hcju/isabelleps huggingface dataset:
preserves_snd_lift_stable: fixes F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and i :: "nat"   and j :: "nat"   and A :: "'c set" assumes "F \<in> preserves snd"   and "i \<noteq> j" shows "lift j F \<in> stable (lift_set i (A \<times> UNIV))"
\<pi>'gen_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option"   and u :: "'v"   and adjs :: "('v \<times> 'a) list"   and v :: "'v" shows "\<pi>'gen Q \<pi> u adjs v = (if v \<notin> fst ` set adjs then \<pi> v else \<pi>' Q \<pi> u v)"
weak_ring_morphism_axioms_def: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and R :: "('d, 'f) ring_scheme" shows "weak_ring_morphism_axioms f I R \<equiv> \<forall>a b. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> (f a = f b) = (a \<ominus>\<^bsub>R\<^esub> b \<in> I)"
sel_13: fixes x31 :: "bool"   and x32 :: "'b" shows "nul (Atom3 x31 x32) = False"
refl: fixes a :: "('a, 'b) Term.term" shows "a \<sqsubseteq> a"
from_cnf_less_\<omega>_exp: fixes ks :: "ordinal list"   and l :: "ordinal" assumes "\<forall>k\<in>set ks. k < l" shows "from_cnf ks < \<omega> ** l"
set_mset_mono: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "A \<subseteq># B" shows "set_mset A \<subseteq> set_mset B"
weak_llessE: fixes R :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "x \<sqsubset>\<^bsub>R\<^esub> y" obtains "x \<sqsubseteq>\<^bsub>R\<^esub> y" and "x .\<noteq>\<^bsub>R\<^esub> y"
Value_LabelFunSem: fixes P :: "'f data \<Rightarrow> bool"   and ST :: "('d, 'e, 'f) status" shows "(VAL P \<in> the (LabelFunSem (HA ST) ST)) = P (Value ST)"
mkId2: fixes h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" shows "Pbij_Dom (mkId h) = Dom h"
size_set_diff: fixes S :: "'b set"   and S' :: "'b set" assumes "finite S"   and "S' \<subseteq> S" shows "size_set (S - S') = size_set S - size_set S'"
l6_7: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet P A B"   and "Tarski_neutral_dimensionless.Out Bet P B C" shows "Tarski_neutral_dimensionless.Out Bet P A C"
runit'_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (category.inv C (monoidal_category.runit C T \<alpha> \<iota> a)) a (T (a, monoidal_category.unity C \<iota>))"
tree\<^sub>i_order: fixes k :: "nat"   and u :: "'a up\<^sub>i" assumes "0 < k"   and "root_order_up\<^sub>i k u" shows "root_order k (tree\<^sub>i u)"
miota_def: fixes R :: "('f, 'g) Ring_scheme"   and M1 :: "('e, 'f, 'h) Module_scheme"   and M :: "('e, 'f, 'h) Module_scheme" shows "miota R M1 M = (\<lambda>x\<in>carrier M1. x)"
subst_type_NF: fixes t :: "dB"   and e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and U :: "LambdaType.type"   and T :: "LambdaType.type"   and u :: "dB" assumes "NF t"   and "e\<langle>i:U\<rangle> \<turnstile> t : T"   and "NF u"   and "e \<turnstile> u : U" shows "\<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
s0_obs_secrecy: shows "oreach s0 \<subseteq> l1_secrecy"
R2_def: fixes msgs :: "bool \<times> bool"   and \<sigma> :: "bool" shows "R2 msgs \<sigma> = (let (b0, b1) = msgs in I \<bind> (\<lambda>(\<alpha>, \<tau>). local.etp.S \<alpha> \<bind> (\<lambda>x\<^sub>\<sigma>. local.etp.S \<alpha> \<bind> (\<lambda>y\<^sub>\<sigma>'. let y\<^sub>\<sigma> = F \<alpha> x\<^sub>\<sigma>; x\<^sub>\<sigma> = F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>; x\<^sub>\<sigma>' = F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>'; \<beta>\<^sub>\<sigma> = B \<alpha> x\<^sub>\<sigma> \<oplus> (if \<sigma> then b1 else b0); \<beta>\<^sub>\<sigma>' = B \<alpha> x\<^sub>\<sigma>' \<oplus> (if \<sigma> then b0 else b1) in return_spmf (\<sigma>, \<alpha>, \<beta>\<^sub>\<sigma>, \<beta>\<^sub>\<sigma>')))))"
