seq :: "nat \<Rightarrow> real \<Rightarrow> real"   and F :: "real \<Rightarrow> real" shows "weak_conv F_seq F \<equiv> \<forall>x. isCont F x \<longrightarrow> (\<lambda>n. F_seq n x) \<longlonglongrightarrow> F x"
word_leI: fixes u :: "'b word"   and v :: "'b word" assumes "\<And>n. \<lbrakk>n < size u; u !! n\<rbrakk> \<Longrightarrow> v !! n" shows "u \<le> v"
wf_disc_sel_common_primitive_6: shows "wf_disc_sel (is_Oiface, oiface_sel) OIface"
fresh_subst_type: fixes \<alpha> :: "Syntax.tvar"   and \<tau> :: "ty"   and \<tau>' :: "ty"   and \<alpha>' :: "Syntax.tvar" shows "atom \<alpha> \<sharp> subst_type \<tau> \<tau>' \<alpha>' = ((\<alpha> = \<alpha>' \<or> atom \<alpha> \<sharp> \<tau>) \<and> (atom \<alpha>' \<sharp> \<tau> \<or> atom \<alpha> \<sharp> \<tau>'))"
infsetsum_of_real: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "(\<Sum>\<^sub>ax\<in>A. of_real (f x)) = of_real (infsetsum f A)"
lower_floor_float_interval: fixes x :: "float interval" shows "lower (floor_float_interval x) = floor_fl (lower x)"
induct: fixes P :: "bexp \<Rightarrow> bool"   and a0 :: "bexp" assumes "\<And>v. P (Var v)"   and "\<And>b b'. \<lbrakk>P b; P b'\<rbrakk> \<Longrightarrow> P (b and b')"   and "\<And>b b'. \<lbrakk>P b; \<And>x. x \<in> cnf b \<Longrightarrow> P b'\<rbrakk> \<Longrightarrow> P (b or b')"   and "\<And>v. P (not (Var v))"   and "\<And>b. P b \<Longrightarrow> P (not (not b))"   and "\<And>b b'. P (not b or not b') \<Longrightarrow> P (not (b and b'))"   and "\<And>b b'. P (not b and not b') \<Longrightarrow> P (not (b or b'))" shows "P a0"
preservation: shows "rel_weakly_preserves_barbs (indRelLTPO TRel) (STCalWB SWB TWB)"
SUP_insert: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'d \<Rightarrow> 'b"   and a :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` insert a A) = sup (f a) (Sup (f ` A))"
erase_graph_aux_def: shows "erase_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>erase_aux. x1 = One \<and> x2 = One) \<or> (\<exists>erase_aux \<tau>\<^sub>1 \<tau>\<^sub>2. x1 = Fun \<tau>\<^sub>1 \<tau>\<^sub>2 \<and> x2 = Fun (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>1) (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>2) \<and> p \<tau>\<^sub>1 (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>1) \<and> p \<tau>\<^sub>2 (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>2)) \<or> (\<exists>erase_aux \<tau>\<^sub>1 \<tau>\<^sub>2. x1 = Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2 \<and> x2 = Syntax.Sum (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>1) (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>2) \<and> p \<tau>\<^sub>1 (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>1) \<and> p \<tau>\<^sub>2 (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>2)) \<or> (\<exists>erase_aux \<tau>\<^sub>1 \<tau>\<^sub>2. x1 = Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2 \<and> x2 = Syntax.Prod (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>1) (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>2) \<and> p \<tau>\<^sub>1 (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>1) \<and> p \<tau>\<^sub>2 (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>\<^sub>2)) \<or> (\<exists>erase_aux \<alpha> \<tau>. x1 = Mu \<alpha> \<tau> \<and> x2 = Mu \<alpha> (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>) \<and> p \<tau> (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>)) \<or> (\<exists>erase_aux \<alpha>. x1 = Alpha \<alpha> \<and> x2 = Alpha \<alpha>) \<or> (\<exists>erase_aux \<tau>. x1 = AuthT \<tau> \<and> x2 = ((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau> \<and> p \<tau> (((\<lambda>uu_. uu_) \<circ> erase_aux) \<tau>)))"
