hcju/isabelleps huggingface dataset:
inst_var_multiple: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and insts :: "((variable \<times> typ) \<times> Core.term) list" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "\<forall>(x, \<tau>)\<in>fst ` set insts. term_ok \<Theta> (Fv x \<tau>)"   and "\<forall>a\<in>snd ` set insts. term_ok \<Theta> a"   and "\<forall>((uu_, \<tau>), a)\<in>set insts. typ_of a = Some \<tau>"   and "\<forall>(v, uu_)\<in>set insts. v \<notin> FV \<Gamma>"   and "distinct (map fst insts)"   and "finite \<Gamma>" shows "\<Theta>,\<Gamma> \<turnstile> subst_term insts B"
arr_dom_iff_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "CCC.arr (CCC.dom f) = CCC.arr f"
subE: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and T :: "ty"   and T' :: "ty"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> T \<le> T'"   and "is_type P T'"   and "wf_prog wf_md P" obtains "T = T'" and "\<forall>C. T \<noteq> Class C"   | C :: "char list" and D :: "char list" where "T = Class C__" and "T' = Class D" and "P \<turnstile> Path C__ to D unique"   | C :: "char list" where "T = NT" and "T' = Class C"
fprod_aux: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'b"   and B :: "'d set" shows "{x \<cdot> y |x y. x \<in> f ` A \<and> y \<in> g ` B} = {f x \<cdot> g y |x y. x \<in> A \<and> y \<in> B}"
vsv_vconverse: shows "vsv ((\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>)"
image_image: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and A :: "'d set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
crel_vs1_alt_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "crel_vs1 R = state_dp_consistency.crel_vs R OO local.rel_state (=)"
rot_x_ivl: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "0 \<le> rot_x x y \<and> rot_x x y \<le> 1"
reduced_run_invisible: fixes q :: "'state"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action stream"   and l :: "'action list"   and w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "set w\<^sub>2 \<subseteq> invisible"
ack_le_mono1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "ack i k \<le> ack j k"
onorm_scaleR_lemma: fixes f :: "'d \<Rightarrow> 'e"   and r :: "real" assumes "bounded_linear f" shows "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
rel_G_witness1: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and tytok :: "('o \<times> ('o \<times> 'p) \<times> 'p \<times> 'q \<times> ('q \<times> 'r) \<times> 'r \<times> 's) itself"   and L1 :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and x :: "('o, 'q, 'b, 'f, 'i, 'l, 's) G"   and y :: "('p, 'r, 'e, 'h, 'k, 'n, 's) G" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok"   and "rel_G L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_G (\<lambda>x (x', y). x' = x \<and> L1 x y) (\<lambda>x (x', y). x' = x \<and> L2 x y) Co1 Co2 Contra1 Contra2 x (rel_G_witness L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y))"
transfers_split: fixes ets :: "'a edge_kind list"   and ets' :: "'a edge_kind list"   and s :: "'a" shows "transfers (ets @ ets') s = transfers ets' (transfers ets s)"
GraphByMap_def: fixes m1_ops :: "('a, 'b, 'c, 'd) map_ops_scheme"   and m2_ops :: "('a, 'e, 'b, 'f) map_ops_scheme"   and s3_ops :: "('g, 'e, 'h) set_ops_scheme"   and m1_mvif :: "('a \<Rightarrow> 'b \<Rightarrow> 'b option) \<Rightarrow> 'c \<Rightarrow> 'c" shows "GraphByMap m1_ops m2_ops s3_ops m1_mvif \<equiv> (StdMap m1_ops \<and> StdMap m2_ops) \<and> StdSet s3_ops \<and> map_value_image_filter (map_op_\<alpha> m1_ops) (map_op_invar m1_ops) (map_op_\<alpha> m1_ops) (map_op_invar m1_ops) m1_mvif"
expands_to_powr_const: fixes F :: "'a"   and basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and p :: "real"   and abort :: "bool" assumes "trimmed_pos F"   and "basis_wf basis"   and "(f expands_to F) basis"   and "p \<equiv> p" shows "((\<lambda>x. f x powr p) expands_to powr_expansion abort F p) basis"
