hcju/isabelleps huggingface dataset:
ex2: fixes C :: "'b set set" assumes "finite_character (C::'b set set)" shows "subset_closed (C::'b set set)"
assms_3: shows "subgroup H (G\<lparr>carrier := normalizer G N\<rparr>)"
sp_in_extends: fixes e :: "'d edge"   and g :: "('d, 'e) rgraph_scheme"   and g' :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and es :: "'d edge list"   and v2 :: "'d" assumes "ArcExt.extends g e g'"   and "Graph.subpath g v1 es v2" shows "Graph.subpath g' v1 es v2"
poly_mapping_eq_iff: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and b :: "'b \<Rightarrow>\<^sub>0 'c" shows "(a = b) = (poly_mapping.lookup a = poly_mapping.lookup b)"
pol_SOME_2: fixes p :: "'a" assumes "p \<in> carrier R" shows "pol_coeff S (SOME c. cf_sol R S X p c) \<and> p = polyn_expr R X (fst (SOME c. cf_sol R S X p c)) (SOME c. cf_sol R S X p c)"
rel_mono: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R3a :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a"   and "R3 \<le> R3a" shows "rel_generat R1 R2 R3 \<le> rel_generat R1a R2a R3a"
mat_adj_mult_sm: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_adj (k *\<^sub>s\<^sub>m A) = cnj k *\<^sub>s\<^sub>m mat_adj A"
stream_space_single_preimage: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and r :: "'b stream"   and n :: "nat" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc F X"   and "r \<in> range (proj_stoch_proc X n)" shows "infinite_cts_filtration.stream_space_single (proj_stoch_proc X n) r \<inter> range (proj_stoch_proc X n) = {r}"
hyps: fixes m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat"   and pending_write :: "bool"   and \<O> :: "nat set" assumes "\<forall>a\<in>A. m' a = m a"   and "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb \<subseteq> A"   and "reads_consistent pending_write \<O> m sb" shows "reads_consistent pending_write \<O> m' sb"
axioms_1: fixes as\<^sub>0 :: "'b"   and alstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and cs\<^sub>0 :: "'e"   and clstep :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R" shows "df_system as\<^sub>0 alstep"
cl5: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<cdot> (z \<cdot> (0::'a))) = x \<cdot> y \<cdot> (z \<cdot> (0::'a))"
elims: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "nat"   and y :: "'e Trace set" assumes "PreEnvironmentJView jkbp jview"   and "PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview x = y" obtains "x = 0" and "y = rel_ext (\<lambda>uu_. \<exists>s. uu_ = tInit s \<and> s \<in> set envInit)"   | n :: "nat" where "x = Suc n" and "y = rel_ext (\<lambda>uu_. \<exists>t eact aact. uu_ = t \<leadsto> envTrans eact aact (tLast t) \<and> t \<in> PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n \<and> eact \<in> set (envAction (tLast t)) \<and> (\<forall>a. aact a \<in> set (JKBP.jAction jkbp (PreEnvironmentJView.mkM envVal jview (PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n)) t a)))"
prod_simps_1: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "arr (ECC.prod f0 f1)"
card_of_ordLeq: fixes A :: "'c set"   and B :: "'d set" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = (|A| \<le>o |B|)"
assms_4: shows "is_canon_rewriter rword (set bs) u a"
phl_skip: fixes x :: "('a \<times> 'a set) set" shows "x \<parallel> 1\<^sub>\<pi> \<subseteq> 1\<^sub>\<pi> \<parallel> x"
mset_left_cancel_elem: fixes a :: "'a"   and b :: "'a"   and A :: "'a multiset" assumes "a \<in># {#b#} + A"   and "a \<noteq> b" shows "a \<in># A"
abs_Gromov_completion_def: shows "abs_Gromov_completion \<equiv> quot_type.abs Gromov_completion_rel Abs_Gromov_completion"
oracle_def: fixes \<sigma> :: "'c \<Rightarrow> bool list option"   and x :: "'c" shows "oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>))"
ipresSwapAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'm, 'h, 'n) model_scheme" shows "ipresSwapAbs hA MOD \<equiv> \<forall>zs z1 z2 us s A. wlsAbs (us, s) A \<longrightarrow> hA (A $[z1 \<and> z2]_zs) = igSwapAbs MOD zs z1 z2 (hA A)"
