hcju/isabelleps huggingface dataset:
two_lists_cases_snoc_Cons: fixes as :: "'c list"   and bs :: "'b list" obtains ys :: "'b list" where "as = []" and "bs = ys__"   | xs :: "'c list" where "as = xs__" and "bs = []"   | xs :: "'c list" and x :: "'c" and y :: "'b" and ys :: "'b list" where "as = xs @ [x]" and "bs = y # ys"
simps_10: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "w \<Turnstile> \<phi> U \<psi> = (\<exists>k. suffix k w \<Turnstile> \<psi> \<and> (\<forall>j<k. suffix j w \<Turnstile> \<phi>))"
supt_irrefl: fixes t :: "('d, 'e) Term.term" shows "(t, t) \<notin> {\<rhd>}"
set_map0_2: fixes f1 :: "'h \<Rightarrow> 'l"   and f2 :: "'i \<Rightarrow> 'k"   and f3 :: "'j \<Rightarrow> 'm" shows "F1set1 \<circ> Fmap f1 f2 f3 = (`) f2 \<circ> F1set1"
heap_is_wellformed_one_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and ptr' :: "(_) object_ptr"   and children' :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr' \<rightarrow>\<^sub>r children'"   and "set children \<inter> set children' \<noteq> {}" shows "ptr = ptr'"
assms_8: shows "p' \<in> sterms \<Gamma> p"
e: assumes "a divides b"   and "b divides a" shows "P"
keys_subset_wf: shows "wfP (\<lambda>p q. keys p \<subset> keys q)"
DagsI: fixes t :: "dag"   and nodes :: "ref set"   and p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" assumes "set_of t \<subseteq> nodes"   and "Dag p low high t"   and "t \<noteq> Tip" shows "t \<in> Dags nodes low high"
intros: fixes R :: "('a \<times> 'a) set" assumes "\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R" shows "asym R"
lstrict_prefix_llength_less: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lstrict_prefix xs ys" shows "llength xs < llength ys"
P'Lin: fixes i :: "indi" shows "P' i \<in> Lin"
test_one_top: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and p :: "'c" assumes "dioid_tests test not" shows "test p \<le> (1::'d)"
eta_par_beta: fixes s :: "dB"   and t :: "dB"   and u :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> t"   and "t => u" shows "\<exists>t'. s => t' \<and> t' \<rightarrow>\<^sub>\<eta>\<^sup>* u"
relpow_Suc_E: fixes x :: "'a"   and z :: "'a"   and n :: "nat"   and R :: "('a \<times> 'a) set" assumes "(x, z) \<in> R ^^ Suc n" obtains y :: "'a" where "(x, y) \<in> R ^^ n" and "(y, z) \<in> R"
exp_conj_1_right: fixes A :: "'b \<Rightarrow> real" assumes "nneg A" shows "A && (\<lambda>_. 1) = A"
Cons: shows "filter P (RBT.keys t) = x # ys"
is_infI: fixes inf :: "'c"   and x :: "'c"   and y :: "'c" assumes "inf \<sqsubseteq> x"   and "inf \<sqsubseteq> y"   and "\<And>z. \<lbrakk>z \<sqsubseteq> x; z \<sqsubseteq> y\<rbrakk> \<Longrightarrow> z \<sqsubseteq> inf" shows "is_inf x y inf"
closedt_mono_1: fixes i :: "nat"   and j :: "nat"   and t :: "'a FOL_Fitting.term" assumes "i \<le> j"   and "closedt i t" shows "closedt j t"
pi1: shows "f bl index {(x, ix), (y, iy)} = 1"
takep: shows "w \<le>p take p w \<cdot> w"
splits_iff: fixes l :: "'a list"   and a :: "'a"   and r :: "'a list"   and ll :: "'a list" shows "((l, a, r) \<in> set (splits ll)) = (ll = l @ a # r)"
pair_sym_digraph_axioms_def: fixes G :: "'b pair_pre_digraph" shows "pair_sym_digraph_axioms G \<equiv> symmetric (with_proj G)"
orderE: fixes a :: "'a upred"   and b :: "'a upred" assumes "b \<sqsubseteq> a" obtains "a = (a \<and> b)"
pp_increasing: fixes x :: "'a" shows "x \<le> - - x"
