hcju/isabelleps huggingface dataset:
padl_inj: fixes m :: "nat"   and n :: "nat" shows "inj_on ((@\<^sub>v) (0\<^sub>v m)) (carrier_vec n)"
to_nat_from_nat_id: fixes x :: "nat" assumes "x < CARD('a)" shows "mod_type_class.to_nat (mod_type_class.from_nat x) = x"
true_cas: fixes \<sigma> :: "'s" assumes "is_ground_subst \<sigma>" shows "I \<Turnstile>m mset CAs \<cdot>cm \<sigma>"
lubg: fixes M' :: "complex mat" shows "(\<forall>n. g n \<le>\<^sub>L M') \<longrightarrow> M \<le>\<^sub>L M'"
gamma4_def: shows "gamma4 = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
beukers_integral3_le: shows "local.beukers_integral3 \<le> 2 * (1 / 27) ^ n * Re (zeta 3)"
degree_map_poly_hom: fixes p :: "'a poly" shows "degree (map_poly hom p) = degree p"
directed: fixes i :: "'a"   and j :: "'a" assumes "i \<in> I"   and "j \<in> I" shows "\<exists>k\<in>I. f i + g j \<le> f k + g k"
mult_imp_less_div_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less (times z y) x" shows "less z (divide x y)"
x_in_set: shows "\<forall>bl\<in>#filter_mset ((\<in>) x) \<B>. {x} \<subseteq> {p \<in> \<V>. p \<in> bl}"
is_sup_related: fixes x :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> y" shows "is_sup x y y"
scale_eq_0_iff: fixes a :: "'a"   and x :: "'a" shows "(a * x = (0::'a)) = (a = (0::'a) \<or> x = (0::'a))"
arctan_eq: fixes x :: "real" shows "(\<Sum>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n) = (\<Sum>k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
fls_lr_inverse_minus_1: fixes f :: "'a fls"   and x :: "'a" shows "fls_left_inverse (- f) (- x) = - fls_left_inverse f x"
CFG_transfer: fixes G :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and \<alpha>e1 :: "'f \<Rightarrow> ('h \<times> 'i \<times> 'h) set"   and \<alpha>e2 :: "'g \<Rightarrow> ('h \<times> 'i \<times> 'h) set"   and \<alpha>n1 :: "'f \<Rightarrow> 'h list"   and \<alpha>n2 :: "'g \<Rightarrow> 'h list"   and invar1 :: "'f \<Rightarrow> bool"   and invar2 :: "'g \<Rightarrow> bool"   and inEdges1 :: "'f \<Rightarrow> 'h \<Rightarrow> ('h \<times> 'i) list"   and inEdges2 :: "'g \<Rightarrow> 'h \<Rightarrow> ('h \<times> 'i) list"   and Entry1 :: "'f \<Rightarrow> 'h"   and Entry2 :: "'g \<Rightarrow> 'h"   and defs1 :: "'f \<Rightarrow> 'h \<Rightarrow> 'j set"   and defs2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'j set"   and uses1 :: "'f \<Rightarrow> 'h \<Rightarrow> 'j set"   and uses2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'j set" assumes "right_total G"   and "(G ===> (=)) \<alpha>e1 \<alpha>e2"   and "(G ===> (=)) \<alpha>n1 \<alpha>n2"   and "(G ===> (=)) invar1 invar2"   and "(G ===> (=)) inEdges1 inEdges2"   and "(G ===> (=)) Entry1 Entry2"   and "(G ===> (=)) defs1 defs2"   and "(G ===> (=)) uses1 uses2" shows "CFG \<alpha>e1 \<alpha>n1 invar1 inEdges1 Entry1 defs1 uses1 \<longrightarrow> CFG \<alpha>e2 \<alpha>n2 invar2 inEdges2 Entry2 defs2 uses2"
assms_1: shows "wf\<^sub>t\<^sub>r\<^sub>m t"
map_scale_single: fixes k :: "'c"   and x :: "'b"   and l :: "'c" shows "k \<cdot> monomial l x = monomial (k * l) x"
pp_arc_expanded: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_arc x = (matrix_idempotent_semiring.coreflexive (x \<odot> mtop \<odot> x\<^sup>t) \<and> matrix_idempotent_semiring.coreflexive (x\<^sup>t \<odot> mtop \<odot> x) \<and> matrix_bounded_idempotent_semiring.total (mtop \<odot> \<ominus> \<ominus> x))"