1: shows "is_poincare_line (moebius_circline (moebius_rotation (pi / 2)) H)"
"1_1": assumes "x = y"   and "sorted xs"   and "sorted (y # ys)" shows "set (subtract_list_sorted xs (y # ys)) = set xs - set (y # ys)"
sign_extend_eq_signed_take_bit: shows "sign_extend = signed_take_bit"
chain_boundary_def: fixes p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p c \<equiv> if p = 0 then 0 else frag_extend (\<lambda>f. \<Sum>k\<le>p. frag_cmul ((- 1) ^ k) (frag_of (singular_face p k f))) c"
finite_LB: fixes xs :: "'a list"   and f :: "atom fm" shows "finite (LB f xs)"
bij: fixes y :: "complex" assumes "y \<in> V" shows "bij ((*) (exp (g y)))"
zero_le_double_add_iff_zero_le_single_add: fixes a :: "'c" shows "((0::'c) \<le> a + a) = ((0::'c) \<le> a)"
sig_red_tail_rtrancl_lc_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg (\<prec>) F)\<^sup>*\<^sup>* p q" shows "punit.lc (rep_list q) = punit.lc (rep_list p)"
simps_1: fixes n :: "nat"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "tighten_bounds_2 (Irrational n (p, l, r)) = (case tighten_poly_bounds p l r (sgn (ipoly p r)) of (l', r', xa) \<Rightarrow> Irrational n (p, l', r'))"
heap_WHILET_def: fixes b :: "'c \<Rightarrow> bool Heap"   and f :: "'c \<Rightarrow> 'c Heap"   and s :: "'c" shows "heap_WHILET b f s \<equiv> heap.fixp_fun (\<lambda>D s. b s \<bind> (\<lambda>bv. if bv then f s \<bind> D else return s)) s"
axiom9_F: fixes x :: "('l, 'k, 'm, 'n) F" shows "|F'set3 x| \<le>o F'bd"
assms_3: shows "r \<cdot> s = s \<cdot> r"
HPT_source_target2: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and s :: "'d"   and l :: "('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate"   and t :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "(s, l, t) \<in> TS"   and "s \<in> States A"   and "A \<in> SAs (HA ST)" shows "t \<in> States A"
expand: fixes nondetT :: "('e, 'd) nondetT"   and nondetT' :: "('e, 'd) nondetT" assumes "run_nondet nondetT = run_nondet nondetT'" shows "nondetT = nondetT'"
Dom_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Dom (Cod t) = Cod t"
insertion_const: fixes f :: "nat \<Rightarrow> real"   and c :: "real" shows "insertion f (Const c) = c"
rvars_def: fixes t :: "(nat \<times> linear_poly) list" shows "rvars t = \<Union> (set (map rvars_eq t))"
hyps: shows "f \<in> singular_simplex_set 0 X"
ord_of_eq_0_iff: fixes n :: "nat" shows "(n\<^sub>\<nat> = []\<^sub>\<circ>) = (n = 0)"
singleton: fixes i :: "nat_int" shows "|i| = 1 \<longrightarrow> (\<exists>n. Rep_nat_int i = {n})"
WT_resource_coinduct: fixes X :: "('c, 'd) resource \<Rightarrow> bool"   and res :: "('c, 'd) resource"   and \<I> :: "('c, 'd) \<I>" assumes "X res"   and "\<And>res q r res'. \<lbrakk>X res; q \<in> outs_\<I> \<I>; (r, res') \<in> set_spmf (run_resource res q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> (X res' \<or> \<I> \<turnstile>res res' \<surd>)" shows "\<I> \<turnstile>res res \<surd>"
small_big_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<omega>[F](g1)"   and "f2 \<in> \<Omega>[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> \<omega>[F](\<lambda>x. g1 x * g2 x)"
Entry_path: fixes n :: "'node" assumes "valid_node n" shows "\<exists>as. (_Entry_) -as\<rightarrow>\<^sub>\<surd>* n"