is_inf_related: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "is_inf x y x"
consis': shows "history_consistent \<theta> p\<^sub>2 (sb\<^sub>1 @ sb')"
all: shows "\<forall>I. finitely_generated_ideal I \<longrightarrow> principal_ideal I"
lr_of_tran_s1_def: fixes rt :: "('a, 'b) routing_rule_scheme list" shows "lr_of_tran_s1 rt = map (\<lambda>r. (route2match r, routing_oiface r)) rt"
mult_eSuc_right: fixes m :: "eint"   and n :: "eint" shows "m * (n + 1) = m * n + m"
cxt_inv_while: fixes E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt" assumes "cxt_to_stmt E c = While e p" shows "c = While e p \<and> E = []"
ptr_kinds_M_ptr_kinds: fixes h :: "(_) heap"   and xa :: "(_) document_ptr list" shows "h \<turnstile> document_ptr_kinds_M \<rightarrow>\<^sub>r xa = (xa = sorted_list_of_set (fset (document_ptr_kinds h)))"
WT_resource_attach: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('aa, 'ab) \<I>"   and conv :: "('y, 'z, 'aa, 'ab) converter"   and res :: "('aa, 'ab) resource" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>' \<turnstile>res res \<surd>" shows "\<I> \<turnstile>res conv \<rhd> res \<surd>"
assms_2: shows "snd (f (k, 0::'c)) = (0::'d)"
continuous_const: fixes F :: "'c filter"   and c :: "'d" shows "continuous F (\<lambda>x. c)"
continuous_polymonial_function: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "polynomial_function f" shows "isCont f x"
lcp_take_eq: fixes u :: "'b list"   and v :: "'b list" shows "take \<^bold>|u \<and>\<^sub>p v\<^bold>| u = take \<^bold>|u \<and>\<^sub>p v\<^bold>| v"
finite_keys: fixes f :: "'b \<Rightarrow>\<^sub>0 'c" shows "finite (Poly_Mapping.keys f)"
v1: shows "M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gs ! i) [0..<k]) \<in> carrier_vec n"
sel_3: fixes x1 :: "'b fset"   and x2 :: "'b"   and x3 :: "'a set" shows "a_fresh (Antecedent x1 x2 x3) = x3"
insert_Diff: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "insert x (A - B) = insert x A - (B - {x})"
lunit_naturality: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "\<l>[S.cod \<mu>] \<cdot>\<^sub>S L \<mu> = \<mu> \<cdot>\<^sub>S \<l>[S.dom \<mu>]"
pnTimes_pnorm: fixes r1 :: "'a rexp"   and r2 :: "'a rexp" shows "pnTimes (pnorm r1) r2 = pnorm (Times r1 r2)"
Loss_L1: fixes t :: "nat"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and lose :: "nat \<Rightarrow> bool list"   and a :: "nat \<Rightarrow> aType list"   and y :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list"   and t2 :: "nat" assumes "\<forall>t2<t. i2 t2 = []"   and "Loss lose a i2 y i"   and "t2 < t"   and "ts lose" shows "i t2 = []"
compositionality: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and list :: "'f list" shows "map f (map g list) = map (f \<circ> g) list"
impl_mp2: fixes P :: "'a upred"   and Q :: "'a upred" shows "((P \<Rightarrow> Q) \<and> P) = (Q \<and> P)"
affine_hull_explicit: fixes p :: "'a set" shows "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
iT_Plus_not_empty: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}" shows "I \<oplus> k \<noteq> {}"
hermitean_distinct_eigenvals: fixes H :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H" shows "(\<exists>k\<^sub>1 k\<^sub>2. k\<^sub>1 \<noteq> k\<^sub>2 \<and> eigenval k\<^sub>1 H \<and> eigenval k\<^sub>2 H) \<or> mat_diagonal H"
