hcju/isabelleps huggingface dataset:
lm017: fixes bids :: "'a \<times> 'b set \<Rightarrow> 'c"   and N :: "'a set"   and G :: "'b set"   and a :: "('a \<times> 'b set) set" shows "{summedBid bids pair |pair. pair \<in> N \<times> (Pow G - {{}})} || a = {summedBid bids pair |pair. pair \<in> N \<times> (Pow G - {{}}) \<inter> a}"
trans: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and i :: "nat"   and s :: "'s"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and s' :: "'s"   and a :: "'m seq_action" assumes "\<sigma> i = fst (sr s)"   and "\<sigma>' i = fst (sr s')"   and "(s, a, s') \<in> automaton.trans (np i)" shows "((\<sigma>, snd (sr s)), a, \<sigma>', snd (sr s')) \<in> automaton.trans (onp i)"
fps_divide_X': fixes f :: "'a fps" assumes "inverse (1::'a) = (1::'a)" shows "f / fps_X = fps_shift 1 f"
Allowed_def: fixes F :: "'d program" shows "Allowed F \<equiv> {G. Acts G \<subseteq> AllowedActs F}"
UNIV: fixes S :: "'b set"   and B :: "'b set set" shows "generated_topology S B S"
edges_are_in_V_2: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "w \<in> V\<^bsub>G\<^esub>"
all_pure_unlift_vars: fixes x :: "nat itrm" assumes "opaque x = []" shows "x \<simeq> Pure (unlift_vars 0 x)"
assms_2: shows "C \<subseteq> U\<^sub>V\<^sub>S"
vconverse_iff: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>) = (r\<lparr>a\<rparr> = b)"
find_path_f_def: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl" shows "find_path_f P ctx cl = find_path_rec_f P ctx cl []"
right_neutral: fixes a :: "'q" shows "a + (0::'q) = a"
simulation1_\<tau>rtrancl3p: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and tls1 :: "'h list"   and s1' :: "'g"   and s2 :: "'i" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "\<tau>trsys.\<tau>rtrancl3p trsys1 \<tau>move1 s1 tls1 s1'"   and "bisim s1 s2" shows "\<exists>tls2 s2'. \<tau>trsys.\<tau>rtrancl3p trsys2 \<tau>move2 s2 tls2 s2' \<and> bisim s1' s2' \<and> list_all2 tlsim tls1 tls2"
bres_def: fixes x :: "'a"   and z :: "'a" shows "x \<rightarrow> z = \<Squnion> {y. x \<cdot> y \<le> z}"
S_transform_bisimilar_iff: fixes P :: "'state"   and Q :: "'state" shows "nominal_ts.bisimilar (\<turnstile>\<^sub>S) S_transform.S_transition P Q = P \<sim>\<cdot>\<^sub>S Q"
liftAll_def: fixes phi :: "'b \<Rightarrow> bool"   and inp :: "'c \<Rightarrow> 'b option" shows "liftAll phi inp \<equiv> \<forall>i v. inp i = Some v \<longrightarrow> phi v"
assms_3: shows "\<infinity>\<^sub>h \<in> circline_set H"
sup_left_idem: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<squnion> y) = x \<squnion> y"
h4: shows "at r \<cdot> tp \<cdot> at r \<cdot> y \<le> at r \<cdot> y \<cdot> tq"
singleton_not_empty: fixes a :: "'b" shows "{a} \<noteq> {}"
assms: fixes C :: "'a set" assumes "C \<in> components S" shows "openin (top_of_set S) C \<and> continuous_on C f"
incrIndexList_help21: fixes l :: "nat"   and ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList (l # ls) m nmax" shows "l = 0"
ii: shows "\<forall>M\<ge>N2. P2 \<in> inferred_clause_sets R S M"
snd_divmod: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "snd (divmod m n) = modulo (numeral.numeral one plus m) (numeral.numeral one plus n)"
1: shows "D1 * S1 + H1 * T1 =m 1"
product_def: fixes A :: "'b set"   and B :: "'d set" shows "Product_Type.product A B = A \<times> B"
