hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "'b triple set"   and x2 :: "'b triple set"   and P :: "'b triple set \<Rightarrow> 'b triple set \<Rightarrow> bool" assumes "x1||-x2"   and "\<And>G. P G {}"   and "\<And>G t ts. \<lbrakk>G|-t; P G {t}; G||-ts; P G ts\<rbrakk> \<Longrightarrow> P G (insert t ts)"   and "\<And>ts G. ts \<subseteq> G \<Longrightarrow> P G ts"   and "\<And>G' ts G. \<lbrakk>G'||-ts; P G' ts; G||-G'; P G G'\<rbrakk> \<Longrightarrow> P G ts"   and "\<And>G ts' ts. \<lbrakk>G||-ts'; P G ts'; ts \<subseteq> ts'\<rbrakk> \<Longrightarrow> P G ts"   and "\<And>P G c Q. \<forall>Z s. P Z s \<longrightarrow> (\<exists>P' Q'. (G|-{P'}. c .{Q'} \<and> P G {{P'}. c .{Q'}}) \<and> (\<forall>s'. (\<forall>Z'. P' Z' s \<longrightarrow> Q' Z' s') \<longrightarrow> Q Z s')) \<Longrightarrow> P G {{P}. c .{Q}}"   and "\<And>G P. P G {{P}. SKIP .{P}}"   and "\<And>G P X a. P G {{\<lambda>Z s. P Z (s[X::=a s])}. X:==a .{P}}"   and "\<And>G P c Q X s' a. \<lbrakk>G|-{P}. c .{\<lambda>Z s. Q Z (s[Loc X::=s'<X>])}; P G {{P}. c .{\<lambda>Z s. Q Z (s[Loc X::=s'<X>])}}\<rbrakk> \<Longrightarrow> P G {{\<lambda>Z s. s' = s \<and> P Z (s[Loc X::=a s])}. LOCAL X:=a IN c .{Q}}"   and "\<And>G P c Q d R. \<lbrakk>G|-{P}. c .{Q}; P G {{P}. c .{Q}}; G|-{Q}. d .{R}; P G {{Q}. d .{R}}\<rbrakk> \<Longrightarrow> P G {{P}. (c;; d) .{R}}"   and "\<And>G P b c Q d. \<lbrakk>G|-{P &> b}. c .{Q}; P G {{P &> b}. c .{Q}}; G|-{P &> Not \<circ> b}. d .{Q}; P G {{P &> Not \<circ> b}. d .{Q}}\<rbrakk> \<Longrightarrow> P G {{P}. IF b THEN c ELSE d .{Q}}"   and "\<And>G P b c. \<lbrakk>G|-{P &> b}. c .{P}; P G {{P &> b}. c .{P}}\<rbrakk> \<Longrightarrow> P G {{P}. WHILE b DO c .{P &> Not \<circ> b}}"   and "\<And>G P Q Procs. \<lbrakk>G \<union> (\<lambda>p. {P p}. BODY p .{Q p}) ` Procs||-(\<lambda>p. {P p}. the (body p) .{Q p}) ` Procs; P (G \<union> (\<lambda>p. {P p}. BODY p .{Q p}) ` Procs) ((\<lambda>p. {P p}. the (body p) .{Q p}) ` Procs)\<rbrakk> \<Longrightarrow> P G ((\<lambda>p. {P p}. BODY p .{Q p}) ` Procs)"   and "\<And>G P pn Q s' X a. \<lbrakk>G|-{P}. BODY pn .{\<lambda>Z s. Q Z (setlocs s (loc s')[X::=s<Res>])}; P G {{P}. BODY pn .{\<lambda>Z s. Q Z (setlocs s (loc s')[X::=s<Res>])}}\<rbrakk> \<Longrightarrow> P G {{\<lambda>Z s. s' = s \<and> P Z (setlocs s newlocs[Loc Arg::=a s])}. X:=CALL pn(a) .{Q}}" shows "P x1 x2"
core_nhds_iff: fixes U :: "'a set"   and x :: "'a" assumes "U \<subseteq> carrier" shows "(x \<in> core U) = (U \<in> nhds x)"
coprime_mult_right_iff: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one c (times a b) = (algebraic_semidom.coprime times one c a \<and> algebraic_semidom.coprime times one c b)"
left_neutral: fixes a :: "'a hrel" shows "false \<sqinter> a = a"
exhaust: fixes y :: "('b, 'a) tllist" obtains x1 :: "'a" where "y = TNil x1"   | x21 :: "'b" and x22 :: "('b, 'a) tllist" where "y = TCons x21 x22"
eventually_happens': fixes F :: "'a filter"   and P :: "'a \<Rightarrow> bool" assumes "F \<noteq> bot"   and "eventually P F" shows "\<exists>x. P x"
finite_Union: fixes X :: "'a set set" assumes "finite X"   and "X \<subseteq> M" shows "\<Union> X \<in> M"
both_mono2': fixes t :: "'a ttree"   and t' :: "'a ttree"   and t'' :: "'a ttree" assumes "t \<sqsubseteq> t'" shows "t'' \<otimes>\<otimes> t \<sqsubseteq> t'' \<otimes>\<otimes> t'"
coprime_mult_self_left_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one (times c a) (times c b) = (dvd.dvd times c one \<and> algebraic_semidom.coprime times one a b)"
chine_assoc_def: shows "\<mu>_\<nu>_\<pi>\<rho>.chine_assoc \<equiv> \<langle>\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>1 \<lbrakk>\<mu>.leg0, \<nu>.leg1 \<cdot> \<nu>_\<pi>\<rho>.prj\<^sub>1\<rbrakk> \<langle>\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>1 \<lbrakk>\<nu>.leg0, \<pi>\<rho>.leg1\<rbrakk> \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<rangle>\<rangle>"
