hcju/isabelleps huggingface dataset:
at_most_one_mset_mset_diff: fixes a :: "'b"   and M :: "'b multiset" assumes "a \<notin># remove1_mset a M" shows "set_mset (remove1_mset a M) = set_mset M - {a}"
finite_ordLess_var: fixes S :: "'f set" shows "(|S| <o |UNIV| \<or> finite S) = (|S| <o |UNIV|)"
lunit_char_2: fixes f :: "'a" assumes "Left.ide f" shows "Left.L (Left.lunit f) = Left.comp (\<i>[trg f] \<star> f) (local.inv \<a>[trg f, trg f, f])"
translate_plus3: fixes a :: "real \<times> real"   and x :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "ccw' (a - x) (b - x) c = ccw' a b (c + x)"
that: fixes C :: "'a set" assumes "C \<in> components S"   and "\<not> connected_component (- C) a b" shows "thesis"
"0_6": shows "\<forall>i\<in>set (L A). \<forall>j\<in>set (L A). i \<noteq> j \<longrightarrow> comp_GSMP_disjoint public arity Ana \<Gamma> (N1 (proj_unl i A)) (N1 (proj_unl j A)) (M i) (M j) C"
sat_precond_as_proj_1: fixes vs :: "'d set"   and s :: "('d, 'e) fmap"   and s' :: "('d, 'e) fmap"   and x :: "('d, 'e) fmap" assumes "fmrestrict_set vs s = fmrestrict_set vs s'" shows "(fmrestrict_set vs x \<subseteq>\<^sub>f s) = (fmrestrict_set vs x \<subseteq>\<^sub>f s')"
Bex_alts: fixes P :: "'alt \<Rightarrow> bool" shows "(\<exists>x\<in>alts. P x) = (P a \<or> P b \<or> P c)"
vim: shows "(\<lambda>w. w !! i) -` opex \<in> sets (vimage_algebra (streams (space borel)) (\<lambda>w. w !! i) borel)"
infnorm_pos_lt: fixes x :: "'a" shows "(0 < infnorm x) = (x \<noteq> (0::'a))"
sg_unit_closed: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "\<one> \<in> H"
sup_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "sup_class.sup f g x = sup_class.sup (f x) (g x)"
SND_LESS: fixes i :: "nat" shows "snd (r' i) < num_acc"
in_pair_iff: fixes a :: "'e"   and b :: "'e"   and A :: "('e \<times> 'e) set" shows "((a, b) \<in>\<^sub>p A) = ((a, b) \<in> A \<or> (b, a) \<in> A)"
continuous_at_compose: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "isCont f a"   and "isCont g (f a)" shows "isCont (g \<circ> f) a"
IH_1: assumes "\<not> (n = 0 \<or> n = 1)"   and "n mod 2 = 0"   and "0 < n div 2"   and "n div 2 \<le> Suc (length kvs)" shows "length (snd (rbtreeify_g (n div 2) kvs)) = Suc (length kvs) - n div 2"
read_data_cache_def: fixes state :: "'b sparc_state"   and va :: "word32" shows "read_data_cache state va \<equiv> let tag = ucast (va >> 12); offset0 = ucast va AND 4092; offset1 = offset0 OR 1; offset2 = offset0 OR 2; offset3 = offset0 OR 3; r0 = dcache_val (tag, offset0) state; r1 = dcache_val (tag, offset1) state; r2 = dcache_val (tag, offset2) state; r3 = dcache_val (tag, offset3) state in if r0 = None \<or> r1 = None \<or> r2 = None \<or> r3 = None then None else let byte0 = case r0 of Some v \<Rightarrow> v; byte1 = case r1 of Some v \<Rightarrow> v; byte2 = case r2 of Some v \<Rightarrow> v; byte3 = case r3 of Some v \<Rightarrow> v in Some ((((ucast byte0 << 24) OR (ucast byte1 << 16)) OR (ucast byte2 << 8)) OR ucast byte3)"
rcs_aux: fixes pc :: "nat"   and bp :: "instr array"   and pc' :: "nat"   and s :: "integer" assumes "pc < pc_bound bp"   and "pc' \<in> set (exec' bp s pc)" shows "pc' < pc_bound bp"
global_reg_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and w :: "word32"   and n :: "nat"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = global_reg_mod w n rd s1 \<and> t2 = global_reg_mod w n rd s2" shows "low_equal t1 t2"
upper_bound: fixes u :: "'v" shows "\<delta> s u \<le> D u"
