hcju/isabelleps huggingface dataset:
stutter_equivI: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> nat"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "stutter_sampler f \<sigma>"   and "stutter_sampler g \<tau>"   and "\<sigma> \<circ> f = \<tau> \<circ> g" shows "\<sigma> \<approx> \<tau>"
equality_option_simps_2: fixes eq\<^sub>'\<^sub>a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" shows "equality_option eq\<^sub>'\<^sub>a None (Some y) = False"
poly_rel_of_int_poly: fixes f' :: "'i list"   and f :: "int poly"   and f'' :: "'a mod_ring poly" assumes "f' = of_int_poly_i ff_ops (Mp f)"   and "f'' = of_int_poly (Mp f)" shows "poly_rel f' f''"
primitive_extractor_correct_4: fixes m :: "'d match_expr"   and disc :: "'d \<Rightarrow> bool"   and sel :: "'d \<Rightarrow> 'e"   and C :: "'e \<Rightarrow> 'd"   and as :: "'e negation_type list"   and ms :: "'d match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "\<forall>disc2. \<not> has_disc disc2 m \<longrightarrow> \<not> has_disc disc2 ms"
wt_bij_finite_tatom_subst_exists_single: fixes S :: "'var set"   and T :: "('fun, 'var) Term.term set"   and a :: "'atom" assumes "finite S"   and "finite T"   and "\<And>x. x \<in> S \<Longrightarrow> \<Gamma> (Var x) = Var a" shows "\<exists>\<sigma>. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subst_range \<sigma> \<subseteq> (\<lambda>c. Fun c []) ` {c. \<Gamma> (Fun c []) = Var a \<and> public c \<and> arity c = 0} - T \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"
fresh_2: fixes a :: "atom"   and name :: "name" shows "a \<sharp> DBVar name = a \<sharp> name"
elims_2: fixes x :: "'b word \<times> nat"   and xa :: "'b word" assumes "simple_match_ip x xa" obtains base :: "'b word" and len :: "nat" and p_ip :: "'b word" where "x = (base, len)" and "xa = p_ip" and "p_ip \<in> ipset_from_cidr base len"
abs_eq: fixes xa :: "int extended \<times> int extended"   and x :: "int extended \<times> int extended" shows "(abs_ivl xa \<le> abs_ivl x) = le_rep xa x"
that: fixes ns' :: "'node list" assumes "g \<turnstile> n-ns'\<rightarrow>m'"   and "strict_prefix ns' ns"   and "m' \<notin> set (butlast ns')" shows "thesis"
inc_dot_def: fixes d :: "nat"   and x :: "item" shows "inc_dot d x = Item (item_rule x) (item_dot x + d) (item_origin x) (item_end x)"
ipcidr_conjunct_word: fixes ips1 :: "'a word \<times> nat"   and ips2 :: "'a word \<times> nat" shows "ipcidr_conjunct ips1 ips2 = (if wordinterval_empty (wordinterval_intersection (ipcidr_tuple_to_wordinterval ips1) (ipcidr_tuple_to_wordinterval ips2)) then None else if wordinterval_subset (ipcidr_tuple_to_wordinterval ips1) (ipcidr_tuple_to_wordinterval ips2) then Some ips1 else Some ips2)"
Lcm_1_iff: fixes A :: "'a set" shows "(Lcm A = (1::'a)) = (\<forall>a\<in>A. is_unit a)"
cadjoint_eqI: fixes F :: "'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'c" assumes "\<And>x y. F x \<bullet>\<^sub>C y = x \<bullet>\<^sub>C G y" shows "G\<^sup>\<dagger> = F"
set_delete: fixes xs :: "['b]"   and a :: "'b" assumes "distinct xs"   and "\<forall>x\<in>set xs. eq\<cdot>a\<cdot>x \<noteq> \<bottom>" shows "set (delete\<cdot>a\<cdot>xs) = set xs - {a}"
valid_imp_PL: fixes p :: "'a"   and q :: "'a" shows "\<tturnstile> (p .\<longrightarrow>. q) = ({p} \<turnstile> q)"
lt_Card_imp_lesspoll: fixes i :: "V"   and a :: "V" assumes "i < a"   and "Card a"   and "Ord i" shows "elts i \<prec> elts a"