take_butlast: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "take n (butlast xs) = take n xs"
subspace_def: fixes S :: "('c, 'd) vec set" shows "vec.subspace S = (0 \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *s x \<in> S))"
UNIV: fixes S :: "'c set"   and B :: "'c set set" shows "generated_topology S B S"
card_fun: shows "CARD('c \<Rightarrow> 'd) = (if CARD('c) \<noteq> 0 \<and> CARD('d) \<noteq> 0 \<or> CARD('d) = 1 then CARD('d) ^ CARD('c) else 0)"
simps_1: shows "match_list_to_match_expr [] = MatchNot MatchAny"
alpha_Tree_induct: fixes t1 :: "('e, 'f, 'g) Tree"   and t2 :: "('e, 'f, 'g) Tree"   and P :: "('e, 'f, 'g) Tree \<Rightarrow> ('e, 'f, 'g) Tree \<Rightarrow> bool" assumes "t1 =\<^sub>\<alpha> t2"   and "\<And>tset1 tset2. \<lbrakk>\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2; a =\<^sub>\<alpha> b\<rbrakk> \<Longrightarrow> P a b; rel_bset (=\<^sub>\<alpha>) tset1 tset2\<rbrakk> \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. \<lbrakk>t1 =\<^sub>\<alpha> t2; P t1 t2\<rbrakk> \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>\<phi>. P (tPred \<phi>) (tPred \<phi>)"   and "\<And>\<alpha>1 t1 \<alpha>2 t2. \<lbrakk>\<And>p. p \<bullet> t1 =\<^sub>\<alpha> t2 \<Longrightarrow> P (p \<bullet> t1) t2; \<And>a b. (a \<rightleftharpoons> b) \<bullet> t1 =\<^sub>\<alpha> t1 \<Longrightarrow> P ((a \<rightleftharpoons> b) \<bullet> t1) t1; \<And>a b. (a \<rightleftharpoons> b) \<bullet> t2 =\<^sub>\<alpha> t2 \<Longrightarrow> P ((a \<rightleftharpoons> b) \<bullet> t2) t2; \<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)\<rbrakk> \<Longrightarrow> P (tAct \<alpha>1 t1) (tAct \<alpha>2 t2)" shows "P t1 t2"
lunit_in_hom_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>PB.lunit f : trg f \<star> f \<Rightarrow> f\<guillemotright>"
Quotient3I: fixes Abs :: "'b \<Rightarrow> 'a"   and Rep :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a. Abs (Rep a) = a"   and "\<And>a. R (Rep a) (Rep a)"   and "\<And>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" shows "Quotient3 R Abs Rep"
continuous_map_limit: fixes X :: "'d topology"   and Y :: "'e topology"   and g :: "'d \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'd"   and l :: "'d"   and F :: "'f filter" assumes "continuous_map X Y g"   and "limitin X f l F" shows "limitin Y (g \<circ> f) (g l) F"
sub_le_flip: fixes A :: "ty"   and B :: "ty"   and f1 :: "funty"   and f2 :: "funty" shows "(A <: B \<longrightarrow> V B \<sqsubseteq> V A) \<and> (f1 <:: f2 \<longrightarrow> Vf f2 \<lesssim> Vf f1)"
A: shows "sinks_aux I D U (x # xs) = sinks_aux I D U xs"
assms_3: fixes \<Psi> :: "'b" shows "(\<Psi>, P, \<tau>.Q) \<in> Rel"
eval_final: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "final e'"
snd_add: fixes x :: "'c \<times> 'b"   and y :: "'c \<times> 'b" shows "snd (x + y) = snd x + snd y"
is_unit_const_poly_iff: fixes c :: "'b" shows "([:c:] dvd 1) = (c dvd (1::'b))"
distinct_disc_2: fixes "term" :: "'b local.term" assumes "is_Prim term" shows "\<not> is_Prim\<^sub>0 term"
refine: shows "(all_left_impl, all_left) \<in> (array_assn int_assn)\<^sup>k \<rightarrow>\<^sub>a array_assn (option_assn nat_assn)"
numeral_2_eq_2: shows "2 = Suc (Suc 0)"
operational_semantics_elim_inv_def: fixes \<C>\<^sub>1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and \<C>\<^sub>2 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" shows "\<C>\<^sub>1 \<hookrightarrow>\<^sub>e\<^sup>\<leftarrow> \<C>\<^sub>2 \<equiv> \<C>\<^sub>2 \<hookrightarrow>\<^sub>e \<C>\<^sub>1"
intro: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" assumes "presheaf_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "presheaf_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str'"   and "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
