hcju/isabelleps huggingface dataset:
fps_compose_divide: fixes g :: "'a fps"   and f :: "'a fps"   and h :: "'a fps" assumes "g dvd f"   and "h $ 0 = (0::'a)" shows "f oo h = (f / g oo h) * (g oo h)"
smult_carrier_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and k :: "'a" assumes "A \<in> carrier_mat nr nc" shows "k \<cdot>\<^sub>m A \<in> carrier_mat nr nc"
less_plus_cases: fixes i :: "nat"   and n1 :: "nat" obtains "i < n1"   | i2 :: "nat" where "i = n1 + i2"
affine_hull_open: fixes S :: "'a set" assumes "open S"   and "S \<noteq> {}" shows "affine hull S = UNIV"
lemma_2_6_21_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a l\<rightarrow> b \<sqinter> c = (a l\<rightarrow> b) \<sqinter> (a l\<rightarrow> c)"
rep_eq: fixes x :: "('a, 'b) lifted_boolean_algebra"   and xa :: "('a, 'b) lifted_boolean_algebra" shows "(x < xa) = ((\<forall>f. Rep_lifted_boolean_algebra x f \<le> Rep_lifted_boolean_algebra xa f) \<and> \<not> (\<forall>f. Rep_lifted_boolean_algebra xa f \<le> Rep_lifted_boolean_algebra x f))"
fls_divide_shift_denom_nonzero: fixes g :: "'b fls"   and f :: "'b fls"   and m :: "int" assumes "g \<noteq> 0" shows "f / fls_shift m g = fls_shift (- m) (f / g)"
subset_box_complex_2: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" shows "(cbox a b \<subseteq> box c d) = (Re a \<le> Re b \<and> Im a \<le> Im b \<longrightarrow> Re c < Re a \<and> Im c < Im a \<and> Re b < Re d \<and> Im b < Im d)"
eq2: shows "ind_in_set A ` ({0..<M} \<inter> A) - ind_in_set A ` B = ind_in_set A ` ({0..<M} \<inter> (A - B))"
Gmult_def: fixes g :: "'g"   and v :: "'v" shows "g *\<cdot> v \<equiv> ((1::'f) \<delta>\<delta> g) \<cdot> v"
leftmost_unaffected_Derivation: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat"   and a :: "symbol list"   and b :: "symbol list" assumes "derivation_ge D (Suc i)"   and "leftmost i a"   and "Derivation a D b" shows "leftmost i b"
Rxa_one: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "R \<diamondsuit>\<^sub>p 1\<^sub>r\<^bsub>R\<^esub> = carrier R"
exhaust: fixes y :: "proj2_line" obtains x :: "proj2" where "y = P2L x"
R_E: obtains "R = (\<le>)"   | "R = (\<lambda>x y. y \<le> x)"
node_deliver_messages_append: fixes xs :: "'msg event list"   and ys :: "'msg event list" shows "node_deliver_messages (xs @ ys) = node_deliver_messages xs @ node_deliver_messages ys"
STA_potential: fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and m :: "nat" assumes "A, n \<turnstile> branch" shows "A, Suc m \<turnstile> branch"
freess_def: shows "freess = ffUnion \<circ> fset_of_list \<circ> map frees"
FVF_finite: fixes e :: "fml" shows "finite (FVF e)"
A6_NSources: shows "\<forall>C\<in>AbstrLevel level0. C \<noteq> sA9 \<and> C \<noteq> sA8 \<and> C \<noteq> sA7 \<longrightarrow> sA6 \<notin> Sources level0 C"
singleton_conv2: fixes a :: "'d" shows "{x. a = x} = {a}"
complex_mvt_line: fixes w :: "complex"   and z :: "complex"   and f :: "complex \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> complex" assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f' u) (at u)" shows "\<exists>u. u \<in> closed_segment w z \<and> Re (f z) - Re (f w) = Re (f' u * (z - w))"
LAss\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls :: "val list"   and i :: "nat"   and ls' :: "val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, ls)\<rangle>"   and "i < length ls"   and "ls' = ls[i := v]" shows "P \<turnstile>\<^sub>1 \<langle>i:=e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h, ls')\<rangle>"
is_rewrite_ordD1: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and a :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "is_rewrite_ord rword" shows "rword a a"
