hcju/isabelleps huggingface dataset:
intro: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and ops :: "'b arith_ops_record" assumes "bi_unique R"   and "right_total R"   and "R (arith_ops_record.zero ops) (0::'c)"   and "R (arith_ops_record.one ops) (1::'c)"   and "(R ===> R ===> R) (arith_ops_record.plus ops) (+)"   and "(R ===> R ===> R) (arith_ops_record.minus ops) (-)"   and "(R ===> R) (arith_ops_record.uminus ops) uminus"   and "(R ===> R ===> R) (arith_ops_record.times ops) (*)"   and "(R ===> R ===> (=)) (=) (=)"   and "Domainp R = arith_ops_record.DP ops" shows "ring_ops ops R"
S_transform_weakly_bisimilar_iff: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and Q :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.weakly_bisimilar nominal_ts.S_satisfies (nominal_ts.S_transition satisfies transition) (S_action.Act \<tau>) P Q = weak_nominal_ts.weakly_bisimilar satisfies transition \<tau> P Q"
sub_m_comm: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme"   and h1 :: "'c"   and h2 :: "'c" assumes "subcring H R"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes>\<^bsub>R\<^esub> h2 = h2 \<otimes>\<^bsub>R\<^esub> h1"
unique_box_desc: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>x. \<phi> x \<^bold>\<rightarrow> \<^bold>\<box>\<phi> x) \<^bold>\<rightarrow> ((\<^bold>\<exists>!x. \<phi> x) \<^bold>\<rightarrow> (\<^bold>\<forall>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<phi> y)) in v]"
span0: shows "span {x. a \<bullet> x = 0} = {x. a \<bullet> x = 0}"
bisimCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> X \<Longrightarrow> insertAssertion (extractFrame R) \<Psi>' \<simeq>\<^sub>F insertAssertion (extractFrame S) \<Psi>'"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> X \<Longrightarrow> \<Psi>' \<rhd> R \<leadsto>[(X \<union> bisim)] S"   and "\<And>\<Psi>' R S \<Psi>''. (\<Psi>', R, S) \<in> X \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', R, S) \<in> X \<or> \<Psi>' \<otimes> \<Psi>'' \<rhd> R \<sim> S"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> X \<Longrightarrow> (\<Psi>', S, R) \<in> X \<or> \<Psi>' \<rhd> S \<sim> R" shows "\<Psi> \<rhd> P \<sim> Q"
pr_strict: fixes fn :: "'f \<times> nat"   and gm :: "'f \<times> nat" shows "pr_strict' fn gm = (pr_weak' fn gm \<and> \<not> pr_weak' gm fn)"
AssertAllState'Induct: fixes t :: "(nat \<times> linear_poly) list"   and P :: "('i \<times> 'a atom) set \<Rightarrow> ('i, 'a) state \<Rightarrow> bool"   and as :: "('i \<times> 'a atom) list" assumes "\<triangle> t"   and "P {} (init t)"   and "\<And>as bs t. \<lbrakk>as \<subseteq> bs; P as t\<rbrakk> \<Longrightarrow> P bs t"   and "\<And>s a as. \<lbrakk>\<not> \<U> s; \<Turnstile> s; \<triangle> (\<T> s); \<nabla> s; P as s; index_valid as s\<rbrakk> \<Longrightarrow> P (insert a as) (assert a s)" shows "P (set as) (assert_all_state t as)"
extract_matrix_flatten_matrix_cong: fixes A :: "'a mat"   and f :: "nat \<Rightarrow> 'a" assumes "\<And>x. x < dim_row A * dim_col A \<Longrightarrow> f x = flatten_matrix A x" shows "extract_matrix f (dim_row A) (dim_col A) = A"
AE_decreasing_then_invariant: fixes f :: "'a \<Rightarrow> 'c" assumes "AE x in M. f (T x) \<le> f x"   and "f \<in> borel_measurable M" shows "AE x in M. f (T x) = f x"
ivl_disj_un_one_6: fixes l :: "'b"   and u :: "'b" assumes "l < u" shows "{l<..<u} \<union> {u..} = {l<..}"
fbox_while: fixes p :: "'a"   and t :: "'a"   and x :: "'a" assumes "d p \<cdot> d t \<le>  |x] p" shows "d p \<le>  |(d t \<cdot> x)\<^sup>\<star> \<cdot> ad t] (d p \<cdot> ad t)"
the_greatest_postfixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "p\<nu> f = (THE x. is_greatest_postfixpoint f x)"
