hcju/isabelleps huggingface dataset:
rem_implicit_pres_ops_goal: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.G (rem_implicit_pres_ops prob) = ast_problem.G prob"
invf2_def: shows "invf2 \<equiv> Abs_fps (\<lambda>n. of_nat (n + 1))"
that: fixes x :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "P x" shows "thesis"
respects_hcomp: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C g = trg\<^sub>C f" shows "(\<rho>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>G (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C g), G g, G f] \<cdot>\<^sub>D (\<rho>\<^sub>1 g \<star>\<^sub>D G f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[H g, \<rho>\<^sub>0 (src\<^sub>C g), G f] \<cdot>\<^sub>D (H g \<star>\<^sub>D \<rho>\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[H g, H f, \<rho>\<^sub>0 (src\<^sub>C f)] = \<rho>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>H (g, f) \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f))"
face_cycle_succ_choose_inter: shows "{H.face_cycle_succ uw, H.face_cycle_succ vw, H.face_cycle_succ (rev_H uw), H.face_cycle_succ (rev_H vw)} \<inter> {uv, rev_G uv} = {}"
assms_11: fixes n :: "nat" shows "static s0 (cruel n)"
Equiv_Exec_equiv_statesI: fixes equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and m :: "'h"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and k1 :: "nat"   and k2 :: "nat" assumes "equiv_states (localState1 c1) (localState2 c2)"   and "Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2" shows "equiv_states (localState1 (f_Exec_Comp trans_fun1 (input_fun1 m # NoMsg\<^bsup>k1 - Suc 0\<^esup>) c1)) (localState2 (f_Exec_Comp trans_fun2 (input_fun2 m # NoMsg\<^bsup>k2 - Suc 0\<^esup>) c2))"
prems: fixes w :: "'a" assumes "w \<in> poly_vars (mc # p)" shows "f w = g w"
indep_subset_carrier: fixes X :: "(nat \<times> int \<times> nat) set" assumes "forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>" shows "X \<subseteq> E"
cadjoint_univ_prop': fixes G :: "'a \<Rightarrow> 'b" assumes "bounded_clinear G" shows "\<forall>x y. x \<bullet>\<^sub>C (G\<^sup>\<dagger>) y = G x \<bullet>\<^sub>C y"
le_imp_inverse_le: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less zero a" shows "less_eq (inverse b) (inverse a)"
subst_apply_pairs_def: fixes F :: "(('d, 'f) Term.term \<times> ('d, 'f) Term.term) list"   and \<theta> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta> \<equiv> map (\<lambda>f. f \<cdot>\<^sub>p \<theta>) F"
ProcModifyReturnSameFaults_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P call init p return' c Q,A"   and "\<forall>\<sigma> n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> {\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P call init p return c Q,A"
le_floor_add: fixes x :: "'b"   and y :: "'b" shows "\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> \<le> \<lfloor>x + y\<rfloor>"
runE': fixes f :: "'c Heap"   and g :: "'b Heap"   and \<sigma> :: "heap option"   and \<sigma>'' :: "heap option"   and res :: "'b" assumes "run (f \<bind> (\<lambda>_. g)) \<sigma> \<sigma>'' res" obtains \<sigma>t :: "heap option" and rt :: "'c" where "run f \<sigma> \<sigma>t rt" and "run g \<sigma>t \<sigma>'' res"
exists_exactly_one_following_state: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ev :: "('d, 'e, 'f) event"   and c :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "distributed_system.can_occur channel trans send recv ev c" shows "\<exists>!c'. distributed_system.next channel trans send recv c ev c'"
"2_3": shows "s \<in>\<^sub>A summary loc2 loc3"
intros: fixes c :: "('a, 'b) sequent"   and R :: "'a"   and Fs :: "('a, 'b) form list"   and ps :: "'c list" assumes "ModalSequents.mset c = \<LM> Compound R Fs  \<RM>"   and "ps \<noteq> []" shows "(ps, c) \<in> upRules"
tail_algebra_def: fixes n :: "nat" shows "tail_algebra n = Pi\<^sub>M UNIV (\<lambda>i. if i < n then trivial_measure (space M) else M)"