cmod_Re: fixes x :: "complex" assumes "0 \<le> x" shows "cmod x = Re x"
DERIV_def: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c" shows "(f has_field_derivative D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'c\<rightarrow> D"
multiset_subset_insert: fixes x :: "'b"   and xs :: "'b multiset" shows "{ps. ps \<subseteq># add_mset x xs} = {ps. ps \<subseteq># xs} \<union> add_mset x ` {ps. ps \<subseteq># xs}"
set_intros_1: fixes x21 :: "'b"   and x22 :: "'b llist" shows "x21 \<in> lset (LCons x21 x22)"
simps_2: fixes n :: "nat"   and nd :: "('a, 'b) Node" shows "FingerTreeStruc.is_leveln_ftree n (Single nd) = FingerTreeStruc.is_leveln_node n nd"
lens_quotient_def: fixes X :: "'b \<Longrightarrow> 'd"   and Y :: "'c \<Longrightarrow> 'd" shows "X /\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. get\<^bsub>X\<^esub> (create\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> v. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> (create\<^bsub>Y\<^esub> \<sigma>) v)\<rparr>"
axioms_1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size" shows "proper_design point_set block_collection"
measure_decreasing: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" assumes "P St"   and "R St St'" shows "(m St', m St) \<in> mR"
s: shows "supp dbtm \<subseteq> atom ` Vs"
t_del_min_bound: fixes t :: "'b tree" assumes "braun t" shows "t_del_min t \<le> 3 * height t"
r_r_into_trancl: fixes a :: "'c"   and b :: "'c"   and R :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> R"   and "(b, c) \<in> R" shows "(a, c) \<in> R\<^sup>+"
absolutely_integrable_Gamma_integral: fixes z :: "complex"   and a :: "real" assumes "0 < Re z"   and "0 < a" shows "complex_set_integrable lebesgue {0<..} (\<lambda>t. complex_of_real t powr (z - 1) / complex_of_real (exp (a * t)))"
uminus_in_intervalI: fixes x :: "'a"   and X :: "'a interval" assumes "x \<in>\<^sub>i X" shows "- x \<in>\<^sub>i - X"
solves_ode_subset_range: fixes x :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "real set"   and X :: "'c set"   and Y :: "'c set" assumes "(x solves_ode f) T X"   and "x ` T \<subseteq> Y" shows "(x solves_ode f) T Y"
add_closed_inverse_right: fixes x :: "'r"   and g :: "'r" assumes "x \<notin> R"   and "g \<in> R" shows "g + x \<notin> R"
simps_2: fixes n :: "nat"   and bs :: "bool list" shows "sieve n (False # bs) = False # sieve (Suc n) bs"
assms_2: shows "snd ` set txs \<subseteq> var"
all_channels_done_in_final_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 init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "channel cid = Some (p, q)" shows "snd (cs final cid) = Done"
simps_2: fixes dims :: "nat list"   and U :: "complex mat"   and P :: "complex mat" shows "state_sig.wlp dims (Utrans U) P = adjoint U * P * U"
bool_similar'_step: fixes s :: "Value \<Rightarrow> CValue \<Rightarrow> bool"   and b :: "bool discr" shows "similar'_step s (B\<cdot>b) (CB\<cdot>b)"
that: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_Groebner_basis G"   and "ideal G = ideal F"   and "\<Union> (indets ` G) \<subseteq> \<Union> (indets ` F)"   and "\<And>g. g \<in> G \<Longrightarrow> \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. indets (q f) \<subseteq> \<Union> (indets ` F) \<and> poly_deg (q f * f) \<le> b \<and> (f \<notin> F' \<longrightarrow> q f = 0))" shows "thesis"