OF_spm3_get_fe: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and ft :: "('d, 'c) flow_entry_match list"   and p :: "'e"   and a :: "'c" assumes "OF_priority_match \<gamma> ft p = Action a" shows "\<exists>fe. ofe_action fe = a \<and> fe \<in> set ft \<and> OF_priority_match_ana \<gamma> ft p = Action fe"
properfactor_nz: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and y :: "'b"   and x :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "y \<noteq> zero" shows "Divisibility.properfactor \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr> x y = (dvd.dvd times x y \<and> \<not> dvd.dvd times y x)"
Invariants_func_is_invariant: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "f \<in> borel_measurable Invariants"   and "x \<in> space M" shows "f (T x) = f x"
mult_zero_right: fixes a :: "'c" shows "bot_homomorphism ((*) a)"
i: fixes i :: "'a" assumes "i \<in> B" shows "i \<bullet> x = i \<bullet> y"
Domainp_apply2I: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h \<Rightarrow> 'i"   and x :: "'d"   and y :: "'e"   and x' :: "'f"   and y' :: "'h" assumes "rel_fun A (rel_fun B C) f g"   and "A x y"   and "B x' y'" shows "Domainp C (f x x')"
goodI: fixes i :: "nat"   and j :: "nat"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c" assumes "i < j"   and "P (f i) (f j)" shows "good P f"
narrow2: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.WN le join Top widen narrow"   and "le y x" shows "le (narrow x y) x"
partition_on_partition_on_unique: fixes A :: "'a set"   and P :: "'a set set"   and x :: "'a" assumes "partition_on A P"   and "x \<in> A" shows "\<exists>!X. x \<in> X \<and> X \<in> P"
dim_subset: fixes S :: "('b, 'c) vec set"   and T :: "('b, 'c) vec set" assumes "S \<subseteq> T" shows "vec.dim S \<le> vec.dim T"
comm_semiring_0_ow: shows "class.comm_semiring_0 = comm_semiring_0_ow UNIV"
solved_def: shows "task.solved = (\<forall>c. c |\<in>| task.conc_forms \<longrightarrow> (\<exists>\<Gamma> t. fst (tree.root t) = (\<Gamma> \<turnstile> c) \<and> \<Gamma> |\<subseteq>| task.ass_forms \<and> task.wf t \<and> tfinite t))"
record_simps_31: fixes g :: "nat \<Rightarrow> nat"   and f :: "tid_or_exn set \<Rightarrow> tid_or_exn set"   and x :: "'b state" shows "update_clock g (update_defined_types f x) = update_defined_types f (update_clock g x)"
g_def: shows "g = (\<lambda>x. real x * ln (real x) / (real (Suc x) * ln (real (Suc x))))"
snd_quot_of_fract_to_fract: fixes x :: "'b" shows "denominator (to_fract x) = (1::'b)"
ext_inject: fixes ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a"   and ext' :: "real_int"   and lan' :: "nat_int"   and own' :: "cars"   and more' :: "'a" shows "(\<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> = \<lparr>ext = ext', lan = lan', own = own', \<dots> = more'\<rparr>) = (ext = ext' \<and> lan = lan' \<and> own = own' \<and> more = more')"
disc_1: fixes x11 :: "nat"   and x12 :: "nat" shows "is_picks (picks x11 x12)"
finsum_rdistr: fixes A :: "'e set"   and a :: "'a"   and f :: "'e \<Rightarrow> 'a" assumes "finite A"   and "a \<in> carrier R"   and "f \<in> A \<rightarrow> carrier R" shows "a \<otimes> finsum R f A = (\<Oplus>i\<in>A. a \<otimes> f i)"
real_of_int_div3: fixes n :: "int"   and x :: "int" shows "real_of_int n / real_of_int x - real_of_int (n div x) \<le> 1"
vproduct_index_vempty: fixes A :: "V \<Rightarrow> V" shows "vproduct 0 A = set {0}"
assms_1: shows "K_GDD (B \<times> I) (f B) \<K>' {{x} \<times> I |x. x \<in> B}"
subst_def: fixes i :: "nat"   and j :: "nat"   and k :: "nat" shows "QEdlo.subst i j k = (if k = 0 then if i = 0 then j else i else k) - 1"
augment_edge_other: fixes e :: "nat \<times> nat"   and e' :: "nat \<times> nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and \<Delta> :: "'capacity" assumes "e \<in> E"   and "e' \<noteq> e" shows "augment_edge f e \<Delta> e' = f e'"