h4: shows "monomials (lowerPoly i j (liftPoly i j (p1 + p2))) = monomials (lowerPoly i j (liftPoly i j p1)) \<union> monomials (lowerPoly i j (liftPoly i j p2))"
finite_reduced_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "finite (reduced_GB F)"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(ext0.Val x2 = ext0.Val y2) = (x2 = y2)"
zor_poly_zero_eq: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and z :: "complex" assumes "f holomorphic_on s"   and "open s"   and "connected s"   and "z \<in> s"   and "\<exists>w\<in>s. f w \<noteq> 0" shows "\<forall>\<^sub>F w in at z. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
bezout: shows "bezout_coefficients (#v) (#w) = (s, t)"
fds_deriv_const: fixes c :: "'a" shows "fds_deriv (fds_const c) = 0"
source_in_not_cycle: fixes \<Delta> :: "('a, 'b) network_scheme"   and p :: "'a list"   and x :: "'a" assumes "flow_attainability \<Delta>"   and "cycle \<Delta> p" shows "(x, source \<Delta>) \<notin> set (cycle_edges p)"
ubounds_def: fixes as :: "atom list" shows "ubounds as = concat (map (\<lambda>x. case x of Less 0 0 \<Rightarrow> [] | Less 0 (Suc i) \<Rightarrow> [i] | Less (Suc nat) nat2 \<Rightarrow> [] | Eq nat1 nat2 \<Rightarrow> []) as)"
And_Not_def: fixes \<phi> :: "'b MFOTL.formula"   and \<psi> :: "'b MFOTL.formula" shows "MFOTL.And_Not \<phi> \<psi> = formula.Neg (formula.Or (formula.Neg \<phi>) \<psi>)"
I2b: fixes s :: "state"   and s' :: "state" assumes "HNext s s'"   and "HInv1 s \<and> HInv2 s" shows "HInv2 s'"
suffs_cond1I: fixes aa :: "nat list"   and aaa :: "nat list"   and b :: "nat list"   and a :: "nat list" assumes "\<forall>y\<in>set aa. y \<le> maxne0_impl aaa b"   and "length aa = length a"   and "a \<bullet> aa = b \<bullet> aaa" shows "suffs (cond1 b aaa) a (aa, b \<bullet> aaa)"
distinct_1: fixes x21 :: "char list"   and x22 :: "aexp" shows "SKIP \<noteq> x21 ::= x22"
mon_ww_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and ww :: "('c, 'f) label list list" shows "mon_ww fg ww \<equiv> foldl (\<union>) {} (map (mon_w fg) ww)"
adjoint_ode_vars: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and ODE :: "('sf, 'sz) ODE"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "ODE_vars (local.adjoint I \<sigma> \<nu>) ODE = ODE_vars (local.adjoint I \<sigma> \<omega>) ODE"
simps: fixes m :: "Body"   and v :: "'a Point" shows "lightcone m v = mkCone v (c m)"
add_right_mono_ge: fixes x :: "'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "x \<le> a + c"   and "a \<le> b" shows "x \<le> b + c"
steps_reachable: fixes xs :: "'a list"   and b :: "'a" assumes "steps xs"   and "hd xs = s\<^sub>0"   and "last xs = b" shows "reachable b"
DataValue_PartDom: fixes i :: "nat"   and D :: "'b data" assumes "i < PartNum (Data.DataSpace D)" shows "DataValue D ! i \<in> Data.DataSpace D !D! i"
is_processT6_S1: fixes X :: "'c event set"   and s :: "'c event list"   and P :: "'c process" assumes "tick \<notin> X"   and "(s @ [tick], {}) \<in> F P" shows "(s, X) \<in> F P"
PO_l2_inv2: shows "reach l2 \<subseteq> l2_inv2"
finite_Keys: fixes A :: "('e \<Rightarrow>\<^sub>0 'f) set" assumes "finite A" shows "finite (Keys A)"
zero_closed: shows "\<zero>\<^bsub>poly_ring L\<^esub> \<in> PIdl\<^bsub>poly_ring L\<^esub> q"
map_sel_3: fixes a :: "('a, 'b) tllist"   and f1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'd" assumes "\<not> is_TNil a" shows "ttl (tmap f1 f2 a) = tmap f1 f2 (ttl a)"
SeqCTermP_imp_OrdP: fixes vf :: "bool"   and s :: "tm"   and k :: "tm"   and t :: "tm" shows "{SeqCTermP vf s k t} \<turnstile> OrdP k"