distinct_length_2_or_more: fixes a :: "'a"   and b :: "'a"   and xs :: "'a list" shows "cycle (a # b # xs) = (a \<noteq> b \<and> cycle (a # xs) \<and> cycle (b # xs))"
simps_2: fixes x :: "'c"   and t :: "('c \<times> 'd) bro" shows "del x (N1 t) = N1 (del x t)"
connected_ivt_hyperplane: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and a :: "'b"   and b :: "real" assumes "connected S"   and "x \<in> S"   and "y \<in> S"   and "a \<bullet> x \<le> b"   and "b \<le> a \<bullet> y" shows "\<exists>z\<in>S. a \<bullet> z = b"
Right_Coset_bij: fixes x :: "'a"   and y :: "'a" assumes "x \<in> G"   and "y \<in> G" shows "bij_betw (inverse x \<cdot> y)\<^sub>R (H |\<cdot> x) (H |\<cdot> y)"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Cod \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Cod t \<^bold>\<star> Cod u) \<^bold>\<star> Cod v"
prereq: fixes u :: "'v"   and v :: "'v"   and s :: "('v, 'es) state_scheme" shows "on_discover param u v s = on_new_root param v s"
DIAndSG2_def: shows "DIAndSG2 = (Pc pid2 \<rightarrow> ([[$\<alpha> vid1]]Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid2)"
pmf_return: fixes x :: "'b"   and y :: "'b" shows "pmf (return_pmf x) y = indicat_real {y} x"
i_set_mult_1_eq_i_set: shows "i_set_mult (Suc 0) = i_set"
alphaAbs_trans_twice: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A'' :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A''' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs A \<or> qGoodAbs A' \<or> qGoodAbs A'' \<or> qGoodAbs A'''"   and "A $= A'"   and "A' $= A''"   and "A'' $= A'''" shows "A $= A'''"
3: shows "[a] @ u =\<^sub>F [b] @ v"
iterate_sel_no_map_def: fixes it :: "('a option \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a option \<Rightarrow> 'a option) \<Rightarrow> 'a option \<Rightarrow> 'a option"   and P :: "'a \<Rightarrow> bool" shows "iterate_sel_no_map it P = iterate_sel it (\<lambda>x. if P x then Some x else None)"
chamber_system_image: shows "f \<turnstile> folding.\<C> = folding.\<C> \<inter> (f \<turnstile> X)"
simps_7: fixes r :: "'c rexp" shows "REV (Star r) = Star (REV r)"
ptr_kinds_M_pure: fixes h :: "(_) heap" shows "pure shadow_root_ptr_kinds_M h"
rel_fun_def: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "A ===> B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
assms_2: fixes X :: "'a upred"   and n :: "nat" shows "(F X \<and> E (n + 1)) = (F (X \<and> E n) \<and> E (n + 1))"
Derivation_unique_dest: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and c :: "symbol list" assumes "Derivation a D b"   and "Derivation a D c" shows "b = c"
skl1_refines_init_l1: shows "init skl1 \<subseteq> R11 `` init l1"
inj_on_from_nat_into: shows "inj_on from_nat_into {A. A \<noteq> {} \<and> countable A}"
is_write_seenI: fixes E :: "('b \<times> ('a, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and P :: "'c prog" assumes "\<And>a ad al v. \<lbrakk>a \<in> read_actions E; action_obs E a = NormalAction (ReadMem ad al v)\<rbrakk> \<Longrightarrow> ws a \<in> write_actions E"   and "\<And>a ad al v. \<lbrakk>a \<in> read_actions E; action_obs E a = NormalAction (ReadMem ad al v)\<rbrakk> \<Longrightarrow> (ad, al) \<in> action_loc P E (ws a)"   and "\<And>a ad al v. \<lbrakk>a \<in> read_actions E; action_obs E a = NormalAction (ReadMem ad al v)\<rbrakk> \<Longrightarrow> value_written P E (ws a) (ad, al) = v"   and "\<And>a ad al v. \<lbrakk>a \<in> read_actions E; action_obs