repv_selectlike_self: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable"   and d :: "real"   and X :: "(variable \<Rightarrow> real) set" shows "(repv \<omega> x d \<in> selectlike X \<omega> {x}) = (d = \<omega> x \<and> \<omega> \<in> X)"
reduced_GB_nonzero_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "0 \<notin> i.punit.reduced_GB F"
typeof_lit_conf: fixes v :: "val"   and T :: "ty"   and P :: "'b prog"   and h :: "heap" assumes "typeof v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
elims_3: fixes x :: "('state, 'effect) L_state"   and xa :: "'pred" assumes "\<not> x \<turnstile>\<^sub>L xa" obtains uu_ :: "'effect" and uv_ :: "'effect fs_set" and P :: "'state" and \<phi> :: "'pred" where "x = AC (uu_, uv_, P)" and "xa = \<phi>__" and "\<not> P \<turnstile> \<phi>__"   | uw_ :: "'effect fs_set \<times> 'state" and \<phi> :: "'pred" where "x = EF uw_" and "xa = \<phi>"
poly_mapping_eq_iff: fixes a :: "'q \<Rightarrow>\<^sub>0 'r"   and b :: "'q \<Rightarrow>\<^sub>0 'r" shows "(a = b) = (lookup a = lookup b)"
equiv_type: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "equiv A r" shows "r \<subseteq> A \<times> A"
idem_normalize: fixes a :: "'a" shows "gcd a a = normalize a"
independent_finite: fixes B :: "'b set" assumes "independent B" shows "finite B"
sym_sum_mset_inter_sets_count: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and blv :: "'a set"   and p :: "'a" assumes "symmetric_bibd point_set block_collection u_block_size index"   and "blv \<in># block_collection"   and "p \<in> blv" shows "int (count (\<Sum>bl\<in>#remove1_mset blv block_collection. mset_set (bl \<inter> blv)) p) = index * (int (card point_set) - 1) div (u_block_size - 1) - 1"
cart_product_add_1_filter2: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and M :: "'a multiset"   and b :: "'b"   and N :: "'b multiset" shows "filter_mset P (M \<times># add_mset b N) = filter_mset P (M \<times># N) + filter_mset P (M \<times># {#b#})"
interleave_hide: fixes A :: "'a event set"   and C :: "'a event set"   and ra :: "'a event list"   and t :: "'a event list"   and u :: "'a event list" assumes "A \<inter> C = {}"   and "ra setinterleaves ((trace_hide t A, trace_hide u A), C)" shows "\<exists>r. ra = trace_hide r A \<and> r setinterleaves ((t, u), C)"
\<R>\<^sub>SI: fixes \<beta> :: "'d \<Rightarrow> 'd"   and \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option"   and \<sigma>' :: "'d \<Rightarrow> ('e, 'd, 'f) val option"   and \<alpha> :: "'e \<Rightarrow> 'e" assumes "bij \<beta>"   and "\<And>l. \<sigma> l = None \<Longrightarrow> \<sigma>' (\<beta> l) = None"   and "\<And>l v. \<sigma> l = Some v \<Longrightarrow> \<sigma>' (\<beta> l) = Some (\<R>\<^sub>V \<alpha> \<beta> v)" shows "\<R>\<^sub>S \<alpha> \<beta> \<sigma> = \<sigma>'"
mtf2_moves_to_front'': fixes xs :: "'b list"   and y :: "'b"   and x :: "'b" assumes "distinct xs"   and "y \<in> set xs"   and "x \<in> set xs"   and "x \<noteq> y" shows "x < y in mtf2 (length xs) x xs = True"
bidir_2: shows "\<exists>rev. bidirected_digraph H rev"
exec_Do_iff: fixes s :: "state"   and f :: "state \<Rightarrow> state set"   and t :: "state" shows "(s -Do f\<rightarrow> t) = (t \<in> f s)"
product_criterion: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'e) set"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'e"   and q :: "'a \<Rightarrow>\<^sub>0 'e" assumes "dickson_grading d"   and "F \<subseteq> punit.dgrad_p_set d m"   and "p \<in> F"   and "q \<in> F"   and "p \<noteq> 0"   and "q \<noteq> 0"   and "gcs (punit.lt p) (punit.lt q) = (0::'a)" shows "punit.crit_pair_cbelow_on d m F p q"