strategy_progression_def: fixes \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list" shows "strategy_progression \<sigma> p = lappend (llist_of (prefixes p)) (ltl (iterates (augment_list \<sigma>) p))"
intro: fixes T :: "'b set set" assumes "\<forall>x\<in>carrier\<^bsub>T\<^esub>. \<forall>y\<in>carrier\<^bsub>T\<^esub>. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds\<^bsub>T\<^esub> x. y \<notin> u) = (\<exists>v\<in>nhds\<^bsub>T\<^esub> y. x \<notin> v)" shows "T1_axioms T"
adm_wf_unique_fixpoint: fixes r :: "('c \<times> 'c) set"   and H :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "wf r"   and "adm_wf r H"   and "f = H f"   and "g = H g" shows "f = g"
linorder_cases: fixes x :: "'d DBMEntry"   and y :: "'d DBMEntry" obtains "x \<prec> y"   | "x = y"   | "y \<prec> x"
intra_path_def: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" shows "CFGExit.intra_path n as n' \<equiv> CFGExit.path n as n' \<and> (\<forall>a\<in>set as. intra_kind (knd a))"
j2: shows "j < dim_col A + dim_col (0\<^sub>m (dim_row B) 0)"
except_eq_zero_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" shows "(except p S = 0) = (keys p \<subseteq> S)"
advantage_nonneg: fixes \<eta> :: "nat"   and \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "0 \<le> local.ind_cca.advantage \<eta> \<A>"
0: shows "C.commutative_square \<nu>.cod.leg0 \<mu>.cod.leg1 (\<nu>.chine \<cdot> \<p>\<^sub>1[\<nu>.dom.leg0, \<mu>.dom.leg1]) (\<mu>.chine \<cdot> \<p>\<^sub>0[\<nu>.dom.leg0, \<mu>.dom.leg1])"
sa'a'_in_L: fixes xa :: "'b" assumes "xa \<in> segment G.face_cycle_succ a' a'" shows "xa \<in> H.face_cycle_set b"
unl_Rall: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and w :: "'b" shows "(\<forall>x. A x) w \<equiv> \<forall>x. A x w"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_before_State_Idle_conv2: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c"   and "\<box> t1 [0\<dots>,k - Suc 0] \<oplus> t0. \<not> (State_Idle localState output_fun trans_fun (localState (s t1)) \<and> output_fun (s t1) \<noteq> NoMsg)" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (\<diamond> t1 [0\<dots>,k - Suc 0] \<oplus> t0. output_fun (s t1) = m \<and> (\<circle> t2 t1 [0\<dots>]. output_fun (s t3) = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t2. output_fun (s t4) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t4))))"
cinner_minus_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (uminus x) y = - cinner x y"
intros_7: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s1 :: "state"   and s2 :: "state"   and T :: "ty" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"   and "s2 = abupd (raise_if (\<not> G,snd s1\<turnstile>v fits T) ClassCast) s1" shows "G\<turnstile>Norm s0 \<midarrow>Cast T e-\<succ>v\<midarrow>n\<rightarrow> s2"
dom_def: fixes i :: "nat"   and j :: "nat" shows "local.dom i j \<equiv> let res = dom_kildall start ! j in i \<in> set res \<or> i = j"
step_4: fixes xs :: "(accent \<times> 'b) list" assumes "snd ` set xs \<subseteq> M"   and "(u', v) \<in> local.lconv xs" shows "thesis"
finite_permutations: fixes S :: "'a set" assumes "finite S" shows "finite {p. p permutes S}"
bin_nth_shiftl: fixes x :: "int"   and n :: "nat"   and m :: "nat" shows "bit (x << n) m = (n \<le> m \<and> bit x (m - n))"
fresh_3: fixes a :: "name"   and x2 :: "'d"   and x1 :: "('d, 'e, 'f) input" shows "a \<sharp> x2\<lparr>x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
assms_2: shows "c \<subseteq>\<^sub>L a"