sdescending_sset_sappend: fixes u :: "'b list"   and v :: "'b stream"   and a :: "'b"   and b :: "'b" assumes "sdescending (u @- v)"   and "a \<in> set u"   and "b \<in> sset v" shows "b \<le> a"
ev_holds_sset: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a stream" shows "ev (holds P) xs = (\<exists>x\<in>sset xs. P x)"
Abs_freeletter_in_FreeGroup_iff: fixes s :: "'b"   and S :: "'b set" shows "(Abs_freeletter s \<in> FreeGroup S) = (s \<in> S)"
set_mset_replicate_mset_subset: fixes n :: "nat"   and x :: "'a" shows "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
second_vertices_def: shows "Q.second_vertices \<equiv> Q.second_vertex ` Q"
dvd_pderiv_iff: fixes p :: "'b poly" shows "(p dvd pderiv p) = (Polynomial.degree p = 0)"
lookup_not_eq_zero_eq_in_keys: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "(lookup f k \<noteq> (0::'c)) = (k \<in> keys f)"
wp_While_False: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<not> bval b s" shows "wp (WHILE b DO c) Q s = Q s"
simple_function_indicator_representation: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and x :: "'b" assumes "simple_function M f"   and "x \<in> space M" shows "f x = (\<Sum>y\<in>f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
bound_domain_and_range_impl_finitely_many_functions: fixes n :: "nat"   and m :: "nat" shows "finite {f. (\<forall>i. f i \<le> n) \<and> (\<forall>i\<ge>m. f i = 0)}"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w"
orthogonal_matrix': fixes Q :: "((real, 'b) vec, 'b) vec" shows "orthogonal_matrix Q = (Q ** Finite_Cartesian_Product.transpose Q = mat 1)"
conj_idem: fixes P :: "'b \<Rightarrow> bool" shows "(P \<and> P) = P"
approx_refl: fixes x :: "'a star" shows "x \<approx> x"
is_OK_def: fixes h :: "'a"   and p :: "('a, 'b, 'c) prog" shows "h \<turnstile> ok p = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr x \<Rightarrow> True)"
Field_def: fixes r :: "('c \<times> 'c) set" shows "Field r = Domain r \<union> Range r"
rep_eq: fixes x :: "'f"   and xa :: "('f, 'e) generator" shows "generator (Cons_trans x xa) = Cons_raw x (generator xa)"
a1: fixes s :: "'a" assumes "s \<in> S" shows "is_orthogonal a s"
emeasure_pair_measure_Times: fixes A :: "'b set"   and N :: "'b measure"   and B :: "('c \<times> real) set" assumes "A \<in> sets N"   and "B \<in> sets (lborel \<Otimes>\<^sub>M M)" shows "emeasure (N \<Otimes>\<^sub>M lborel \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure (lborel \<Otimes>\<^sub>M M) B"
simps_13: fixes x :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "isDERIV x (Powr a b) vs = (isDERIV x a vs \<and> isDERIV x b vs \<and> 0 < interpret_floatarith a vs)"
Legendre_in_cong_eq: fixes p :: "int"   and b :: "int"   and a :: "int"   and m :: "int" assumes "2 < p"   and "b \<in> {- 1, 0, 1}" shows "[Legendre a m = b] (mod p) = (Legendre a m = b)"
act: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "finite {nid. \<parallel>nid\<parallel>\<^bsub>t n\<^esub>}"
pnet_init_in_net_ips_in_net_tree_ips: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree"   and i :: "nat" assumes "s \<in> init (pnet np p)"   and "i \<in> net_ips s" shows "i \<in> net_tree_ips p"