Sup_img_rule_complete: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" shows "(\<forall>x. x \<in> S \<longrightarrow> f x \<le> SPEC \<Phi>) = (Sup (f ` S) \<le> SPEC \<Phi>)"
2: shows "\<forall>i j. i < j \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row j A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ j $ n \<noteq> (0::'b))"
scast_up_scast_id: fixes w :: "'b word" assumes "is_up scast" shows "scast (scast w) = w"
subset_singleton_iff_lepoll: fixes S :: "'a set" shows "(\<exists>x. S \<subseteq> {x}) = (S \<lesssim> {()})"
n: shows "maximally_connected \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"
assms_4: shows "offs_num n xs index key mi ma i = 0"
gt_tpoly_irrefl: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "\<not> tkbo_coefs.gt_tpoly ground_heads_var \<delta> arity_sym wt_sym p p"
\<Delta>\<Phi>_pass1: fixes hs :: "'b hp list" assumes "hs \<noteq> []" shows "\<Phi>_hps (pass\<^sub>1 hs) - \<Phi>_hps hs \<le> 2 * log 2 (real (size_hps hs)) - real (length hs) + 2"
cast_below_cast: fixes A :: "'a defl"   and B :: "'a defl" shows "(cast\<cdot>A \<sqsubseteq> cast\<cdot>B) = (A \<sqsubseteq> B)"
reversepath_part_circlepath: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" shows "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
saturated: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent" assumes "valid_in_port (v, p)" shows "\<exists>e\<in>edges. snd e = (v, p)"
extended_distance_middle_formula: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" assumes "x \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b)" shows "extended_distance x (middle a b) = extended_distance a b / 2 + (SUP w\<in>underS Bonk_Schramm_extension_unfolded_wo (middle a b) \<inter> extended_distance_set. extended_distance x w - max (extended_distance a w) (extended_distance b w))"
list_fast_rec_all_Leaf: fixes ts :: "'b tree list" assumes "\<forall>t\<in>set ts. t = \<langle>\<rangle>" shows "list_fast_rec ts = []"
remove_key_def: fixes k0 :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'c" shows "remove_key k0 f = Abs_poly_mapping (\<lambda>k. lookup f k when k \<noteq> k0)"
tensor_mat_dim_col: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_col (tensor_mat m1 m2) = d"
arr_char: fixes f :: "nat" shows "J.arr f = (f \<in> {m})"
abs_eq: fixes x :: "32 word" shows "int_of_uint32 (Abs_uint32 x) = uint x"
simps_5: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_siso tval aval indis (c1 | c2) = (PL_Indis.SC_siso tval aval indis c1 \<and> PL_Indis.SC_siso tval aval indis c2)"
uminus_inf_iff: fixes z :: "complex_homo" shows "(~\<^sub>h z = \<infinity>\<^sub>h) = (z = \<infinity>\<^sub>h)"
hensel_axioms_def: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" shows "hensel_axioms p f a \<equiv> (f \<in> carrier (UP (padic_int p)) \<and> a \<in> carrier (padic_int p)) \<and> UP_cring.to_fun (padic_int p) f a \<noteq> \<zero>\<^bsub>padic_int p\<^esub> \<and> eint 2 * padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) a) < padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) f a)"
to_tree_order_is_l_to_tree_order: shows "l_to_tree_order to_tree_order"
A: shows "A = Ax * MPoly_Type.monom (monomial (Suc 0) x') 1 + A'"
zeta_partial_sum_le: fixes x :: "real"   and m :: "nat" assumes "x \<in> {0<..1}" shows "(\<Sum>k = 1..m. real k powr (x - 1)) \<le> real m powr x / x"