partial_state2_axioms: shows "partial_state2 {Suc k} (vars1 - {Suc k})"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b" shows "Infinity \<noteq> Point x21 x22"
mk_next_def: fixes x :: "'b ltln" shows "mk_next x \<equiv> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> X\<^sub>n x"
consistency_def: fixes C :: "'a fm set set" shows "consistency C \<equiv> \<forall>S\<in>C. (\<forall>p. \<not> (Pro p \<in> S \<and> (\<^bold>\<not> Pro p) \<in> S)) \<and> \<^bold>\<bottom> \<notin> S \<and> (\<forall>Z. (\<^bold>\<not> \<^bold>\<not> Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<and> B) \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<or> B)) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A, \<^bold>\<not> B} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<or> B) \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<and> B)) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A} \<in> C \<or> S \<union> {\<^bold>\<not> B} \<in> C) \<and> (\<forall>A B. (A \<^bold>\<longrightarrow> B) \<in> S \<longrightarrow> S \<union> {\<^bold>\<not> A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. (\<^bold>\<not> (A \<^bold>\<longrightarrow> B)) \<in> S \<longrightarrow> S \<union> {A, \<^bold>\<not> B} \<in> C) \<and> (\<forall>A. tautology A \<longrightarrow> S \<union> {A} \<in> C) \<and> (\<forall>A i. \<not> (K i A \<in> S \<and> (\<^bold>\<not> K i A) \<in> S))"
ladder_\<gamma>_simp1: fixes u :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L' :: "(nat \<times> nat \<times> nat) list" assumes "u < length L" shows "ladder_\<gamma> a D (L @ L') u = ladder_\<gamma> a D L u"
zero: fixes V :: "'c set" assumes "vectorspace V" shows "(0::'c) \<in> V"
cong_le_nat: fixes y :: "nat"   and x :: "nat"   and n :: "nat" assumes "y \<le> x" shows "[x = y] (mod n) = (\<exists>q. x = q * n + y)"
extension_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "extension_on A r s = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> s \<longrightarrow> (x, y) \<in> r)"
try_gpv_bind_lossless: fixes gpv :: "('d, 'b, 'c) gpv"   and f :: "'d \<Rightarrow> ('e, 'b, 'c) gpv"   and gpv' :: "('e, 'b, 'c) gpv" assumes "lossless_gpv \<I>_full gpv" shows "TRY gpv \<bind> f ELSE gpv' = gpv \<bind> (\<lambda>x. TRY f x ELSE gpv')"
Invariants_quasi_Invariants_functions: fixes f :: "'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable M" shows "(\<exists>g\<in>borel_measurable Invariants. AE x in M. f x = g x) = (AE x in M. f (T x) = f x)"
simps_1: fixes m :: "'b message_af" shows "last_message_hold_init [] m = []"
path_connected_space_imp_path_component_of: fixes X :: "'b topology"   and a :: "'b"   and b :: "'b" assumes "path_connected_space X"   and "a \<in> topspace X"   and "b \<in> topspace X" shows "path_component_of X a b"
arr_char: fixes f :: "'c" shows "Cop.arr f = C.arr f"
tllist_all2_tnthD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist"   and n :: "nat" assumes "tllist_all2 P Q xs ys"   and "enat n < tlength xs" shows "P (tnth xs n) (tnth ys n)"
sum_diff_nat_ivl: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "m \<le> n"   and "n \<le> p" shows "sum f {m..<p} - sum f {m..<n} = sum f {n..<p}"
Do: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and f :: "state \<Rightarrow> state set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool" shows "C \<turnstile>\<^sub>t {\<lambda>z s. (\<forall>t\<in>f s. P z t) \<and> f s \<noteq> {}} Do f {P}"