urel64_normalize: fixes pp :: "uint64"   and x :: "uint64"   and y :: "int" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295"   and "urel64 x y" shows "urel64 (if x = 0 then 0 else 1) (if y = 0 then 0 else 1)"
set_of_def: fixes S :: "'b" shows "set_of S = {x. is_node x \<and> memb x S}"
intro: fixes T :: "'b set set" assumes "\<forall>x\<in>carrier\<^bsub>T\<^esub>. \<forall>y\<in>carrier\<^bsub>T\<^esub>. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds\<^bsub>T\<^esub> x. \<exists>v\<in>nhds\<^bsub>T\<^esub> y. u \<inter> v = {})" shows "T2_axioms T"
R_disj: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is R healthy"   and "Q is R healthy" shows "P \<or> Q is R healthy"
suffix: fixes i :: "nat" assumes "i < Suc n" shows "suffix k w \<Turnstile>\<^sub>n af (if i < n then \<phi> else \<psi>) (w [min i n \<rightarrow> k])[X]\<^sub>\<nu>"
tableau_valuated_def: fixes s :: "('a, 'b) state" shows "\<nabla> s \<equiv> \<forall>x\<in>tvars (\<T> s). Mapping.lookup (\<V> s) x \<noteq> None"
c5: assumes "\<pi>\<^sub>3 r = unk" shows "P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r)))"
set_less_eq_finite_iff: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "(A \<sqsubseteq> B) = (A \<sqsubseteq>' B)"
Un_iff: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" shows "((c::'b) \<in> (A::'b set) \<union> (B::'b set)) = (c \<in> A \<or> c \<in> B)"
\<Delta>_simps_4: shows "local.dom ((\<mu> \<star> tab\<^sub>0 (local.dom \<mu>)) \<cdot> \<mu>.r.tab) = tab\<^sub>1 (local.dom \<mu>)"
eq_ivl_empty: fixes p :: "int extended \<times> int extended" shows "eq_ivl p empty_rep = is_empty_rep p"
induct: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and P :: "'a stream \<Rightarrow> bool" assumes "(\<phi> suntil \<psi>) x"   and "\<And>\<omega>. \<psi> \<omega> \<Longrightarrow> P \<omega>"   and "\<And>\<omega>. \<lbrakk>\<phi> \<omega>; (\<phi> suntil \<psi>) (stl \<omega>); P (stl \<omega>)\<rbrakk> \<Longrightarrow> P \<omega>" shows "P x"
qtable_def: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" shows "qtable n A P Q X = (table n A X \<and> (\<forall>x. (x \<in> X \<and> P x \<longrightarrow> Q x) \<and> (wf_tuple n A x \<and> P x \<and> Q x \<longrightarrow> x \<in> X)))"
intros_2: fixes t :: "('e, 'f, 'g) Tree" shows "(t, tNot t) \<in> Tree_wf"
nonzero_root_set_def: fixes les :: "(real \<times> real \<times> real) set" shows "nonzero_root_set les \<equiv> {x. \<exists>(a, b, c)\<in>les. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
simps: fixes A' :: "('b \<times> bal) tree alt"   and a :: "'b"   and ba :: "bal"   and BC :: "('b \<times> bal) tree" shows "baldL A' a ba BC = (case A' of Same A \<Rightarrow> Same \<langle>A, (a, ba), BC\<rangle> | Diff A \<Rightarrow> case ba of Lh \<Rightarrow> Diff \<langle>A, (a, Bal), BC\<rangle> | Bal \<Rightarrow> Same \<langle>A, (a, Rh), BC\<rangle> | Rh \<Rightarrow> case BC of \<langle>B, (c, Lh), C\<rangle> \<Rightarrow> Diff (rot2 A a B c C) | \<langle>B, (c, Bal), C\<rangle> \<Rightarrow> Same \<langle>\<langle>A, (a, Rh), B\<rangle>, (c, Lh), C\<rangle> | \<langle>B, (c, Rh), C\<rangle> \<Rightarrow> Diff \<langle>\<langle>A, (a, Bal), B\<rangle>, (c, Bal), C\<rangle>)"
ai: fixes i :: "nat" shows "a i \<in> sigma_sets sp A"
1: assumes "xs prefix of i" shows "apply_operations xs \<noteq> None"
fdomain_fconverse: fixes r :: "V" shows "\<D>\<^sub>\<bullet> (r\<inverse>\<^sub>\<bullet>) = \<R>\<^sub>\<bullet> r"