iv: fixes t :: "('a, 'b) SkewBinomialTree" assumes "tree_invar t"   and "queue_invar q" shows "queue_invar (ins t q)"
dg_Cod_vempty_if_Arr_vempty: assumes "\<CC>\<lparr>Arr\<rparr> = []\<^sub>\<circ>" shows "\<CC>\<lparr>Cod\<rparr> = []\<^sub>\<circ>"
mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](g1)"   and "f2 \<in> \<Theta>[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> \<Theta>[F](\<lambda>x. g1 x * g2 x)"
cont2cont_fun: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i"   and y :: "'h" assumes "cont f" shows "cont (\<lambda>x. f x y)"
f_join_f_expand_iMODb: fixes k :: "nat"   and xs :: "'b message_af list"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "xs \<odot> k \<Join>\<^sub> [ n * k, mod k, d ] = xs \<Join>\<^sub> [n\<dots>,d]"
AnnSeq: fixes c0 :: "'b ann_com"   and c1 :: "'b ann_com"   and q :: "'b set" assumes "\<turnstile> c0  pre c1"   and "\<turnstile> c1  q" shows "\<turnstile> AnnSeq c0 c1  q"
seqstar_unfoldl: fixes x :: "'a" shows "(1::'a) + x ; x\<^sup>; \<le> x\<^sup>;"
alphaEnv_refl: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "qGoodEnv rho" shows "rho &= rho"
GatewaySystem_L1aux: fixes req :: "nat \<Rightarrow> reqType list"   and stop :: "nat \<Rightarrow> 'a list"   and a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and d :: "nat"   and i :: "nat \<Rightarrow> 'b list"   and dt :: "nat \<Rightarrow> 'b list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat" assumes "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "msg (Suc 0) req \<and> msg (Suc 0) a \<and> msg (Suc 0) stop \<and> ts lose \<longrightarrow> (\<forall>t. (ack t = [init_state] \<and> req (Suc t) = [init] \<and> lose (Suc t) = [False] \<and> lose (Suc (Suc t)) = [False] \<longrightarrow> ack (Suc (Suc t)) = [connection_ok]) \<and> (ack t = [connection_ok] \<and> req (Suc t) = [send] \<and> (\<forall>k\<le>Suc d. lose (t + k) = [False]) \<longrightarrow> i (Suc (t + d)) = inf_last_ti dt t \<and> ack (Suc t) = [sending_data]) \<and> (ack (t + d) = [sending_data] \<and> a (Suc t) = [sc_ack] \<and> (\<forall>k\<le>Suc d. lose (t + k) = [False]) \<longrightarrow> vc (Suc (t + d)) = [vc_com]))" shows "ack (t + 3 + k + d + d) = [sending_data] \<and> a (Suc (t + 3 + k + d)) = [sc_ack] \<and> (\<forall>ka\<le>Suc d. lose (t + 3 + k + d + ka) = [False]) \<longrightarrow> vc (Suc (t + 3 + k + d + d)) = [vc_com]"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>UP f : UP a \<Rightarrow> UP b\<guillemotright>"
less_induct: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "class.wellorder less_eq less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
mult_2: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus (num.Bit0 num.One)) z = plus z z"