write_state_reg_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "write_state_reg_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rs1 = get_operand_w5 (op_list ! 1); rd = get_operand_w5 (op_list ! 3) in do curr_win \<leftarrow> get_curr_win (); psr_val \<leftarrow> gets (cpu_reg_val PSR); s_val \<leftarrow> gets (\<lambda>s. get_S psr_val); op2 \<leftarrow> gets (get_operand2 op_list); rs1_val \<leftarrow> gets (user_reg_val curr_win rs1); result \<leftarrow> gets (\<lambda>s. rs1_val XOR op2); if instr_name = sreg_type WRY then do modify (delayed_pool_add (DELAYNUM, result, Y)); return () od else if instr_name = sreg_type WRASR then if privileged_ASR rd \<and> s_val = 0 then do raise_trap privileged_instruction; return () od else if illegal_instruction_ASR rd then do raise_trap illegal_instruction; return () od else do modify (delayed_pool_add (DELAYNUM, result, ASR rd)); return () od else if instr_name = sreg_type WRPSR then if s_val = 0 then do raise_trap privileged_instruction; return () od else if NWINDOWS \<le> uint (ucast result) then do raise_trap illegal_instruction; return () od else do pil_val \<leftarrow> gets (\<lambda>s. get_PIL result); et_val \<leftarrow> gets (\<lambda>s. get_ET result); new_psr_val \<leftarrow> gets (\<lambda>s. update_PSR_et_pil et_val pil_val psr_val); write_cpu new_psr_val PSR; modify (delayed_pool_add (DELAYNUM, result, PSR)); return () od else if instr_name = sreg_type WRWIM then if s_val = 0 then do raise_trap privileged_instruction; return () od else do result_f \<leftarrow> gets (\<lambda>s. result << nat (32 - NWINDOWS) >> nat (32 - NWINDOWS)); modify (delayed_pool_add (DELAYNUM, result_f, WIM)); return () od else if s_val = 0 then do raise_trap privileged_instruction; return () od else do tbr_val \<leftarrow> gets (cpu_reg_val TBR); tbr_val_11_0 \<leftarrow> gets (\<lambda>s. tbr_val AND 4095); result_tmp \<leftarrow> gets (\<lambda>s. result AND 4294963200); result_f \<leftarrow> gets (\<lambda>s. tbr_val_11_0 OR result_tmp); modify (delayed_pool_add (DELAYNUM, result_f, TBR)); return () od od"
gAbsIndif_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gAbsIndif MOD \<equiv> \<forall>xs s x X1' X2' X. isInBar (xs, s) \<and> wls s X1' \<and> wls s X2' \<and> gWls MOD s X \<longrightarrow> gAbs MOD xs x X1' X = gAbs MOD xs x X2' X"
natural_transformation_axioms: shows "natural_transformation D D Adj.D.map Adj.GF.map \<eta>"
set_restriction_Int: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set"   and B :: "'a set" assumes "set_restriction f" shows "f (A \<inter> B) = f A \<inter> f B"
accept\<^sub>G\<^sub>R_def: fixes R :: "('c \<Rightarrow> 'd \<Rightarrow> 'c) \<times> 'c \<times> (('c \<times> 'd \<times> 'c) set \<times> ('c \<times> 'd \<times> 'c) set set) set"   and w :: "nat \<Rightarrow> 'd" shows "accept\<^sub>G\<^sub>R R w \<equiv> \<exists>(Fin, Inf)\<in>snd (snd R). accepting_pair\<^sub>G\<^sub>R (fst R) (fst (snd R)) (Fin, Inf) w"
degree_msum_pdevs_ge2: fixes f :: "'b pdevs"   and n :: "nat"   and g :: "'b pdevs" assumes "degree f \<le> n" shows "degree g \<le> degree (msum_pdevs n f g) - n"
get_parent_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_parent ptr' \<rightarrow>\<^sub>r Some parent" shows "(parent \<in> set c) = (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr' \<in> set c)"
surjD: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "surj f" shows "\<exists>x. y = f x"
the_coext_def: fixes x' :: "'d"   and f :: "'c" shows "the_coext x' f = (THE g. is_coext x' f g)"
simps: fixes s :: "Store"   and a1 :: "Location"   and a2 :: "Value" shows "s\<turnstile> a1 reachable_from a2 = ((\<exists>l. a1 = l \<and> a2 = ref l \<and> ref l \<noteq> nullV) \<or> (\<exists>l k. a1 = l \<and> a2 = ref k \<and> s\<turnstile> l reachable_from (s@@k) \<and> ref k \<noteq> nullV))"
sat_mono: fixes A :: "constraint list"   and B :: "constraint list"   and X :: "(nat, rat) mapping" assumes "set A \<subseteq> set B"   and "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set B" shows "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set A"
case_2: fixes f1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and f2 :: "'d set \<Rightarrow> 'e \<Rightarrow> 'c"   and x21 :: "'d set"   and x22 :: "'e" shows "(case Rem x21 x22 of Add x xa \<Rightarrow> f1 x xa | Rem x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
