hcju/isabelleps huggingface dataset:
assoc_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.arr \<a>\<^sub>C[f, g, h]"
simps: fixes G :: "'b list set"   and u :: "'b list" shows "Dec G u = (SOME us. us \<in> lists G\<^sub>+ \<and> concat us = u)"
refl_trans_closure_of_indRelST: fixes TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "refl TRel" shows "indRelSTEQ SRel TRel = (indRelST SRel TRel)\<^sup>*"
wf_rcv_append': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t V (S @ S')"   and "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S')"
analytic: shows "f' analytic_on {s. 1 \<le> Re s}"
finite_bounded_functions: fixes S :: "nat set"   and k :: "nat" assumes "finite S" shows "finite {f. (\<forall>i\<in>{1..k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1..k} \<longrightarrow> f i = i)}"
wordinterval_Union: fixes ws :: "'a wordinterval list" shows "wordinterval_to_set (wordinterval_Union ws) = \<Union> (wordinterval_to_set ` set ws)"
create_graph\<^sub>T_def: fixes \<phi> :: "'b ltlr" shows "create_graph\<^sub>T \<phi> \<equiv> expand\<^sub>T (\<lparr>name = expand_new_name expand_init, incoming = {expand_init}, new = {\<phi>}, old = {}, next = {}\<rparr>, {}) \<bind> (\<lambda>(uu_, nds). RETURN nds)"
cltn2_abs_def: fixes B :: "((real, 3) vec, 3) vec" shows "cltn2_abs B \<equiv> Abs_cltn2 (invertible_proportionality `` {B})"
le_trans: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "order.greater_eq j i"   and "order.greater_eq k j" shows "order.greater_eq k i"
Vfrom_rank_eq: fixes A :: "V"   and x :: "V" shows "Vfrom A (rank x) = Vfrom A x"
build_lin_independent_seq: fixes us :: "'b list"   and n :: "nat" assumes "set us \<subseteq> V"   and "R_lin_independent us" shows "\<exists>ws. set ws \<subseteq> V \<and> R_lin_independent (ws @ us) \<and> (RSpan (ws @ us) = V \<or> length ws = n)"
subgraph_refl: fixes G :: "nat set \<times> nat set set" shows "subgraph G G"
weight_spmf_le_1: fixes p :: "'b spmf" shows "weight_spmf p \<le> 1"
path_entry0: fixes E :: "(nat \<times> nat) set" shows "path_entry E [] 0"
fib_mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "fib m \<le> fib n"
nat_cut_greater_infinite_not_empty: fixes I :: "nat set"   and t :: "nat" assumes "infinite I" shows "I \<down>> t \<noteq> {}"
borel_measurable_implies_simple_function_sequence: fixes u :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "u \<in> borel_measurable M" shows "\<exists>f. incseq f \<and> (\<forall>i. (\<forall>x. f i x < top) \<and> simple_function M (f i)) \<and> u = Sup (range f)"
simps_1: fixes r :: "rat" shows "sgn_2 (Rational r) = sgn r"
inject_5: fixes x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com"   and y61 :: "'d set"   and y62 :: "('d, 'e, 'f) com" shows "(While x61 x62 = While y61 y62) = (x61 = y61 \<and> x62 = y62)"
one_neq_neg_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.ring_char_0 one times plus zero minus uminus" shows "one \<noteq> uminus one"
fold_A_opp_fold_A_chamber_systems_2: shows "ChamberComplex.chamber_system (fold_A \<turnstile> A) \<inter> ChamberComplex.chamber_system (opp_fold_A \<turnstile> A) = {}"
uminus_p32_def: fixes p :: "uint32"   and x :: "uint32" shows "uminus_p32 p x = (if x = 0 then 0 else p - x)"
dualization_rew_4: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<not> \<diamond>\<langle>A\<rangle>_v \<equiv> \<box>[\<not> A]_v"
inject: fixes x1 :: "'c match_expr"   and x2 :: "action"   and y1 :: "'c match_expr"   and y2 :: "action" shows "(Rule x1 x2 = Rule y1 y2) = (x1 = y1 \<and> x2 = y2)"
DBM_reset_sound: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and c :: "'c"   and u :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "v c \<le> n"   and "u \<in> [reset M n (v c) d]\<^bsub>v,n\<^esub>" obtains d' :: "'d" where "u(c := d') \<in> [M]\<^bsub>v,n\<^esub>"
assms_2: shows "constructive_security_aux_obsf real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 adv2"
sel_6: fixes x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm" shows "arg (App x21 x22) = x22"
SLN_left: fixes P :: "assn" shows "SLN * P = P"
intro: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c" assumes "finite \<Sigma>"   and "finite (reach \<Sigma> \<delta> q\<^sub>0)"   and "range w \<subseteq> \<Sigma>" shows "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"