not_in_neg: fixes var :: "nat"   and p :: "real mpoly" shows "(var \<notin> vars p) = (var \<notin> vars (- p))"
enn2real_plus: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < top"   and "b < top" shows "enn2real (a + b) = enn2real a + enn2real b"
mem_twist_other: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and mem :: "'a \<Rightarrow> 'b"   and v :: "'b" assumes "z \<noteq> x"   and "z \<noteq> y" shows "mem(x := mem y, z := v) = mem(z := v, x := (mem(z := v)) y)"
assms_2: shows "m2 \<in> carrier_mat (2 ^ n) (2 ^ n)"
Zero_ring: assumes "1\<^sub>r = \<zero>" shows "zeroring R"
triangular_to_jnf_steps_dims_1: fixes A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_1 A) = dim_row A"
transI: fixes r :: "('e \<times> 'e) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r" shows "trans r"
subset_entailed: fixes N2 :: "'a set"   and N1 :: "'a set" assumes "N2 \<subseteq> N1" shows "N1 \<Turnstile> N2"
wf_disc_sel_common_primitive_5: shows "wf_disc_sel (is_Iiface, iiface_sel) IIface"
prems: shows "j \<le> i \<and> j \<le> n"
distinct_81: fixes x5 :: "load_store_type"   and x8 :: "shift_type" shows "load_store_type x5 \<noteq> shift_type x8"
abs_pred_trivI: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "P t" shows "abs_pred P t"
uminus_def: fixes x :: "'c" shows "- x = (if x = bot then top else bot)"
allowed_delete: fixes i :: "'a"   and x :: "'a"   and j :: "nat" assumes "Broadcast (i, Delete x) \<in> set (history j)" shows "\<exists>i' n' v b. Deliver (i', Insert (x, v, b) n') \<sqsubset>\<^sup>j Broadcast (i, Delete x)"
tendsto_add_ereal: fixes x :: "ereal"   and y :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and F :: "'a filter"   and g :: "'a \<Rightarrow> ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "\<bar>y\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
assms: fixes I :: "'a interval" shows "split I \<noteq> []"
well_ordered_strategy_def: shows "well_ordered_strategy \<equiv> override_on \<sigma>_arbitrary (\<lambda>v. choose v v) S"
CHECK_rule_iff: fixes c :: "bool"   and e :: "'c"   and \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "unit \<Rightarrow> bool" shows "(CHECK c e \<le> ESPEC \<Phi> \<Psi>) = ((c \<longrightarrow> \<Psi> ()) \<and> (\<not> c \<longrightarrow> \<Phi> e))"
de_Morgan_disj: fixes x :: "'a"   and y :: "'a" shows "NOT (x OR y) = NOT x AND NOT y"
distinct_9: fixes x2 :: "('a, 'b, 'c, 'd) Tree"   and x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree" shows "tNot x2 \<noteq> tAct x41 x42 x43"
externals_def: fixes asig :: "'d set \<times> 'd set \<times> 'd set" shows "externals asig \<equiv> inputs asig \<union> outputs asig"
distinct_5: fixes name1 :: "name"   and name2 :: "name" shows "name1[name2] \<noteq> \<tau>"
codomains_def: fixes f :: "'a" shows "codomains f \<equiv> {b. ide b \<and> b \<cdot> f \<noteq> null}"
bisimSubstResNil: fixes \<Psi> :: "'b"   and x :: "name" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>\<zero> \<sim>\<^sub>s \<zero>"
elims: fixes x :: "'a list"   and xa :: "'a"   and y :: "'a list" assumes "close x xa = y" obtains L :: "'a list" and xb :: "'a" where "x = L" and "xa = xb" and "y = filter (\<lambda>y. y \<noteq> xb) L"
l_new_element_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<equiv> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
D: fixes X :: "msg"   and K :: "nat" assumes "P X" shows "P (Decrypt K X)"