lin_independent_imp_hd_independent_from_Span: fixes m :: "'v"   and ms :: "'v list" assumes "m \<in> V"   and "set ms \<subseteq> V"   and "R_lin_independent (m # ms)" shows "m \<notin> RSpan ms"
elims_3: fixes x :: "('c \<times> nat) tree" assumes "\<not> sngl x" obtains "x = \<langle>\<rangle>"   | uw_ :: "('c \<times> nat) tree" and ux_ :: "'c" and lva :: "nat" and uy_ :: "('c \<times> nat) tree" and uz_ :: "'c" and lvb :: "nat" and va_ :: "('c \<times> nat) tree" where "x = \<langle>uw_, (ux_, lva), \<langle>uy_, (uz_, lvb), va_\<rangle>\<rangle>" and "\<not> lvb < lva"
bind_spmf_def: fixes x :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "x \<bind> f = x \<bind> (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
that_1: assumes "x \<noteq> y"   and "{x--y} \<subseteq> X"   and "\<And>z. z \<in> {x--y} \<Longrightarrow> 0 < f z \<bullet> rot (x - y)" shows "thesis"
simps_3: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg" shows "bicategorical_language.Ide V src trg (t \<^bold>\<star> u) = (bicategorical_language.Ide V src trg t \<and> bicategorical_language.Ide V src trg u \<and> bicategorical_language.Src src t = bicategorical_language.Trg trg u)"
UN_simps_10: fixes B :: "'b \<Rightarrow> 'a set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "\<Union> (B ` f ` A) = (\<Union>a\<in>A. B (f a))"
coinduct: fixes R :: "'a process \<Rightarrow> 'a process \<Rightarrow> bool"   and process :: "'a process"   and process' :: "'a process" assumes "R process process'"   and "\<And>process process'. R process process' \<Longrightarrow> isAction process = isAction process' \<and> (isAction process \<longrightarrow> isAction process' \<longrightarrow> prefOf process = prefOf process' \<and> R (contOf process) (contOf process')) \<and> (isChoice process \<longrightarrow> isChoice process' \<longrightarrow> R (ch1Of process) (ch1Of process') \<and> R (ch2Of process) (ch2Of process'))" shows "process = process'"
is_map_fst_inj: fixes S :: "('a \<times> 'b) set" assumes "is_map S" shows "inj_on fst S"
hmem_succ_ne: fixes x :: "hf"   and y :: "hf" assumes "succ x \<^bold>\<in> y" shows "x \<noteq> y"
step_cfunc_usr: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and iu :: "'iusr_core" assumes "S p q"   and "iu \<in> IU" shows "p \<bind> (\<lambda>s1. map_spmf fst (cfunc_usr core1 s1 iu)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_usr core2 s2 iu))"
all_lifted_cons_rel: fixes q :: "'q" assumes "q \<in> Q" shows "consequence_relation Bot_FL (\<lambda>N1 N2. entails_q q (\<Union> (\<G>_F_L_q q ` N1)) (\<Union> (\<G>_F_L_q q ` N2)))"
proved_imp_proved_PfP: fixes \<alpha> :: "fm" assumes "{} \<turnstile> \<alpha>" shows "{} \<turnstile> PfP \<guillemotleft>\<alpha>\<guillemotright>"
parts_keys_of: fixes A :: "agent" shows "parts (keys_of A) = keys_of A"
f_def: shows "f = (\<lambda>p i. A $$ (i, p i))"
right_unique_cr_spmf_gpv: shows "right_unique spmf_to_gpv.cr_spmf_gpv"
assms_2: assumes "\<And>\<phi>. \<phi> \<in> \<phi>s \<Longrightarrow> \<phi> \<pi>l" shows "\<chi>"
LTL_prop_entailment_monotonI: fixes S :: "'c ltl set"   and \<phi> :: "'c ltl"   and S' :: "'c ltl set" assumes "S \<Turnstile>\<^sub>P \<phi>"   and "S \<subseteq> S'" shows "S' \<Turnstile>\<^sub>P \<phi>"
cong3_diff2: fixes B :: "'p"   and C :: "'p"   and A :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "B \<noteq> C"   and "A B C Cong3 A' B' C'" shows "B' \<noteq> C'"