biquadratic_solution: fixes y :: "'a"   and p :: "'a"   and q :: "'a" shows "(y ^ 4 + p * y\<^sup>2 + q = (0::'a)) = (\<exists>z. z\<^sup>2 + p * z + q = (0::'a) \<and> z = y\<^sup>2)"
relcompp_exists: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and r' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'d"   and y :: "'f" shows "(r OO r') x y = (\<exists>u. r x u \<and> r' u y)"
arcs_slim_in_G: shows "parcs H.slim \<subseteq> parcs slim"
LAss_reds: fixes V :: "char list"   and e :: "expr"   and s :: "char list \<Rightarrow> val option"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>V:=e,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>" shows "s' = s(V := \<lbrakk>e\<rbrakk>s)"
homeomorphic_open_imp_same_dimension: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "open S"   and "S \<noteq> {}"   and "open T"   and "T \<noteq> {}" shows "DIM('a) = DIM('b)"
linear_def: fixes f :: "'d \<Rightarrow> 'e" shows "linear f \<equiv> Vector_Spaces.linear (*\<^sub>R) (*\<^sub>R) f"
valuate_ex: fixes x :: "'a"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'a" shows "(x \<in> valuate f) = (\<exists>e. (\<forall>i. e i \<in> {- 1..1}) \<and> x = f e)"
phiDefs_def: fixes g :: "'g"   and n :: "'node" shows "step.phiDefs g n \<equiv> {v. (n, v) \<in> dom (p_g g)}"
context_sharing_Cut: fixes \<Gamma> :: "'a formula multiset"   and A :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> A, \<Delta>"   and "A, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
alpha_Tree_FL_valid_Tree: fixes t1 :: "('b, 'pred, 'act, 'effect) Tree"   and t2 :: "('b, 'pred, 'act, 'effect) Tree"   and P :: "'state" assumes "t1 =\<^sub>\<alpha> t2" shows "FL_valid_Tree P t1 = FL_valid_Tree P t2"
sorted_on_linord_of_list: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "Linear_order r"   and "set xs = Field r"   and "sorted_on r xs"   and "distinct xs" shows "linord_of_list (rev xs) = r"
nths_upt_eq_take: fixes l :: "'a list"   and n :: "nat" shows "nths l {..<n} = take n l"
llexord_LCons_left: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist"   and ys :: "'c llist" shows "llexord r (LCons x xs) ys = (\<exists>y ys'. ys = LCons y ys' \<and> (x = y \<and> llexord r xs ys' \<or> r x y))"
submonoid_inverse_closed: fixes u :: "'a" assumes "IA.additive.sub.invertible u"   and "u \<in> I" shows "IA.additive.inverse u \<in> I"
vlrestriction_vsubset_vrange: fixes U :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V" assumes "\<D>\<^sub>\<circ> (VLambda U f) \<subseteq>\<^sub>\<circ> A" shows "VLambda U f \<restriction>\<^sup>l\<^sub>\<circ> A = VLambda U f"
igSwapIPresIGWlsSTR_def: fixes MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'p, 'q, 'r) model_scheme" shows "igSwapIPresIGWlsSTR MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s (igSwap MOD zs z1 z2 X) = igWls MOD s X"
chart_last_nonzero_scaleR: fixes c :: "real"   and n :: "('b \<times> real) nonzero" assumes "c \<noteq> 0" shows "chart_last_nonzero (c *\<^sub>R n) = chart_last_nonzero n"
assms_2: shows "filterlim g G (at x within {x. \<not> P x})"
convert_extTA_simps_2: fixes f :: "'i \<Rightarrow> 'e"   and ta :: "('c, 'd, 'i, 'f, 'g, 'h) thread_action" shows "\<lbrace>convert_extTA f ta\<rbrace>\<^bsub>l\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>"
Field_insert: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" shows "Field (insert (a, b) r) = {a, b} \<union> Field r"
bit_twiddle_min: fixes y :: "'b word"   and x :: "'b word" shows "y XOR (x XOR y) AND (if x < y then - 1 else 0) = min x y"
subring_is_ring: fixes R :: "('c, 'd) ring_scheme"   and H :: "'c set" assumes "ring R"   and "subring H R" shows "ring (R\<lparr>carrier := H\<rparr>)"