isint_def: fixes t :: "MIR.num"   and bs :: "real list" shows "isint t bs \<equiv> real_of_int \<lfloor>Inum bs t\<rfloor> = Inum bs t"
\<theta>_eq: fixes x :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int"   and y :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "x \<in> singular_relcycle_set p X S"   and "y \<in> singular_relcycle_set p X S"   and "singular_relboundary_set p X S #>\<^bsub>relcycle_group p X S\<^esub> x = singular_relboundary_set p X S #>\<^bsub>relcycle_group p X S\<^esub> y" shows "\<theta> x = \<theta> y"
apply_operations_def: fixes es :: "('a \<times> ('a, 'b) operation) event list" shows "apply_operations es \<equiv> hb.apply_operations (node_deliver_messages es) []"
MWLfSteps_det_cases_4: fixes b :: "'exp"   and c1 :: "('exp, 'id) MWLfCom"   and c2 :: "('exp, 'id) MWLfCom"   and m :: "'id \<Rightarrow> 'val"   and W :: "('exp, 'id) MWLfCom list"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (if b then c1 else c2 fi) m W m'" obtains "W = [c1]" and "m' = m" and "BMap (E b m)"   | "W = [c2]" and "m' = m" and "\<not> BMap (E b m)"
valid_successElem: fixes M :: "'b \<Rightarrow> (bool \<times> 'b) option"   and \<sigma> :: "'b"   and f :: "'b \<Rightarrow> bool" assumes "M \<sigma> = Some (f \<sigma>, \<sigma>)" shows "(\<sigma> \<Turnstile> M) = f \<sigma>"
local_help1: fixes vs :: "'a list"   and c :: "'a list"   and r :: "'a"   and d :: "'a list"   and a :: "'a list"   and b :: "'a list" assumes "vs = c @ r # d"   and "vs = a @ r # b"   and "r \<notin> set a"   and "r \<notin> set b" shows "a = c"
word_set_bit_def: fixes a :: "'b word"   and n :: "nat"   and x :: "bool" shows "set_bit_class.set_bit a n x = word_of_int (bin_sc n x (uint a))"
R: shows "real R = pi / (2 * \<theta>) - 1 / 2"
Standard_numeral: fixes k :: "num" shows "numeral k \<in> Standard"
isPath_append_edge: fixes v :: "nat"   and p :: "(nat \<times> nat) list"   and v' :: "nat"   and v'' :: "nat" assumes "isPath v p v'"   and "(v', v'') \<in> E" shows "isPath v (p @ [(v', v'')]) v''"
fract_collapse_1: fixes k :: "'a" shows "Fraction_Field.Fract (0::'a) k = 0"
case_eq_if: fixes f :: "('b \<Rightarrow> ('c \<times> ('b, 'c, 'd, 'e) converter, 'd, 'e) gpv) \<Rightarrow> 'a"   and converter :: "('b, 'c, 'd, 'e) converter" shows "(case converter of Converter x \<Rightarrow> f x) = f (run_converter converter)"
that: fixes d :: "'a \<Rightarrow> 'a set" assumes "gauge d"   and "\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p1 \<and> p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k} \<and> d fine p2 \<longrightarrow> norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" shows "thesis"
f4: shows "ucast (get_CWP (cpu_reg_val PSR s1)) = fst (fst (get_curr_win () s1)) \<and> s1 = snd (fst (get_curr_win () s1))"
assms_6: fixes Cs :: "'b set list" assumes "SimplicialComplex.min_maxsimpchain W (C # Cs)" shows "pgallery (g \<Turnstile> (C # Cs))"
scale_add3: fixes a :: "'a"   and x :: "real"   and b :: "'a" shows "ccw (0::'a) a (x *\<^sub>R a + b) = ccw (0::'a) a b"
neq: fixes numnoabs :: "Ferrack.num \<Rightarrow> bool"   and t :: "Ferrack.num"   and bs :: "real list" assumes "numnoabs t" shows "Ifm bs (case rsplit0 t of (x, xa) \<Rightarrow> neq x xa) = Ifm bs (NEq t) \<and> isrlfm (case rsplit0 t of (x, xa) \<Rightarrow> neq x xa)"