tri_quad_le_degree: fixes g :: "graph"   and v :: "nat" shows "tri g v + quad g v \<le> degree g v"
inducts: fixes x1 :: "('procS, 'procT) Proc"   and x2 :: "('procS, 'procT) Proc"   and P :: "('procS, 'procT) Proc \<Rightarrow> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "x1 \<R>\<lbrakk>\<cdot>\<rbrakk>R x2"   and "\<And>S. P (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))" shows "P x1 x2"
has_derivative_at_alt: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "(f has_derivative f') (at x) = (bounded_linear f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>y. norm (y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)))"
ground_iff_no_vars: fixes M :: "('b, 'a) Term.term set" shows "ground M = (\<forall>v. Var v \<notin> subterms\<^sub>s\<^sub>e\<^sub>t M)"
proj_def: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b" shows "dynamic_component.proj tCMP active c = lmap (tCMP c) \<circ> lfilter (active c)"
sum_clauses_1: fixes f :: "'c \<Rightarrow> 'b" shows "sum f {} = (0::'b)"
lsu_top: shows "top_pres \<nu>\<^sup>\<natural>"
inj: shows "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
xor_top: fixes x :: "'g" shows "x \<oplus> \<top> = - x"
invKey_pubK_image_priK_image: fixes AS :: "agent set" shows "invKey ` pubK ` AS = priK ` AS"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" shows "bounded_antilinear_axioms f"
col: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and i :: "nat" assumes "mat nr nc m"   and "i < nc" shows "vec nr (col m i)"
execn_mark_guards_to_execn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>mark_guards f c,s\<rangle> =n\<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' = Fault f \<longrightarrow> t' = t) \<and> (isFault t' \<longrightarrow> isFault t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
rep_eq: shows "blinfun_apply snd_blinfun = snd"
injective: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d"   and y :: "'d" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A"   and "x \<noteq> y" shows "f x \<noteq> f y"
GS_initial_impl_def: fixes I :: "'b \<Rightarrow> node_state option"   and v0 :: "'b"   and succs :: "'b set" shows "GS_initial_impl I v0 succs \<equiv> ([v0], [0], I(v0 \<mapsto> STACK 0), if succs = {} then [] else [(0, succs)])"
msig_qmdecl_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "msig (qmdecl sig m) = sig"
crel_vs_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and v :: "'c"   and f :: "['d]" shows "crel_vs R v f \<equiv> \<forall>heap. P heap \<and> Q heap \<and> state_dp_consistency.cmem heap \<longrightarrow> (case execute f heap of None \<Rightarrow> False | Some (v', heap') \<Rightarrow> P heap' \<and> Q heap' \<and> R v v' \<and> state_dp_consistency.cmem heap')"
subst_5: fixes y :: "name"   and v :: "trm"   and s :: "trm" shows "[s][y::=v] = [s[y::=v]]"
assms_3: fixes \<eta> :: "nat" shows "interaction_any_bounded_converter (sim_outer \<eta>) (bound_sim \<eta>)"
ZFfunComp_def: fixes f :: "ZF"   and g :: "ZF" shows "f |o| g \<equiv> ZFfun (|dom|f) (|cod|g) (\<lambda>x. g |@| (f |@| x))"
eq_\<I>_converterD_WT: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv1 :: "('a, 'b, 'c, 'd) converter"   and conv2 :: "('a, 'b, 'c, 'd) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd> = \<I>, \<I>' \<turnstile>\<^sub>C conv2 \<surd>"
frac_le_eq: fixes y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "y \<noteq> (0::'a)"   and "z \<noteq> (0::'a)" shows "(x / y \<le> w / z) = ((x * z - w * y) / (y * z) \<le> (0::'a))"