in_set_conv_decomp: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in>. xs) = (\<exists>ys zs. xs = ys @ x # zs)"
intro: fixes r :: "V" assumes "fpairs r = r" shows "fbrelation r"
iWeakUntil_iUntil_conj_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<W> t2 I. Q t2) = (\<not> (\<not> Q t1. t1 \<U> t2 I. \<not> P t2 \<and> \<not> Q t2))"
Tree2_list_to_rbt: fixes xs :: "int list" shows "Tree2.bst (list_to_rbt xs)"
le_sup_iff: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "less_eq (sup x y) z = (less_eq x z \<and> less_eq y z)"
wf_exists_wls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and R :: "('e \<times> 'e) set"   and f :: "'e \<Rightarrow> 'f"   and s :: "'e" assumes "FixSyn TYPE('a) Delta"   and "wf R"   and "\<And>s. (\<exists>xs. s = varSortAsSort Delta xs) \<or> FixSyn.witOpS Delta s (f s) R" shows "\<exists>X. FixSyn.wls Delta s X"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "Right.seq g f" shows "Right.comp g f = g \<cdot> f"
from_Q_to_Q_inf: fixes Gs :: "'a literal multiset set llist"   and C :: "'a literal multiset"   and D :: "'a literal multiset"   and i :: "nat" assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "Gs = lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts"   and "C \<in> Liminf_llist Gs - sr.Rf (Liminf_llist Gs)"   and "D \<in> Q_of_state (lnth Sts i)"   and "enat i < llength Sts"   and "subsumes D C"   and "\<forall>E\<in>{E \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts). subsumes E C}. \<not> strictly_subsumes E D" shows "D \<in> Q_of_state (Liminf_state Sts)"
cases: fixes a1 :: "'b set"   and a2 :: "'b list"   and a3 :: "'b set"   and a4 :: "'b list"   and R :: "('b \<times> 'b) set" assumes "((a1, a2), a3, a4) \<in> dfs_step R" obtains W :: "'b list" and \<sigma> :: "'b" and Wtl :: "'b list" and Wn :: "'b list" and \<Sigma> :: "'b set" and W' :: "'b list" and \<Sigma>' :: "'b set" where "a1 = \<Sigma>" and "a2 = W" and "a3 = \<Sigma>'" and "a4 = W'" and "W = \<sigma> # Wtl" and "distinct Wn" and "set Wn = R `` {\<sigma>} - \<Sigma>" and "W' = Wn @ Wtl" and "\<Sigma>' = R `` {\<sigma>} \<union> \<Sigma>"
finfun_const_True_conv_top: shows "finfun_apply (finfun_const True) = top"
sup_bot_left: fixes x :: "'b" shows "\<bottom> \<squnion> x = x"
mat_of_rows_list_impl: fixes n :: "nat"   and rs :: "'a list list"   and A :: "'a mat_impl" assumes "mat_of_rows_list_impl n rs = Some A" shows "mat_impl A = mat_of_rows_list n rs"
create_name_igba_def: fixes \<phi> :: "'b ltlr" shows "create_name_igba \<phi> \<equiv> create_name_gba \<phi> \<bind> (\<lambda>A. gba_to_idx A \<bind> (\<lambda>A'. stat_set_data_nres (card (g_V A)) (card (g_V0 A')) (igbg_num_acc A') \<bind> (\<lambda>_. RETURN A')))"
welltypedE_4: fixes e1 :: "expr"   and e2 :: "expr"   and T :: "Unary_PCF.type" assumes "e1 ? e2 ::: T" obtains "T = \<B>" and "e1 ::: \<B>" and "e2 ::: \<B>"
wf\<^sub>s\<^sub>t\<^sub>s_def: fixes \<S> :: "('fun, 'var) strand_step list set" shows "wf\<^sub>s\<^sub>t\<^sub>s \<S> \<equiv> (\<forall>S\<in>\<S>. wf\<^sub>s\<^sub>t {} (dual\<^sub>s\<^sub>t S)) \<and> (\<forall>S\<in>\<S>. \<forall>S'\<in>\<S>. fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S' = {})"
impl_def: fixes a :: "'b G"   and b :: "'b G" shows "a l\<rightarrow> b \<equiv> Abs_G ((Rep_G b - Rep_G a + u) \<sqinter> u)"
atMostTwoLanes: fixes ts :: "traffic"   and c :: "cars" shows "|res ts c| + |clm ts c| \<le> 2"