E a = NormalAction (ReadMem ad al v)\<rbrakk> \<Longrightarrow> \<not> P,E \<turnstile> a \<le>hb ws a"   and "\<And>a ad al v. \<lbrakk>a \<in> read_actions E; action_obs E a = NormalAction (ReadMem ad al v); is_volatile P al\<rbrakk> \<Longrightarrow> \<not> P,E \<turnstile> a \<le>so ws a"   and "\<And>a ad al v a'. \<lbrakk>a \<in> read_actions E; action_obs E a = NormalAction (ReadMem ad al v); a' \<in> write_actions E; (ad, al) \<in> action_loc P E a'; P,E \<turnstile> ws a \<le>hb a'; P,E \<turnstile> a' \<le>hb a\<rbrakk> \<Longrightarrow> a' = ws a"   and "\<And>a ad al v a'. \<lbrakk>a \<in> read_actions E; action_obs E a = NormalAction (ReadMem ad al v); a' \<in> write_actions E; (ad, al) \<in> action_loc P E a'; is_volatile P al; P,E \<turnstile> ws a \<le>so a'; P,E \<turnstile> a' \<le>so a\<rbrakk> \<Longrightarrow> a' = ws a" shows "is_write_seen P E ws"
alt_nondet_def: fixes return :: "'i \<Rightarrow> 'h"   and bind :: "'h \<Rightarrow> ('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and union :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and m1 :: "('g, 'h) nondetT"   and m2 :: "('g, 'h) nondetT" shows "nondetM_base.alt_nondet return bind union m1 m2 = NondetT (bind (run_nondet m1) (\<lambda>A. bind (run_nondet m2) (\<lambda>B. return (union A B))))"
fdia_star_induct_var: fixes x :: "'a"   and y :: "'a" assumes " |x\<rangle> y \<le> d y" shows " |x\<^sup>\<star>\<rangle> y \<le> d y"
to_tree_order_ptr_in_result: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list" assumes "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes" shows "ptr \<in> set nodes"
ordinal_1_times: fixes x :: "ordinal" shows "oSuc 0 * x = x"
fast_bv_to_nat_Cons: fixes b :: "bit"   and bs :: "bit list"   and k :: "num" shows "fast_bv_to_nat_helper (b # bs) k = fast_bv_to_nat_helper bs ((case b of \<zero> \<Rightarrow> num.Bit0 | \<one> \<Rightarrow> num.Bit1) k)"
Liminf_llist_LCons: fixes X :: "'a set"   and Xs :: "'a set llist" shows "Liminf_llist (LCons X Xs) = (if lnull Xs then X else Liminf_llist Xs)"
ldropWhile_K_False: shows "ldropWhile (\<lambda>_. False) = id"
PLam_stable: fixes i :: "nat"   and I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and A :: "'c set" assumes "i \<in> I"   and "\<forall>j. F j \<in> preserves snd" shows "(PLam I F \<in> stable (lift_set i (A \<times> UNIV))) = (F i \<in> stable (A \<times> UNIV))"
sorted_list_of_set_greaterThanLessThan: fixes i :: "nat"   and j :: "nat" assumes "Suc i < j" shows "list_of {i<..<j} = Suc i # list_of {Suc i<..<j}"
simps_12: fixes x31 :: "name"   and x32 :: "nterm"   and x2 :: "name" shows "(\<Lambda>\<^sub>n x31. x32) \<noteq> Nvar x2"
naturality: fixes f :: "'c" assumes "C.arr f" shows "C.map (C.cod f) \<cdot> C.map f = C.map f \<cdot> C.map (C.dom f)"
case_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a" shows "(case BOOL of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f2"
size_mset_SucE: fixes A :: "'a multiset"   and n :: "nat" assumes "size A = Suc n" obtains a :: "'a" and B :: "'a multiset" where "A = {#a#} + B" and "size B = n"
simps_1: shows "rlh \<langle>\<rangle> = 0"
inv_exec: fixes ss :: "'b heap list"   and f :: "'b op" assumes "Ball (set ss) is_root'"   and "length ss = arity f" shows "is_root' (Pairing_Heap_List1_Analysis.exec f ss)"
sqrtu_upper: fixes x :: "real"   and n :: "nat" shows "x \<le> (sqrtu x n)\<^sup>2"
