hcju/isabelleps huggingface dataset:
setops\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('a, 'b) stateful_strand_step list"   and B :: "('a, 'b) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t (A @ B) = setops\<^sub>s\<^sub>s\<^sub>t A \<union> setops\<^sub>s\<^sub>s\<^sub>t B"
cond_cases_2: fixes h :: "heap" assumes "success p h"   and "\<not> cond p h" shows "execute p h = Some (False, h)"
hom: shows "homotopic_with (\<lambda>h. h ` T \<subseteq> V) S U f g"
left_idem: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<inter># (a \<inter># b) = a \<inter># b"
S3Fail: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCFail crCh rmCh rst p \<and> $S3 rmhist p \<and> HNext rmhist p \<and> unchanged (e p, c p, m p) \<longrightarrow> S6 rmhist p$"
list_all2_Nil2: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e list" shows "list_all2 P xs [] = (xs = [])"
unfold: fixes s :: "'c"   and h :: "'c \<rightarrow> ('d, 'c) Step" assumes "s \<noteq> \<bottom>" shows "unfold\<cdot>h\<cdot>s = unfold2\<cdot>(unfold\<cdot>h)\<cdot>(h\<cdot>s)"
cofactor_make_mat_0k: fixes k :: "nat"   and n :: "nat"   and B :: "'a mat" assumes "k < n"   and "k \<noteq> 0"   and "1 < n" shows "Determinant.cofactor (make_mat n k B) 0 k = - B $$ (1, 0)"
dom_deny_pfun: fixes f :: "'c \<Rightarrow> 'd option" shows "dom (D\<^sub>p f) = dom f"
eq: shows "row_iarray (mod_type_class.to_nat x) (Gauss_Jordan_iarrays (matrix_to_iarray A)) = row_iarray (mod_type_class.to_nat y) (Gauss_Jordan_iarrays (matrix_to_iarray A))"
lossless_response: fixes r :: "nat \<times> nat"   and w :: "nat \<times> nat"   and e :: "nat" shows "lossless_spmf (response r w e)"
prefs_from_table_wfD_3: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list" assumes "prefs_from_table_wf agents alts xss" shows "distinct (map fst xss)"
mcont_bind_spmf: fixes luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e spmf"   and g :: "'e \<Rightarrow> 'd \<Rightarrow> 'f spmf" assumes "mcont luba orda lub_spmf (ord_spmf (=)) f"   and "\<And>y. mcont luba orda lub_spmf (ord_spmf (=)) (g y)" shows "mcont luba orda lub_spmf (ord_spmf (=)) (\<lambda>x. f x \<bind> (\<lambda>y. g y x))"
subset_insert: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A" shows "(A \<subseteq> x \<triangleright> B) = (A \<subseteq> B)"
s_def: fixes i :: "nat" shows "s i = (if i = 1 then S else T)"
bit_set_bit_iff: fixes m :: "nat"   and a :: "'a"   and n :: "nat" shows "bit (set_bit m a) n = (bit a n \<or> m = n \<and> (2::'a) ^ n \<noteq> (0::'a))"
arithmetic_simps_57: fixes a :: "'a" shows "a - (0::'a) = a"
Parallel_AllNone: fixes Ts :: "('a ann_com option \<times> 'a set) list"   and X :: "'a set" assumes "All_None Ts" shows "SEM (Parallel Ts) X = X"
sup_option_assoc: fixes x :: "'c option"   and y :: "'c option"   and z :: "'c option" shows "sup_option (sup_option x y) z = sup_option x (sup_option y z)"
union_open: fixes T :: "'b set set"   and M :: "'b set set" assumes "topology T"   and "\<forall>m\<in>M. m open\<^bsub>T\<^esub>" shows "\<Union> M open\<^bsub>T\<^esub>"
nth_trailing_zeros: fixes xs :: "'b list"   and n :: "nat" shows "Poly_Mapping.nth (xs @ replicate n (0::'b)) = Poly_Mapping.nth xs"
type_max_dma_type: fixes x :: "'Var"   and mem :: "'Var \<Rightarrow> 'Val" shows "type_max (dma_type x) mem = dma mem x"
rtrancl_comp_trancl_conv: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>* O r = r\<^sup>+"
dBC_2: shows "dim_col (FB B C) = dim_col B + dim_col C"
Suc_lessI: fixes m :: "nat"   and n :: "nat" assumes "order.greater n m"   and "Suc m \<noteq> n" shows "order.greater n (Suc m)"
llexord_LNil: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c llist" shows "llexord r LNil ys"
Im_complex_div_lt_0: fixes a :: "complex"   and b :: "complex" shows "(Im (a / b) < 0) = (Im (a * cnj b) < 0)"
L_def: shows "L = (\<lambda>S. {x. jumpF h (at_left x) \<noteq> 0 \<and> x \<in> S})"
assms_3: shows "\<forall>\<^sub>F x in F. c1 * norm (g x) \<le> norm (f x) \<and> norm (f x) \<le> c2 * norm (g x)"