struct_spec_def: fixes sel :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" shows "punit.struct_spec sel ap ab compl = (sel_spec sel \<and> punit.ap_spec ap \<and> ab_spec ab \<and> punit.compl_struct compl)"
decompose_Max: fixes A :: "'a set" assumes "A \<subseteq> Aa__ - M" shows "restrict_relation A (of_weak_ranking (weak_ranking_aux (Aa__ - M))) = (\<lambda>x y. x \<in> A \<and> y \<in> Max_wrt_among (of_weak_ranking (weak_ranking_aux (Aa__ - M))) A \<or> y \<notin> Max_wrt_among (of_weak_ranking (weak_ranking_aux (Aa__ - M))) A \<and> restrict_relation (A - Max_wrt_among (of_weak_ranking (weak_ranking_aux (Aa__ - M))) A) (of_weak_ranking (weak_ranking_aux (Aa__ - M))) x y)"
intro: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c, 'd) sa_rec_scheme"   and B :: "('b, 'c, 'e) sa_rec_scheme" assumes "\<And>a b. (a, b) \<in> R \<Longrightarrow> sa_L A a = sa_L B b" shows "lsimulation_axioms R A B"
bex_Un: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>A \<union> B. P x) = ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x))"
zero_less_divide_1_iff: fixes a :: "'d" shows "((0::'d) < (1::'d) / a) = ((0::'d) < a)"
leftmost_Derives1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and i :: "nat"   and a :: "symbol list"   and j :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftmost \<NN> \<TT> i a"   and "CFG.Derives1 \<NN> \<TT> \<RR> a j r b" shows "i \<le> j"
nextl_eps: fixes q :: "hf"   and Q :: "hf set"   and u :: "'a list"   and q' :: "hf" assumes "q \<in> ST.nextl Q u"   and "(q, q') \<in> eps \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>"   and "q' \<in> nfa.states \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>" shows "q' \<in> ST.nextl Q u"
NTCatDom_def: fixes \<eta> :: "('d, 'g, 'e, 'h, 'f, 'i, 'j) NatTrans_scheme" shows "NTCatDom \<eta> \<equiv> CatDom (NTDom \<eta>)"
indefinite_ivl_integral_continuous_2: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real" assumes "f integrable_on closed_segment a b" shows "continuous_on (closed_segment b a) (\<lambda>x. ivl_integral a x f)"
f: fixes x :: "'a" shows "norm (f x) \<le> norm x * Kf"
converse_tranclE: fixes x :: "'e"   and z :: "'e"   and r :: "('e \<times> 'e) set" assumes "(x, z) \<in> r\<^sup>+" obtains "(x, z) \<in> r"   | y :: "'e" where "(x, y) \<in> r" and "(y, z) \<in> r\<^sup>+"
jmpl_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "jmpl_instr instr \<equiv> let op_list = snd instr; rd = get_operand_w5 (op_list ! 3) in do curr_win \<leftarrow> get_curr_win (); jmp_addr \<leftarrow> gets (get_addr op_list); if jmp_addr AND 3 \<noteq> 0 then do raise_trap mem_address_not_aligned; return () od else do rd_next_val \<leftarrow> gets (\<lambda>s. if rd \<noteq> 0 then cpu_reg_val PC s else user_reg_val curr_win rd s); write_reg rd_next_val curr_win rd; npc_val \<leftarrow> gets (cpu_reg_val nPC); write_cpu npc_val PC; write_cpu jmp_addr nPC; return () od od"
cases: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'d + 'f"   and a2 :: "'e + 'g" assumes "rel_sum R1 R2 a1 a2" obtains a :: "'d" and c :: "'e" where "a1 = Inl a" and "a2 = Inl c" and "R1 a c"   | b :: "'f" and d :: "'g" where "a1 = Inr b" and "a2 = Inr d" and "R2 b d"
fresh_var_def: fixes e :: "exp" shows "fresh_var e = (SOME v. v \<notin> fv e)"
real_root_inverse: fixes n :: "nat"   and x :: "real" shows "root n (inverse x) = inverse (root n x)"
emeasure_eq_measure: fixes A :: "real set" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
in_fst_image: fixes a :: "'a"   and AB :: "('a \<times> 'b) set" shows "(a \<in> fst ` AB) = (\<exists>b. (a, b) \<in> AB)"
hyps_3: shows "u \<cdot> \<sigma> \<unrhd> t \<cdot> \<sigma>"
wlconf_ext_list: fixes G :: "prog"   and s :: "st"   and l :: "'a \<Rightarrow> val option"   and L :: "'a \<Rightarrow> ty option"   and vns :: "'a list"   and Ts :: "ty list"   and vs :: "val list" assumes "G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L"   and "distinct vns"   and "length Ts = length vns"   and "list_all2 (conf G s) vs Ts" shows "G,s\<turnstile>l(vns [\<mapsto>] vs)[\<sim>\<Colon>\<preceq>]L(vns [\<mapsto>] Ts)"