sorted_list_of_set_eq_nil: fixes A :: "'a set" assumes "finite A" shows "(sorted_list_of_set A = []) = (A = {})"
complete_measure_axioms: fixes M :: "'c measure" shows "complete_measure (completion M)"
assms_2: shows "\<forall>s as. sat_precond_as s as \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> card (traversed_states s as) \<le> k"
nat_power_def: shows "nat_power = (\<lambda>n. \<lambda>a\<in>carrier R. a [^] n)"
Q: shows "wp\<^sub>3\<^sub>' c (wp\<^sub>3\<^sub>' (WHILE b DO c) Q) (ps, n - 1)"
minusinf_ex: fixes p :: "fm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list" assumes "islin p"   and "Ifm vs (x # bs) (minusinf p)" shows "\<exists>x. Ifm vs (x # bs) p"
simps_1: fixes A :: "'a E.term"   and uu :: "'a E.term"   and uv :: "'a" shows "Dom (MkArr A uu uv) = A"
assms_10: shows "bijective (x * top) \<and> bijective (x[[top]])"
Cons_1: assumes "(f, g) \<in> set ps" shows "monom_mult ((1::'b) / lc (fst g)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst g)) (fst g) \<in> set (pdata_pairs_to_list ps)"
coinduct: fixes X :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool"   and \<phi> :: "'a stream \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> (\<exists>xs. x = xs \<and> \<psi> xs) \<or> (\<exists>xs. x = xs \<and> \<phi> xs \<and> (X (stl xs) \<or> (\<phi> until \<psi>) (stl xs)))" shows "(\<phi> until \<psi>) x"
sup_ge1: fixes x :: "'b"   and y :: "'b" shows "x \<le> x \<squnion> y"
Fun_def: fixes f :: "'t" shows "S'.Fun f = restrict (S'.some_img \<circ> (\<cdot>\<acute>) f \<circ> inv_into (S'.hom S'.unity (S'.dom f)) S'.some_img) (S'.set (S'.dom f))"
simps_8: fixes uu :: "checker_type" shows "produce uu Bot = Bot"
mult_imp_le_div_pos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less_eq (times z y) x" shows "less_eq z (divide x y)"
filter_filter: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a fset" shows "ffilter P (ffilter Q xs) = ffilter (\<lambda>x. Q x \<and> P x) xs"
in_outs_\<I>_iff_responses_\<I>: fixes out :: "'d"   and \<I> :: "('d, 'e) \<I>" shows "(out \<in> outs_\<I> \<I>) = (responses_\<I> \<I> out \<noteq> {})"
streams_iff_sset: fixes s :: "'a stream"   and A :: "'a set" shows "(s \<in> streams A) = (sset s \<subseteq> A)"
atoms_pltl_sugar_1: fixes \<phi> :: "'b pltl" shows "atoms_pltl (not\<^sub>p \<phi>) = atoms_pltl \<phi>"
infinite_cyclic_subgroup_int: fixes x :: "'a" assumes "x \<in> carrier G" shows "infinite (carrier (subgroup_generated G {x})) = (\<forall>i j. x [^] i = x [^] j \<longrightarrow> i = j)"
E: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b" assumes "yoneda_lemma_fixed_e C S \<phi> F a e" shows "e \<in> set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F a)"
\<tau>'_def: shows "\<tau>' = \<tau>\<^sup>2"
new_shadow_root_no_disconnected_nodes: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_disconnected_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr) \<rightarrow>\<^sub>r []"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "trm"   and x3 :: "trm"   and x4 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 \<equiv> x3 : x4"   and "\<And>\<Gamma> t T y.  \<Gamma> \<turnstile> t : T  \<Longrightarrow> P y \<Gamma> t t T"   and "\<And>\<Gamma> t s T y. \<lbrakk>\<Gamma> \<turnstile> t \<equiv> s : T; \<And>z. P z \<Gamma> t s T\<rbrakk> \<Longrightarrow> P y \<Gamma> s t T"   and "\<And>\<Gamma> s t T u y. \<lbrakk>\<Gamma> \<turnstile> s \<equiv> t : T; \<And>z. P z \<Gamma> s t T; \<Gamma> \<turnstile> t \<equiv> u : T; \<And>z. P z \<Gamma> t u T\<rbrakk> \<Longrightarrow> P y \<Gamma> s u T"   and "\<And>x \<Gamma> T\<^sub>1 s\<^sub>2 t\<^sub>2 T\<^sub>2 y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>; (x, T\<^sub>1) # \<Gamma> \<turnstile> s\<^sub>2 \<equiv> t\<^sub>2 : T\<^sub>2; \<And>z. P z ((x, T\<^sub>1) # \<Gamma>) s\<^sub>2 t\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (Lam [x].s\<^sub>2) (Lam [x].t\<^sub>2) (T\<^sub>1\<rightarrow>T\<^sub>2)"   and "\<And>\<Gamma> s\<^sub>1 t\<^sub>1 T\<^sub>1 T\<^sub>2 s\<^sub>2 t\<^sub>2 y. \<lbrakk>\<Gamma> \<turnstile> s\<^sub>1 \<equiv> t\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2; \<And>z. P z \<Gamma> s\<^sub>1 t\<^sub>1 (T\<^sub>1\<rightarrow>T\<^sub>2); \<Gamma> \<turnstile> s\<^sub>2 \<equiv> t\<^sub>2 : T\<^sub>1; \<And>z. P z \<Gamma> s\<^sub>2 t\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (App s\<^sub>1 s\<^sub>2) (App t\<^sub>1 t\<^sub>2) T\<^sub>2"   and "\<And>x \<Gamma> s\<^sub>2 t\<^sub>2 T\<^sub>1 s\<^sub>1 t\<^sub>1 T\<^sub>2 y. \<lbrakk>x \<sharp> y; x \<sharp> (\<Gamma>, s\<^sub>2, t\<^sub>2); (x, T\<^sub>1) # \<Gamma> \<turnstile> s\<^sub>1 \<equiv> t\<^sub>1 : T\<^sub>2; \<And>z. P z ((x, T\<^sub>1) # \<Gamma>) s\<^sub>1 t\<^sub>1 T\<^sub>2; \<Gamma> \<turnstile> s\<^sub>2 \<equiv> t\<^sub>2 : T\<^sub>1; \<And>z. P z \<Gamma> s\<^sub>2 t\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (App (Lam [x].s\<^sub>1) s\<^sub>2) (t\<^sub>1[x::=t\<^sub>2]) T\<^sub>2"   and "\<And>x \<Gamma> s t T\<^sub>1 T\<^sub>2 y. \<lbrakk>x \<sharp> (\<Gamma>, s, t); (x, T\<^sub>1) # \<Gamma> \<turnstile> App s (Var x) \<equiv> App t (Var x) : T\<^sub>2; \<And>z. P z ((x, T\<^sub>1) # \<Gamma>) (App s (Var x)) (App t (Var x)) T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> s t (T\<^sub>1\<rightarrow>T\<^sub>2)"   and "\<And>\<Gamma> s t y. \<lbrakk> \<Gamma> \<turnstile> s : TUnit ;  \<Gamma> \<turnstile> t : TUnit \<rbrakk> \<Longrightarrow> P y \<Gamma> s t TUnit" shows "P z x1 x2 x3 x4"
Ssegment_mono: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec>\<^bsub>D\<^esub> b = (Ssegment D a \<subset> Ssegment D b)"
vimage_restr_def: fixes f :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"   and A :: "(nat \<Rightarrow> 'a) set" shows "vimage_restr f A \<equiv> f -` (A \<inter> space P) \<inter> space P"
stateful_strand_step_subst_inI_1: fixes t :: "('e, 'f) Term.term"   and A :: "('e, 'f) stateful_strand_step list"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" assumes "send\<langle>t\<rangle> \<in> set A" shows "send\<langle>t \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
continuous_imp_tendsto: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "nat \<Rightarrow> 'b" assumes "isCont f x0"   and "x \<longlonglongrightarrow> x0" shows "(f \<circ> x) \<longlonglongrightarrow> f x0"
sets_measure_spmf: fixes p :: "'a spmf" shows "sets (measure_spmf p) = sets (count_space UNIV)"
