hcju/isabelleps huggingface dataset:
extension_from_clopen: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and U :: "'b set" assumes "openin (top_of_set S) T"   and "closedin (top_of_set S) T"   and "continuous_on T f"   and "f ` T \<subseteq> U"   and "U = {} \<Longrightarrow> S = {}" obtains g :: "'a \<Rightarrow> 'b" where "continuous_on S g" and "g ` S \<subseteq> U" and "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
prefix_imp_take_eq: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list" assumes "n \<le> length xs"   and "prefix xs ys" shows "xs \<down> n = ys \<down> n"
KBasic2_6: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<diamond>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> \<^bold>\<diamond>\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi> in v]"
rel_Grp: fixes A1 :: "'h set"   and f1 :: "'h \<Rightarrow> 'j"   and A2 :: "'i set"   and f2 :: "'i \<Rightarrow> 'k" shows "rel_sum (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. Basic_BNFs.setl x \<subseteq> A1 \<and> Basic_BNFs.setr x \<subseteq> A2} (map_sum f1 f2)"
inj_on_opp_chambers'': fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexFolding X f"   and "C \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "f ` C = f ` D" shows "C = D"
abs_summable_on_Sigma_project1: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B"   and "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)" shows "(\<lambda>x. \<Sum>\<^sub>ay\<in>B x. norm (f x y)) abs_summable_on A"
assms: shows "trace_callee_eq callee1 callee2 A p q"
abs_eq: fixes xa :: "'b ptrm"   and x :: "'b ptrm" shows "SimplyTyped.Pair (abs_trm xa) (abs_trm x) = abs_trm (PPair xa x)"
cf_get_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and e :: "nat \<times> nat" shows "cf_get cf e \<equiv> assert (e \<in> E \<union> E\<inverse>) \<bind> (\<lambda>_. return (cf e))"
pigeonhole: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "card (f ` A) < card A" shows "\<not> inj_on f A"
HAInitState_HAStates: fixes A :: "('a, 'b, 'c) hierauto" shows "HAInitState A \<in> HAStates A"
B_2: fixes n :: "nat" assumes "1 \<le> n" shows "\<bar>F (real n)\<bar> \<le> B * sqrt (real n)"
key_sets_union: fixes X :: "msg"   and H :: "msg set"   and H' :: "msg set" shows "key_sets X (H \<union> H') = key_sets X H \<union> key_sets X H'"
less_eqE: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" obtains c :: "'c" where "b = plus a c"
put_inef: fixes \<sigma> :: "'b"   and v :: "'a" shows "put \<sigma> v = \<sigma>"
representation_of_cones_functor_axioms: shows "representation_of_cones_functor (\<cdot>\<^sub>J) (\<cdot>) D (\<cdot>\<^sub>S) \<phi> \<iota> a \<Phi>"
on_curve_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p :: "'c point" assumes "ell_field R" shows "ell_field.on_curve R a b p = (case p of Infinity \<Rightarrow> True | Point x y \<Rightarrow> x \<in> carrier R \<and> y \<in> carrier R \<and> y [^]\<^bsub>R\<^esub> 2 = x [^]\<^bsub>R\<^esub> 3 \<oplus>\<^bsub>R\<^esub> a \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> b)"
ideal_prod_mono2: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "ideal R A"   and "ideal R B"   and "ideal R C"   and "A \<subseteq> B" shows "C \<diamondsuit>\<^sub>r A \<subseteq> C \<diamondsuit>\<^sub>r B"
internalize_ordLess: fixes r' :: "('d \<times> 'd) set"   and r :: "('e \<times> 'e) set" shows "(r' <o r) = (\<exists>p. Field p \<subset> Field r \<and> r' =o p \<and> p <o r)"
assms_2: shows "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b) \<or> g x \<noteq> (0::'b)"