Osum_def: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" shows "r \<union>o r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"
crdt_ops_ref_less: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and oper :: "'d"   and r :: "'c" assumes "crdt_ops xs deps"   and "(oid, oper) \<in> set xs"   and "r \<in> deps oper" shows "r < oid"
sub_rel_of_def: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "sub_rel_of g subs \<equiv> SubRel.vertices subs \<subseteq> Graph.vertices g"
filtercomap_ident: fixes F :: "'b filter" shows "filtercomap (\<lambda>x. x) F = F"
point_rep_singleton_inval: fixes x :: "'b"   and b :: "'b set" assumes "x \<notin> b" shows "{#b#} rep x = 0"
Ex_less_Suc2: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<exists>i<Suc n. P i) = (P 0 \<or> (\<exists>i<n. P (Suc i)))"
a_def: shows "a \<equiv> (t - y) / (x - y)"
mult_row_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f"   and q :: "'d"   and i :: "'f" shows "mult_row_row A a q i = (\<chi>j. if i = a then q * A $ a $ j else A $ i $ j)"
cong_DirProds_IDirProds: fixes S :: "'c \<Rightarrow> 'a set"   and I :: "'c set" assumes "is_idirprod (carrier G) S I"   and "finite I" shows "DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I \<cong> G"
is_expansion_aux_imp_smallo': fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux xs f basis" obtains e :: "real" where "f \<in> o(\<lambda>x. hd basis x powr e)"
ideal_inc_unit1: fixes a :: "'a"   and I :: "'a set" assumes "a \<in> carrier R"   and "Unit R a"   and "ideal R I"   and "a \<in> I" shows "I = carrier R"
pheap_merge: fixes h1 :: "'b heap"   and h2 :: "'b heap" assumes "pheap h1"   and "pheap h2" shows "pheap (merge h1 h2)"
shunting_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> - z) = (x \<le> - z \<squnion> - y)"
inverse_composition_commute: fixes x :: "'a"   and y :: "'a" assumes "additive.invertible x"   and "additive.invertible y"   and "x \<in> R"   and "y \<in> R" shows "- (x + y) = - y - x"
prod_list_normalize: fixes P :: "'b poly list" shows "normalize (prod_list P) = prod_list (map normalize P)"
asAbs_qSwapAbs_swapAbs: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGoodAbs qA" shows "asAbs (qA $[[x1 \<and> x2]]_xs) = asAbs qA $[x1 \<and> x2]_xs"
ded_thm_cor_3: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<psi> \<^bold>\<rightarrow> \<chi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<chi> in v]"
graph_union_preserves_restrict: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" assumes "graph G\<^sub>1"   and "graph G\<^sub>2" shows "graph (graph_union G\<^sub>1 G\<^sub>2)"
zero_and_eq: fixes a :: "'a" shows "(0::'a) AND a = (0::'a)"
fun_upd_upd: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'f"   and z :: "'f" shows "f(x := y, x := z) = f(x := z)"
bind_pure_returns_result_I: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and x :: "'z"   and g :: "'z \<Rightarrow> ('x, 'y, 'aa) prog"   and y :: "'aa" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
bind_map_pmf: fixes f :: "'j \<Rightarrow> 'i"   and M :: "'j pmf"   and g :: "'i \<Rightarrow> 'h pmf" shows "map_pmf f M \<bind> g = M \<bind> (\<lambda>x. g (f x))"
assms: shows "\<bar>x\<bar> \<le> 1 / 2"