exec_moves_def: fixes addr2thread_id :: "'b \<Rightarrow> 'c"   and thread_id2addr :: "'c \<Rightarrow> 'b"   and spurious_wakeups :: "bool"   and empty_heap :: "'a"   and allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and heap_read :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> bool"   and heap_write :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a \<Rightarrow> bool"   and ci :: "('b, 'a) check_instr"   and P :: "'b J1_prog"   and t :: "'c"   and es :: "'b expr1 list" shows "JVM_heap_base.exec_moves addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t es \<equiv> JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci (compP2 P) (compEs2 es) (compxEs2 es 0 0) t"
elims_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x" obtains "xa__ = []"   | x :: "'b" and ys :: "'b list" where "xa__ = x # ys" and "Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys"
that: fixes w :: "'action stream"   and u :: "'action stream" assumes "R.run w q"   and "v =\<^sub>I u"   and "u \<preceq>\<^sub>I w"   and "lproject visible (llist_of_stream u) = lproject visible (llist_of_stream w)" shows "thesis"
deg_gtE: fixes p :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "p \<in> carrier P"   and "deg R p < i" shows "p i = \<zero>"
assms: fixes num' :: "nat"   and sid' :: "nat" assumes "m = Pkt num' sid'" shows "num' \<le> no (\<sigma> sid')"
append_listrel1I: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set"   and us :: "'g list"   and vs :: "'g list" assumes "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r" shows "(xs @ us, ys @ vs) \<in> listrel1 r"
that: fixes s :: "'a" assumes "s \<in> S"   and "s \<in> frontier T" shows "thesis"
pair_less_prev: fixes u :: "nat \<times> nat"   and j :: "nat"   and i :: "nat"   and k :: "nat" assumes "(u, j, i) \<in> pair_less"   and "u \<in> IJ k" shows "prev j i = Some u \<or> (\<exists>x. (u, x) \<in> pair_less \<and> prev j i = Some x)"
inj_eq: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "inj f" shows "(f x = f y) = (x = y)"
get_obj_def: fixes s :: "state"   and a :: "loc" shows "get_obj s a \<equiv> the (heap s a)"
must_sync_def2: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong>2 = (\<exists>ta x' m' s. r2.r_syntax t x m ta x' m' \<and> r2.actions_ok s t ta)"
insert_mod2_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and s :: "'s" shows "insert_mod2 P f x s = (if P x then insert_mod (f x) s else s)"
rtrancl_subset: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R \<subseteq> S"   and "S \<subseteq> R\<^sup>*" shows "S\<^sup>* = R\<^sup>*"
get_dom_component_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c" shows "ptr \<in> set c"
term_struct_rel_strong_axioms_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "term_struct_rel_strong_axioms P \<equiv> (\<forall>name t. P (const name) t \<longrightarrow> t = const name) \<and> (\<forall>u\<^sub>1 u\<^sub>2 t. P (app u\<^sub>1 u\<^sub>2) t \<longrightarrow> (\<exists>t\<^sub>1 t\<^sub>2. t = app t\<^sub>1 t\<^sub>2 \<and> P u\<^sub>1 t\<^sub>1 \<and> P u\<^sub>2 t\<^sub>2))"
add_mset: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and x :: "'k"   and N :: "'k multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (add_mset x N) = f x (comm_monoid_mset.F f z N)"
transp_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "transp_on P A = (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. P x y \<and> P y z \<longrightarrow> P x z)"
valid_path_join_eq: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "pathfinish g1 = pathstart g2" shows "valid_path (g1 +++ g2) = (valid_path g1 \<and> valid_path g2)"
1: shows "to_polynomial R a (deg R (to_polynomial R a)) \<otimes> (g [^]\<^bsub>P\<^esub> n) (deg R (g [^]\<^bsub>P\<^esub> n)) \<noteq> \<zero>"
hseq_char: fixes \<nu> :: "'b"   and \<mu> :: "'b" shows "D.hseq \<nu> \<mu> = (D.arr \<mu> \<and> D.arr \<nu> \<and> src\<^sub>D \<nu> = trg\<^sub>D \<mu>)"
real_le_affinity: fixes m :: "'b"   and y :: "'b"   and x :: "'b"   and c :: "'b" assumes "(0::'b) < m" shows "(y \<le> m * x + c) = (inverse m * y + - (c / m) \<le> x)"
