hcju/isabelleps huggingface dataset:
WTrtBinOp: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e1 :: "'addr expr"   and T1 :: "ty"   and e2 :: "'addr expr"   and T2 :: "ty"   and bop :: "bop"   and T :: "ty" assumes "P,E,h \<turnstile> e1 : T1"   and "P,E,h \<turnstile> e2 : T2"   and "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T" shows "P,E,h \<turnstile> e1 \<guillemotleft>bop\<guillemotright> e2 : T"
simps_1: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b"   and y21a :: "'a"   and y22a :: "'a"   and y23a :: "'b" shows "(\<Phi>.Ya.Cop_S.MkArr x21 x22 x23 = \<Phi>.Ya.Cop_S.MkArr y21a y22a y23a) = (x21 = y21a \<and> x22 = y22a \<and> x23 = y23a)"
complementTA_correct_1: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "complete_tree_automaton TA A" shows "ta_lang (complete_tree_automaton.complementTA TA) = ranked_trees A - ta_lang TA"
H_1: shows "f \<in> space\<^sub>N N1"
rcos_disjoint: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "pairwise disjnt (rcosets\<^bsub>G\<^esub> H)"
nz: fixes x :: "'a" assumes "x \<in> set xs" shows "x \<noteq> (0::'a)"
Rats_no_top_le: fixes x :: "real" shows "\<exists>q\<in>\<rat>. x \<le> q"
map_is_disj: fixes f :: "'c formula"   and F :: "'c \<Rightarrow> 'd" assumes "is_disj f" shows "is_disj (map_formula F f)"
that: fixes p1 :: "(nat \<times> nat) list"   and u :: "nat"   and w :: "nat"   and p2 :: "(nat \<times> nat) list" assumes "p = p1 @ (u, v) # (v, w) # p2"   and "incoming v \<inter> set p = {(u, v)}"   and "outgoing v \<inter> set p = {(v, w)}" shows "thesis"
absorb_iff1: fixes a :: "'a upred"   and b :: "'a upred" shows "(b \<sqsubseteq> a) = ((a \<and> b) = a)"
outer_invar_loc_def: fixes G :: "('a, 'b) graph_rec_scheme"   and it :: "'a set"   and D :: "'a set" shows "outer_invar_loc G it D \<equiv> fr_graph G \<and> outer_invar_loc_axioms G it D"
is_process9: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set" assumes "is_process P" shows "\<forall>s. s @ [tick] \<notin> DIVERGENCES P \<or> s \<in> DIVERGENCES P"
uncountable_ball: fixes r :: "real"   and a :: "'a" assumes "0 < r" shows "uncountable (ball a r)"
suffix_suffix: fixes m :: "nat"   and k :: "nat"   and x :: "nat \<Rightarrow> 'c" shows "suffix m (suffix k x) = suffix (k + m) x"
join_hlp1: fixes a :: "bdd"   and s :: "bddi"   and b :: "bdd" shows "is_bdd_impl a s * is_bdd_impl b s \<Longrightarrow>\<^sub>A is_bdd_impl a s * is_bdd_impl b s * \<up> (a = b)"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = vec n (\<lambda>j. if j = i then 1::'c else (0::'c))"
takeWhile_neq_rev: fixes xs :: "'g list"   and x :: "'g" assumes "distinct xs"   and "x \<in> set xs" shows "takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
Red1While: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and b :: "'addr expr1"   and c :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>while (b) c,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>if (b) (c;; while (b) c) else unit,s\<rangle>"
t_stop_def: fixes v :: "real"   and a :: "real" shows "t_stop v a \<equiv> - v / a"
resr1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<rightarrow> y) \<cdot> z \<le> x \<rightarrow> y \<cdot> z"
canon_ang_id: fixes \<alpha> :: "real" assumes "- pi < \<alpha> \<and> \<alpha> \<le> pi" shows "\<downharpoonright>\<alpha>\<downharpoonleft> = \<alpha>"
sequence_to_failure_ob: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM" assumes "\<not> M1 \<preceq> M2"   and "well_formed M1"   and "well_formed M2" obtains io :: "('d \<times> 'e) list" where "sequence_to_failure M1 M2 io"