skip_eval_elim: fixes annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Skip \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = Stop \<and> mds' = mds \<oplus> annos \<and> mem' = mem"
rjoin_deterministic: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and r' :: "'r"   and \<sigma>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and v :: "('r, 'l, 'v) val"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = Some (\<sigma>', \<tau>', VE v)" shows "revision_step r s s' = (s' = (s(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r' := None))"
simps_11: fixes \<C> :: "t_context"   and ts :: "checker_type" shows "check_single \<C> Unreachable ts = type_update ts [] (TopType [])"
lookup_Trie_Mapping: fixes t :: "('b, 'c) trie" shows "Mapping.lookup (Trie_Mapping t) = lookup t"
GreenThm_typeI_divisible: assumes "gen_division s (cubeImage ` two_chain_typeI)" shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} (two_chain_boundary two_chain_typeI)"
separation_t1: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.t1_space open" shows "(x \<noteq> y) = (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
obvious_forall: fixes X :: "'b set"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x\<in>X. P x"   and "x \<in> X" shows "P x"
partCompat_partJoin: fixes theta :: "('a \<times> 'a) set"   and P :: "'b set set"   and f :: "'b \<Rightarrow> 'a"   and Q :: "'b set set" assumes "sym theta"   and "trans theta"   and "partCompat P theta f"   and "partCompat Q theta f" shows "partCompat (partJoin P Q) theta f"
3: fixes x :: "'a" shows "x \<squnion> x = x"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'a" shows "x\<^bsup>n + m\<^esup> = x\<^bsup>n\<^esup> @ x\<^bsup>m\<^esup>"
is_sublist_hd: fixes a :: "'a"   and as :: "'a list"   and vs :: "'a list" assumes "is_sublist (a # as) vs" shows "is_sublist [a] vs"
Gcd_empty: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd {} = zero"
l_adds: shows "lcs (lp g1) (lp g2) adds lp (q g1 \<odot> g1)"
mono_SucI2: fixes X :: "nat \<Rightarrow> 'a" assumes "\<forall>n. X (Suc n) \<le> X n" shows "monoseq X"
sg16b: shows "Suc (3 + k) = 4 + k"
pr_strict'_def: fixes f :: "'f \<times> nat"   and g :: "'f \<times> nat" shows "kbo'.pr_strict' f g = (f \<notin> kbo'.lConsts \<and> (pr_strict' f g \<or> g \<in> kbo'.lConsts))"
proj_minus: fixes k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (p - q) = proj_poly k p - proj_poly k q"
distinct_sorted_strict_mono_iff: fixes l :: "'a list"   and i :: "nat"   and j :: "nat" assumes "distinct l"   and "sorted l"   and "i < length l"   and "j < length l" shows "(l ! i < l ! j) = (i < j)"
upper_right_eq: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set" assumes "finite X"   and "Regions.valid_region X k I r" shows "(\<forall>x\<in>X. Regions.isGreater (I x)) = (\<forall>t\<ge>0. \<forall>u\<in>Regions.region X I r. u \<oplus> t \<in> Regions.region X I r)"
iso: shows "g \<circ> g' \<in> Group.iso C C"
lappend_strict: shows "lappend\<cdot>\<bottom> = \<bottom>"
design_iso_block_size_eq: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "card bl = card (\<pi> ` bl)"
hypsy: shows "y \<in> \<rat> \<and> y ^ 3 - 3 * y = 1"
real_LIM_sandwich_zero: fixes f :: "'b \<Rightarrow> real"   and a :: "'b"   and g :: "'b \<Rightarrow> real" assumes "f \<midarrow>a\<rightarrow> 0"   and "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"   and "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x" shows "g \<midarrow>a\<rightarrow> 0"
mset_points_distinct_diff_one: fixes x :: "'a" shows "distinct_mset (mset_set (\<V> - {x}))"
\<tau>'_def: fixes t :: "real" shows "\<tau>' t = 1"
Asm: fixes P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" shows "{(P, CALL, Q)} \<turnstile>\<^sub>t {P} CALL {Q}"