insert_remdups: fixes x :: "'b"   and xs :: "'b list" shows "List.insert x (remdups xs) = remdups (List.insert x xs)"
1: fixes x' :: "'c"   and g :: "'d" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y x' g" shows "\<eta>y.is_ext x' g (\<psi> x' g) \<and> (\<forall>f'. \<eta>y.is_ext x' g f' \<longrightarrow> f' = \<psi> x' g)"
zip_Nil: fixes ys :: "'e list" shows "zip [] ys = []"
LIMSEQ_le_const: fixes X :: "nat \<Rightarrow> 'c"   and x :: "'c"   and a :: "'c" assumes "X \<longlonglongrightarrow> x"   and "\<exists>N. \<forall>n\<ge>N. a \<le> X n" shows "a \<le> x"
elims: fixes x :: "('d, 'e) fmap"   and xa :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list"   and y :: "('d, 'e) fmap" assumes "exec_plan x xa = y" obtains s :: "('d, 'e) fmap" where "x = s__" and "xa = []" and "y = s__"   | s :: "('d, 'e) fmap" and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" where "x = s" and "xa = a # as" and "y = exec_plan (state_succ s a) as"
mono_Ndet: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq> Q" shows "(P |-| S) \<sqsubseteq> (Q |-| S)"
G_fcs_eq: fixes a :: "'b" assumes "a \<in> arcs G" shows "GM.face_cycle_set a = proj_arcs_H ` H.face_cycle_set (proj_arcs_G a)"
replicate_app_Cons_same: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "replicate n x @ x # xs = x # replicate n x @ xs"
P: shows "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"
HAInitState_HARoot2: fixes A :: "('d, 'e, 'f) hierauto" shows "HAInitState A \<in> States (Root (SAs A) (CompFun A))"
nth_Cons_pos: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" assumes "0 < n" shows "(x # xs) ! n = xs ! (n - 1)"
join_below: fixes x :: "'b"   and y :: "'b"   and a :: "'b" assumes "compatible x y"   and "x \<sqsubseteq> a"   and "y \<sqsubseteq> a" shows "x \<squnion> y \<sqsubseteq> a"
distance0_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.distance0 v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> - v\<^sub>o * \<delta> - a\<^sub>o * \<delta>\<^sup>2 / 2"
prems_2: shows "wellformed_venv \<Gamma>\<^sub>2"
NextPerformOpD_2: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and \<Delta>neg :: "('loc \<times> 't) multiset"   and \<Delta>mint_msg :: "('b \<times> 'loc \<times> 't) multiset"   and \<Delta>mint_self :: "('loc \<times> 't) multiset" assumes "NextPerformOp' c0 c1 p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self" shows "prop_config c1 = prop_config c0"
antisym_conv1: fixes x :: "'a word"   and y :: "'a word" assumes "\<not> x <s y" shows "(x \<le>s y) = (x = y)"
span_minimal: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "S \<subseteq> T"   and "vec.subspace T" shows "vec.span S \<subseteq> T"
null_ccsqrt: fixes x :: "complex" shows "(ccsqrt x = 0) = (x = 0)"
val_Integer_to_MkI: fixes i :: "Integer" assumes "defined i" shows "i = MkI\<cdot>\<lbrakk>i\<rbrakk>"
sterms_psimps_2: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and pn :: "'a" assumes "wellformed \<Gamma>" shows "sterms \<Gamma> (call(pn)) = sterms \<Gamma> (\<Gamma> pn)"
linordered_field_no_ub: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less x y"
map_le_iff_map_add_commute: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "(f \<subseteq>\<^sub>m f ++ g) = (f ++ g = g ++ f)"
trans_lifted: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "(SOME y. y \<in> A, SOME y. y \<in> B) \<in> r"   and "(SOME y. y \<in> B, SOME y. y \<in> C) \<in> r" shows "(SOME y. y \<in> A, SOME y. y \<in> C) \<in> r"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) FG.map"