a: shows "abc_final (abc_steps_l (0, args) A stp) A"
map_add_restrict_nonmember_right: fixes x :: "'c"   and m' :: "'c \<Rightarrow> 'd option"   and m :: "'c \<Rightarrow> 'd option" assumes "x \<notin> dom m'" shows "(m ++ m') |` {x} = m |` {x}"
isStaticLoc_simps_1: fixes cf :: "CAttId"   and a :: "nat" shows "isStaticLoc (objLoc cf a) = False"
dom_eq_SomeD2: fixes m :: "'b \<Rightarrow> 'c option"   and n :: "'b \<Rightarrow> 'd option"   and x :: "'b"   and y :: "'d" assumes "dom m = dom n"   and "n x = Some y" shows "m x \<noteq> None"
step: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and E'' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'"   and "UNIF ss E' E''" shows "UNIF (\<sigma> # ss) E E''"
bind_advantage_def: fixes \<eta> :: "nat"   and \<A> :: "'grp \<Rightarrow> ('grp \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "Schnorr_\<Sigma>_commit.abstract_com.bind_advantage \<eta> \<A> \<equiv> spmf (Schnorr_\<Sigma>_commit.abstract_com.bind_game \<eta> \<A>) True"
\<section>: fixes y :: "'a"   and u :: "real" assumes "y \<in> S"   and "0 \<le> u \<and> u \<le> 1" shows "(1 - u) *\<^sub>R a + u *\<^sub>R y \<in> S"
numneg_def: fixes t :: "Ferrack.num" shows "numneg t = nummul t (- 1)"
front_tickFree_single: fixes a :: "'c event" shows "front_tickFree [a]"
reflT: shows "rel_reflects_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
path_connected_eq_homotopic_points: fixes S :: "'a set" shows "path_connected S = (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
lbl: shows "(lb, l) \<in> set (identify_blocks A k)"
simps: fixes x :: "'b extC"   and y :: "'b extC" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
bij: shows "bij_betw (ind_in_set A) ({0..<M} \<inter> A) {0..<card ({0..<M} \<inter> A)}"
assms_4: shows "t \<noteq> \<^bold>\<I>"
cmult_1: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "1 \<otimes> \<kappa> = \<kappa>"
lhopital_right_at_top_at_top: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and g :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "filterlim f at_top (at_right a)"   and "filterlim g at_top (at_right a)"   and "\<forall>\<^sub>F x in at_right a. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_right a. (g has_real_derivative g' x) (at x)"   and "LIM x at_right a. f' x / g' x :> at_top" shows "LIM x at_right a. f x / g x :> at_top"
C1: fixes \<phi> :: "'c"   and P :: "('a, 'b, 'c) psi" assumes "(\<phi>, P) mem CsP" shows "\<exists>Q. (\<phi>, Q) mem CsQ \<and> guarded Q \<and> P \<doteq>\<^sub>c Q"
Rep_lam_inject: fixes x :: "lam"   and y :: "lam" shows "(Rep_lam x = Rep_lam y) = (x = y)"
P4: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[F]_v"
Step2: fixes g1 :: "'a"   and g2 :: "'a" assumes "g1 \<in> carrier G"   and "g2 \<in> carrier G" shows "(\<lambda>h\<in>carrier G. g1 \<otimes> g2 \<otimes> h \<otimes> inv (g1 \<otimes> g2)) = (\<lambda>h\<in>carrier G. g1 \<otimes> h \<otimes> inv g1) \<otimes>\<^bsub>BijGroup (carrier G)\<^esub> (\<lambda>h\<in>carrier G. g2 \<otimes> h \<otimes> inv g2)"
functor_implies_equals_dom: assumes "functor (\<cdot>\<^sub>J) (\<cdot>) \<alpha>.map" shows "\<alpha>.F.map = \<alpha>.map"
Quotient_rep_abs_fold_unmap: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x' :: "'b"   and x :: "'a"   and Rep' :: "'b \<Rightarrow> 'a" assumes "Quotient R Abs Rep T"   and "x' \<equiv> Abs x"   and "R x x"   and "Rep x' \<equiv> Rep' x'" shows "R (Rep' x') x"