rawpsubstT_notIn: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and x :: "'a"   and txs :: "('b \<times> 'a) list"   and t :: "'b" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "x \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "t \<in> trm"   and "FvarsT t \<inter> snd ` set txs = {}" shows "Generic_Syntax.rawpsubstT substT t txs = t"
strict_coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a < c" shows "a \<sqinter> b < c"
minus_fun_def: fixes A :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b" shows "A - B = (\<lambda>x. A x - B x)"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "ModuleHom smult M smult T"   and "ModuleEnd_axioms M T" shows "ModuleEnd smult M T"
assms_1: shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"
subseq_append': fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" shows "subseq (zs @ xs) (zs @ ys) = subseq xs ys"
card_of_Plus_infinite2: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "|B| \<le>o |A|" shows "|B <+> A| =o |A|"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and ternaryvalue :: "ternaryvalue" shows "P (case ternaryvalue of TernaryTrue \<Rightarrow> f1 | TernaryFalse \<Rightarrow> f2 | TernaryUnknown \<Rightarrow> f3) = ((ternaryvalue = TernaryTrue \<longrightarrow> P f1) \<and> (ternaryvalue = TernaryFalse \<longrightarrow> P f2) \<and> (ternaryvalue = TernaryUnknown \<longrightarrow> P f3))"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat"   and y :: "three" assumes "h1 x xa xb = y" obtains k :: "nat" and i :: "nat" and j :: "nat" where "x = k" and "xa = i" and "xb = j" and "y = (if i = j then One else if j \<le> k then Zero else Two)"
simple_rule_dtor_def: fixes r :: "'c simple_rule" shows "simple_rule_dtor r \<equiv> case r of SimpleRule m a \<Rightarrow> (m, a)"
scaleR2_eq: fixes e :: "real"   and xr :: "'a"   and P :: "'a" assumes "0 < e" shows "ccw (0::'a) xr (e *\<^sub>R P) = ccw (0::'a) xr P"
intros_12: fixes c\<^sub>2 :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node"   and b :: "expr"   and c\<^sub>1 :: "cmd" assumes "c\<^sub>2 \<turnstile> n -et\<rightarrow> n'"   and "n \<noteq> (_Entry_)" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> n \<oplus> #:c\<^sub>1 + 1 -et\<rightarrow> n' \<oplus> #:c\<^sub>1 + 1"
snoc: shows "tvs (mk_all_list vs B) = tvs B \<union> tvsT_Set (snd ` set vs)"
seq_semI: fixes \<Delta> :: "('sf, 'sc, 'sz) formula list"   and \<psi> :: "('sf, 'sc, 'sz) formula"   and \<Gamma> :: "('sf, 'sc, 'sz) formula list"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "List.member \<Delta> \<psi>"   and "(\<And>\<phi>. List.member \<Gamma> \<phi> \<Longrightarrow> \<nu> \<in> fml_sem I \<phi>) \<Longrightarrow> \<nu> \<in> fml_sem I \<psi>" shows "\<nu> \<in> seq_sem I (\<Gamma>, \<Delta>)"
line_in: fixes x :: "real" assumes "x \<in> {0..1}" shows "a + x *\<^sub>R u \<in> D"
close_provable_sound: 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 \<phi> :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "sound (SG, C)"   and "sound (close SG \<phi>, \<phi>)" shows "sound (close SG \<phi>, C)"
ik\<^sub>s\<^sub>tD': fixes t :: "('d, 'e) Term.term"   and S :: "('d, 'e) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "t \<in> trms\<^sub>s\<^sub>t S"
mplus_strict: fixes m :: "'r match" shows "\<bottom> +++ m = \<bottom>"
ip0: fixes I :: "'c set"   and R :: "('c, 'd) Ring_scheme" shows "I \<^bsup>\<diamondsuit>R 0\<^esup> = carrier R"
has_derivative_linear: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "(f has_derivative f') F" shows "linear f'"
rgs: shows "x = (\<Sum>(s, h)\<leftarrow>rgs. s \<delta>\<delta> h)"
stk_price: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b" assumes "CRR_hyps p M geom_proc u d init Mkt r risk_free_asset stk" shows "prices Mkt stk = geom_proc"
K: shows "\<forall>n<K. f $$ n = (0::'a)"
extremum_uniqueI: fixes a :: "'a" assumes "top_class.top \<le> a" shows "a = top_class.top"
proj_unl_cons_1: fixes l :: "'e"   and a :: "'d"   and A :: "('e strand_label \<times> 'd) list" shows "proj_unl l ((ln l, a) # A) = a # proj_unl l A"