rotate6: fixes A6 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A6 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 H))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H))))) \<turnstile> B"
peano_def: fixes zero :: "'a"   and succ :: "'a \<Rightarrow> 'a" shows "peano zero succ \<equiv> (\<forall>m. succ m \<noteq> zero) \<and> (\<forall>m n. (succ m = succ n) = (m = n)) \<and> (\<forall>P n. P zero \<longrightarrow> (\<forall>n. P n \<longrightarrow> P (succ n)) \<longrightarrow> P n)"
PiE_mono: fixes A :: "'f set"   and B :: "'f \<Rightarrow> 'g set"   and C :: "'f \<Rightarrow> 'g set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi\<^sub>E A B \<subseteq> Pi\<^sub>E A C"
simps_1: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree"   and y21 :: "'a tree"   and y22 :: "'a"   and y23 :: "'a tree" shows "(\<langle>x21, x22, x23\<rangle> = \<langle>y21, y22, y23\<rangle>) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
fun_le_trans: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list"   and t3 :: "(val \<times> val) list" assumes "t1 \<lesssim> t2"   and "t2 \<lesssim> t3" shows "t1 \<lesssim> t3"
complex_of_real_strict_mono_iff: fixes x :: "real"   and y :: "real" shows "(complex_of_real x < complex_of_real y) = (x < y)"
exe_wf_osig_imp_wf_osig: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "exe_wf_osig oss" shows "wf_osig (translate_osig oss)"
mono2mono_TCons: fixes orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and b :: "'h"   and t :: "'f \<Rightarrow> ('g, 'h) tllist"   and x :: "'g" assumes "monotone orda (tllist_ord b) t" shows "monotone orda (tllist_ord b) (\<lambda>xa. TCons x (t xa))"
hyps_3: fixes e\<^sub>1 :: "expr" assumes "e = \<lparr>C\<rparr>e\<^sub>0"   and "e'' = \<lparr>C\<rparr>e\<^sub>1" shows "P,E \<turnstile> \<langle>e\<^sub>0,s\<rangle> \<rightarrow>* \<langle>e\<^sub>1,s''\<rangle>"
us_mono: fixes i :: "nat"   and j :: "nat"   and y :: "nat list" assumes "i < j"   and "j < sum_list y" shows "\<^bold>u y i <\<^sub>v \<^bold>u y j"
simps_5: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "next_free (\<phi> U\<^sub>p \<psi>) = (next_free \<phi> \<and> next_free \<psi>)"
order_A: fixes ws :: "nat \<Rightarrow> real" shows "order (A ws) = 2 * N_half"
Re_winding_number_tendsto_part_circlepath: fixes z0 :: "complex"   and a :: "complex" shows "((\<lambda>r. Re (winding_number (part_circlepath z0 r 0 pi) a)) \<longlongrightarrow> 1 / 2) at_top"
A3: shows "\<not> (C = {} \<or> C = ce_sets)"
redundant_infer_def: fixes N :: "'f set"   and \<iota> :: "'f inference" shows "redundant_infer N \<iota> = (\<exists>DD\<subseteq>N. DD \<union> set (side_prems_of \<iota>) \<Turnstile> {concl_of \<iota>} \<and> (\<forall>D\<in>DD. D < main_prem_of \<iota>))"
ffmember_filter: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and A :: "'a fset" shows "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)"
axioms_2: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" assumes "SN_order_pair S NS" shows "SN_ars S"
prj_simps_5: shows "C.dom trg_trg_f.Prj\<^sub>0\<^sub>1 = Chn ((trg f \<star> trg f) \<star> f)"
R1: fixes X :: "('c, 'd) ipurge_rec"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "ipurge_tr_rev_t_setp X \<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr>"   and "D x \<in> sources I D u (x # xs)" shows "ipurge_tr_rev_t_setp X \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = ys @ [x]\<rparr>"