rel_witness_resource_def: fixes A :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_witness_resource A A' B = corec_resource (\<lambda>(res1, res2). map_spmf (map_prod id Inr \<circ> rel_witness_prod) \<circ> rel_witness_spmf (rel_prod B (rel_resource (A OO A') B)) \<circ> rel_witness_fun A A' (run_resource res1, run_resource res2))"
rep_eq: fixes x :: "('b, 'a) aezfun" shows "aezfun (- x) = - aezfun x"
unfoldr_append_generator_Inr: fixes g1 :: "('a, 'b) generator"   and g2 :: "('a, 'c) generator"   and s2 :: "'c" shows "list.unfoldr (append_generator g1 g2) (Inr s2) = list.unfoldr g2 s2"
monoI: fixes f :: "'a \<Rightarrow> 'c" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "mono f"
ide_in_hom: fixes a :: "'a" shows "C.ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
tuple_simps_2: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "local.dom (ECC.tuple f g) = local.dom f"
nat_term_compare_inv_conv: fixes to :: "'a nat_term_order" shows "nat_term_compare_inv to = key_compare (key_order_of_nat_term_order_inv to)"
assms_2: shows "eval c t t' n (\<circle>\<^sub>b\<gamma>)"
bind_gpv_sel: fixes r :: "('g, 'e, 'f) gpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "the_gpv (r \<bind> f) = the_gpv r \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> the_gpv (f x) | IO out c \<Rightarrow> return_spmf (IO out (\<lambda>input. c input \<bind> f)))"
contr: shows "continuous_on (W \<union> (S \<union> T)) r"
hom_one: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "group_hom G H h" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
right_neutral: fixes a :: "('a \<times> 'a) set" shows "a O Id = a"
eq_alive: shows "\<forall>x. alive x s1 = alive x s2"
assms: shows "\<forall>i\<le>n. 0 \<le> f i"
vec_plus: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "dims A = dims B" shows "vec (A + B) = vec_plus (vec A) (vec B)"
aux10: fixes a :: "'b"   and p :: "('b, 'c) Combinators list" assumes "a \<in> set (net_list p)" shows "a \<in> set (net_list_aux p)"
acc_prsv: shows "\<forall>ppred p q f x. okRead1 f x ppred \<and> np - maxfaults \<le> count ppred np \<and> ppred p \<and> ppred q \<longrightarrow> \<bar>cfn p f - f q\<bar> \<le> \<alpha> x"
leadsTo_weaken_L: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set" assumes "F \<in> A \<longmapsto> A'"   and "B \<subseteq> A" shows "F \<in> B \<longmapsto> A'"
finite_imp_bounded_convex_hull: fixes s :: "'a set" assumes "finite s" shows "bounded (convex hull s)"
trans: fixes a :: "('c, 'd) Term.term"   and b :: "('c, 'd) Term.term"   and c :: "('c, 'd) Term.term" assumes "a \<sqsubseteq> b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
Collect_empty_eq_bot: fixes P :: "'a \<Rightarrow> bool" shows "(Collect P = {}) = (P = bot_class.bot)"
B: shows "F (lim_proc (range Y) [+] S) \<subseteq> F (lim_proc (range (\<lambda>i. Y i [+] S)))"
resr_comp1: fixes y :: "'a"   and x :: "'a" shows "y \<cdot> inv_resr y (y \<cdot> x) = y \<cdot> x"
ord_strict_p_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<prec>\<^sub>p q"   and "keys r \<inter> keys q = {}" shows "p + r \<prec>\<^sub>p q + r"
simps_13: fixes f1 :: "'a"   and f2 :: "BitList \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "BitList \<Rightarrow> 'a \<Rightarrow> 'a"   and x2 :: "BitList" shows "rec_BitList f1 f2 f3 (Bit0 x2) = f2 x2 (rec_BitList f1 f2 f3 x2)"
delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_pointer_removed: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = Some h'" shows "ptr |\<notin>| object_ptr_kinds h'"