key_ring_morphism: fixes V :: "'a set set" assumes "is_zariski_open V"   and "\<pp> \<in> V" shows "\<exists>\<phi>. ring_homomorphism \<phi> st.carrier_stalk st.add_stalk st.mult_stalk (st.zero_stalk V) (st.one_stalk V) pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at \<and> (\<forall>U\<in>top.neighborhoods \<pp>. \<forall>s\<in>\<O> U. (\<phi> \<circ> st.canonical_fun U) s = local.key_map U s)"
right_inverse: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and inverse :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "group f z inverse" shows "f a (inverse a) = z"
inj_map_strong: fixes x :: "'a list"   and xa :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and fa :: "'a \<Rightarrow> 'b" assumes "\<And>z za. \<lbrakk>z \<in> set x; za \<in> set xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map f x = map fa xa" shows "x = xa"
eq: shows "set (ab gs bs hs data') = set bs \<union> set hs"
lprefix_imp_llexord: fixes xs :: "'c llist"   and ys :: "'c llist"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "xs \<sqsubseteq> ys" shows "llexord r xs ys"
encode_problem_parallel_complete_i: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "is_valid_problem_strips \<Pi>"   and "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi>"   and "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True) \<and> (\<not> \<A> (State k (index (\<Pi>\<^sub>\<V>) v))) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v \<noteq> Some True)" shows "\<A> \<Turnstile> \<Phi>\<^sub>I \<Pi>"
R39_a: shows "pmf (sds R39) a = 1 / 2"
Iagree_def: fixes I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp"   and V :: "('d + 'e + 'f) set" shows "Iagree I J V \<equiv> \<forall>i\<in>V. (\<forall>x. i = Inl x \<longrightarrow> Functions I x = Functions J x) \<and> (\<forall>x. i = Inr (Inl x) \<longrightarrow> Contexts I x = Contexts J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> Predicates I x = Predicates J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> Programs I x = Programs J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> ODEs I x = ODEs J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> ODEBV I x = ODEBV J x)"
af_G_letter_abs_def: shows "\<up>af\<^sub>G \<equiv> af_G_abs.f_abs"
is_pure_conv: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> assn" shows "is_pure R = (\<exists>R'. R = pure R')"
ext_inject: fixes nothingb :: "unit"   and mode :: "shadow_root_mode"   and child_nodes :: "(_) node_ptr list"   and more :: "'p"   and nothingb' :: "unit"   and mode' :: "shadow_root_mode"   and child_nodes' :: "(_) node_ptr list"   and more' :: "'p" shows "(\<lparr>RShadowRoot.nothing = nothingb, mode = mode, child_nodes = child_nodes, \<dots> = more\<rparr> = \<lparr>RShadowRoot.nothing = nothingb', mode = mode', child_nodes = child_nodes', \<dots> = more'\<rparr>) = (nothingb = nothingb' \<and> mode = mode' \<and> child_nodes = child_nodes' \<and> more = more')"
split_stream_first': fixes x :: "'b"   and xs :: "'b stream" assumes "x \<in> sset xs" obtains ys :: "'b list" and zs :: "'b stream" where "xs = ys @- x ## zs" and "x \<notin> set ys"
gf_U'_def: fixes x :: "'s"   and y :: "'s"   and z :: "real" shows "gf_U' x y z = (\<Sum>n. u x y n * real (Suc n) * z ^ n)"
glb_common: fixes s1 :: "'a"   and s :: "'a"   and rs1 :: "('b \<times> 'c) list"   and s2 :: "'a"   and rs2 :: "('b \<times> 'c) list" assumes "s1 = s \<star> rs1"   and "s2 = s \<star> rs2" shows "\<exists>rs. s1 \<sqinter> s2 = s \<star> rs \<and> set rs \<subseteq> set rs1 \<union> set rs2"