mult_im_sheaf_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" assumes "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.mult_im_sheaf S mult_str f \<equiv> \<lambda>V. mult_str (f \<^sup>\<inverse> S V)"
ts_ord_mult_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "n x \<sqsubseteq> n y" shows "n z \<cdot> n x \<sqsubseteq> n z \<cdot> n y"
nxt: fixes M :: "'b dfa"   and q :: "hf"   and x :: "'b" assumes "dfa M"   and "q \<in> dfa.states M" shows "dfa.nxt M q x \<in> dfa.states M"
sq_norm_row_sylvester_mat1: fixes i :: "nat"   and g :: "'a poly"   and f :: "'a poly" assumes "i < degree g" shows "\<parallel>row (sylvester_mat f g) i\<parallel>\<^sup>2 = \<parallel>f\<parallel>\<^sup>2"
isomorphic_implies_same_tab_2: fixes r' :: "'a" assumes "r \<cong> r'" shows "tab\<^sub>1 r = tab\<^sub>1 r'"
exec_meth_ci_appD: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and fr' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t h (stk, loc, pc, None) ta h' fr'" shows "ci_app ci (ins ! pc) P h stk loc undefined undefined pc []"
splitTree_def: fixes p :: "'d \<Rightarrow> bool"   and i :: "'d"   and t :: "('c, 'd) FingerTree" shows "splitTree p i t \<equiv> extract_splitres (splitTree_aux p i t)"
intros_5: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Log], Undecided\<rangle> \<Rightarrow> Undecided"
dma_control_var_def: fixes v :: "nat" shows "dma_control_var v \<equiv> if v = 0 then Low else High"
assms_4: shows "\<Psi> \<rhd> Q \<approx> P"
upper_asymptotic_density_in_01_3: fixes A :: "nat set" shows "0 \<le> upper_asymptotic_density A"
arr_char: fixes f :: "'a \<times> 'a" shows "B.VxV.arr f = (B.arr (fst f) \<and> B.arr (snd f))"
Sbis_matchC_C_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>s t" shows "matchC_C Sbis t s"
bisim_def: shows "Stream_bisim \<equiv> \<lambda>R. \<forall>x y. R x y \<longrightarrow> x = \<bottom> \<and> y = \<bottom> \<or> (\<exists>a Stream a' Stream'. a = a' \<and> R Stream Stream' \<and> x = a && Stream \<and> y = a' && Stream')"
extf_ext_compat_list_strong: fixes f :: "'s" shows "ext_compat_list_strong (extf f)"
simps_2: fixes n :: "nat" shows "enat_of_option (Some n) = enat n"
index_minus_mat_2: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_row (A - B) = dim_row B"
set_iterator_def: fixes it :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and S0 :: "'f set" shows "set_iterator it S0 \<equiv> set_iterator_genord it S0 (\<lambda>_ _. True)"
del_order: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and t :: "'b btree"   and x :: "'b" assumes "split split"   and "0 < k"   and "root_order k t"   and "bal t" shows "almost_order k (split.del split k x t)"
is_swap_def: shows "is_swap = (\<forall>x\<in>{0, 1}. f x = 1 - x)"
dim_basis: fixes A :: "'a vec set" assumes "finite A"   and "KD.basis A" shows "KD.dim = card A"
path_valid_node_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "CFG.valid_node sourcenode targetnode valid_edge n"
left_neutral: fixes a :: "'b" shows "\<top> \<sqinter> a = a"
subset_UnE: fixes C :: "'e set"   and A :: "'e set"   and B :: "'e set" assumes "C \<subseteq> A \<union> B" obtains A' :: "'e set" and B' :: "'e set" where "A' \<subseteq> A" and "B' \<subseteq> B" and "C = A' \<union> B'"
sbelows_rel_def: fixes A :: "('a \<times> 'c) set" shows "\<langle>A\<rangle>sbelows_rel = \<langle>\<langle>A\<rangle>sctn_rel\<rangle>list_set_rel O br sbelow_halfspaces top"
HVZK_unfold2: assumes "\<Sigma>_protocol" shows "\<forall>h e out. e \<in> challenge_space \<longrightarrow> h \<in> valid_pub \<longrightarrow> out \<in> set_spmf (S_raw h e) \<longrightarrow> check h (fst out) e (snd out)"