rep_eq: fixes x :: "float" shows "real_of_float \<bar>x\<bar> = \<bar>real_of_float x\<bar>"
gen_lower_bound_increasing: fixes a :: "real"   and x :: "real"   and fl :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "a \<le> x"   and "\<And>y. \<lbrakk>a \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> ((\<lambda>x. fl x - f x) has_real_derivative g y) (at y)"   and "\<And>y. \<lbrakk>a \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> g y \<le> 0"   and "fl a = f a" shows "fl x \<le> f x"
rank_mono: fixes a :: "V"   and b :: "V" assumes "a \<le> b" shows "rank a \<le> rank b"
fres_bres: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "prod.bres x (prod.fres y z) = prod.fres (prod.bres x y) z"
permutation_insert_inserted: fixes i :: "nat"   and j :: "nat"   and p :: "nat \<Rightarrow> nat" shows "permutation_insert i j p i = j"
sum_2: fixes f :: "2 \<Rightarrow> 'c" shows "sum f UNIV = f 1 + f 2"
l: shows "l = (LEAST n. (a, b) \<in> set rel ^^ n)"
replace_col_def: fixes A :: "'b mat"   and b :: "'b vec"   and k :: "nat" shows "replace_col A b k = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if j = k then b $ i else A $$ (i, j))"
nonzero_positions_def: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "nonzero_positions A = {pos. A (fst pos) (snd pos) \<noteq> (0::'b)}"
trcl_pair_unconsE: fixes s :: "'f"   and c :: "'g"   and e :: "'h"   and w :: "'h list"   and s' :: "'f"   and c' :: "'g"   and S :: "(('f \<times> 'g) \<times> 'h \<times> 'f \<times> 'g) set" assumes "((s, c), e # w, s', c') \<in> trcl S" obtains sh :: "'f" and ch :: "'g" where "((s, c), e, sh, ch) \<in> S" and "((sh, ch), w, s', c') \<in> trcl S"
assms: shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
ssubseqs_subset: fixes as :: "'b list"   and bs :: "'b list" assumes "as \<in> ssubseqs bs" shows "order.greater_eq (ssubseqs bs) (ssubseqs as)"
snd_comp_apsnd: fixes f :: "'b \<Rightarrow> 'c" shows "(snd \<circ>\<circ> apsnd) f = f \<circ> snd"
d: fixes w :: "complex" assumes "w \<in> ball \<xi> d" shows "g w \<noteq> 0"
closed_segment_real: fixes a :: "real"   and b :: "real" shows "{a--b} = (if a \<le> b then {a..b} else {b..a})"
span_insert_zero: fixes B :: "'b set" shows "ideal (insert (0::'b) B) = ideal B"
intro: fixes point_set :: "'b set"   and grouping :: "int"   and block_collection :: "'b set multiset"   and min_index :: "int"   and u_block_size :: "int" assumes "\<And>ps. \<lbrakk>ps \<subseteq> point_set; int (card ps) = grouping\<rbrakk> \<Longrightarrow> int (block_collection index ps) \<le> min_index"   and "grouping \<le> u_block_size"   and "1 \<le> grouping" shows "t_packing_design_axioms point_set block_collection u_block_size grouping min_index"
inj_group_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_group_add_hom hom \<equiv> group_add_hom hom \<and> zero_hom_0 hom"
normalized_corthogonal_mat_is_unitary: fixes ws :: "complex vec list"   and n :: "nat" assumes "set ws \<subseteq> carrier_vec n"   and "corthogonal ws"   and "length ws = n" shows "unitary (mat_of_cols n (map vec_normalize ws))"
cutoff_l: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt"   and p :: "(real, 'b) vec"   and c :: "(real, 'b) vec" assumes "invar (Node k v l r)"   and "v \<le> p $ k"   and "dist p c \<le> dist v (p $ k)" shows "\<forall>q\<in>set_kdt l. dist p c \<le> dist p q"