n_galois1': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<le> !y + !z) = (!x \<cdot> t y \<le> !z)"
hoare_assume_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and p :: "'c Assertion"   and b :: "'c Assertion"   and x :: "'c"   and q :: "'c Assertion" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "hoare p ([\<cdot> b ] * x) q = hoare (p \<sqinter> b) x q"
semicategory_smc_1: fixes \<alpha> :: "V"   and \<aa> :: "V"   and \<ff> :: "V" assumes "\<Z> \<alpha>"   and "\<aa> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<ff> \<in>\<^sub>\<circ> Vset \<alpha>" shows "semicategory \<alpha> (smc_1 \<aa> \<ff>)"
double_compl: fixes x :: "'c" shows "NOT (NOT x) = x"
RedCall: fixes h :: "heap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and P :: "(char list list \<times> char list exp) prog"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and vs :: "val list"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" assumes "h a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "length Ts = length pns" shows "P \<turnstile> \<langle>addr a\<bullet>M(map Val vs),(h, l, sh),b\<rangle> \<rightarrow> \<langle>blocks (this # pns, Class D # Ts, Addr a # vs, body),(h, l, sh),b\<rangle>"
Max_triangle_greatest: fixes z :: "nat" shows "Max_triangle z = (GREATEST k. triangle k \<le> z \<and> k \<le> z \<or> k = 0)"
quad_part_1_neq: fixes var :: "nat"   and L :: "real list"   and p :: "real mpoly"   and deg :: "nat"   and D :: "real"   and a :: "real mpoly"   and A :: "real"   and b :: "real mpoly"   and B :: "real"   and d :: "real mpoly"   and C :: "real" assumes "var < length L"   and "MPoly_Type.degree p var = deg"   and "D \<noteq> 0"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = A"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = B"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = D" shows "aEval (Neq p) (L[var := (A + B * C) / D]) = aEval (Neq (quadratic_part_1 var a b d (Neq p))) (L[var := C])"
scaleC: fixes r :: "complex"   and b :: "'a" shows "g (r *\<^sub>C b) = r *\<^sub>C g b"
min_correct_1: fixes s :: "'s"   and P :: "'u \<times> 'v \<Rightarrow> bool" assumes "invar s"   and "rel_of (\<alpha> s) P \<noteq> {}" shows "min s P \<in> Some ` rel_of (\<alpha> s) P"
unfolding_is_subst: fixes \<phi> :: "'b ltln" shows "Unf \<phi> = subst \<phi> (\<lambda>\<psi>. Some (Unf \<psi>))"
reflexive: fixes p :: "'c"   and A :: "('d, 'c) nba" shows "p \<in> reachable A p"
locally_finite_onD: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set"   and p :: "'c" assumes "locally_finite_on X I U"   and "p \<in> X" shows "finite {i \<in> I. p \<in> U i}"
lNAct_def: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.lNAct active c t n \<equiv> LEAST n'. n = n' \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. k < n \<and> active c (t k))"
cbig_def2: fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and combine :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and collect_id :: "'c"   and \<sigma>' :: "'b"   and coll :: "'c"   and collect :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c"   and P :: "'a"   and \<sigma> :: "'b" assumes "CollectionSemantics small endset combine collect_id" shows "((\<sigma>', coll) \<in> CollectionSemantics.cbig small endset collect combine collect_id P \<sigma>) = (\<exists>n. (\<sigma>', coll) \<in> CollectionSemantics.csmall_nstep small collect combine collect_id P \<sigma> n \<and> \<sigma>' \<in> endset)"
less_num_nodes_edge_Exit: fixes prog :: "cmd" obtains l :: "nat" and et :: "(char list \<Rightarrow> val option) edge_kind" where "l < #:prog" and "prog \<turnstile> (_ l _) -et\<rightarrow> (_Exit_)"
set_update_subset_insert: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b" shows "set (xs[i := x]) \<subseteq> insert x (set xs)"
assms: shows "A \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)\<lparr>Obj\<rparr>"