reachable_from_ObjLoc_impl_Obj: fixes s :: "Store"   and l :: "Location"   and cf :: "CAttId"   and a :: "nat"   and C :: "CTypeId" assumes "s\<turnstile> l reachable_from (s@@objLoc cf a)"   and "C = cls cf" shows "s\<turnstile> l reachable_from objV C a"
induct: fixes P :: "'b heap \<Rightarrow> bool"   and a0 :: "'b heap" assumes "\<And>x uu_. P (Hp x uu_)"   and "P Empty" shows "P a0"
simps_6: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_X_enat (\<phi> or\<^sub>n \<psi>) = combine mk_or (rewrite_X_enat \<phi>) (rewrite_X_enat \<psi>)"
minus_equality: fixes y :: "'c"   and x :: "'c" assumes "y \<oplus>\<^bsub>S\<^esub> x = \<zero>\<^bsub>S\<^esub>"   and "x \<in> carrier S"   and "y \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> x = y"
has_derivative_on_open1: fixes t :: "real"   and s :: "'a" assumes "0 < t"   and "t \<in> T"   and "s \<in> S" obtains B :: "real set" where "t \<in> B" and "open B" and "B \<subseteq> T" and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B"
gval_fold_equiv_gval_foldr: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gAnd G (Bc True)) s = gval (foldr gAnd G (Bc True)) s"
NEGc_decreasing: fixes X :: "ntrm set"   and Y :: "ntrm set"   and B :: "ty" assumes "X \<subseteq> Y" shows "NEGc B Y \<subseteq> NEGc B X"
field_def: fixes R :: "('e, 'f) ring_scheme" shows "field R \<equiv> domain R \<and> field_axioms R"
subst_syntactic_and_axioms_def: fixes P :: "'c \<Rightarrow> bool" shows "subst_syntactic_and_axioms P \<equiv> \<forall>t env. P t \<longrightarrow> fmpred (\<lambda>_. P) env \<longrightarrow> P (subst t env)"
diff: shows "real_of_1 (p1, l1, r1) \<noteq> real_of_1 (p2, l2, r2)"
read_only_reads_unowned_axioms: shows "read_only_reads_unowned ts"
simple_packet_tag_def: fixes ct_state :: "ctstate"   and p :: "('a, 'b) simple_packet_scheme" shows "simple_packet_tag ct_state p \<equiv> \<lparr>p_iiface = p_iiface p, p_oiface = p_oiface p, p_src = p_src p, p_dst = p_dst p, p_proto = p_proto p, p_sport = p_sport p, p_dport = p_dport p, p_tcp_flags = p_tcp_flags p, p_payload = p_payload p, p_tag_ctstate = ct_state, \<dots> = simple_packet.more p\<rparr>"
zip_append: fixes xs :: "'k list"   and us :: "'l list"   and ys :: "'k list"   and vs :: "'l list" assumes "length xs = length us" shows "zip (xs @ ys) (us @ vs) = zip xs us @ zip ys vs"
Polys_UNIV: shows "P[UNIV] = UNIV"
Null_def: shows "Null \<equiv> \<lambda>x. \<lparr>A!,x\<rparr> \<^bold>& \<^bold>\<not>(\<^bold>\<exists>F. \<lbrace>x,F\<rbrace>)"
psolutions_usolves_ode: fixes t1 :: "real" assumes "t1 \<in> cball t et" shows "(psolutions t1 usolves_ode f from t) {t--t1} (cball (flow t0 x0 t) ex)"
set_bset_eqvt: fixes p :: "perm"   and B :: "'e set['f]" shows "p \<bullet> set_bset B = set_bset (p \<bullet> B)"
inorder_delete_splay: fixes t :: "('c \<times> 'd) tree"   and x :: "'c" assumes "sorted1 (inorder t)" shows "inorder (Splay_Map.delete x t) = AList_Upd_Del.del_list x (inorder t)"
AE_iff_null_sets: fixes N :: "'c set"   and M :: "'c measure" assumes "N \<in> sets M" shows "(N \<in> null_sets M) = (AE x in M. x \<notin> N)"
intp: shows "integrable M (\<lambda>x. fp (T x) * g x)"
Invariants_sets: shows "sets Invariants = {A \<in> local.P.events. T -` A \<inter> space P = A}"
QS_def: shows "QS = (\<lambda>tab. semiJoin tab (st, tt)) ` Q"
union_insert_fset: fixes x :: "'b"   and S :: "'b fset"   and T :: "'b fset" shows "insert_fset x S |\<union>| T = insert_fset x (S |\<union>| T)"