wfactors_ee_cong_l: fixes as :: "'a list"   and bs :: "'a list"   and b :: "'a" assumes "essentially_equal (Ring_Divisibility.mult_of R) as bs"   and "wfactors (Ring_Divisibility.mult_of R) bs b"   and "b \<in> carrier (Ring_Divisibility.mult_of R)"   and "set as \<subseteq> carrier (Ring_Divisibility.mult_of R)"   and "set bs \<subseteq> carrier (Ring_Divisibility.mult_of R)" shows "wfactors (Ring_Divisibility.mult_of R) as b"
rep_eq: fixes x :: "'a ccsubspace"   and xa :: "'a ccsubspace" shows "(x < xa) = (space_as_set x \<subset> space_as_set xa)"
inj_Var: fixes x :: "'var"   and y :: "'var" assumes "x \<in> var"   and "y \<in> var" shows "(Var x = Var y) = (x = y)"
sheaf_spec_morphisms_def: fixes U :: "('a \<times> 'a) set set set"   and V :: "('a \<times> 'a) set set set" shows "local.local.sheaf_spec_morphisms U V \<equiv> \<lambda>s\<in>local.local.sheaf_spec U. restrict s V"
seqstar_inf_parstar: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "less_eq (seqstar a) (star a)"
renormalize: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ccw (0::'a) (a - b) (c - a)" shows "ccw b a c"
enabled_def: fixes K :: "'a \<Rightarrow> 'a pmf" shows "MC_syntax.enabled K \<equiv> gfp (\<lambda>p x1 x2. \<exists>\<omega> s. x1 = s \<and> x2 = \<omega> \<and> p (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
lub_eq_bottom_iff: fixes Y :: "nat \<Rightarrow> 'c" assumes "chain Y" shows "((\<Squnion>i. Y i) = \<bottom>) = (\<forall>i. Y i = \<bottom>)"
calc: shows "statically_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) no_labels.Red_I_\<G> no_labels.Red_F_\<G>_empty"
vars_poly_of_vars: fixes a :: "char list multiset" shows "vars (poly_of_vars a) \<subseteq> \<phi> ` set_mset a"
bhv2cnf_cnf2bhv: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "dynamic_component.lActive active c t \<le> n" shows "dynamic_component.bhv2cnf tCMP active c t (dynamic_component.cnf2bhv tCMP active c t n) = n"
SUCC_inject: fixes x :: "tm"   and y :: "tm"   and H :: "fm set" shows "insert (SUCC x EQ SUCC y) H \<turnstile> x EQ y"
Map_in_Hom: fixes f :: "('obj, 'obj) concrete_category.arr" assumes "arr f" shows "Map f \<in> (if Dom f = Cod f then {Dom f} else {})"
l_neg: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<ominus>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
cases: fixes x :: "('d, 'e, 'f) generat" obtains uu_ :: "'d" where "x = Pure uu_"   | out :: "'e" and c :: "'f" where "x = IO out c"
lcm_abs2_int: fixes x :: "int"   and y :: "int" shows "lcm x \<bar>y\<bar> = lcm x y"
degree_m_le_degree: fixes f :: "int poly" shows "degree_m f \<le> degree f"
Gcd_eq_1_I: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a one"   and "a \<in> A" shows "Gcd A = one"
upd_space: fixes i :: "nat" assumes "i < n" shows "upd i < n"
Schwarz_Lemma': fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on ball 0 1"   and "f 0 = 0"   and "\<And>z. cmod z < 1 \<Longrightarrow> cmod (f z) < 1" shows "((\<forall>\<xi>. cmod \<xi> < 1 \<longrightarrow> cmod (f \<xi>) \<le> cmod \<xi>) \<and> cmod (deriv f 0) \<le> 1) \<and> ((\<exists>z. cmod z < 1 \<and> z \<noteq> 0 \<and> cmod (f z) = cmod z) \<or> cmod (deriv f 0) = 1 \<longrightarrow> (\<exists>\<alpha>. (\<forall>z. cmod z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> cmod \<alpha> = 1))"
Exs_def: shows "Exs = {z. \<exists>A x Gamma. z = (FAll Neg A # Gamma, {instanceF x A # Gamma})}"
select_object_any_exec0\<^sub>S\<^sub>e\<^sub>q: fixes \<tau> :: "'d state \<times> 'd state"   and f :: "'f \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'e"   and s_set :: "'f list" assumes "\<tau> \<Turnstile> \<delta> select_object_any\<^sub>S\<^sub>e\<^sub>q f s_set" shows "\<tau> \<Turnstile> select_object_any\<^sub>S\<^sub>e\<^sub>q f s_set \<triangleq> f (hd s_set)"