eval_is_hom: fixes K :: "'a set"   and a :: "'a" assumes "subring K R"   and "a \<in> carrier R" shows "(\<lambda>p. local.eval p a) \<in> ring_hom (K [X]) R"
fresh_fun_app: fixes x :: "'a"   and F :: "'a \<Rightarrow> 'b option"   and y :: "'a"   and a :: "'b" assumes "atom x \<sharp> F"   and "x \<noteq> y"   and "F y = Some a" shows "atom x \<sharp> a"
simps_9: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Abs a) vs = False"
Rep_resumption_Terminal: fixes a :: "'h" shows "Rep_resumption (Terminal a) = TERMINAL a"
roots_of_real_main_def: fixes p :: "real poly" shows "roots_of_real_main p \<equiv> let n = degree p in if n = 0 then [] else if n = 1 then [roots1 p] else if n = 2 then rroots2 p else real_roots_of_rat_poly (map_poly to_rat p)"
Polygamma_approx_0: fixes m :: "nat"   and x :: "'a" shows "Polygamma_approx 0 m x = (x - (1::'a) / (2::'a)) * ln x - x + of_real (ln (2 * pi)) / (2::'a) + stirling_sum 0 m x"
assms: shows "\<forall>x\<in>set xs. P\<cdot>x = TT"
normalise_eq: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim> normalise \<phi>"
isMaxFlow_alt: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "Network c s t" shows "Network.isMaxFlow c s t f = (NFlow c s t f \<and> (\<forall>f'. NFlow c s t f' \<longrightarrow> Flow.val c s f' \<le> Flow.val c s f))"
induct: fixes P4 :: "(cell \<times> interface) list \<Rightarrow> bool"   and P1 :: "sdiagram \<Rightarrow> bool"   and P2 :: "cell \<Rightarrow> bool"   and P3 :: "cell \<times> interface \<Rightarrow> bool"   and sdiagram :: "sdiagram" assumes "\<And>x. P4 x \<Longrightarrow> P1 (SDiagram x)"   and "\<And>x. P2 (Filler x)"   and "\<And>x1a x2 x3. P2 (Basic x1a x2 x3)"   and "\<And>x1a x2. P1 x2 \<Longrightarrow> P2 (Exists_sdia x1a x2)"   and "\<And>x1a x2 x3 x4. \<lbrakk>P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P2 (Choose_sdia x1a x2 x3 x4)"   and "\<And>x1a x2 x3. P1 x2 \<Longrightarrow> P2 (Loop_sdia x1a x2 x3)"   and "\<And>x1a x2. P2 x1a \<Longrightarrow> P3 (x1a, x2)"   and "P4 []"   and "\<And>x1a x2. \<lbrakk>P3 x1a; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1a # x2)" shows "P1 sdiagram"
DERIV_quotient: fixes f :: "'a \<Rightarrow> 'a"   and d :: "'a"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and e :: "'a" assumes "(f has_field_derivative d) (at x within s)"   and "(g has_field_derivative e) (at x within s)"   and "g x \<noteq> (0::'a)" shows "((\<lambda>y. f y / g y) has_field_derivative (d * g x - e * f x) / g x ^ Suc (Suc 0)) (at x within s)"
church_rosser_to_confluence: fixes x :: "'a"   and y :: "'a" assumes "(x + y)\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>" shows "y\<^sup>\<star> ; x\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>"
assms_2: fixes c :: "'a" assumes "\<And>a. a \<in> A \<Longrightarrow> a dvd c" shows "b dvd c"
wd_dests_1: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "well_def prog" shows "healthy (wp prog)"
T: shows "\<forall>c'\<in>set (c\<^sub>2 # T). stack c\<^sub>1 \<lesssim> stack c'"
point_minus_def: fixes A :: "point" shows "- A = Point (- abscissa A) (- ordinate A)"
learn_consE_1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_cons \<psi> U s" shows "environment \<psi> U s"
assms_1: fixes A :: "real tensor" assumes "A \<in> set\<^sub>v Ts" shows "dims A = ds"
interior: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and p :: "'b"   and q :: "'b"   and r :: "'b"   and t :: "'b" assumes "ccw_system4 ccw S"   and "distinct4 p q r t"   and "in4 S p q r t"   and "ccw t q r"   and "ccw p t r"   and "ccw p q t" shows "ccw p q r"
updVal_commute: fixes xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and sX :: "'b"   and sY :: "'b" assumes "xs \<noteq> ys \<or> x \<noteq> y" shows "val (x := sX)_xs (y := sY)_ys = val (y := sY)_ys (x := sX)_xs"
nonEqualitySetOfSets: fixes X :: "'b set"   and Y :: "'b set" assumes "X \<noteq> Y" shows "{{x} |x. x \<in> X} \<noteq> {{x} |x. x \<in> Y}"