InvariantCEntailed_def: fixes conflictFlag :: "bool"   and F :: "Literal list list"   and C :: "Literal list" shows "InvariantCEntailed conflictFlag F C \<equiv> conflictFlag \<longrightarrow> formulaEntailsClause F C"
B_Said_YM2: fixes B :: "agent"   and A :: "agent"   and NA :: "nat"   and nb :: "msg"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> yahalom"   and "B \<notin> bad" shows "Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace> \<in> set evs"
bij_A_carrier: shows "bij_betw (Poly \<circ> list_of_vec) (carrier_vec (degree f)) carrier_irr"
ffold_rec: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'e"   and A :: "'e fset"   and z :: "'f" assumes "comp_fun_commute f"   and "x |\<in>| A" shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
lipschitz_on_cong: fixes C :: "real"   and D :: "real"   and U :: "'b set"   and V :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "C = D"   and "U = V"   and "\<And>x. x \<in> V \<Longrightarrow> g x = f x" shows "C-lipschitz_on U g = D-lipschitz_on V f"
ordLeq_cexp1: fixes r :: "('c \<times> 'c) set"   and q :: "('d \<times> 'd) set" assumes "(r, czero) \<notin> ordIso \<and> Card_order r"   and "Card_order q" shows "q \<le>o q ^c r"
simps: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'b \<times> 'd"   and a2 :: "'c \<times> 'e" shows "rel_prod R1 R2 a1 a2 = (\<exists>a b c d. a1 = (a, c) \<and> a2 = (b, d) \<and> R1 a b \<and> R2 c d)"
falsifies\<^sub>c\<^sub>s_empty: fixes C :: "fterm literal set" assumes "falsifies\<^sub>c [] C" shows "C = {}"
filter_def: fixes F :: "'b set set"   and T :: "'b set set" shows "Topology.filter F T \<equiv> F \<in> Filters\<^bsub>T\<^esub>"
subgroup_action: fixes H :: "'a set" assumes "subgroup H G" shows "group_action (G\<lparr>carrier := H\<rparr>) \<phi> (rcosets P)"
Zp_continuous_is_closed: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_continuous f" shows "f \<in> carrier (function_ring (carrier Zp) Zp)"
induct: fixes P :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> 'b rule list \<Rightarrow> state \<Rightarrow> bool"   and a0 :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a1 :: "'c"   and a2 :: "'b rule list"   and a3 :: "state" assumes "\<And>\<gamma> p s. P \<gamma> p [] s"   and "\<And>\<gamma> p v va X. P \<gamma> p (v # va) (Decision X)"   and "\<And>\<gamma> p m a rs. \<lbrakk>\<not> matches \<gamma> m a p \<Longrightarrow> P \<gamma> p rs Undecided; \<lbrakk>\<not> \<not> matches \<gamma> m a p; a = Log\<rbrakk> \<Longrightarrow> P \<gamma> p rs Undecided; \<lbrakk>\<not> \<not> matches \<gamma> m a p; a = Empty\<rbrakk> \<Longrightarrow> P \<gamma> p rs Undecided\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided" shows "P a0 a1 a2 a3"
coeff_f_0_code: fixes f :: "'b poly" shows "coeff f 0 = (case coeffs f of [] \<Rightarrow> 0::'b | x # xa \<Rightarrow> x)"
"case": fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x1 :: "'c"   and x2 :: "'d" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
simps_1: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'e trm"   and T1 :: "'e trm"   and T2 :: "'e trm" shows "substA \<pi> (Eq T1 T2) = Eq (subst \<pi> T1) (subst \<pi> T2)"
VectorSpaceHomSetD_Im: fixes T :: "'a \<Rightarrow> 'b"   and fsmult :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and V :: "'a set"   and fsmult' :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and W :: "'b set" assumes "T \<in> VectorSpaceHomSet fsmult V fsmult' W" shows "T ` V \<subseteq> W"
units_of_pow: fixes x :: "'a"   and n :: "nat" assumes "x \<in> Units (add_monoid R)" shows "x [^]\<^bsub>units_of (add_monoid R)\<^esub> n = [n] \<cdot> x"
IncomingInfo_def: fixes c :: "('d, 'c, 'e) configuration_scheme"   and k :: "nat"   and p :: "'d"   and q :: "'d" shows "IncomingInfo c k p q = sum_list (drop k (c_msg c p q)) + c_temp c p"
wp3EndMT: fixes p :: "((int \<times> int) set set, int) Combinators list"   and xs :: "((int \<times> int) set set, int) Combinators"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and po :: "int" assumes "wellformed_policy3 (p @ [xs])"   and "AllowPortFromTo a b po \<in> set p" shows "dom (C (AllowPortFromTo a b po)) \<inter> dom (C xs) = {}"