mat_of_cols_carrier_2: fixes n :: "nat"   and vs :: "'a Matrix.vec list" shows "dim_row (mat_of_cols n vs) = n"
smallo_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "o[F](g) = {f. \<forall>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
assms_2: shows "((\<le>), (\<le>)) \<in> B \<rightarrow> B \<rightarrow> bool_rel"
no_overlaps_lroft_hlp3: fixes amr :: "('b \<times> 32 simple_match \<times> char list \<times> simple_action) list"   and aa :: "'b"   and ab :: "of_match_field set"   and ac :: "of_action list"   and ifs :: "char list list"   and ba :: "'b"   and bb :: "of_match_field set"   and bc :: "of_action list" assumes "distinct (map fst amr)"   and "(aa, ab, ac) \<in> set (lr_of_tran_s3 ifs amr)"   and "(ba, bb, bc) \<in> set (lr_of_tran_s3 ifs amr)"   and "ac \<noteq> bc" shows "aa \<noteq> ba"
mem_cong_ax_def: shows "mem_cong_ax = Var X1 EQ Var X2 AND Var X3 EQ Var X4 IMP Var X1 IN Var X3 IMP Var X2 IN Var X4"
unl_lift2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'d \<Rightarrow> 'b"   and y :: "'d \<Rightarrow> 'c"   and w :: "'d" shows "(f<x, y>) w \<equiv> f (x w) (y w)"
continuous_frac: fixes x :: "real" assumes "x \<notin> \<int>" shows "isCont frac x"
assms: shows "(k, \<gamma>) \<in> boundary rot_diamond_cube"
simps_1: fixes edge :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and capacity :: "'k \<times> 'k \<Rightarrow> ennreal"   and source :: "'k"   and sink :: "'k"   and more :: "'l" shows "capacity \<lparr>edge = edge, capacity = capacity, source = source, sink = sink, \<dots> = more\<rparr> = capacity"
exec_step_ind_IfFalse_False: fixes stk :: "val list"   and i :: "int"   and P :: "jvm_prog"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "hd stk = Bool False" shows "exec_step_ind (StepI (IfFalse i)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (tl stk, loc, C\<^sub>0, M\<^sub>0, nat (int pc + i), ics) # frs, sh)"
overrides_t_Some_iff: fixes x :: "'a"   and s :: "'b \<Rightarrow> 'a set"   and t :: "'b \<Rightarrow> 'a set"   and k :: "'b" shows "(x \<in> (s \<oplus>\<oplus> t) k) = (x \<in> t k \<or> t k = {} \<and> x \<in> s k)"
gjs_and_take_cols_var: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc"   and "n \<le> nc" shows "take_cols A (map snd (pivot_positions (gauss_jordan_single A))) = take_cols_var A (map snd (pivot_positions (gauss_jordan_single A)))"
case_rews_1: fixes f :: "C \<rightarrow> 'a" shows "C_case\<cdot>f\<cdot>\<bottom> = \<bottom>"
dp_aux2: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_p_fun z"   and "is_total w"   and "z\<^sup>\<smile> ; x = 1" shows "(w ; x\<^sup>\<smile> \<cdot> y ; z\<^sup>\<smile>) ; z = y"
dropWhile_eq_Cons_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "(dropWhile P xs = y # ys) = (xs = takeWhile P xs @ y # ys \<and> \<not> P y)"
assms: shows "p \<noteq> 0 \<or> q \<noteq> 0"
reflect_poly_0_iff: fixes p :: "'a poly" shows "(reflect_poly p = 0) = (p = 0)"
m: fixes m :: "int" assumes "poly_mod_type TYPE('a) m" shows "m = int CARD('a)"
outs_\<I>_mono: fixes \<I> :: "('c, 'd) \<I>"   and \<I>' :: "('c, 'd) \<I>" assumes "\<I> \<le> \<I>'" shows "outs_\<I> \<I> \<subseteq> outs_\<I> \<I>'"
eval_in_hom_1: fixes t :: "'a E.term" assumes "E.Arr t" shows "\<guillemotleft>EVAL t : EVAL (E.Src t) \<rightarrow>\<^sub>B EVAL (E.Trg t)\<guillemotright>"
ord_isom_def: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e" shows "ord_isom D E f = (ord_inj D E f \<and> surj_to f (carrier D) (carrier E))"
leD: fixes x :: "'d"   and y :: "'d" assumes "x \<le> y" shows "\<not> y < x"
assms_2: shows "G2 \<in> {g - {x} |g. g \<in> \<G> \<and> g \<noteq> {x}}"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) Cop.H Cop.\<a> Cop.\<i> Cop.src Cop.trg g f' \<eta>' \<epsilon>'"