graphI: fixes x :: "'b"   and F :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "x \<in> F" shows "(x, f x) \<in> graph F f"
ring_axioms: shows "ring stfx.carrier_direct_lim stfx.add_rel stfx.mult_rel (stfx.class_of V (zero_str\<^sub>Y V)) (stfx.class_of V (one_str\<^sub>Y V))"
set_finfun_to_list: fixes f :: "('a, 'b) finfun" shows "set (finfun_to_list f) = {x. finfun_apply (finfun_dom f) x}"
hdiff_zero: fixes x :: "hf" shows "x - 0 = x"
lang_eq_ext: fixes r :: "'a rexp"   and s :: "'a rexp" shows "(lang r = lang s) = (\<forall>w\<in>lists (atoms r \<union> atoms s). (w \<in> lang r) = (w \<in> lang s))"
meet_compl_bot: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "\<^bold>-\<^bsub>L\<^esub> x \<sqinter>\<^bsub>L\<^esub> x = \<bottom>\<^bsub>L\<^esub>"
is_nextElem_between_empty: fixes vs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "distinct vs"   and "is_nextElem vs a b" shows "between vs a b = []"
filter_empty_ivls_def: fixes X :: "'b" shows "filter_empty_ivls X = X"
\<beta>'': fixes Y :: "complex multiset" assumes "Y \<in> Roots_ms" shows "\<beta>' Y = of_rat (\<beta>'' Y)"
abel_semigroup_axioms: shows "abel_semigroup (\<cdot>)"
distinct_1: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "Leaf \<noteq> Node x21 x22 x23 x24 x25"
rep_number_alt_def_all: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int" assumes "constant_rep_design point_set block_collection design_rep_number" shows "\<forall>x\<in>point_set. block_collection rep x = design_rep_number"
aezfun_inverse: fixes x :: "('a, 'b) aezfun" shows "Abs_aezfun (aezfun x) = x"
a: shows "( \<Gamma> \<Rightarrow>* E, n) \<in> derivable R*"
prefix_length_prefix: fixes ps :: "'d list"   and xs :: "'d list"   and qs :: "'d list" assumes "prefix ps xs"   and "prefix qs xs"   and "length ps \<le> length qs" shows "prefix ps qs"
ord_antisym: assumes "ord s t"   and "ord t s" shows "s = t"
lprefix_ltakeWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "ltakeWhile P xs \<sqsubseteq> xs"
inject: fixes x1 :: "'d"   and x2 :: "'e"   and y1 :: "'d"   and y2 :: "'e" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
simps: fixes c :: "'b ann_com option"   and q :: "'b set" shows "OG_Hoare.post (c, q) = q"
assms_5: shows "h \<turnstile> get_elements_by_class_name ptr idd \<rightarrow>\<^sub>h h'"
assms_3: shows "p (deg R p) \<otimes> q (deg R q) \<noteq> \<zero>"
zip_eq_ConsE: fixes xs :: "'g list"   and ys :: "'h list"   and xy :: "'g \<times> 'h"   and xys :: "('g \<times> 'h) list" assumes "zip xs ys = xy # xys" obtains x :: "'g" and xs' :: "'g list" and y :: "'h" and ys' :: "'h list" where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = zip xs' ys'"
pth_2: fixes x :: "'b"   and y :: "'b" shows "x - y \<equiv> x + - y"
elims_2: fixes x :: "nat"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "wadjust_stop x xa xb" obtains m :: "nat" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = m" and "xa = rs" and "xb = (l, r)" and "\<exists>rn. l = [Bk] \<and> r = Oc \<up> Suc m @ Bk # Oc \<up> Suc (Suc rs) @ Bk \<up> rn"
16: shows "w\<^sup>+ \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>"
ExistImpliesMinEnabled: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list"   and p :: "'a"   and msg :: "('a, 'c) message" assumes "execution trans sends start exec trace"   and "isReceiverOf p msg"   and "enabled (last exec) msg" shows "\<exists>msg'. execution.minimalEnabled exec trace msg'"
simps_5: fixes dims :: "nat list"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and \<rho> :: "complex mat" shows "state_sig.denote dims (While M S) \<rho> = state_sig.denote_while dims (M 0) (M 1) (state_sig.denote dims S) \<rho>"
fv_safe_get_and: fixes \<phi> :: "formula" assumes "safe_formula \<phi>" shows "fv \<phi> \<subseteq> \<Union> (fv ` set (filter safe_formula (get_and_list \<phi>)))"
iso_unit: fixes a :: "'a" assumes "obj a" shows "local.iso \<i>[a]"