calculus_with_standard_redundancy_axioms: shows "calculus_with_standard_redundancy Inf Bot (\<Turnstile>)"
nonpos_Reals_divide_I2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>\<^sub>\<le>\<^sub>0"   and "b \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "a / b \<in> \<real>\<^sub>\<le>\<^sub>0"
pseudo_range_stake: fixes f :: "bool stream \<Rightarrow> 'b"   and g :: "bool list \<Rightarrow> 'b"   and n :: "nat" assumes "\<And>w. f w = g (stake n w)" shows "sum f (range (pseudo_proj_True n)) = sum g (gener_bool_list n)"
field_differentiable_within_Arcsin: fixes z :: "complex"   and s :: "complex set" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "Arcsin field_differentiable at z within s"
path_connectedin_absolute: fixes X :: "'b topology"   and S :: "'b set" shows "path_connectedin (subtopology X S) S = path_connectedin X S"
carrier_Fexp': fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and e :: "exp"   and a :: "Arity" assumes "CoCallArityEdom Aexp" shows "carrier (CoCallArity.Texp ccExp Aexp e\<cdot>a) \<subseteq> fv e"
containsDuplicateEdge'_def: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and "is" :: "nat list" shows "containsDuplicateEdge' g f v is \<equiv> 2 \<le> |is| \<and> ((\<exists>k<|is| - 2. let i0 = is ! k; i1 = is ! (k + 1); i2 = is ! (k + 2) in duplicateEdge g f f\<^bsup>i1\<^esup> \<bullet> v f\<^bsup>i2\<^esup> \<bullet> v \<and> i0 < i1 \<and> i1 < i2) \<or> (let i0 = is ! 0; i1 = is ! 1 in duplicateEdge g f f\<^bsup>i0\<^esup> \<bullet> v f\<^bsup>i1\<^esup> \<bullet> v \<and> i0 < i1))"
prepend_in_R1: fixes f :: "nat \<Rightarrow> nat option"   and vs :: "nat list" assumes "f \<in> \<R>" shows "vs \<odot> f \<in> \<R>"
set: fixes A :: "'a set" assumes "A \<in> sets M" shows "P (indicator A)"
span_pad_1: fixes U :: "'b vec set"   and n :: "nat"   and m :: "nat" assumes "U \<subseteq> carrier_vec n" shows "(\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` LinearCombinations.module.span class_ring (module_vec TYPE('b) n) U = LinearCombinations.module.span class_ring (module_vec TYPE('b) (n + m)) ((\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` U)"
W_def: fixes db :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and dc :: "((char list \<Rightarrow> int) \<times> (char list \<Rightarrow> int)) set" shows "W db dc = (\<lambda>dw. {(s, t). if db s then (s, t) \<in> dc O dw else s = t})"
ide_cod: fixes f :: "'a" assumes "R.arr f" shows "R.ide (R.cod f)"
ospec: fixes A :: "'c option"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<forall>x\<in>set_option A. P x"   and "A = Some x" shows "P x"
nhds_greater: fixes V :: "'a set"   and U :: "'a set"   and x :: "'a" assumes "V \<subseteq> carrier"   and "U \<subseteq> V"   and "U \<in> nhds x" shows "V \<in> nhds x"
run_writer_def: fixes writerT :: "('e, 'f, 'd) writerT" shows "run_writer writerT = (case writerT of WriterT x \<Rightarrow> x)"
map_fset_image: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c fset" shows "fset (map_fset f S) = f ` fset S"
lookupNotIn: fixes x :: "'a"   and m :: "'a \<Rightarrow>\<^sub>0 'b" assumes "x \<notin> keys m" shows "lookup m x = (0::'b)"
IH: fixes p :: "nat"   and k :: "nat" assumes "Suc p \<le> Suc n" shows "smap_stoch_proc (Suc n) f k w ! p = f (k + p) w"
map_1: fixes f :: "'d \<Rightarrow> 'c" shows "lmap f LNil = LNil"
purgeIdle_replicate_Idle: fixes n :: "nat"   and st :: "'a" shows "GM_sec_model.purgeIdle (replicate n (Idle st)) = []"
emN_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "emN P = (\<lambda>l s. P s < \<infinity>)"
one_is_power_of_any: fixes a :: "nat" shows "\<exists>i. 1 = a ^ i"
convex_rel_interior_only_if: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "S \<noteq> {}"   and "\<forall>x\<in>S. \<exists>e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" shows "z \<in> rel_interior S"