Liminf_eqI: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "\<And>P. eventually P F \<Longrightarrow> Inf (f ` Collect P) \<le> x"   and "\<And>y. (\<And>P. eventually P F \<Longrightarrow> Inf (f ` Collect P) \<le> y) \<Longrightarrow> x \<le> y" shows "Liminf F f = x"
empty_set_less_eq: fixes A :: "'a set" shows "{} \<sqsubseteq> A"
sqrt_case: fixes SQ :: "real"   and i :: "nat" assumes "0 \<le> SQ" shows "SQ ^ (i div 2) * real (i mod 2) * sqrt SQ + SQ ^ (i div 2) * (1 - real (i mod 2)) = sqrt SQ ^ i"
derivable_def: fixes x :: "'a" shows "derivable x \<equiv> \<exists>X. derivation X \<and> x \<in> X"
mod_emp_simp: fixes h :: "heap" shows "(h, {}) \<Turnstile> emp"
circ_mult_sub_sup: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>\<circ> \<le> (x \<squnion> y)\<^sup>\<circ>"
VSLEG_def: shows "VSLEG = VSGeneral \<circ> VSEquality \<circ> VSLucky"
that: fixes i :: "nat" assumes "i < length (fst aux)"   and "f = (fst (fst aux ! i), fst data + i, snd (fst aux ! i))" shows "thesis"
le_numeral_extra_3: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and less_eq :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and less :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
IH: assumes "m $$ k = Some v" shows "{k} \<lhd> m = {k $$:= v}"
mset_right_cancel_union: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "a \<in># A + B"   and "a \<notin># B" shows "a \<in># A"
plus_p_def: fixes p :: "int"   and x :: "int"   and y :: "int" shows "plus_p p x y \<equiv> let z = x + y in if p \<le> z then z - p else z"
prv_nprvI: fixes \<phi> :: "'fmla"   and F :: "'fmla set" assumes "prv \<phi>"   and "\<phi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F \<phi>"
x: shows "x = c 0 \<cdot>\<^sub>v v + lincomb_list (c \<circ> Suc) Vs"
simps_7: fixes t :: "'a local.term" shows "Dom \<^bold>\<r>\<^bold>[t\<^bold>] = Dom t \<^bold>\<otimes> \<^bold>\<I>"
X4_UNIV: shows "UNIV = set [X4.Xd1, X4.Xd1', X4.Xd2, Xd2']"
domains_comp: fixes g :: "'b"   and f :: "'b" assumes "\<phi>.B.seq g f" shows "\<phi>.B.domains (g \<cdot>\<^sub>B f) = \<phi>.B.domains f"
elims_3: fixes x :: "('i \<times> constraint) list"   and xa :: "'i set"   and xb :: "'a ns_constraint list \<times> 's" assumes "\<not> invariant_cs x xa xb" obtains cs :: "('i \<times> constraint) list" and J :: "'i set" and ds :: "'a ns_constraint list" and s :: "'s" where "x = cs" and "xa = J" and "xb = (ds, s)" and "\<not> (ds = map snd (to_ns cs) \<and> invariant_nsc (to_ns cs) J s)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> int list \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and atom :: "atom" shows "P (case atom of Le x xa \<Rightarrow> f1 x xa | Dvd x xa xb \<Rightarrow> f2 x xa xb | NDvd x xa xb \<Rightarrow> f3 x xa xb) = ((\<forall>x11 x12. atom = Le x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22 x23. atom = Dvd x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)) \<and> (\<forall>x31 x32 x33. atom = NDvd x31 x32 x33 \<longrightarrow> P (f3 x31 x32 x33)))"
SBHWriteNonVolatile: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Write False a (D, f) A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, sb @ [Write\<^sub>s\<^sub>b False a (D, f) (f \<theta>) A L R W], m, \<D>, \<O>, \<R>, \<S>)"
g_newind: shows "\<forall>y<x. \<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything}"