exec_move_AAssI1: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "'a J1_prog"   and t :: "'b"   and e :: "'a expr1"   and h :: "'c"   and s :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and e' :: "'a expr1"   and e'' :: "'a expr1" assumes "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t e h s ta h' s'" shows "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t (e\<lfloor>e'\<rceil> := e'') h s ta h' s'"
realizable_Append_ics_SDG_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n'' :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.realizable sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''"   and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns' n'" obtains ns'' :: "'b SDG_node list" where "SDG.realizable sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ ns'') n'"
nn_integral_monotone_convergence_SUP_nat: fixes f :: "'c \<Rightarrow> nat \<Rightarrow> ennreal"   and Y :: "'c set" assumes "Complete_Partial_Order.chain (\<le>) (f ` Y)"   and "Y \<noteq> {}" shows "\<integral>\<^sup>+ x. (SUP i\<in>Y. f i x) \<partial>count_space UNIV = (SUP i\<in>Y. integral\<^sup>N (count_space UNIV) (f i))"
assms: fixes v :: "'t" assumes "v \<in> keys r" shows "component_of_term v < j"
Ri_mono: fixes N :: "'a literal multiset set"   and N' :: "'a literal multiset set" assumes "N \<subseteq> N'" shows "sr.Ri N \<subseteq> sr.Ri N'"
distinct_39: shows "ToBooleanOp \<noteq> ToRealOp"
intro: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" assumes "strategyproof_sds agents alts sds"   and "an_sds agents alts sds" shows "strategyproof_an_sds agents alts sds"
face_of_trans: fixes S :: "'b set"   and T :: "'b set"   and u :: "'b set" assumes "S face_of T"   and "T face_of u" shows "S face_of u"
util_def: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "ordinal_utility carrier relation u"   and "x \<in> carrier"   and "y \<in> carrier" shows "x \<succeq>[relation] y = (u y \<le> u x)"
interval_integral_norm: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" assumes "interval_lebesgue_integrable lborel a b f"   and "a \<le> b" shows "norm (interval_lebesgue_integral lborel a b f) \<le> LBINT t=a..b. norm (f t)"
bipartite_web_of_sel_1: fixes \<Gamma> :: "('a, 'b) web_scheme"   and u :: "'a"   and v :: "'a" shows "edge (bipartite_web_of \<Gamma>) (Inl u) (Inr v) = (edge \<Gamma> u v \<or> u = v \<and> u \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub> \<and> u \<notin> A \<Gamma> \<and> v \<notin> B \<Gamma>)"
elims: fixes x :: "'b tag_const"   and y :: "'b tag_const" assumes "inverse x = y" obtains t :: "'b" where "x = \<tau>\<^sub>c\<^sub>s\<^sub>t t" and "y = \<tau>\<^sub>c\<^sub>s\<^sub>t (inverse t)"
inca_axioms: shows "inca F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"
shiftl1_2t: fixes w :: "'b word" shows "shiftl1 w = 2 * w"
simps_5: fixes c :: "int"   and e :: "Ferrack.num" shows "isrlfm (Lt (CN 0 c e)) = (0 < c \<and> numbound0 e)"
wf_genericJoin: fixes n :: "nat"   and V :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set" assumes "rwf_query n V Q Qn"   and "1 \<le> card V" shows "table n V (genericJoin V Q Qn)"
induct: fixes P1 :: "'a exp \<Rightarrow> bool"   and P2 :: "'a exp list \<Rightarrow> bool"   and exp_list :: "'a exp list" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3. P1 (x1\<bullet>\<^sub>sx2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2{x3} := x4)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2(x3))"   and "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (x1\<bullet>\<^sub>sx2(x3))"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (try x1 catch(x2 x3) x4)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (INIT x1 (x2,x3) \<leftarrow> x4)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x2; P1 x4\<rbrakk> \<Longrightarrow> P1 (RI (x1,x2) ; x3 \<leftarrow> x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 exp_list"
sup_Un_eq2: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set" shows "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
rel_bset_False: fixes x :: "'c set['e]"   and y :: "'d set['e]" shows "rel_bset (\<lambda>x y. False) x y = (x = bempty \<and> y = bempty)"
dghm_proj_components_4: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>HomCod\<rparr> = \<AA> i"