prog_configs_mono: fixes xs :: "'a memref list"   and ys :: "'a memref list" assumes "set xs \<subseteq> set ys" shows "prog_configs xs \<subseteq> prog_configs ys"
assms: shows "prefix\<cdot>xs\<cdot>ys = TT"
isLam_Lam: fixes x :: "Vars.var"   and e :: "exp" shows "isLam (Lam [x]. e)"
subset_dfs_bij: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and q :: "bool list"   and q0 :: "bool list"   and v :: "nat" assumes "subset_DFS A n"   and "nfa_is_node A q"   and "nfa_is_node A q0" shows "(bdd_lookup (fst (subset_dfs A q0)) q = Some v) = (v < length (snd (subset_dfs A q0)) \<and> snd (subset_dfs A q0) ! v = q)"
poly_as_sum_of_monoms: fixes p :: "'a poly" shows "(\<Sum>i\<le>degree p. Polynomial.monom (poly.coeff p i) i) = p"
division_of_tagged_division: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" assumes "s tagged_division_of i" shows "snd ` s division_of i"
l8_20_1_R1: fixes A :: "'p"   and B :: "'p"   and P :: "'p" assumes "A = B" shows "Per B A P"
a_remove_child_locs_def: fixes ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" shows "local.a_remove_child_locs ptr owner_document = set_child_nodes_locs ptr \<union> set_disconnected_nodes_locs owner_document"
size_neq_size_imp_neq: fixes x :: "'b"   and y :: "'b" assumes "size x \<noteq> size y" shows "x \<noteq> y"
trancl_induct: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (a, y) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>+; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
power_minus_even: fixes n :: "nat"   and a :: "'c" assumes "even n" shows "(- a) ^ n = a ^ n"
apath_awalk_to_apath: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "apath u (awalk_to_apath p) v"
simps_2: fixes c1 :: "('a, 'atom) com"   and c2 :: "('a, 'atom) com" shows "SC_discr (c1 ;; c2) = (SC_discr c1 \<and> SC_discr c2)"
Cons: fixes n :: "'node"   and ns :: "'node list"   and n' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "P n ns m"   and "n' \<in> set (predecessors g n)" shows "P n' (n' # ns) m"
fake_mono_ik: fixes IK :: "msg set"   and IK' :: "msg set"   and bad :: "agent set"   and chan :: "chan set" assumes "IK \<subseteq> IK'" shows "fake bad IK chan \<subseteq> fake bad IK' chan"
allT: fixes F :: "'c \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (\<forall>x. \<box>F x) = \<box>(\<forall>x. F x)"
content_times_primitive_part: fixes p :: "'d poly" shows "smult (content p) (primitive_part p) = p"
S1_def: fixes msgs :: "'grp \<times> 'grp"   and out1 :: "unit" shows "S1 msgs out1 = (let (m0, m1) = msgs in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. sample_uniform (order \<G>) \<bind> (\<lambda>c. return_spmf (msgs, \<^bold>g [^] a, \<^bold>g [^] b, \<^bold>g [^] c, \<^bold>g [^] (a * b))))))"
skip_def: fixes i :: "nat" shows "skip i = (\<lambda>x. if i = 0 then Suc x else if x \<in> {j. j \<le> i - Suc 0} then x else Suc x)"
cone_decomp_perm: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "cone_decomp T ps"   and "ps <~~> qs" shows "cone_decomp T qs"
cobounded1: fixes a :: "'d"   and b :: "'d" shows "inf a b \<le> a"
assms_3: fixes u :: "'s \<rightarrow> 'a LList" assumes "P u" shows "P (unfoldF\<cdot>h\<cdot>u)"
lmember_strict: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a" shows "lmember\<cdot>eq\<cdot>x\<cdot>\<bottom> = \<bottom>"