apply_power_eq_self_iff: fixes f :: "'b perm"   and n :: "nat"   and a :: "'b" shows "(apply (f ^ n) a = a) = (order f a dvd n)"
le_approx2T: fixes P :: "'a process"   and Q :: "'a process"   and s :: "'a event list" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "(s \<in> T Q) = (s \<in> T P)"
assms: fixes a :: "Arity" shows "ccExp_syn a e[x::=y] G|` S = ccExp_syn a e G|` S"
SN_on_trancl_imp_SN_on: fixes R :: "('b \<times> 'b) set"   and T :: "'b set" assumes "SN_on (R\<^sup>+) T" shows "SN_on R T"
assms: fixes b :: "'a"   and a :: "'a" assumes "b \<in> B"   and "a \<in> A"   and "a + b = (0::'a)" shows "a = (0::'a)"
empty_le: fixes A :: "'k multiset" shows "{#} \<subseteq># A"
finite: fixes P :: "'a \<Rightarrow> bool" assumes "finite {x. P x}" shows "\<not> eventually P F"
\<delta>_states_finite: fixes \<delta> :: "('d, 'e) ta_rule set" assumes "finite \<delta>" shows "finite (\<delta>_states \<delta>)"
diff_diff_ennreal'': fixes z :: "ennreal"   and y :: "ennreal"   and x :: "ennreal" assumes "z \<le> y" shows "x - (y - z) = (if y - z \<le> x then x + z - y else 0)"
perfect_sec_P2_def: fixes m1 :: "nat"   and m2 :: "nat" shows "local.sim_non_det_def.perfect_sec_P2 m1 m2 \<equiv> R2 m1 m2 = funct m1 m2 \<bind> (\<lambda>(s1, s2). local.sim_non_det_def.Ideal2 m2 m1 s2)"
count_empty: fixes a :: "'a" shows "count (\<Empt>) a = 0"
rel: shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q"
fps_nth_inject: fixes x :: "'a fps"   and y :: "'a fps" shows "(($) x = ($) y) = (x = y)"
cring_simprules_26: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<otimes> x = \<zero>"
fold_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('c, 'd) rbt" shows "RBT_Impl.fold f t = fold (\<lambda>(x, y). f x y) (RBT_Impl.entries t)"
g''_def: shows "g'' = snd (snd (splitFace g' v a f' [countVertices g'..<countVertices g' + n]))"
associatedD1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" shows "dvd.dvd times a b"
y_3: shows "\<forall>i<dim_vec y. (0::'a) \<le> y $ i"
IH: assumes "cs' \<noteq> []"   and "upd_cs cs' as = []"   and "\<forall>xs ys. as = xs @ ys \<and> ys \<noteq> [] \<longrightarrow> upd_cs cs' xs \<noteq> []" shows "last as \<in> get_return_edges (last cs')"
rep_eq: fixes x :: "'b dense"   and xa :: "'b dense" shows "(x \<le> xa) = (Rep_dense x \<le> Rep_dense xa)"
Suc_5: shows "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m m"
lapp_inf: fixes s :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "s \<in> A\<^sup>\<omega>" shows "s @@ t = s"
bintrunc_n_0: fixes n :: "nat" shows "take_bit n 0 = 0"
star_sim3: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> x = y \<cdot> z" shows "z \<cdot> x\<^sup>\<star> = y\<^sup>\<star> \<cdot> z"
x_zero_le_c: fixes x :: "'a" shows "x \<cdot> (0::'a) \<le> 1\<^sub>\<pi>"
simps_2: fixes x :: "'a"   and r :: "'a list"   and n :: "nat" shows "delblockhelp (x # r) n = (if n \<le> 0 then x # r else delblockhelp r (n - 1))"
NotL_inv: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> F, \<Delta>"
rn_price_eq: fixes N :: "bool stream measure"   and q :: "real"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "der \<in> borel_measurable (G matur)"   and "n \<le> matur" shows "rn_price N der matur n w = rn_price_ind N der matur n w"
discI_2: fixes tllist :: "('f, 'g) tllist"   and x21 :: "'f"   and x22 :: "('f, 'g) tllist" assumes "tllist = TCons x21 x22" shows "\<not> is_TNil tllist"
remove_vertex_V': fixes G :: "('a, 'b) Graph_scheme"   and x :: "'a" assumes "Digraph G" shows "V\<^bsub>Digraph.remove_vertex G x\<^esub> \<subseteq> V\<^bsub>G\<^esub>"
a2: shows "N \<in> Cignorabimus \<alpha> V"
type_low_vars_low_b: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'BExp"   and mds :: "Mode \<Rightarrow> 'Var set"   and x :: "'Var" assumes "\<Gamma> \<turnstile>\<^sub>b e \<in> Low "   and "mds_consistent mds \<Gamma>"   and "x \<in> bexp_vars e" shows "to_total \<Gamma> x = Low"