assert_neg: fixes p :: "'b Assertion" shows "{\<cdot> - p } = neg_assert {\<cdot> p }"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_erase1 ires rs (l, r) = (\<exists>ln rn. l = Oc # ires \<and> tl r = Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
more_arith_simps_6: fixes a :: "'b" shows "a * (1::'b) = a"
ideal_generated_singleton_dvd: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and d :: "'b"   and x :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "ring.ideal_generated plus zero uminus times S = ring.ideal_generated plus zero uminus times {d}"   and "x \<in> S" shows "dvd.dvd times d x"
fst_edge_at: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and "is" :: "nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "fst (Solved_Task.edge_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is) = Solved_Task.edge_from freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is"
dvd_add_times_triv_right_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus b (times c a)) = dvd.dvd times a b"
ball_less_Suc_eq: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>i<Suc n. P i) = (P 0 \<and> (\<forall>i<n. P (Suc i)))"
map_graph_inv: fixes f :: "'a \<Rightarrow> 'b option" shows "graph_map (map_graph f) = f"
NoReceivingNoChange: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and m :: "('a, 'c) message"   and cfg2 :: "('a, 'c, 'b) configuration"   and p :: "'a" assumes "asynchronousSystem.steps trans sends cfg1 m cfg2"   and "\<not> isReceiverOf p m" shows "states cfg1 p = states cfg2 p"
length_removeKeyList: fixes ws :: "'c list"   and ps :: "('c \<times> 'd) list" shows "|removeKeyList ws ps| \<le> |ps|"
TS_non_containment_causes_final: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<notin> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "set xs \<subseteq> inputs M2"   and "final_iteration M2 M1 \<Omega> V m i"   and "OFSM M2" shows "\<exists>xr j. xr \<noteq> xs \<and> prefix xr xs \<and> j \<le> i \<and> vs @ xr \<in> RM M2 M1 \<Omega> V m j"
eq_\<I>_gpvI: fixes A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('d, 'b, 'c) gpv" assumes "rel_spmf (eq_\<I>_generat A \<I> (eq_\<I>_gpv A \<I>)) (the_gpv gpv) (the_gpv gpv')" shows "eq_\<I>_gpv A \<I> gpv gpv'"
supset: shows "set (reduction_signs p qs signs subsets (M_mat signs subsets)) \<subseteq> set (characterize_consistent_signs_at_roots_copr p qs)"
to_fun_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_cring R" shows "UP_cring.to_fun R f \<equiv> to_function R f"
mplus_strict: fixes m :: "'a match" shows "\<bottom> +++ m = \<bottom>"
bit_set_bits_word_iff: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" shows "bit (set_bits P) n = (n < LENGTH('a) \<and> P n)"
times_eint_simps_2: shows "\<infinity> * \<infinity> = \<infinity>"
not_in_FV_in_fv_not_in: fixes x :: "variable"   and \<tau> :: "typ"   and \<Gamma> :: "Core.term set"   and t :: "Core.term" assumes "(x, \<tau>) \<notin> FV \<Gamma>"   and "(x, \<tau>) \<in> fv t" shows "t \<notin> \<Gamma>"
set_iterator_foldli_correct: fixes xs :: "'c list" assumes "distinct xs" shows "set_iterator (foldli xs) (set xs)"
pelims: fixes x :: "int \<times> nat"   and y :: "int \<times> nat" assumes "incr2 x = y"   and "incr2_dom x" obtains xa :: "int" and n :: "nat" where "x = (xa, n)" and "y = (xa, n + 1)" and "incr2_dom (xa, n)"
invertible_left_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "local.inverse u \<cdot> u = \<one>"
guha_equal_hlp: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list"   and p :: "'d" assumes "no_overlaps \<gamma> ft" shows "guha_table_semantics \<gamma> ft p (ftb_to_option (OF_priority_match \<gamma> ft p))"