simps_2: fixes edge :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and weight :: "'a \<Rightarrow> ennreal"   and A :: "'a set"   and B :: "'a set"   and more :: "'b" shows "A \<lparr>edge = edge, weight = weight, A = A, B = B, \<dots> = more\<rparr> = A"
Lbot: shows "Axp [] ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>)"
inj_expr_4: fixes e :: "expr" shows "inj (Add e)"
distinct_3: fixes x1 :: "'a"   and x3 :: "'b" shows "Assumption x1 \<noteq> Rule x3"
flip_flip: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" shows "flip (flip f) = f"
simps_2: fixes x :: "nat option"   and xs :: "nat option list"   and i :: "nat"   and j :: "nat"   and l :: "nat"   and T :: "bool list list" shows "mk_eqcl' (x # xs) i j l T = (if tr_lookup T j i \<or> x \<noteq> None then x else Some l) # mk_eqcl' xs i (Suc j) l T"
one_less_power: fixes a :: "'d"   and n :: "nat" assumes "(1::'d) < a"   and "0 < n" shows "(1::'d) < a ^ n"
obtain_nonzero: assumes "V \<noteq> 0" obtains g :: "'c" where "g \<in> V" and "g \<noteq> (0::'c)"
HSigma_empty1: fixes B :: "hf \<Rightarrow> hf" shows "HSigma 0 B = 0"
completely_multiplicative_function_axioms: shows "completely_multiplicative_function liouville_lambda"
decomp_set_unfold: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list" assumes "Ana t = (K, M)" shows "set (decomp t) = {send\<langle>t\<rangle>\<^sub>s\<^sub>t} \<union> Send ` set K \<union> Receive ` set M"
simps_2: fixes instT :: "((variable \<times> String.literal set) \<times> typ) list"   and insts :: "((variable \<times> typ) \<times> Core.term) list"   and idn :: "variable"   and T :: "typ" shows "subst_type_term instT insts (Fv idn T) = (let T' = subst_typ instT T in the_default (Fv idn T') (lookup (\<lambda>x. x = (idn, T')) insts))"
pure_assn_proper: fixes b :: "bool" shows "proper (pure_assn_raw b)"
tfr_subset_2: fixes B :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and A :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "tfr\<^sub>s\<^sub>e\<^sub>t B"   and "A \<subseteq> B" shows "tfr\<^sub>s\<^sub>e\<^sub>t A"
whenE_def: fixes P :: "bool"   and f :: "'d \<Rightarrow> (('e + unit) \<times> 'd) \<times> bool" shows "whenE P f \<equiv> if P then f else returnOk ()"
genby_lcoset_el_reduce: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> \<langle>S\<rangle>" shows "a +o \<langle>S\<rangle> = \<langle>S\<rangle>"
assms_3: shows "Marker \<notin> set (msgs c cid)"
dim_zero_subspace_eq: fixes A :: "('a, 'b) vec set" assumes "vec.subspace A" shows "(vec.dim A = 0) = (A = {0})"
l_insert_before_wf2_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" shows "l_insert_before_wf2_axioms type_wf known_ptrs insert_before heap_is_wellformed \<equiv> (\<forall>h ptr child ref h'. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> insert_before ptr child ref \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h') \<and> (\<forall>h ptr child ref h'. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> insert_before ptr child ref \<rightarrow>\<^sub>h h' \<longrightarrow> known_ptrs h') \<and> (\<forall>h ptr child ref h'. type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> heap_is_wellformed h \<longrightarrow> h \<turnstile> insert_before ptr child ref \<rightarrow>\<^sub>h h' \<longrightarrow> heap_is_wellformed h')"
rt: shows "returns_to {x \<in> R. x \<bullet> n = c} x"
step_1: shows "Infer CC' D E \<in> \<Gamma>"
module_hom_smult: fixes f :: "'h \<Rightarrow> 'i"   and R :: "('j, 'k) ring_scheme"   and M :: "('j, 'h, 'l) module_scheme"   and N :: "('j, 'i, 'm) module_scheme"   and r :: "'j"   and m :: "'h" assumes "f \<in> module_hom R M N"   and "r \<in> carrier R"   and "m \<in> carrier M" shows "f (r \<odot>\<^bsub>M\<^esub> m) = r \<odot>\<^bsub>N\<^esub> f m"
graph_union_def: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" shows "graph_union G\<^sub>1 G\<^sub>2 = LG (edges G\<^sub>1 \<union> edges G\<^sub>2) (vertices G\<^sub>1 \<union> vertices G\<^sub>2)"
sumPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "Rel \<subseteq> Rel'"   and "Id \<subseteq> Rel'" shows "P \<oplus> R \<leadsto>\<guillemotleft>Rel'\<guillemotright> Q \<oplus> R"
add_carrier_mat: fixes B :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and A :: "'b mat" assumes "B \<in> carrier_mat nr nc" shows "A + B \<in> carrier_mat nr nc"
subseq_singleton_left: fixes x :: "'b"   and ys :: "'b list" shows "subseq [x] ys = (x \<in> set ys)"
word_plus_and_or_coroll2: fixes x :: "'b word"   and w :: "'b word" shows "(x AND w) + (x AND NOT w) = x"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "inf (dual xa) (dual x) = dual (sup xa x)"
hyps: shows "pochhammer ((2::'a) * z) (2 * n) = of_nat (2 ^ (2 * n)) * pochhammer z n * pochhammer (z + (1::'a) / (2::'a)) n"
w_simps_4: shows "trg \<mu>.chine = src s\<^sub>0"
rel_pmf_bij_betw: fixes f :: "'c \<Rightarrow> 'd"   and p :: "'c pmf"   and q :: "'d pmf" assumes "bij_betw f (set_pmf p) (set_pmf q)"   and "\<And>x. x \<in> set_pmf p \<Longrightarrow> pmf p x = pmf q (f x)" shows "rel_pmf (\<lambda>x. (=) (f x)) p q"
prems_1: shows "snd (cs (S t i) cid) = Done"
godel_code'_set: fixes i :: "nat"   and nl :: "nat list" shows "{u. Pi (Suc i) ^ u dvd Suc (Suc 0) ^ length nl * godel_code' nl (Suc 0)} = {u. Pi (Suc i) ^ u dvd godel_code' nl (Suc 0)}"
Gamma_of_int: fixes n :: "int" shows "Gamma (of_int n) = (if 0 < n then fact (nat (n - 1)) else (0::'a))"
f5: shows "x \<le> word_of_int (uint (word_of_int (- 1 + 2 ^ LENGTH('b))))"
reduction_doesnt_break_things_invertibility: fixes subsets :: "nat list list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list" assumes "length subsets = length signs"   and "invertible_mat (M_mat signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs" shows "invertible_mat (get_matrix (reduce_system p (qs, M_mat signs subsets, subsets, signs)))"
foldSetD_imp_finite: fixes A :: "'b set"   and x :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "(A, x) \<in> foldSetD D f e" shows "finite A"
member_le_sum_list: fixes x :: "'a multiset"   and xs :: "'a multiset list" assumes "x \<in> set xs" shows "x \<subseteq># monoid_add.sum_list (+) {#} xs"
valid_Cn_not: fixes p :: "'a" shows "\<tturnstile> (.\<not> p) = (Cn {p} = UNIV)"
deduct_2: fixes p :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list"   and q :: "('a, 'b) fm" assumes "p # ps \<turnstile> q" shows "ps \<turnstile> p \<^bold>\<longrightarrow> q"
sndTSdet: fixes n :: "nat"   and xs :: "nat list"   and initH :: "nat list"   and init :: "nat list" assumes "n \<le> length xs" shows "snd (TSdet init initH xs n) = rev (take n xs) @ initH"
dom_minus: fixes f :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and A :: "'b set" assumes "f x = None" shows "dom f - insert x A = dom f - A"
simps_2: fixes c :: "'a list" shows "maxi [c] = (c \<noteq> [])"
that: fixes \<sigma> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and s0 :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and \<theta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t0 :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"   and "s0 \<in> set M"   and "is_Fun s0"   and "s = s0 \<cdot> \<sigma>"   and "\<Gamma> s = \<Gamma> s0"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "t0 \<in> set M"   and "is_Fun t0"   and "t = t0 \<cdot> \<delta> \<cdot> \<theta>"   and "\<Gamma> t = \<Gamma> t0" shows "thesis"
simps_3: fixes n :: "nat"   and f :: "'a itrm"   and x :: "'a itrm"   and i :: "nat" shows "unlift' n (f \<diamondop> x) i = unlift' n f (i + iorder x) \<degree> unlift' n x i"
xor_bot: fixes x :: "'b" shows "x \<oplus> \<bottom> = x"