sum_list_pair_relator_list: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and s :: "'b"   and t :: "'b" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero (alternating_list (2 * monoid_add.add_order plus zero (plus s t)) s t) = zero"
Interp_imp_INTERP: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<le> D"   and "Interp D \<Turnstile> C" shows "INTERP \<Turnstile> C"
finite_subset_wf: shows "wf {(X, Y). X \<subset> Y \<and> finite Y}"
soundnessForSat: fixes F0 :: "Literal list list"   and Vbl :: "nat set"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list" assumes "vars F0 \<subseteq> Vbl"   and "InvariantVarsF F F0 Vbl"   and "InvariantConsistent M"   and "InvariantEquivalentZL F M F0"   and "\<not> formulaFalse F (elements M)"   and "Vbl \<subseteq> vars (elements M)" shows "model (elements M) F0"
pred_spmf_return: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "pred_spmf P (return_spmf x) = P x"
assms_1: fixes x :: "complex"   and t :: "'b" assumes "x \<in> U"   and "t \<in> cbox a b" shows "((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
is_sublist_tl: fixes a :: "'a"   and as :: "'a list"   and vs :: "'a list" assumes "is_sublist (a # as) vs" shows "is_sublist as vs"
redT_silent_move_invariant: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and Is :: "'t \<Rightarrow> 'i option" assumes "\<tau>mredT s s'"   and "ts_inv P Is (thr s) (shr s)" shows "ts_inv P Is (thr s') (shr s')"
add_mset: fixes x :: "'a"   and N :: "'a multiset" shows "\<Prod>\<^sub># (add_mset x N) = x * \<Prod>\<^sub># N"
snd_Gauss_Jordan_upt_k_PA: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "snd (Gauss_Jordan_upt_k_PA A k) = Gauss_Jordan_upt_k A k"
swapEnv_psubstEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "goodEnv rho"   and "goodEnv rho'" shows "rho &[rho'] &[z1 \<and> z2]_zs = rho &[z1 \<and> z2]_zs &[rho' &[z1 \<and> z2]_zs]"
H: shows "\<Gamma> \<turnstile> (\<lambda><:S. s) : T"
Suc_less: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and i :: "'b"   and j :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less i j"   and "plus i one \<noteq> j" shows "less (plus i one) j"
length_to_bl_eq: fixes w :: "'b word" shows "length (to_bl w) = LENGTH('b)"
delete_edge_valid': fixes v :: "'v"   and e :: "'w"   and v' :: "'v" shows "valid_graph (delete_edge v e v' G)"
of_list_def: fixes xs :: "'a list" shows "Array.of_list xs = Heap_Monad.heap (Array.alloc xs)"
r_constant: fixes x :: "'a" assumes "x \<in> \<V>" shows "\<B> rep x * (\<k> - 1) = \<Lambda> * (\<v> - 1)"
p_antitone_sup: fixes uminus :: "'c \<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 x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "less_eq (uminus (sup x y)) (uminus x)"
dagger_unfoldl_distr: fixes y :: "'a"   and x :: "'a" shows "y + x \<cdot> x\<^sup>\<star> \<cdot> y = x\<^sup>\<star> \<cdot> y"
WhileTrue: fixes b :: "bexp"   and s\<^sub>1 :: "char list \<Rightarrow> int"   and c :: "com"   and s\<^sub>2 :: "char list \<Rightarrow> int"   and s\<^sub>3 :: "char list \<Rightarrow> int" assumes "bval b s\<^sub>1"   and "(c, s\<^sub>1) \<Rightarrow> s\<^sub>2"   and "(WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3" shows "(WHILE b DO c, s\<^sub>1) \<Rightarrow> s\<^sub>3"
Increasing_def: fixes f :: "'b \<Rightarrow> 'c" shows "Increasing f \<equiv> \<Inter>z. Stable {s. z \<le> f s}"
cdf_cts_step_2: fixes x :: "real"   and y :: "real" assumes "x < y" shows "expectation (cts_step x y) \<le> cdf M y"