elims: fixes x :: "Core.term"   and y :: "typ list" assumes "strip_all_vars x = y" obtains alla :: "String.literal" and S :: "typ" and T :: "typ" and t :: "Core.term" where "x = Ct alla S $ Abs T t" and "y = (if alla = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT then T # strip_all_vars t else [])"   | v :: "String.literal" and va :: "typ" where "x = Ct v__ va__" and "y = []"   | v :: "variable" and va :: "typ" where "x = Fv vb__ vaa__" and "y = []"   | v :: "nat" where "x = Bv vc__" and "y = []"   | v :: "typ" and va :: "Core.term" where "x = Abs vd__ vab__" and "y = []"   | vb :: "variable" and vc :: "typ" and va :: "Core.term" where "x = Fv vba__ vca__ $ vac__" and "y = []"   | vb :: "nat" and va :: "Core.term" where "x = Bv vbb__ $ vad__" and "y = []"   | vb :: "typ" and vc :: "Core.term" and va :: "Core.term" where "x = Abs vbc__ vcb__ $ vae__" and "y = []"   | vb :: "Core.term" and vc :: "Core.term" and va :: "Core.term" where "x = vbd__ $ vcc__ $ va" and "y = []"   | v :: "Core.term" and vb :: "String.literal" and vc :: "typ" where "x = ve__ $ Ct vbe__ vcd__" and "y = []"   | v :: "Core.term" and vb :: "variable" and vc :: "typ" where "x = vf__ $ Fv vbf__ vce__" and "y = []"   | v :: "Core.term" and vb :: "nat" where "x = vg__ $ Bv vbg__" and "y = []"   | v :: "Core.term" and vb :: "Core.term" and vc :: "Core.term" where "x = v $ (vb $ vc)" and "y = []"
cases: fixes x :: "'d DBMEntry" obtains c :: "'d" where "x = Le c__"   | c :: "'d" where "x = Lt c"   | "x = \<infinity>"
restrict_IF_id: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and v' :: "'a"   and val :: "bool" assumes "ifex_ordered (IF v t e)"   and "v' < v" shows "restrict (IF v t e) v' val = IF v t e"
src_dom: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "src\<^sub>B (B.dom \<mu>) = src\<^sub>B \<mu>"
mono_step': fixes S :: "'a Abs_State.st option"   and S' :: "'a Abs_State.st option"   and c :: "'a Abs_State.st option acom"   and c' :: "'a Abs_State.st option acom" assumes "S \<sqsubseteq> S'"   and "c \<sqsubseteq> c'" shows "step' S c \<sqsubseteq> step' S' c'"
sorted_on_sort_on: fixes xs :: "'a list" assumes "set xs \<subseteq> Field r" shows "sorted_on r (sort_on r xs)"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" obtains "a = a \<sqinter> b"
assms_3: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq x y"
residual_network_sel_3: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "source (res.residual_network f) = source (residual_network f)"
one_closed: shows "\<zero>\<^bsub>R Quot I\<^esub> \<in> carrier (R Quot I)"
eval_cases_17: 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>throw e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains a :: "nat" where "e' = Throw a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>addr a,s'\<rangle>"   | "e' = THROW NullPointer" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>null,s'\<rangle>"   | e'a :: "char list exp" where "e' = throw e'a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
subterm_wpo_s: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" assumes "s \<rhd> t" shows "s \<succ> t"
SPLIT_1: shows "((s, c), ee, sh, ch) \<in> ntrs fg"
frechet_derivative_works_on_open: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and x :: "'b" assumes "f differentiable_on X"   and "open X"   and "x \<in> X" shows "(f has_derivative frechet_derivative f (at x)) (at x)"
pw_conc_nofail: fixes R :: "('a \<times> 'b) set"   and S :: "'b nres" shows "nofail (\<Down> R S) = nofail S"
R_assign: fixes P :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "\<forall>s. P s \<longrightarrow> Q (s(v := e s))" shows "v ::= e \<subseteq> rel_R \<lceil>P\<rceil> \<lceil>Q\<rceil>"
execute_parallel_operator_cons_equals: fixes s :: "'a \<Rightarrow> bool option"   and a :: "'a strips_operator"   and ops :: "'a strips_operator list" assumes "are_all_operators_applicable s (a # ops)"   and "are_all_operator_effects_consistent (a # ops)"   and "are_all_operators_non_interfering (a # ops)" shows "execute_parallel_operator s (a # ops) = execute_parallel_operator (s ++ map_of (effect_to_assignments a)) ops"