prod_set_simps_1: fixes x :: "'a"   and y :: "'b" shows "fsts (x, y) = {x}"
assms_4: shows "h \<turnstile> remove_shadow_root ptr \<rightarrow>\<^sub>h h'"
Dag_upd_same_l_lemma: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "p \<noteq> Null" shows "\<not> Dag p (l(p := p)) r t"
prems_2: shows "Fun f T \<sqsubseteq> \<Gamma> (Var x)"
f4: shows "(r \<otimes> r'', s \<otimes> s'') \<in> carrier rel"
EndPhase0_44: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and bk :: "DiskBlock" assumes "EndPhase0 s s' p"   and "bk \<in> blocksOf s p"   and "HInv4d s p"   and "Inv2c_inner s p" shows "\<exists>d. \<exists>rb\<in>blocksRead s p d. bal bk \<le> mbal (block rb)"
bounded_linear_left: fixes b :: "'b" shows "bounded_linear (\<lambda>a. a ** b)"
subterm_inj_on_imp_inj_on: fixes f :: "'a \<Rightarrow> ('b, 'c) Term.term"   and A :: "'a set" assumes "subterm_inj_on f A" shows "inj_on f A"
rep_eq: fixes x :: "('a, 'b) pp" shows "keys_pp x = keys (pp.mapping_of x)"
frag_extend_0: fixes b :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int" shows "frag_extend b 0 = 0"
closure_of_memE: fixes a :: "'d"   and S :: "('d, 'e) eq_object_scheme"   and A :: "'d set" assumes "a \<in> closure_of\<^bsub>S\<^esub> A" obtains "a \<in> carrier S" and "a .\<in>\<^bsub>S\<^esub> A"
assms_6: shows "(aaa, aaaa, baa) \<in> set les"
rDown: assumes "invar l \<and> invar r \<and> lv = Suc (Suc (lvl r)) \<and> lv = Suc (lvl l)" shows "thesis"
simps_4: fixes a :: "MIR.num" shows "numbound0 (Neg a) = numbound0 a"
enabled_iff: fixes s :: "'s"   and \<omega> :: "'s stream" shows "enabled s \<omega> = (enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
assms_3: fixes e :: "'b" assumes "e \<in> arcs G" shows "0 \<le> c e"
allpowers_def: fixes x :: "'a"   and n :: "nat" shows "allpowers x n = {y. \<exists>i<n. y = x ^ i}"
in_sccs_subset_imp_eq: fixes c :: "('a, 'b) pre_digraph"   and d :: "('a, 'b) pre_digraph" assumes "c \<in> sccs"   and "d \<in> sccs"   and "verts c \<subseteq> verts d" shows "c = d"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (x \<sqsubseteq> y \<and> y \<sqsubseteq> x)"
value_Zl_mI_pos: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat"   and l :: "nat" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)"   and "j \<le> n"   and "l \<le> n" shows "0 \<le> (\<nu>\<^bsub>K P j\<^esub>) (Zl_mI K P I l)"
inversion_number_between_Cons_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" assumes "sorted_wrt (\<le>) (x # xs)"   and "sorted_wrt (\<le>) (y # ys)" shows "inversion_number_between (x # xs) (y # ys) = (if \<not> y < x then inversion_number_between xs (y # ys) else inversion_number_between (x # xs) ys + length (x # xs))"
distinct_append: fixes xs :: "'d list"   and ys :: "'d list" shows "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
eq: shows "(reduced (Suc n) \<circ> lab) ` f = {..n}"
regular_closed_star: fixes x :: "'a" assumes "regular x" shows "regular (x\<^sup>\<star>)"
dghm_dag_Rel_def: fixes \<alpha> :: "V" shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha> = [vid_on (dg_Rel \<alpha>\<lparr>Obj\<rparr>), VLambda (dg_Rel \<alpha>\<lparr>Arr\<rparr>) converse_Rel, op_dg (dg_Rel \<alpha>), dg_Rel \<alpha>]\<^sub>\<circ>"
assms: shows "(u \<longlongrightarrow> \<infinity>) F"
bot_Inf: fixes A :: "'a set" assumes "bot \<in> A" shows "Inf A = bot"
ide_dom: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.arr f" shows "VVV.ide (VVV.dom f)"
cases_2: fixes z :: "'t \<Rightarrow>\<^sub>0 'b"   and y :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(red F)\<^sup>*\<^sup>* (p - q) z"   and "red F z y"   and "P p (q + z)" shows "P p (q + y)"
dsr5: fixes x :: "'b"   and y :: "'b" shows "ad (ad (x + y)) = ad (ad x) + ad (ad y)"