card_set_mset: fixes M :: "'b multiset"   and A :: "'b set" assumes "M \<subseteq># mset_set A" shows "card (set_mset M) = size M"
simps_1: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "pairs (x # y # xs) = (x, y) # pairs (y # xs)"
divides_zero: fixes a :: "'a list" assumes "a \<in> carrier (K [X])" shows "a divides\<^bsub>K [X]\<^esub> \<zero>\<^bsub>K [X]\<^esub>"
atU_union_cases: fixes U :: "'a set"   and c1 :: "'a list multiset"   and c2 :: "'a list multiset" assumes "atU U (c1 + c2)" obtains "atU U c1"   | "atU U c2"
Ord_ordertype: fixes A :: "V set" shows "Ord (ordertype A VWF)"
that: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "homeomorphism (cbox a b) (cbox c d) f g"   and "f a = c"   and "f b = d" shows "thesis"
dag_less_le: fixes x :: "dag"   and y :: "dag" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
frontier_subset_closed: fixes S :: "'b set" assumes "closed S" shows "frontier S \<subseteq> S"
intros_5: fixes X :: "cmsg" assumes "X \<in> cpayload" shows "cHash X \<in> cpayload"
less_imp_neq: fixes x :: "'c list"   and y :: "'c list" assumes "strict_prefix x y" shows "x \<noteq> y"
run_fail_nondet: shows "run_nondet fail_nondet = return empty"
identity_\<kappa>_def: shows "(\<^bold>=) \<equiv> (\<^bold>=\<^sub>\<kappa>)"
completeE: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and s :: "'c set"   and f :: "nat \<Rightarrow> 'c" assumes "class.metric_space dist uniformity open"   and "uniform_space.complete open uniformity s"   and "\<forall>n. f n \<in> s"   and "uniform_space.Cauchy uniformity f" obtains l :: "'c" where "l \<in> s" and "filterlim f (topological_space.nhds open l) sequentially"
canceling_neighbor: fixes i :: "nat"   and l :: "(bool \<times> 'c) list"   and a :: "(bool \<times> 'c) list"   and b :: "(bool \<times> 'c) list" assumes "cancels_to_1_at i l a"   and "cancels_to_1_at (Suc i) l b" shows "a = b"
pochhammer_minus: fixes b :: "'a"   and k :: "nat" shows "pochhammer (- b) k = (- (1::'a)) ^ k * pochhammer (b - of_nat k + (1::'a)) k"
reflects_left_adjoint: fixes f :: "'a" assumes "ide f"   and "S.is_left_adjoint (S.UP f)" shows "is_left_adjoint f"
a_2: shows "b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>"
pli_lemma: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "extend h F \<squnion> G \<in> stable C"   and "F \<squnion> project h C G \<in> project_set h C \<inter> project_set h A \<longmapsto> project_set h B" shows "F \<squnion> project h C G \<in> project_set h C \<inter> project_set h A \<longmapsto> project_set h C \<inter> project_set h B"
f_Exec_Stream_expand_map_aggregate_snoc: fixes k :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and x :: "'f message_af"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun ((xs @ [x]) \<odot> k) c)) k ag = f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c)) k ag @ [ag (map f (f_Exec_Comp_Stream trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (xs \<odot> k) c)))]"
cl3: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set"   and T :: "('a \<times> 'a set) set" shows "R \<cdot> (S \<parallel> T) \<subseteq> R \<cdot> S \<parallel> (R \<cdot> T)"
assms_4: shows "edom ae \<inter> domA \<Delta> = {}"
ap_specD1: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "ap_spec ap" shows "set (ap gs bs ps hs data) \<subseteq> set ps \<union> set hs \<times> (set gs \<union> set bs \<union> set hs)"