assms: shows "\<forall>x\<in>set xs. 1 \<le> x"
closed_cdi_subset: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "closed_cdi \<Omega> M" shows "M \<subseteq> Pow \<Omega>"
erase_l_bal: fixes l :: "'a tree"   and r :: "'a tree"   and n :: "'a" assumes "hinv l"   and "hinv r"   and "height (erase l) = height (erase r) + 2" shows "erase (l_bal n l r) = l_bal\<^sub>0 n (erase l) (erase r)"
sees_wf_mdecl: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "wf_mdecl wf_md P D (M, Ts, T, m)"
card_characters: shows "card (characters (Characters (Characters G))) = order (Characters (Characters G))"
one_dim_apply_is_times_def: fixes \<psi> :: "'b"   and \<phi> :: "'b" shows "\<psi> * \<phi> = (\<langle>1::'b, \<psi>\<rangle> * \<langle>1::'b, \<phi>\<rangle>) *\<^sub>C (1::'b)"
weakly_bisimilar_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and Q :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.weakly_bisimilar satisfies transition \<tau> P Q \<equiv> \<exists>R. weak_nominal_ts.is_weak_bisimulation satisfies transition \<tau> R \<and> R P Q"
ide_cod: fixes f :: "'a" assumes "B.arr f" shows "B.ide (B.cod f)"
cones_functor_axioms: shows "cones_functor (\<cdot>\<^sub>J) (\<cdot>) D (\<cdot>\<^sub>S) \<iota>"
fpxs_compose_power_power: fixes r :: "rat"   and f :: "'b fpxs"   and n :: "nat" assumes "0 < r" shows "fpxs_compose_power (f ^ n) r = fpxs_compose_power f r ^ n"
max_min_same_3: fixes x :: "'a"   and y :: "'a" shows "max (min x y) y = y"
cinfinite_mono: fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set" assumes "r1 \<le>o r2"   and "cinfinite r1" shows "cinfinite r2"
rel_refl_strong: fixes x :: "('d, 'e) optionT"   and R1a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> phantom_optionT x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> set_optionT x \<Longrightarrow> R2a z2 z2" shows "rel_optionT' R1a R2a x x"
abs_wpt_def: fixes x :: "'c" shows "abs_wpt x = Abs_Assertion (wpt x)"
assms: shows "x \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<circ> A"
bits_1_div_exp: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide one (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = zero_neq_one.of_bool one zero (n = 0)"
orbit_unfold_image: fixes f :: "'b perm"   and a :: "'b" shows "orbit f a = (\<lambda>n. apply (f ^ n) a) ` {..<order f a}"
clientK_not_spied: fixes A :: "agent"   and B :: "agent"   and PMS :: "nat"   and evs :: "event list"   and Na :: "nat"   and Nb :: "nat"   and NA :: "nat"   and NB :: "nat" assumes "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "Says A Spy (Key (clientK (Na, Nb, PRF (PMS, NA, NB)))) \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> tls" shows "Key (clientK (Na, Nb, PRF (PMS, NA, NB))) \<notin> parts (knows Spy evs)"
measurable_fun_upd: fixes I :: "'a set"   and J :: "'a set"   and i :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and N :: "'b measure"   and M :: "'a \<Rightarrow> 'c measure"   and h :: "'b \<Rightarrow> 'c" assumes "I = J \<union> {i}"   and "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M J M"   and "h \<in> N \<rightarrow>\<^sub>M M i" shows "(\<lambda>x. (f x)(i := h x)) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
rec_nonstop_def: shows "rec_nonstop = Cn 3 rec_NSTD [rec_conf]"
dom_llp: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "(x \<subseteq> rdom y ; x) = (rdom x \<subseteq> rdom y)"
elims_2: fixes x :: "'a"   and xa :: "'a edge list"   and xb :: "'a" assumes "ces x xa xb" obtains v1 :: "'a" and v2 :: "'a" where "x = v1__" and "xa = []" and "xb = v2__" and "v1__ = v2__"   | v1 :: "'a" and e :: "'a edge" and es :: "'a edge list" and v2 :: "'a" where "x = v1" and "xa = e # es" and "xb = v2" and "src e = v1 \<and> ces (tgt e) es v2"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>C g\<guillemotright>" shows "src\<^sub>C \<mu> = src\<^sub>C f"