gen_common_subdivision_imp_eq_line_integral_1: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "common_sudiv_exists one_chain1 one_chain2"   and "boundary_chain one_chain1"   and "boundary_chain one_chain2"   and "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"   and "finite one_chain1"   and "finite one_chain2"   and "finite basis" shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
braun_insert: fixes t :: "'b tree"   and x :: "'b" assumes "braun t" shows "braun (insert x t)"
AuthResS: fixes evsAResS :: "event list"   and C :: "agent"   and k :: "nat"   and KP :: "nat"   and KC2 :: "nat"   and KM :: "nat"   and P_I :: "msg"   and PIData :: "msg"   and HOIData :: "msg"   and P :: "agent"   and PIHead :: "msg"   and PANData :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and HOD :: "msg"   and PurchAmt :: "nat"   and M :: "agent" assumes "evsAResS \<in> set_pur"   and "C = Cardholder k"   and "Key KP \<notin> used evsAResS"   and "KP \<in> symKeys"   and "CardSecret k \<noteq> 0"   and "KC2 \<in> symKeys"   and "KM \<in> symKeys"   and "P_I = \<lbrace>sign (priSK C) \<lbrace>Hash PIData, HOIData\<rbrace>, EXcrypt KC2 (pubEK P) \<lbrace>PIHead, HOIData\<rbrace> PANData\<rbrace>"   and "PANData = \<lbrace>Pan (pan C), Nonce (PANSecret k)\<rbrace>"   and "PIData = \<lbrace>PIHead, PANData\<rbrace>"   and "PIHead = \<lbrace>Number LID_M, Number XID, HOD, Number PurchAmt, Agent M, Hash \<lbrace>Number XID, Nonce (CardSecret k)\<rbrace>\<rbrace>"   and "Gets P (EncB (priSK M) KM (pubEK P) \<lbrace>Number LID_M, Number XID, HOIData, HOD\<rbrace> P_I) \<in> set evsAResS" shows "Says P M (EncB (priSK P) KP (pubEK M) \<lbrace>Number LID_M, Number XID, Number PurchAmt\<rbrace> authCode) # evsAResS \<in> set_pur"
mark_object_def: fixes p :: "'a process_name"   and p_ph_enabled :: "('a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state) \<Rightarrow> bool" shows "mark_object p p_ph_enabled \<equiv> eq_imp (\<lambda>_ s. s p) p_ph_enabled"
hyps_2: shows "\<not> (prime x \<longrightarrow> odd x \<longrightarrow> (\<forall>xa xb. coprime xa xb \<longrightarrow> x dvd xa\<^sup>2 + 3 * xb\<^sup>2 \<longrightarrow> is_qfN x 3))"
compser_nsg: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and i :: "nat" assumes "Group G"   and "0 < n"   and "compseries G n f"   and "i \<le> n - 1" shows "\<natural>\<^bsub>G\<^esub>f i \<triangleright> f (Suc i)"
t_list_of_A_induct: fixes t :: "'b tree"   and n :: "nat" assumes "height t \<le> n" shows "t_list_of_A t \<le> 2 * n * size t"
normal_invE_2: fixes N :: "'a set"   and x :: "'a"   and h :: "'a" assumes "N \<lhd> G"   and "x \<in> carrier G"   and "h \<in> N" shows "x \<otimes> h \<otimes> inv x \<in> N"
poly_eq_poly_eq_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(poly p = poly q) = (p = q)"
OF_unsafe_safe_match_linear_eq: fixes oft :: "(of_match_field, 'b) flow_entry_match list" assumes "list_all (all_prerequisites \<circ> ofe_fields) oft" shows "OF_match_linear OF_match_fields_unsafe oft = OF_match_linear OF_match_fields_safe oft"
Diff1_foldSetD: fixes A :: "'b set"   and x :: "'b"   and y :: "'a"   and e :: "'a" assumes "(A\<setminus>{x}, y) \<in> foldSetD D (\<cdot>) e"   and "x \<in> A"   and "A \<subseteq> B" shows "(A, x \<cdot> y) \<in> foldSetD D (\<cdot>) e"
SeqReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e;; e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>e';; e\<^sub>2,s',b'\<rangle>"