linear_order_on_singleton: fixes x :: "'d" shows "linear_order_on {x} {(x, x)}"
rev_ordsetmap: fixes a :: "'a"   and b :: "'a" assumes "a \<in> P"   and "b \<in> P"   and "codomain.greater_eq (f b) (f a)" shows "b \<succeq> a"
simps_47: fixes x4 :: "('a, 'b) fm" shows "params_fm (\<^bold>\<forall> x4) = params_fm x4"
quad_part_2_less: fixes var :: "nat"   and L :: "real list"   and SQ :: "real"   and p :: "real mpoly"   and deg :: "nat"   and sq :: "real mpoly" assumes "var < length L"   and "0 \<le> SQ"   and "MPoly_Type.degree p var = deg"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) sq = SQ" shows "aEval (Less p) (L[var := sqrt SQ]) = aEval (Less (quadratic_part_2 var sq p)) (L[var := sqrt SQ])"
T2_complete: fixes v :: "Vertices"   and w :: "Vertices" assumes "v \<in> V\<^bsub>T2\<^esub>"   and "w \<in> V\<^bsub>T2\<^esub>"   and "v \<noteq> w" shows "v \<rightarrow>\<^bsub>T2\<^esub> w"
unity_root_uminus: fixes k :: "nat"   and m :: "int" shows "unity_root k (- m) = cnj (unity_root k m)"
wf\<^sub>s\<^sub>t_simple_induct: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and P :: "('b, 'a) strand_step list \<Rightarrow> bool" assumes "wf\<^sub>s\<^sub>t V S"   and "simple S"   and "P []"   and "\<And>v S. \<lbrakk>wf\<^sub>s\<^sub>t V S; simple S; P S\<rbrakk> \<Longrightarrow> P (S @ [send\<langle>Var v\<rangle>\<^sub>s\<^sub>t])"   and "\<And>t S. \<lbrakk>wf\<^sub>s\<^sub>t V S; simple S; P S; fv t \<subseteq> V \<union> \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S))\<rbrakk> \<Longrightarrow> P (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"   and "\<And>X F S. \<lbrakk>wf\<^sub>s\<^sub>t V S; simple S; P S\<rbrakk> \<Longrightarrow> P (S @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])" shows "P S"
F2set2_JF1set_incl_JF2set: fixes a' :: "'d JF1"   and a :: "'d JF2" assumes "a' \<in> F2set2 (dtor2 a)" shows "JF1set a' \<subseteq> JF2set a"
rel_fun_iff_geq_image2p: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" shows "(R ===> S) f g = (BNF_Greatest_Fixpoint.image2p f g R \<le> S)"
emeasure_prod_count_space_single: fixes x :: "'c \<times> 'd" shows "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) {x} = 1"
D_imp_CR: fixes ars :: "('c \<times> 'd \<times> 'c) set"   and r :: "('d \<times> 'd) set" assumes "\<forall>P. peak ars P \<longrightarrow> (\<exists>\<sigma>' \<tau>'. DD ars r (fst P, snd P, \<sigma>', \<tau>'))" shows "CR (unlabel ars)"
sgs_sb: shows "sigma_sets \<Omega> M \<subseteq> ls"
distinct_join_raw: fixes xs :: "('b \<times> 'c) list"   and f :: "'b \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c"   and ys :: "('b \<times> 'c) list" assumes "distinct (map fst xs)" shows "distinct (map fst (join_raw f xs ys))"
simps_1: fixes x21 :: "'b"   and x22 :: "'b llist"   and y21 :: "'b"   and y22 :: "'b llist" shows "(x21 ## x22 = y21 ## y22) = (x21 = y21 \<and> x22 = y22)"
map: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b list"   and x2 :: "nat" shows "map_fundef f (Fundef x1 x2) = Fundef (map f x1) x2"
rep_eq: fixes x :: "'a" shows "(*\<^sub>V) (cblinfun_mult_right x) = (*) x"
const_list_def: fixes xs :: "e list" shows "const_list xs = list_all is_const xs"
subset_fst_imageI: fixes A :: "'b set"   and B :: "'c set"   and S :: "('b \<times> 'c) set"   and y :: "'c" assumes "A \<times> B \<subseteq> S"   and "y \<in> B" shows "A \<subseteq> fst ` S"
assms_1: shows "c \<turnstile> ev \<mapsto> c'"