upper_times: fixes A :: "'a interval"   and B :: "'a interval" shows "upper (A * B) = Max {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
fimage_mono: fixes A :: "'a fset"   and B :: "'a fset"   and f :: "'a \<Rightarrow> 'b" assumes "A |\<subseteq>| B" shows "f |`| A |\<subseteq>| f |`| B"
in_mset_fold_plus_iff: fixes x :: "'a"   and M :: "'a multiset"   and NN :: "'a multiset multiset" shows "(x \<in># fold_mset (+) M NN) = (x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N))"
isomorphic_unit_left: fixes f :: "'a" assumes "B.ide f" shows "trg f \<star> f \<cong>\<^sub>B f"
is_singletonI': fixes A :: "'g set" assumes "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "is_singleton A"
supp_3: fixes fmla :: "fmla" shows "supp (neg fmla) = supp fmla"
partial_magma_axioms: shows "partial_magma (\<cdot>)"
ip_cidr_set_notsubset_empty_inter: fixes i1 :: "'b word"   and r1 :: "nat"   and i2 :: "'b word"   and r2 :: "nat" assumes "\<not> ip_cidr_set i1 r1 \<subseteq> ip_cidr_set i2 r2"   and "\<not> ip_cidr_set i2 r2 \<subseteq> ip_cidr_set i1 r1" shows "ip_cidr_set i1 r1 \<inter> ip_cidr_set i2 r2 = {}"
interpret_slp_nth: fixes slp :: "floatarith list"   and n :: "nat"   and xs :: "real list" assumes "length slp \<le> n" shows "interpret_slp slp xs ! n = xs ! (n - length slp)"
topological_tendstoD: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter"   and S :: "'a set" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open l) F"   and "open S"   and "l \<in> S" shows "\<forall>\<^sub>F x in F. f x \<in> S"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x \<otimes> y \<in> A"
bounded_hashcode_nat_def: fixes def_hashmap_size :: "'a itself \<Rightarrow> nat"   and hashcode :: "'a \<Rightarrow> uint32"   and n :: "nat"   and x :: "'a" assumes "class.hashable def_hashmap_size" shows "hashable.bounded_hashcode_nat hashcode n x = nat_of_hashcode (hashcode x) mod n"
star_left_induct: fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "z \<squnion> y * x \<le> x \<longrightarrow> y\<^sup>\<star> * z \<le> x"
boundOutputInduct: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"   and "x \<sharp> P"   and "\<And>P a x P' b C. \<lbrakk>P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a x P'"   and "\<And>P a x P' b c C. \<lbrakk>P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'; \<And>C. F C P a x P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a x P'"   and "\<And>P a x P' C. \<lbrakk>P \<longmapsto>\<^sub>e a[x] \<prec>\<^sub>e P'; a \<noteq> x\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a x P'"   and "\<And>P Q a x P' C. \<lbrakk>P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x P'"   and "\<And>P Q a x Q' C. \<lbrakk>Q \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e Q'; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x Q'"   and "\<And>P P' Q a x C. \<lbrakk>P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'; x \<sharp> Q; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P' \<parallel> Q)"   and "\<And>P Q Q' a x C. \<lbrakk>Q \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e Q'; x \<sharp> P; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P \<parallel> Q')"   and "\<And>P P' a x y C. \<lbrakk>P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'; y \<noteq> a; y \<noteq> x; y \<sharp> C; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (<\<nu>y>P) a x (<\<nu>y>P')"   and "\<And>P a x P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'; \<And>C. F C (P \<parallel> !P) a x P'\<rbrakk> \<Longrightarrow> F C (!P) a x P'" shows "F C P a x P'"
pt_eq_eqvt: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> (x = y) = (pi \<bullet> x = pi \<bullet> y)"
arr_dom_iff_arr: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c" assumes "category C" shows "partial_magma.arr C (partial_magma.dom C f) = partial_magma.arr C f"