map: fixes f :: "'c \<Rightarrow> 'b"   and x1 :: "'c"   and x2 :: "'c tree"   and x3 :: "'c tree" shows "map_tree f (Node x1 x2 x3) = Node (f x1) (map_tree f x2) (map_tree f x3)"
basic: fixes t :: "'a llist set" assumes "t \<in> B" shows "t open"
remdups_append2: fixes xs :: "'b list"   and ys :: "'b list" shows "remdups (xs @ remdups ys) = remdups (xs @ ys)"
s_mul_extI: fixes A :: "'a multiset"   and A1 :: "'a multiset"   and A2 :: "'a multiset"   and B :: "'a multiset"   and B1 :: "'a multiset"   and B2 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "A = A1 + A2"   and "B = B1 + B2"   and "(A1, B1) \<in> multpw ns"   and "A2 \<noteq> {#}"   and "\<And>b. b \<in># B2 \<Longrightarrow> \<exists>a. a \<in># A2 \<and> (a, b) \<in> s" shows "(A, B) \<in> s_mul_ext ns s"
integral_Dirichlet_kernel_mul_expand: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and n :: "nat" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "LINT x|lebesgue_on S. Dirichlet_kernel n x * f x = LINT x|lebesgue_on S. sin ((real n + 1 / 2) * x) * f x / (2 * sin (x / 2))"
Inf_le_Sup: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "less_eq (Inf A) (Sup A)"
ide_char: fixes f :: "(bool, unit) arr" shows "ide f = (f = MkIde False \<or> f = MkIde True)"
inj_of_char: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "inj (comm_semiring_1.of_char times one plus zero)"
If_THE_False: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c2 :: "com"   and c1 :: "com" assumes "\<not> bval b s"   and "\<down> (c2, s)" shows "Suc (\<down>\<^sub>t (c2, s)) = \<down>\<^sub>t (IF b THEN c1 ELSE c2, s)"
Iff_trans: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> A IFF B"   and "H \<turnstile> B IFF C" shows "H \<turnstile> A IFF C"
runit'_in_hom_2: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<guillemotleft>\<r>\<^sup>-\<^sup>1[f] : f \<Rightarrow> f \<star> src f\<guillemotright>"
left_inv_of_unitary_transpose: fixes U :: "complex mat" assumes "unitary U" shows "U\<^sup>t\<^sup>\<dagger> * U\<^sup>t = 1\<^sub>m (dim_row U)"
in_set_conv_decomp: fixes x :: "'c"   and xs :: "'c list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
exec_c: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
cong_transitivity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D"   and "Cong C D E F" shows "Cong A B E F"
shext_trans: fixes sh :: "sheap"   and sh' :: "sheap"   and sh'' :: "sheap" assumes "sh \<unlhd>\<^sub>s sh'"   and "sh' \<unlhd>\<^sub>s sh''" shows "sh \<unlhd>\<^sub>s sh''"
iwlsFSw_termFSwImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e" assumes "iwlsFSw MOD" shows "termFSwImorph h hA MOD = ipresCons h hA MOD"
Nil: fixes A :: "'b set" shows "List.list.Nil \<in> List.lists A"
clock_jview_def: shows "clock_jview \<equiv> \<lambda>a t. (tLength t, envObs a (tLast t))"
lQuots_simps_2: fixes c :: "'c"   and s :: "'c list"   and A :: "'c list set" shows "lQuots (c # s) A = lQuots s (lQuot c A)"
verit_minus_simplify_1: fixes a :: "'c" shows "a - a = (0::'c)"
homeomorphic_def: fixes s :: "'c set"   and t :: "'d set" shows "s homeomorphic t \<equiv> \<exists>f g. homeomorphism s t f g"