gtrp_env_5: shows "((s_, add_mset s c_), e_, s'_, add_mset s c'_) \<in> ntrs fg"
l2_nostep_defs_3: shows "l2_trans \<equiv> \<Union>m M X Rb Ra A B K Y. (l2_step1 Ra A B \<union> l2_step2 Rb A B X Y \<union> l2_step3 Ra A B X Y \<union> l2_step4 Rb A B X Y \<union> l2_dy_fake_chan M \<union> l2_dy_fake_msg m \<union> l2_lkr_others A \<union> l2_lkr_after A \<union> l2_skr Ra K)\<^sup>="
higher_differentiable_on_SucD: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "higher_differentiable_on S f (Suc n)" shows "higher_differentiable_on S f n"
intro: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Sup :: "'b set \<Rightarrow> 'b" assumes "\<And>A x. \<lbrakk>Complete_Partial_Order.chain less_eq A; x \<in> A\<rbrakk> \<Longrightarrow> less_eq x (Sup A)"   and "\<And>A z. \<lbrakk>Complete_Partial_Order.chain less_eq A; \<And>x. x \<in> A \<Longrightarrow> less_eq x z\<rbrakk> \<Longrightarrow> less_eq (Sup A) z" shows "class.ccpo_axioms Sup less_eq"
degeneralize_ext_def: fixes ecnv :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" shows "degeneralize_ext ecnv \<equiv> if num_acc = 0 then \<lparr>g_V = V \<times> {0}, g_E = {((q, 0), q', 0) |q q'. (q, q') \<in> E}, g_V0 = V0 \<times> {0}, bg_F = V \<times> {0}, \<dots> = ecnv G\<rparr> else \<lparr>g_V = V \<times> {0..<num_acc}, g_E = {((q, i), q', i') |i i' q q'. i < num_acc \<and> (q, q') \<in> E \<and> i' = (if i \<in> acc q then (i + 1) mod num_acc else i)}, g_V0 = V0 \<times> {0}, bg_F = {(q, 0) |q. 0 \<in> acc q}, \<dots> = ecnv G\<rparr>"
code: fixes x :: "complex" shows "inverse x = Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))"
assms_4: fixes p1 :: "perm"   and p2 :: "perm" assumes "P p1"   and "P p2"   and "supp p1 \<subseteq> S"   and "supp p2 \<subseteq> S" shows "P (p1 + p2)"
simps_2: fixes x :: "nat"   and A :: "ty"   and \<Gamma> :: "(nat \<times> ty) list" shows "Venv ((x, A) # \<Gamma>) = (x, V A) # Venv \<Gamma>"
add_le_same_cancel1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus b a) b = less_eq a zero"
x_coord_bounds: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "- 1 / 2 \<le> x_coord x \<and> x_coord x \<le> 1 / 2"
max_x_le_take: fixes y :: "nat list"   and b :: "nat list"   and a :: "nat list"   and i :: "nat"   and l :: "nat" assumes "length y \<le> length b" shows "hlde_ops.max_x a b y i \<le> hlde_ops.max_x a b (take l y) i"
af_decompose_2: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and w :: "'a set list" shows "af (\<phi> or \<psi>) w = af \<phi> w or af \<psi> w"
lim_imp_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and f0 :: "'c" assumes "F \<noteq> \<bottom>"   and "(f \<longlongrightarrow> f0) F" shows "Limsup F f = f0"
assms_5: shows "\<^bold>\<lfloor>u\<^bold>\<rfloor> = \<^bold>\<lfloor>v\<^bold>\<rfloor>"
iterate_to_list_genord_correct: fixes it :: "('c list \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'c list \<Rightarrow> 'c list) \<Rightarrow> 'c list \<Rightarrow> 'c list"   and S0 :: "'c set"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "set_iterator_genord it S0 R" shows "set (iterate_to_list it) = S0 \<and> distinct (iterate_to_list it) \<and> sorted_wrt R (rev (iterate_to_list it))"
columnvector_row_code: fixes v :: "('b, 'c) vec"   and i :: "'c" shows "($) (columnvector_row v i) = (\<lambda>j. v $ i)"