no_spoofing_def: fixes ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and rs :: "'c common_primitive rule list" shows "no_spoofing TYPE('b) ipassmt rs \<equiv> \<forall>iface\<in>dom ipassmt. \<forall>p. (common_matcher, in_doubt_allow),p\<lparr>p_iiface := iface_sel iface\<rparr>\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<longrightarrow> p_src p \<in> ipcidr_union_set (set (the (ipassmt iface)))"
induct: fixes P :: "Formula.trm list \<Rightarrow> event_data list \<Rightarrow> bool"   and a0 :: "Formula.trm list"   and a1 :: "event_data list" assumes "P [] []"   and "\<And>x ts y ys. (x = y \<Longrightarrow> P ts ys) \<Longrightarrow> P (trm.Const x # ts) (y # ys)"   and "\<And>x ts y ys. P ts ys \<Longrightarrow> P (trm.Var x # ts) (y # ys)"   and "\<And>vb va. P (trm.Var vb # va) []"   and "\<And>vb vc va uv_. P (trm.Plus vb vc # va) uv_"   and "\<And>vb vc va uv_. P (trm.Minus vb vc # va) uv_"   and "\<And>vb va uv_. P (trm.UMinus vb # va) uv_"   and "\<And>vb vc va uv_. P (trm.Mult vb vc # va) uv_"   and "\<And>vb vc va uv_. P (trm.Div vb vc # va) uv_"   and "\<And>vb vc va uv_. P (trm.Mod vb vc # va) uv_"   and "\<And>vb va uv_. P (trm.F2i vb # va) uv_"   and "\<And>vb va uv_. P (trm.I2f vb # va) uv_"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)"   and "\<And>vd ve vc v va. P (trm.Plus vd ve # vc) (v # va)"   and "\<And>vd ve vc v va. P (trm.Minus vd ve # vc) (v # va)"   and "\<And>vd vc v va. P (trm.UMinus vd # vc) (v # va)"   and "\<And>vd ve vc v va. P (trm.Mult vd ve # vc) (v # va)"   and "\<And>vd ve vc v va. P (trm.Div vd ve # vc) (v # va)"   and "\<And>vd ve vc v va. P (trm.Mod vd ve # vc) (v # va)"   and "\<And>vd vc v va. P (trm.F2i vd # vc) (v # va)"   and "\<And>vd vc v va. P (trm.I2f vd # vc) (v # va)" shows "P a0 a1"
V_subseteq_R1: fixes i :: "nat" shows "V_bclim i \<subseteq> \<R>"
of_nat_1: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 1 = one"
a0: shows "\<one> \<odot>\<^bsub>M\<^esub> v = v"
rep_eq: fixes ode_ops :: "ode_ops"   and x :: "'e" shows "blinfun_apply (ode_operations.ode_d2 ode_ops x) = (if x \<in> ode_operations.Csafe ode_ops then ode_operations.ode_d ode_ops 1 x else (\<lambda>_. 0))"
AbsS: fixes v :: "i"   and x :: "\<kappa>" shows "[\<lparr>A!,x\<rparr> in v] = (\<exists>o\<^sub>1 y. Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<alpha>\<nu> y)"
map_simp_1: shows "local.map J.Zero = C.dom f0"
f_join_Nil_conv: fixes xs :: "'b list"   and I :: "nat set" shows "(xs \<Join>\<^sub> I = []) = (I \<down>< length xs = {})"
ir_valid_Seq1': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c1 :: "com"   and c2 :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P (c1;; c2) c' Q" shows "ir_valid (\<lambda>t t'. \<exists>s s'. P s s' \<and> (c1, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t' \<and> (\<exists>u. (c2, t) \<Rightarrow> u \<and> Q u t')) c2 SKIP Q"
cfcomp2: fixes f :: "'t \<rightarrow> 'r"   and g :: "'s \<rightarrow> 't"   and x :: "'s" shows "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
assms_1: shows "S = {(=), (<), (\<le>), (\<noteq>)}"
hyp_UnOp: fixes e' :: "expr"   and unop :: "unop" assumes "P e'" shows "P (UnOp unop e')"
foldr_conv_fold: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list" shows "foldr f xs = fold f (rev xs)"
WhileHighCompositionality: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and b :: "expr"   and c' :: "com" assumes "\<Gamma>,High \<turnstile> while (b) c'" shows "nonInterference \<Gamma> (while (b) c')"
absolutely_integrable_component: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "f absolutely_integrable_on A" shows "(\<lambda>x. f x \<bullet> b) absolutely_integrable_on A"