Q_rel_H_zipping: fixes x :: "'a H"   and y :: "'c H"   and L :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "('a \<times> 'c) H" assumes "Q x"   and "Q y"   and "rel_H L x y"   and "rel_H (\<lambda>x (x', y). x' = x \<and> L x y) x z"   and "rel_H (\<lambda>(x, y') y. y' = y \<and> L x y) z y" shows "Q z"
labeled_sat_eqs_list_all: fixes i :: "'b"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and D :: "('b \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and M :: "('fun, 'var) Term.term set"   and ac :: "poscheckvariant"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>(j, p)\<in>{(i, t, s)} \<union> set D. \<forall>(k, q)\<in>{(i, t, s)} \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"   and "\<lbrakk>M; map (\<lambda>d. \<langle>ac: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t) D\<rbrakk>\<^sub>d \<I>" shows "list_all (\<lambda>d. fst d = i) D"
diff_const_axiom_def: shows "diff_const_axiom \<equiv> Equals (Differential ($f fid1 local.empty)) (Const 0)"
fcs_a_neq_a: shows "G.face_cycle_succ a \<noteq> a"
l_cancel_one: fixes x :: "'a list"   and a :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "a \<in> carrier (poly_ring R)" shows "(x \<oplus>\<^bsub>poly_ring R\<^esub> a = x) = (a = \<zero>\<^bsub>poly_ring R\<^esub>)"
effect_makeE: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and h :: "heap"   and h' :: "heap"   and r :: "'b array" assumes "effect (Array.make n f) h h' r" obtains "r = fst (Array.alloc (map f [0..<n]) h)" and "h' = snd (Array.alloc (map f [0..<n]) h)" and "Array.get h' r = map f [0..<n]" and "Array.present h' r" and "\<not> Array.present h r"
tranclp_unfold: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" shows "r\<^sup>+\<^sup>+ a b = ((a, b) \<in> {(x, y). r x y}\<^sup>+)"
arr_char: fixes f :: "'a" shows "arr f = (C.arr f \<and> Ide (C.dom f) \<and> Ide (C.cod f))"
left_idem: fixes a :: "'d"   and b :: "'d" shows "min a (min a b) = min a b"
fps_lr_one_over_one_minus_fps_X_squared_2: shows "fps_right_inverse ((1 - fps_X)\<^sup>2) (1::'b) = Abs_fps (\<lambda>n. of_nat (n + 1))"
diff_single_eq_union: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x \<in># M" shows "(M - (x, {#}) = N) = (M = x, N)"
intros_7: fixes s :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" assumes "\<forall>t. (s, t) \<notin> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> \<Rightarrow> Stuck"
\<delta>'_2': shows "\<delta>' \<upharpoonleft> V\<^bsub>\<V>i\<^esub> = \<delta> \<upharpoonleft> V\<^bsub>\<V>i\<^esub>"
externals_def: fixes asig :: "'c signature" shows "externals asig \<equiv> inputs asig \<union> outputs asig"
match_pre_def: shows "match_pre \<equiv> \<lambda>(x, v) s. s x = Some v"
simps_16: fixes Vs :: "char list list"   and e :: "char list exp"   and c :: "char list exp" shows "compE\<^sub>1 Vs (while (e) c) = while (compE\<^sub>1 Vs e) compE\<^sub>1 Vs c"
HC_2: shows "H \<subseteq> Bounded_vec (fact (n + 1) * max (1::'a) Bnd ^ n)"
assms: shows "y \<in> fv ((\<theta> \<circ>\<^sub>s \<sigma>) x)"
fps_nth_inverse: fixes x :: "('c, 'd) fps" shows "Abs_fps (($) x) = x"
sum_list_hd_tl: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "sum_list (tl xs) = - hd xs + sum_list xs"
infsetsum'_def: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "infsetsum' f A = (if infsetsum'_converges f A then Lim (finite_subsets_at_top A) (sum f) else (0::'c))"