proj_mult_scalar: fixes k :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (p \<odot> q) = p * proj_poly k q"
power_mod: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (power.power one times (modulo a b) n) b = modulo (power.power one times a n) b"
limit_mat_const: fixes A :: "complex mat"   and m :: "nat" assumes "A \<in> carrier_mat m m" shows "limit_mat (\<lambda>k. A) A m"
ae_filter_bot: fixes s :: "'s" shows "ae_filter (T s) \<noteq> \<bottom>"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end3 n (l, r) = (\<exists>i j. 0 < n \<and> i + j = n \<and> l = Oc \<up> i @ [Bk] \<and> r = Oc # Bk \<up> j @ Oc \<up> n)"
mono_g: shows "mono_on g {1..} \<or> mono_on (\<lambda>x. - g x) {1..}"
small_all_dghms: fixes \<alpha> :: "V" shows "small {\<FF>. \<exists>\<AA> \<BB>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>}"
norm_of_real_diff: fixes b :: "real"   and a :: "real" shows "norm (of_real b - of_real a) \<le> \<bar>b - a\<bar>"
distinct_9: shows "ConcatOp \<noteq> string_binop.LessEqOp"
freshOutputTransition: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sup>^a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
exists_complex_root: fixes n :: "nat"   and z :: "complex" assumes "n \<noteq> 0" obtains w :: "complex" where "z = w ^ n"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "poly_mapping.lookup (x * xa) = prod_fun (poly_mapping.lookup x) (poly_mapping.lookup xa)"
T_soundness: shows "CM.Model TE_wtFsym wtPsym TE_arOf TE_resOf parOf tPB intT TE_intF intP"
less_eq_imp_interp_subseteq_Interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and D :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "C \<le> D" shows "ground_resolution_with_selection.interp S N C \<subseteq> ground_resolution_with_selection.Interp S N D"
diamond_to_confluence: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and T :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "diamond R"   and "T \<le> R"   and "R \<le> T\<^sup>*\<^sup>*" shows "confluent T"
expandNonTauFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'e"   and C' :: "'f" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>P \<sharp>* C'"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* C'"   and "\<alpha> \<noteq> \<tau>" obtains p :: "(name \<times> name) list" and \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P'' :: "'b" where "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))" and "(p \<bullet> \<Psi>\<^sub>P) \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P''" and "distinctPerm p" and "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P''\<rangle>" and "A\<^sub>P' \<sharp>* P'" and "A\<^sub>P' \<sharp>* \<alpha>" and "A\<^sub>P' \<sharp>* (p \<bullet> \<alpha>)" and "A\<^sub>P' \<sharp>* C" and "bn (p \<bullet> \<alpha>) \<sharp>* C'" and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>" and "bn (p \<bullet> \<alpha>) \<sharp>* P'" and "distinct A\<^sub>P'"
simps_4: fixes K :: "nat"   and X :: "freemsg" shows "freenonces (DECRYPT K X) = freenonces X"
locally_sound_mode_use_preservation: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" assumes "secure_refinement \<R>\<^sub>A \<R> P"   and "preserves_local_guarantee_compliance \<R>" shows "preserves_locally_sound_mode_use \<R>"
sorted_snoc_le: fixes xs :: "'c list"   and x :: "'c"   and y :: "'c" assumes "ASSUMPTION (sorted (xs @ [x]))"   and "x \<le> y" shows "sorted (xs @ [y])"
runit_simps_5: fixes f :: "'a" assumes "B.ide f" shows "B.cod (B.runit f) = f"
arrI: fixes f :: "'a" assumes "right f f" shows "R.arr f"
drop_bit_take_bit: fixes m :: "nat"   and n :: "nat"   and a :: "'a" shows "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
length_shuffles: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "zs \<in> shuffles xs ys" shows "length zs = length xs + length ys"
ringS: shows "ring (S\<lparr>carrier := h ` K\<rparr>)"
alw_next_imp_exchange_next: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "alw Next s" shows "alw cri.next (smap exchange_config s)"