ceiling_diff_of_int: fixes x :: "'a"   and z :: "int" shows "\<lceil>x - of_int z\<rceil> = \<lceil>x\<rceil> - z"
pt_freshs_freshs: fixes pi :: "('b \<times> 'b) list"   and Xs :: "'b set"   and Ys :: "'b set"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "set pi \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* x"   and "Ys \<sharp>* x" shows "pi \<bullet> x = x"
exec_to_exec_strip_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "\<not> isFault t" shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> \<Rightarrow> t"
sees_method_fun: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and M :: "char list"   and TS :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and TS' :: "ty list"   and T' :: "ty"   and m' :: "'a"   and D' :: "char list" assumes "P \<turnstile> C sees M: TS\<rightarrow>T = m in D"   and "P \<turnstile> C sees M: TS'\<rightarrow>T' = m' in D'" shows "TS' = TS \<and> T' = T \<and> m' = m \<and> D' = D"
prv_imp_cnj3R: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (imp \<phi>2 \<chi>) (imp (cnj \<phi>1 \<phi>2) \<chi>))"
nfoldli_assert': fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli l c (\<lambda>x s. ASSERT (x \<in> set l) \<bind> (\<lambda>_. f x s)) s = nfoldli l c f s"
assms_2: shows "AE x in M. f x \<le> c"
DERIV_local_min: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real"   and d :: "real" assumes "(f has_real_derivative l) (at x)"   and "0 < d"   and "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x \<le> f y" shows "l = 0"
hyps_3: shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
eq_on_sym: fixes s :: "'b \<Rightarrow> 'c"   and t :: "'b \<Rightarrow> 'c"   and X :: "'b set" assumes "s = t on X" shows "t = s on X"
continuous_on_inverse: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "\<forall>x\<in>s. f x \<noteq> (0::'b)" shows "continuous_on s (\<lambda>x. inverse (f x))"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (f, cod \<iota>)) local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (f, cod \<iota>))"
leq_trans: fixes x :: "'h spmf"   and y :: "'h spmf"   and z :: "'h spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y z" shows "ord_spmf (=) x z"
disc_1: fixes x1 :: "'b" shows "is_TNil (TNil x1)"
of_discrete_inject: fixes x :: "'b discrete"   and y :: "'b discrete" shows "(of_discrete x = of_discrete y) = (x = y)"
57: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> - y) \<sqinter> y = x \<sqinter> y"
lp: shows "lpfp\<^sub>c f c0 = Some c"
simps_1: fixes uu :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "remdups_on uu [] = []"
simps_1: fixes r :: "'b rexp" shows "lexer r [] = (if nullable r then Some (mkeps r) else None)"
plus_tree_simps_3: fixes t :: "'b tree"   and t' :: "'b tree" shows "right (t + t') = right t + right t'"
homotopy_eqv_homotopic_triviality_imp: fixes S :: "'b set"   and T :: "'c set"   and U :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "S homotopy_eqv T"   and "continuous_on U f"   and "f ` U \<subseteq> T"   and "continuous_on U g"   and "g ` U \<subseteq> T"   and "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S; continuous_on U g; g ` U \<subseteq> S\<rbrakk> \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) U S f g" shows "homotopic_with_canon (\<lambda>x. True) U T f g"
Q'Chain: shows "(\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>R)) \<otimes> \<Psi>' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q''"