measure_finite_Union: fixes S :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "finite S"   and "A ` S \<subseteq> sets M"   and "disjoint_family_on A S"   and "\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" shows "Sigma_Algebra.measure M (\<Union> (A ` S)) = (\<Sum>i\<in>S. Sigma_Algebra.measure M (A i))"
assms_1: shows "OrderingSetMap (\<^bold>\<le>*) (\<^bold><*) less_eq'' less'' Q g"
join_idem: fixes x :: "'b"   and y :: "'b" assumes "compatible x y" shows "x \<squnion> (x \<squnion> y) = x \<squnion> y"
zero: fixes m :: "nat"   and j :: "nat" assumes "m \<le> j"   and "j < n" shows "(\<Sum>x\<in>set (unit_vecs_first n m). a x * x $ j) = (0::'a)"
poly_to_mpoly_inverse: fixes v :: "nat"   and p :: "'b poly" shows "mpoly_to_poly v (poly_to_mpoly v p) = p"
select_convs_4: fixes counter :: "nat"   and discovered :: "'d \<Rightarrow> nat option"   and finished :: "'d \<Rightarrow> nat option"   and pending :: "('d \<times> 'd) set"   and stack :: "'d list"   and tree_edges :: "('d \<times> 'd) set"   and back_edges :: "('d \<times> 'd) set"   and cross_edges :: "('d \<times> 'd) set"   and sccs :: "'d set set"   and lowlink :: "'d \<Rightarrow> nat option"   and tj_stack :: "'d list"   and more :: "'c" shows "tarjan_state.more \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = more"
eq_bottom_iff: fixes x :: "'b" shows "(x = \<bottom>) = (x \<sqsubseteq> \<bottom>)"
intros_1: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and v :: "val" shows "\<Gamma> \<turnstile>\<^sub>c CVal v : val_type v"
net_ips_is_dom_netmap: fixes s :: "'c net_state" shows "net_ips s = dom (netmap s)"
iPP: shows "bprv (wrepr.PP \<langle>\<phi>\<rangle>)"
af_congruentness: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and xs :: "'a set list" assumes "af_congruent eq"   and "eq \<phi> \<psi>" shows "eq (af \<phi> xs) (af \<psi> xs)"
bhd: shows "eval (Cn 1 r_hd [r_pdec1]) [arg] \<down>= e_hd b"
A: fixes X :: "('a, 'b) action" assumes "X \<preceq>S Q" shows "P X \<preceq>S Q"
sum_set_upt_conv_sum_list_nat: fixes f :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum f (set [m..<n]) = sum_list (map f [m..<n])"
in_open_segment_iff_line: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> {a<--<b}) = (\<exists>c\<in>{0<..<1}. a \<noteq> b \<and> x = {a -- b}\<^bsub>c\<^esub>)"
o_eq_dest: fixes a :: "'e \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'e"   and c :: "'f \<Rightarrow> 'd"   and d :: "'c \<Rightarrow> 'f"   and v :: "'c" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
assms_8: fixes exp :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and var :: "Vars.var" assumes "P1 exp"   and "P2 \<Gamma>" shows "P2 ((var, exp) # \<Gamma>)"
entries_rbtreeify_g_append: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)" shows "entries (fst (rbtreeify_g n kvs)) @ snd (rbtreeify_g n kvs) = kvs"
member_singD: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "P x"   and "List.member [x] y" shows "P y"
subset_val_process': fixes A :: "'b set"   and p :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market"   and n :: "nat"   and w :: "'c" assumes "finite A"   and "support_set p \<subseteq> A" shows "val_process Mkt p n w = (\<Sum>x\<in>A. prices Mkt x n w * p x (Suc n) w)"
insert_compr: fixes a :: "'c"   and B :: "'c set" shows "insert a B = {x. x = a \<or> x \<in> B}"
sg_l_i: fixes H :: "'a set"   and x :: "'a" assumes "G \<guillemotright> H "   and "x \<in> H" shows "\<rho> x \<cdot> x = \<one>"
distinct_9: fixes x3 :: "bool" shows "Null \<noteq> Bool x3"
padd_commut: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b list"   and a :: "'b list" assumes "class.comm_semiring_0 plus zero times" shows "semiring_0.padd plus b a = semiring_0.padd plus a b"
sums_iff: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" shows "f sums x = (summable f \<and> suminf f = x)"