assms: shows "powr_nat \<bar>x\<bar> y \<le> u"
disj_zero: fixes x :: "'a" shows "x ## (0::'a)"
has_limits_of_shape_def: fixes J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "B.has_limits_of_shape J \<equiv> \<forall>D. diagram J (\<cdot>\<^sub>B) D \<longrightarrow> (\<exists>a \<chi>. limit_cone J (\<cdot>\<^sub>B) D a \<chi>)"
allUses_def: fixes \<alpha>n :: "'b \<Rightarrow> 'c list"   and inEdges' :: "'b \<Rightarrow> 'c \<Rightarrow> ('c \<times> 'd) list"   and "uses" :: "'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and phis :: "'b \<Rightarrow> 'c \<times> 'a \<Rightarrow> 'a list option"   and g :: "'b"   and n :: "'c" shows "CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g n \<equiv> uses g n \<union> CFG_SSA_base.phiUses \<alpha>n inEdges' phis g n"
globEA: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool"   and n' :: "nat" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "sb.eval c t t' n (\<box>\<^sub>b\<gamma>)"   and "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'" shows "sb.eval c t t' n' \<gamma>"
assms_1: shows "qGoodInp qinp \<or> qGoodInp qinp'"
simps_1: fixes \<alpha> :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "eval_tpoly \<alpha> (PVar x) = \<alpha> x"
lead_coeff_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "Polynomial.lead_coeff (p * q) = Polynomial.lead_coeff p * Polynomial.lead_coeff q"
thm_2_3_6_pp: fixes fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list"   and b :: "nat" assumes "pp_pm.is_GB_cofactor_bound (Poly_Mapping.map_key PP ` set fs) b" shows "punit.is_Groebner_basis (set (punit.Macaulay_list (deg_shifts_pp (Indets_pp fs) b fs)))"
FG_advice_b2_helper: fixes Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n G\<^sub>n \<psi>"   and "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<Sigma>\<^sub>1" shows "w \<Turnstile>\<^sub>n \<phi>"
intro: fixes \<U> :: "'b set"   and \<V> :: "'b set"   and \<A> :: "'b set multiset"   and \<B> :: "'b set multiset" assumes "\<U> \<subseteq> \<V>"   and "\<A> \<subseteq># \<B>" shows "sub_set_system_axioms \<U> \<A> \<V> \<B>"
prj_simps_7: shows "C.cod doms.Prj\<^sub>1\<^sub>1 = doms.\<mu>.apex"
accimethdsD: fixes im :: "qtname \<times> mhead"   and G :: "prog"   and pack :: "pname"   and I :: "qtname"   and sig :: "sig" assumes "im \<in> accimethds G pack I sig" shows "im \<in> DeclConcepts.imethds G I sig \<and> G \<turnstile> Iface I accessible_in pack"
prems: shows "insert a_ (insert b_ S_) \<inter> B = {}"
min_less_iff_conj: fixes z :: "'a word"   and x :: "'a word"   and y :: "'a word" shows "(z <s signed.min x y) = (z <s x \<and> z <s y)"
Next'_preserves_invs_1: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration" assumes "cri.InvMsgInGlob (exchange_config c0)"   and "inv_init_imp_prop_safe c0"   and "InvGlobPointstampsEq c0"   and "Next' c0 c1"   and "\<forall>p. prop_invs (prop_config c0 p)" shows "inv_init_imp_prop_safe c1"
select_convs_2: fixes c_sinvar :: "'b graph \<Rightarrow> bool"   and c_offending_flows :: "'b graph \<Rightarrow> ('b \<times> 'b) set set"   and c_isIFS :: "bool"   and more :: "'c" shows "c_offending_flows \<lparr>c_sinvar = c_sinvar, c_offending_flows = c_offending_flows, c_isIFS = c_isIFS, \<dots> = more\<rparr> = c_offending_flows"
pull_up_token_run_tokens: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "token_run x n = token_run y n" shows "\<exists>q. x \<in> configuration q n \<and> y \<in> configuration q n"
diagonal: fixes \<alpha> :: "fm"   and i :: "name" obtains \<delta> :: "fm" where "{} \<turnstile> \<delta> IFF \<alpha>(i::=\<guillemotleft>\<delta>\<guillemotright>)" and "supp \<delta> = supp \<alpha> - {atom i}"
not_in_segment2D: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> segment f a b" shows "x \<noteq> b"
degree_prod_sum_lt_n: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b poly"   and i :: "nat" assumes "\<And>i. i < n \<Longrightarrow> degree (f i) \<le> 1"   and "i < n"   and "degree (f i) = 0" shows "degree (prod f {0..<n}) < n"