esqrt_simps_2: shows "esqrt 1 = 1"
rel_game_def: fixes \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf" shows "okamoto_\<Sigma>_commit.rel_game \<A> = TRY G \<bind> (\<lambda>(h, w). \<A> h \<bind> (\<lambda>w'. return_spmf ((h, w') \<in> R))) ELSE return_spmf False"
of_nat_less_iff: fixes one :: "'f"   and times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m < n)"
not_less: fixes x :: "'b Word.word"   and y :: "'b Word.word" shows "(\<not> x <s y) = (y \<le>s x)"
cases: fixes x :: "tm \<times> tm" obtains s :: "name" and x :: "tm" and y :: "tm" where "atom s \<sharp> (x, y)" and "xa__ = (x, y)"
arev_arev: fixes a :: "'b" shows "rev_G (rev_G a) = a"
hyps_3: assumes "a \<sharp> t1" shows "a \<sharp> t2"
greaterThanAtMost_iff: fixes i :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "(i \<in> ord.greaterThanAtMost less_eq less l u) = (less l i \<and> less_eq i u)"
af\<^sub>G_simps_4: fixes \<phi> :: "'b ltl"   and x :: "'b set"   and xs :: "'b set list" shows "af\<^sub>G (X \<phi>) (x # xs) = af\<^sub>G \<phi> xs"
strict_fun_eq: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> local.F_strict"   and "g \<in> local.F_strict"   and "f ` {0..<n} = g ` {0..<n}" shows "f = g"
collapse: fixes phantom :: "('a, 'b) phantom" shows "phantom (of_phantom phantom) = phantom"
dc_on_LEXEQ: shows "dc_on (LEXEQ strict_suffix) (SEQ (lists A))"
set_shadow_root_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and tag :: "(_) shadow_root_ptr option" assumes "ShadowRootClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok set_shadow_root element_ptr tag"
config_config_distinct: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and init :: "'a list"   and qs :: "'a list" shows "distinct (fst (config\<^sub>p A init qs)) = distinct init"
rtrancl_mono_rightI: fixes S :: "('a \<times> 'a) set"   and S' :: "('a \<times> 'a) set" assumes "S \<subseteq> S'" shows "S \<subseteq> S'\<^sup>*"
H3': fixes x12 :: "nat"   and q :: "int mpoly"   and r :: "int mpoly"   and Aa :: "nat multiset" assumes "the (fmlookup A x12) * q - r \<in> More_Modules.ideal polynomial_bool"   and "r - spec \<in> More_Modules.ideal polynomial_bool" shows "r \<in> pac_ideal (insert (the (fmlookup A x12)) ((\<lambda>x. the (fmlookup A x)) ` set_mset Aa))"
axioms_1: fixes \<Omega> :: "'b set"   and M :: "'b set set" assumes "algebra \<Omega> M" shows "ring_of_sets \<Omega> M"
stable_Int: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set" assumes "F \<in> stable A"   and "F \<in> stable A'" shows "F \<in> stable (A \<inter> A')"
arr_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "\<F>C.arr (f \<otimes> g)"
var\<^sub>C_of_inj: shows "inj Eg2_var\<^sub>C_of_Eg1"
jkbp_def: fixes a :: "Agent" shows "jkbp a = [\<lparr>guard = Kor \<^bold>K\<^sub>a Kprop (Dirty a) \<^bold>K\<^sub>a Knot (Kprop (Dirty a)), action = SayIKnow\<rparr>, \<lparr>guard = Kand (Knot \<^bold>K\<^sub>a Kprop (Dirty a)) (Knot \<^bold>K\<^sub>a Knot (Kprop (Dirty a))), action = SayNothing\<rparr>]"
mdl_is_module: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "submodule R M H" shows "R module mdl M H"
cod_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b"   and "\<F>C.ide c" shows "\<F>C.cod \<a>[a, b, c] = a \<otimes> b \<otimes> c"
MkArr_Map: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "MkArr (Dom f) (Cod f) (Map f) = f"
multpw_def: shows "multpw \<equiv> \<lambda>ns. {(xa, x). multpwp (\<lambda>x xa. (x, xa) \<in> ns) xa x}"
ereal_complete_Inf: fixes S :: "ereal set" shows "\<exists>x. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)"
assms_2: shows "poly (p * q) a \<noteq> 0"
insert_lepoll_insertD: fixes u :: "'a"   and A :: "'a set"   and v :: "'b"   and B :: "'b set" assumes "insert u A \<lesssim> insert v B"   and "u \<notin> A"   and "v \<notin> B" shows "A \<lesssim> B"