lossless_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> bool spmf" shows "local.ddh.lossless \<A> = (\<forall>\<alpha> \<beta> \<gamma>. lossless_spmf (\<A> \<alpha> \<beta> \<gamma>))"
min_shift: fixes S :: "nat set"   and f :: "nat \<Rightarrow> real"   and x :: "real" assumes "S \<noteq> {}"   and "finite S" shows "Min (f ` S) + x = (MIN p\<in>S. f p + x)"
event_stays_valid_if_no_occurrence: 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 c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and ev' :: "('d, 'e, 'f) event" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "occurs_on ev \<noteq> occurs_on ev'"   and "distributed_system.can_occur channel trans send recv ev' c" shows "distributed_system.can_occur channel trans send recv ev' c'"
diff_localI: fixes U :: "'a \<Rightarrow> 'a set"   and f :: "'a \<Rightarrow> 'c" assumes "\<And>x. x \<in> src.carrier \<Longrightarrow> diff k (src.charts_submanifold (U x)) charts2 f"   and "\<And>x. x \<in> src.carrier \<Longrightarrow> open (U x)"   and "\<And>x. x \<in> src.carrier \<Longrightarrow> x \<in> U x" shows "diff k charts1 charts2 f"
embed_underS2: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set"   and g :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "Well_order r"   and "embed r s g"   and "a \<in> Field r" shows "g ` underS r a = underS s (g a)"
spray_event: fixes x :: "'a" assumes "spray x \<noteq> {}" shows "x \<in> \<E>"
strict_trans2: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "strict_prefix a b"   and "prefix b c" shows "strict_prefix a c"
identify_blocks_def: fixes A :: "'b mat"   and i :: "nat" shows "??.Jordan_Normal_Form_Existence.identify_blocks A i = ??.Jordan_Normal_Form_Existence.identify_blocks_main A i []"
counit_simps_2: shows "local.dom r.\<epsilon> = tab\<^sub>0 r \<star> (tab\<^sub>0 r)\<^sup>*"
1: shows "\<r>[f \<star> g] = (f \<star> \<r>[g]) \<cdot> \<a>[f, g, src g]"
assms_4: shows "\<not> sig_crit bs (new_syz_sigs ss bs p) p"
exec_move_ALengthI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e\<bullet>length) h s ta h' s'"
transfers_split: fixes ets :: "('var, 'val, 'ret, 'pname) edge_kind list"   and ets' :: "('var, 'val, 'ret, 'pname) edge_kind list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "transfers (ets @ ets') s = transfers ets' (transfers ets s)"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "nth_el (x # xs) (Suc n) = nth_el xs n"
build_match: fixes s :: "'s"   and v :: "'v" assumes "match s = Some v" shows "s = build v"
lexLess_def: shows "lexLess = {(M1, M2). (M2, M1) \<in> lexord decisionLess}"
g_append_cong: fixes f :: "'d \<Rightarrow> 'e"   and merge :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d list \<Rightarrow> 'f"   and xs1 :: "'d list"   and xs2 :: "'d list"   and ys1 :: "'d list"   and ys2 :: "'d list" assumes "groupsort f merge g"   and "g xs1 = g xs2"   and "g ys1 = g ys2" shows "g (xs1 @ ys1) = g (xs2 @ ys2)"
sys_block_sizes_obtain_bl: fixes x :: "int" assumes "x \<in> sys_block_sizes" shows "\<exists>bl\<in>#\<B>. int (card bl) = x"
prems_5: shows "\<alpha> = \<omega>_sum \<alpha>s"
SUP_sup_distrib: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'd" shows "\<Squnion>(f ` A) \<squnion> \<Squnion>(g ` A) = (SUP a\<in>A. f a \<squnion> g a)"
ipresIGFresh_def: fixes h :: "'h \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'h, 'p, 'q) model_scheme"   and MOD' :: "('r, 's, 'l, 't, 'u, 'o, 'i, 'v, 'w) model_scheme" shows "ipresIGFresh h MOD MOD' \<equiv> \<forall>ys y s X. igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFresh MOD' ys y (h X)"
min_product_idem: fixes A :: "'b fset set" shows "A \<otimes>\<^sub>m A = min_set A"
map\<^sub>0_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (\<sigma>\<^sub>0 a) = F.map\<^sub>0 a"