WTrt_binop_Eq: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty" shows "P \<turnstile> T1\<guillemotleft>Eq\<guillemotright>T2 : Boolean"
butlast_prefix_suffix_not_empty: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix (butlast xs) ys" shows "ys \<noteq> []"
lr_1: shows "l \<in> \<omega>_limit_set x"
Exit_source: fixes a :: "'edge" assumes "valid_edge a"   and "sourcenode a = (_Exit_)" shows "False"
gb_schema_aux_term2_def: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "gb_schema_aux_term2 d gs = {(a, b). dgrad_p_set_le d (args_to_set (gs, a)) (args_to_set (gs, b)) \<and> component_of_term ` Keys (args_to_set (gs, a)) \<subseteq> component_of_term ` Keys (args_to_set (gs, b))}"
vw': shows "v \<cdot> s = s \<cdot> w"
assms_2: shows "fv t \<subseteq> fv\<^sub>s\<^sub>t S"
bit_mask_iff: fixes m :: "nat"   and n :: "nat" shows "bit (mask m) n = (n < min LENGTH('a) m)"
map_getminE: fixes m :: "'q"   and k :: "'v"   and p :: "nat" assumes "Q_getmin m = (k, p)"   and "Q_invar m"   and "Q_lookup m \<noteq> Map.empty" obtains "Q_lookup m k = Some p" and "\<forall>k' p'. Q_lookup m k' = Some p' \<longrightarrow> p \<le> p'"
ARUN: shows "igb.is_acc_run = is_acc_run"
DS: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"
generate_valid_topology_subseteq_edges: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "edges (generate_valid_topology M G) \<subseteq> edges G"
apex_def: shows "SPN_fgh.\<nu>\<pi>.apex \<equiv> Maps.dom SPN_fgh.\<nu>\<pi>.leg0"
assms_3: shows "trace (complex_of_real c \<cdot>\<^sub>m \<rho>) \<le> 1"
myss2_def: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a" shows "myss2 x y a z \<equiv> myss x y \<and> myss' a \<and> myss2_axioms x y a z"
wp_seq_var: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "p \<le> wp x r"   and "r \<le> wp y q" shows "p \<le> wp (x \<cdot> y) q"
simps_3: fixes c :: "int"   and t :: "MIR.num"   and s :: "MIR.num" shows "maxcoeff (CF c t s) = max \<bar>c\<bar> (maxcoeff s)"
finite_fix: fixes \<T> :: "'a stateful_policy" assumes "wf_stateful_policy \<T>" shows "finite (flows_fix \<T>)"
size_4: fixes x21 :: "'g"   and x22 :: "'g list" shows "length (x21 # x22) = length x22 + Suc 0"
nth_map_conv: fixes xs :: "'c list"   and ys :: "'d list"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "length xs = length ys"   and "\<forall>i<length xs. f (xs ! i) = g (ys ! i)" shows "map f xs = map g ys"
unique_connecting_path_def: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" shows "v \<leadsto>\<^bsub>G\<^esub> w \<equiv> THE xs. v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
abc_bcd_abd: shows "\<forall>a b c d. ord a b c \<and> ord b c d \<longrightarrow> ord a b d"
induced_modulusD: fixes d :: "nat"   and a :: "nat" assumes "\<Phi>.induced_modulus d"   and "coprime a d__"   and "[a = 1] (mod d)" shows "\<Phi> a = 1"
silent_moves_equal_length_1: 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 S :: "'b SDG_node set"   and f :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and ms :: "'b list"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and as :: "'a list"   and ms' :: "'b list"   and s' :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.silent_moves sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S f ms s as ms' s'" shows "length ms = length s"
gr_implies_not_zero: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "'g"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less m n" shows "n \<noteq> zero"
lActive_less: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat"   and n :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<forall>n'>n. \<not> \<parallel>c\<parallel>\<^bsub>t n'\<^esub>" shows "\<langle>c \<and> t\<rangle> \<le> n"