imageE: fixes b :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "s2p (\<P> f A) b" obtains x :: "'d" where "b = f x" and "s2p A x"
BSIA_implies_SIA_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<rho>' :: "'b V_rec \<Rightarrow> 'b set"   and \<V>' :: "'b V_rec"   and \<rho> :: "'b V_rec \<Rightarrow> 'b set" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "BSIA \<rho>' \<V>' Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>"   and "\<rho> \<V> = \<rho>' \<V>'" shows "SIA \<rho> \<V> Tr\<^bsub>ES\<^esub>"
fresh_delete: fixes a :: "atom"   and \<Gamma> :: "('b \<times> 'c) list"   and v :: "'b" assumes "a \<sharp> \<Gamma>" shows "a \<sharp> delete v \<Gamma>"
LNil_lprefix: fixes xs :: "'c llist" shows "LNil \<sqsubseteq> xs"
ring_iso_imp_img_domain: fixes h :: "'a \<Rightarrow> 'd"   and S :: "('d, 'e) ring_scheme" assumes "h \<in> ring_iso R S" shows "domain (S\<lparr>zero := h \<zero>\<rparr>)"
bv_length_extend: fixes w :: "bit list"   and i :: "nat"   and b :: "bit" assumes "length w \<le> i" shows "length (bv_extend i b w) = i"
red_inf_impl: fixes \<iota> :: "('f \<times> 'l) inference"   and NL :: "('f \<times> 'l) set" assumes "\<iota> \<in> Red_I NL" shows "to_F \<iota> \<in> no_labels.Red_I_\<G> (fst ` NL)"
e1: assumes "irreducible G a"   and "b \<in> Units G" shows "P"
intro: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "h \<in> hom G H" shows "group_hom_axioms G H h"
Cons_2: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
right_refines_AC: fixes b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "b \<sqsubseteq> (a \<Squnion> b)"
acyclic_impl_antisym_rtrancl: fixes r :: "('a \<times> 'a) set" assumes "acyclic r" shows "antisym (r\<^sup>*)"
in_PLS_leq_2_pow_n: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>x. x \<in> PLS s as \<and> x \<le> 2 ^ card (prob_dom PROB) - 1"
VSeq_implies_HPar: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "VSeq t u" shows "HPar t u"
assms_3: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R"   and "I \<cdot> J \<subseteq> P" shows "I \<subseteq> P \<or> J \<subseteq> P"
mem_limsup_iff: fixes x :: "'a"   and A :: "nat \<Rightarrow> 'a set" shows "(x \<in> limsup A) = (\<exists>\<^sub>F n in sequentially. x \<in> A n)"
explicationGoodness1: shows "\<forall>c x xa. \<lfloor>ActsOnPurpose x xa \<^bold>\<rightarrow> Good x xa\<rfloor>\<^sub>c"
under_zero: assumes "Field r \<noteq> {}" shows "local.under zero = {zero}"
hyps: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set" assumes "sharing_consistent \<S> \<O> sb"   and "a \<in> \<O>"   and "a \<notin> all_unshared sb" shows "a \<in> dom (share sb \<S>) \<union> acquired True sb \<O>"
span_span: fixes A :: "('b, 'c) vec set" shows "vec.span (vec.span A) = vec.span A"
longest_common_suffix: fixes xs :: "'b list"   and ys :: "'b list" shows "\<exists>ss xs' ys'. xs = xs' @ ss \<and> ys = ys' @ ss \<and> (xs' = [] \<or> ys' = [] \<or> last xs' \<noteq> last ys')"
induct: fixes P :: "'c trm \<Rightarrow> bool"   and trm :: "'c trm" assumes "\<And>x. P (Var x)"   and "\<And>x. P (trm.Const x)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (x1a \<cdot> x2a)" shows "P trm"
Decrypt: fixes K :: "nat"   and U :: "freemsg" shows "Decrypt K (Abs_Msg (msgrel `` {U})) = Abs_Msg (msgrel `` {DECRYPT K U})"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
propos_subset: fixes \<phi> :: "'b ltl" shows "propos \<phi> \<subseteq> nested_propos \<phi>"
bigo_const_mult6: fixes c :: "'d"   and f :: "'c \<Rightarrow> 'd" shows "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
msubstlt_nb: fixes a :: "poly"   and r :: "tm"   and t :: "tm"   and s :: "tm"   and c :: "poly"   and d :: "poly" assumes "islin (Lt (CNP 0 a r))"   and "tmbound0 t"   and "tmbound0 s" shows "bound0 (msubstlt c t d s a r)"
mk_acc_impl_def: fixes Flist :: "'e set list" shows "mk_acc_impl Flist \<equiv> let acc = Map.empty in nfoldli Flist (\<lambda>_. True) (\<lambda>A (i, acc). FOREACH\<^bsup>\<lambda>it acc'. acc' = (\<lambda>v. if v \<in> A - it then Some (insert i (the_default {} (acc v))) else acc v)\<^esup> A (\<lambda>v acc. RETURN (acc(v \<mapsto> insert i (the_default {} (acc v))))) acc \<bind> (\<lambda>acc. RETURN (Suc i, acc))) (0, acc) \<bind> (\<lambda>(uu_, acc). RETURN (\<lambda>x. the_default {} (acc x)))"