in_set_takeD: fixes x :: "'b"   and n :: "nat"   and xs :: "'b list" assumes "x \<in> set (take n xs)" shows "x \<in> set xs"
substAbs_preserves_freshAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "freshAbs zs z A"   and "fresh zs z Y" shows "freshAbs zs z (A $[Y / y]_ys)"
wlp_while_split: fixes dims :: "nat list"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and P :: "complex mat" assumes "state_sig.well_com dims (While M S)"   and "state_sig.is_quantum_predicate dims P" shows "state_sig.wlp dims (While M S) P = adjoint (M 0) * P * M 0 + adjoint (M 1) * state_sig.wlp dims S (state_sig.wlp dims (While M S) P) * M 1"
fresh_fset_fminus: fixes x :: "'a"   and A :: "'a fset" assumes "atom x \<sharp> A" shows "A |-| {|x|} = A"
permutes_compose: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and q :: "'e \<Rightarrow> 'e" assumes "p permutes S"   and "q permutes S" shows "q \<circ> p permutes S"
conn: fixes T :: "'a set" assumes "closed T"   and "T \<subset> S" shows "connected (- T)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "plus_hom hom" shows "semigroup_add_hom hom"
cong_pow_I: fixes a :: "nat"   and b :: "nat"   and x :: "'a"   and n :: "'a" assumes "a = b" shows "[x ^ a = x ^ b] (mod n)"
LstSeq_trunc: fixes s :: "hf"   and k :: "hf"   and y :: "hf"   and l :: "hf" assumes "LstSeq s k y"   and "l \<^bold>\<in> k" shows "LstSeq s l (app s l)"
ide_hcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "ide \<nu>"   and "ide \<mu>"   and "src \<nu> = trg \<mu>" shows "ide (\<nu> \<star> \<mu>)"
derivation_ge_def: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat" shows "derivation_ge D i = (\<forall>d\<in>set D. i \<le> fst d)"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom (src f) = src (local.dom f)"
ipresIGSubstAll_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'd, 'e, 'j) model_scheme" shows "ipresIGSubstAll h hA termMOD MOD = ipresSubstAll h hA MOD"
trcl_one_elem: fixes c :: "'f"   and e :: "'g"   and c' :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set" assumes "(c, e, c') \<in> t" shows "(c, [e], c') \<in> trcl t"
l2_inv3D_HashNonce1: fixes s :: "l2_state"   and R :: "fid_t"   and N :: "nat"   and X :: "msg" assumes "s \<in> l2_inv3"   and "hmac (NonceF (R $ N)) X \<in> synth (analz (extr (bad s) (ik s) (chan s)))" shows "R \<in> bad_runs s"
par_beta_body: fixes L :: "char list set"   and t :: "sterm"   and t' :: "sterm" assumes "finite L"   and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> t' = \<sigma>[s,p] t'')" shows "body t \<and> body t'"
double_compl: fixes x :: "'b" shows "NOT (NOT x) = x"
sum_list_S_in_W: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "sum_list ss \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"
cases: fixes a :: "('b, 'c) Kruskal_Examples.term" assumes "a \<in> terms" obtains ts :: "('b, 'c) Kruskal_Examples.term list" and f :: "'b" where "a = Fun f ts" and "\<forall>t\<in>set ts. t \<in> terms"
index_mat_swapcols_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "swapcols k l A $$ (i, j) = (if k = j then A $$ (i, l) else if l = j then A $$ (i, k) else A $$ (i, j))"
pseudo_proj_True_def: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat" assumes "infinite_coin_toss_space p M" shows "pseudo_proj_True n = (\<lambda>w. stake n w @- sconst True)"
comp_apply: fixes f :: "'y \<Rightarrow> 'x"   and g :: "'z \<Rightarrow> 'y"   and x :: "'z" shows "(f \<circ> g) x = f (g x)"
AllowedActs_totalize: fixes F :: "'b program" shows "AllowedActs (totalize F) = AllowedActs F"
fin_dg_Arr_vfinite: shows "vfinite (\<AA>\<lparr>Arr\<rparr>)"