Bang_def: fixes x1 :: "ccs" shows "!x1 \<equiv> Abs_ccs (Bang_Rep (Rep_ccs x1))"
or_eq_not_not_and: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "or a b = not (and (not a) (not b))"
leftderives1_eq_leftderivations1: fixes x :: "symbol list"   and y :: "symbol list" shows "leftderives1 x y = ((x, y) \<in> leftderivations1)"
trans: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<preceq> b"   and "b \<preceq> c" shows "a \<preceq> c"
Step_CQ: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and \<theta>\<^sub>1 :: "('sf, 'sz) trm"   and \<theta>\<^sub>2 :: "('sf, 'sz) trm"   and \<sigma> :: "('sf, 'sc, 'sz) subst" assumes "step_result_dom ((SG, C), i, CQ \<theta>\<^sub>1 \<theta>\<^sub>2 \<sigma>)" shows "step_result (SG, C) (i, CQ \<theta>\<^sub>1 \<theta>\<^sub>2 \<sigma>) = (closeI SG i, C)"
neg_intro: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "\<phi> \<pi>l \<Longrightarrow> False" shows "Shallow.neg \<phi> \<pi>l"
fill_offs_enum_count_none: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and n :: "nat" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "0 < n" shows "count (mset (fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma)) None = 0"
substitute_above_arg: fixes t :: "'a ttree"   and f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set" shows "t \<sqsubseteq> substitute f T t"
arity_def: fixes fundef :: "'b fundef" shows "arity fundef = (case fundef of Fundef x1 x2 \<Rightarrow> x2)"
sup_bot_left: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "mbot \<oplus> x = x"
assms: shows "wfP (\<rightarrow>)\<inverse>\<inverse>"
complement_sum_character: fixes x :: "i" shows "\<forall>y. P y (x \<oplus> \<midarrow> x)"
spr_action_jaction: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "set (local.spr_simAction a ec) = set (jAction (spr_repMC (toSet (fst ec))) (tFirst t, tLast t) a)"
phiArg_in_allVars: fixes g :: "'g"   and v :: "'val"   and v' :: "'val" assumes "phiArg g v v'" shows "v' \<in> allVars g"
cast_PrimT2: fixes G :: "prog"   and S :: "ty"   and pt :: "prim_ty" assumes "G\<turnstile>S\<preceq>? PrimT pt" shows "\<exists>t. S = PrimT t \<and> G\<turnstile>PrimT t\<preceq>PrimT pt"
rep_eq: fixes x :: "\<Pi>\<^sub>3" shows "Semantics.d\<^sub>3 x = Some (eval\<Pi>\<^sub>3 x)"
reducing_sys_satisfies_properties_R: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list"   and matrix :: "rat mat" assumes "p \<noteq> 0"   and "0 < length qs"   and "satisfies_properties_R p qs subsets signs matrix" shows "satisfies_properties_R p qs (get_subsets_R (reduce_system_R p (qs, matrix, subsets, signs))) (get_signs_R (reduce_system_R p (qs, matrix, subsets, signs))) (get_matrix_R (reduce_system_R p (qs, matrix, subsets, signs)))"
tla_defs_2: fixes f :: "'b \<Rightarrow> 'c" shows "$f \<equiv> %s. f (first s)"
gfh: fixes x :: "'a" assumes "x \<in> S" shows "(g \<circ> f) x = exp (\<i> * complex_of_real (h x))"
size_Diff_submset: fixes M :: "'k multiset"   and M' :: "'k multiset" assumes "M \<subseteq># M'" shows "size (M' - M) = size M' - size M"
hyp: shows "\<forall>s. s \<noteq> \<dd> \<longrightarrow> s \<circ> A = A"
distinct_27: fixes x31 :: "('a, 'b, 'c) expr"   and x32 :: "('a, 'b, 'c) expr"   and x33 :: "('a, 'b, 'c) expr"   and x4 :: "('a, 'b, 'c) expr" shows "Ite x31 x32 x33 \<noteq> Ref x4"
complex_poly_decompose_rsquarefree: fixes p :: "complex poly" assumes "rsquarefree p" shows "Polynomial.smult (Polynomial.lead_coeff p) (\<Prod>z\<in>{z. poly p z = 0}. [:- z, 1:]) = p"
P1_S2_def: fixes \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)"   and M :: "'grp \<times> 'grp"   and z :: "'e"   and out1 :: "unit"   and s' :: "'f" shows "P1_S2 \<A> M z out1 s' = (let (\<A>1, \<A>2, \<A>3) = \<A> in \<A>3 s')"
analz_parts_subset: fixes H :: "msg set" shows "analz H \<subseteq> parts H"