strict_order_iff: fixes b :: "'t"   and a :: "'t" shows "(b \<prec>\<^sub>t a) = (a = ord_term_lin.max a b \<and> a \<noteq> b)"
fps_divide_times_eq: fixes g :: "'a fps"   and f :: "'a fps" assumes "g \<noteq> 0" shows "f * g / g = f"
F_unfold: fixes i :: "name" shows "F i = (if i \<in> V then Var (p \<bullet> i) else Var i)"
rel_of_enex_def: fixes enex :: "('b \<Rightarrow> 'e set) \<times> ('e \<Rightarrow> 'b \<Rightarrow> 'd)" shows "rel_of_enex enex \<equiv> let (en, ex) = enex in {(s, ex a s) |s a. a \<in> en s}"
nonempty_size_mset: fixes M :: "('d \<times> 'e) multiset" assumes "M \<noteq> {#}" shows "0 < size_mset M"
induct: fixes P :: "'form \<times> nat list \<Rightarrow> bool"   and a0 :: "'form \<times> nat list" assumes "\<And>pf. P (pf, [])"   and "\<And>pf i is. P (pf, i # is)" shows "P a0"
best_worst_indiff_all_indiff_2: fixes b :: "'a pmf"   and w :: "'a pmf" assumes "b \<in> local.best"   and "w \<in> local.worst"   and "b \<approx>[\<R>] w" shows "\<forall>e\<in>local.\<P>. e \<approx>[\<R>] b"
FreshExt: fixes x :: "name"   and S :: "pi"   and T :: "pi"   and U :: "pi" assumes "x \<sharp> S" shows "(<\<nu>x>((S \<parallel> T) \<parallel> U), S \<parallel> <\<nu>x>(T \<parallel> U)) \<in> Rel"
to_nat_split: fixes x :: "'b"   and y :: "'b" shows "(to_nat x = to_nat y) = (x = y)"
parPres: fixes P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "P \<approx> Q" shows "P \<parallel> R \<approx> Q \<parallel> R"
preservation: shows "rel_weakly_preserves_barbs (indRelRSTPO SRel TRel) (STCalWB SWB TWB)"
mono_bal_d: fixes n :: "nat"   and dl :: "nat"   and n' :: "nat" assumes "bal_d n dl"   and "n \<le> n'" shows "bal_d n' dl"
impl: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G"   and "is_canon_rewriter rword G v g" shows "is_sig_red (\<prec>\<^sub>t) (=) G (monom_mult (1::'b) (pp_of_term v - lp g) g)"
right: fixes i :: "'a" assumes "i \<in> I"   and "a \<le> i"   and "i < b" shows "(f \<longlongrightarrow> u i) (at_right i)"
combine_rabin_pairs: fixes I :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and f :: "'b \<Rightarrow> ('c \<times> 'd \<times> 'c) set"   and g :: "'b \<Rightarrow> ('c \<times> 'd \<times> 'c) set"   and w :: "nat \<Rightarrow> 'd" assumes "\<And>x. x \<in> I \<Longrightarrow> accepting_pair\<^sub>R \<delta> q\<^sub>0 (f x, g x) w" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (\<Union> {f x |x. x \<in> I}, {g x |x. x \<in> I}) w"
assms_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z"
comm_semiring_ow_axioms_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "comm_semiring_ow_axioms U plus times \<equiv> \<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times (plus a b) c = plus (times a c) (times b c)"
conv_abscissa_add_leI: fixes f :: "'a fds"   and d :: "ereal"   and g :: "'a fds" assumes "conv_abscissa f \<le> d"   and "conv_abscissa g \<le> d" shows "conv_abscissa (f + g) \<le> d"
irreducible_dvd_prod_mset: fixes p :: "'a poly"   and as :: "'a poly multiset" assumes "irreducible p"   and "p dvd \<Prod>\<^sub># as" shows "\<exists>a\<in>#as. p dvd a"
card_map_elide: fixes n :: "nat" assumes "n \<le> CARD('b word)" shows "card (word_of_nat ` {0..<n}) = card {0..<n}"
star_sum_var: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> + y\<^sup>\<star>)\<^sup>\<star>"
AE_count_space: fixes P :: "'c \<Rightarrow> bool"   and A :: "'c set" shows "almost_everywhere (count_space A) P = (\<forall>x\<in>A. P x)"