prems_1: shows "i < Monitor.progress \<sigma> P (formula.MatchF I r) (plen \<pi>)"
strict_trans1: fixes b :: "'b multiset"   and a :: "'b multiset"   and c :: "'b multiset" assumes "b \<subseteq># a"   and "c \<subset># b" shows "c \<subset># a"
assms_6: shows "f'' : a'' \<mapsto>\<^bsub>\<CC>\<^esub> b''"
le_Sup_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and A :: "'a set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq x (Sup A) = (\<forall>y. less y x \<longrightarrow> (\<exists>a\<in>A. less y a))"
intersects_spec_def: fixes X :: "'c set"   and sctn :: "'c sctn" shows "intersects_spec X sctn = SPEC (\<lambda>b. b \<or> X \<inter> plane_of sctn = {})"
in_JF2rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d JF2"   and y :: "'e JF2" shows "JF2rel R x y = (\<exists>z. z \<in> JF2in {(x, y). R x y} \<and> JF2map fst z = x \<and> JF2map snd z = y)"
single_bind: fixes x :: "'b"   and P :: "'b \<Rightarrow> 'a pred" shows "bind (single x) P = P x"
makeFaceFinalFaceList_def: fixes f :: "face"   and fs :: "face list" shows "makeFaceFinalFaceList f fs \<equiv> replace f [setFinal f] fs"
densh: shows "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure dc_False.\<Y> \<rho> \<bind> h) R \<delta>h"
capacity_pos: fixes e :: "'v \<times> 'v" assumes "e \<in> \<^bold>E" shows "0 < capacity \<Delta> e"
X_def: fixes xs :: "'b list"   and j :: "nat" shows "X xs j = [:- x xs j, 1::'b:]"
ce_set_lm_1: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "p \<in> PrimRec2" shows "fn_to_set p \<in> ce_sets"
that_1: fixes n :: "nat" assumes "\<And>m. n \<le> m \<Longrightarrow> f n = f m" shows "thesis"
Union_rel_internal: fixes L :: "('a \<times> 'c set) set"   and S :: "('c \<times> 'b set) set" shows "Union_rel L S = L O \<langle>S\<rangle>set_rel O br \<Union> top"
gcd_fib_add: fixes m :: "nat"   and n :: "nat" shows "gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n)"
cil_cases: fixes w :: "'d list"   and wa :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and wb :: "'d list" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb" obtains "w = []" and "wa = []" and "wb = []"   | "wa = []" and "w = wb"   | "w = wa" and "wb = []"   | ea :: "'d" and wa' :: "'d list" and w' :: "'d list" where "w = ea # w'a__" and "wa = ea # wa'" and "w'a__ \<in> wa' \<otimes>\<^bsub>\<alpha>\<^esub> wb" and "fst (\<alpha> ea) \<inter> mon_pl (map \<alpha> wb) = {}"   | eb :: "'d" and wb' :: "'d list" and w' :: "'d list" where "w = eb # w'" and "wb = eb # wb'" and "w' \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb'" and "fst (\<alpha> eb) \<inter> mon_pl (map \<alpha> wa) = {}"
SUCC_SUBS_lemma: fixes x :: "tm"   and y :: "tm" shows "{SUCC x SUBS SUCC y} \<turnstile> x SUBS y"
lconst_def: fixes a :: "'b" shows "lconst a \<equiv> repeat a"
frontier: fixes u :: "'v"   and v :: "'v" assumes "\<pi> u = Some v" shows "v \<in> S (A Q \<pi>)"
incrIndexList_help2: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "hd ls = 0"
preserves_cod: fixes f :: "'j" assumes "J.arr f" shows "A_B.cod (cones_map f \<chi> f) = D (J.cod f)"
phils_update_take: fixes N :: "nat"   and n :: "nat"   and ps :: "nat list"   and e :: "dining_event" assumes "DiningPhilosophers N" shows "take n (DiningPhilosophers.phils_state_update N ps e) = DiningPhilosophers.phils_state_update N (take n ps) e"
simps: fixes a1 :: "'sort"   and a2 :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" shows "wls a1 a2 = ((\<exists>xs x. a1 = asSort xs \<and> a2 = Var xs x) \<or> (\<exists>delta inp binp. a1 = stOf delta \<and> a2 = Op delta inp binp \<and> wlsInp delta inp \<and> wlsBinp delta binp))"