hcju/isabelleps huggingface dataset:
Q: shows "((\<mu> R\<^sub>r \<eta> \<in> old (fst (n, ns)) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst (n, ns)) \<union> new (fst (n, ns)) \<or> \<eta> \<in> old (fst (n, ns)) \<union> new (fst (n, ns)) \<and> \<mu> R\<^sub>r \<eta> \<in> next (fst (n, ns))) \<and> (\<forall>nd\<in>snd (n, ns). \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst (n, ns)) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old (fst (n, ns)) \<union> new (fst (n, ns)))) \<and> (\<forall>nd\<in>snd (n, ns). \<forall>nd'\<in>snd (n, ns). \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst (n, ns)). nm < name (fst (n, ns))) \<and> 0 < name (fst (n, ns)) \<and> (\<forall>q\<in>snd (n, ns). name q < name (fst (n, ns)) \<and> (\<forall>nm\<in>incoming q. nm < name (fst (n, ns))))) \<and> expand_name_ident (snd (n, ns))"
spacexy: shows "space2 x y = xsep * xsep + ysep * ysep + zsep * zsep"
nat_of_hashcode_def: shows "nat_of_hashcode \<equiv> nat_of_uint32"
mk_linear_orders_decisive_on_set_r: fixes r :: "'a list"   and C :: "'a set"   and B :: "'a set" assumes "r \<in> local.mk_linear_orders C B"   and "decisive_on A f"   and "B \<subseteq> A" shows "set r = B"
fds_const_mult: fixes a :: "'b"   and b :: "'b" shows "fds_const (a * b) = fds_const a * fds_const b"
S2: shows "c_assoc_have_key (pr_gr (loc_upb (c_snd (c_snd n)) x)) (c_pair (c_snd (c_snd n)) x) = 0"
generate_into_G: fixes G :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and a :: "'b"   and H :: "'b set" assumes "Group_Theory.group G composition unit"   and "a \<in> group.generate G composition unit (G \<inter> H)" shows "a \<in> G"
l_inv_ex: fixes x :: "cltn2" assumes "x \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "\<exists>y\<in>carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>. y \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = \<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"
eval_eqI: fixes env :: "v sem_env"   and st1 :: "'a state"   and e1 :: "exp0"   and st2 :: "'a state"   and e2 :: "exp0" assumes "\<And>r. evaluate True env st1 e1 r = evaluate True env st2 e2 r" shows "eval env e1 st1 = eval env e2 st2"
assms_2: fixes \<alpha> :: "freeRes"   and Q' :: "pi" assumes "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
hyps_2: assumes "p \<noteq> 0" shows "card (proots_within p s) \<le> proots_count p s"
pheap_insert: fixes h :: "'b tree"   and x :: "'b" assumes "is_root h"   and "pheap h" shows "pheap (Pairing_Heap_Tree.insert x h)"
eq_diff_eq: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a = c - b) = (a + b = c)"
intros: fixes c :: "('test, 'atom) com" assumes "\<And>s c' s'. (c, s) \<rightarrow>c (c', s') \<Longrightarrow> s \<approx> s' \<and> discr c'"   and "\<And>s s'. (c, s) \<rightarrow>t s' \<Longrightarrow> s \<approx> s'" shows "discr c"
dfa_of_pf'_well_formed: fixes n :: "nat"   and p :: "pf" shows "wf_dfa (dfa_of_pf' n p) n"
not_eq_extremum: fixes a :: "'e" shows "(a \<noteq> bot) = (bot < a)"
deadlockD3: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x" assumes "deadlock s"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" shows "\<forall>w. wset s t \<noteq> \<lfloor>PostWS w\<rfloor>"
card_insert_if: fixes A :: "'g set"   and x :: "'g" assumes "finite A" shows "card (insert x A) = (if x \<in> A then card A else Suc (card A))"
assms_4: shows "\<forall>\<^sub>F x in F. f x \<in> {l1 x..u1 x}"
fake_synth_analz_extr: fixes X :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set" assumes "X \<in> fake bad (synth (analz (extr bad IK CH))) CH" shows "synth (analz (extr bad IK (insert X CH))) = synth (analz (extr bad IK CH))"
step: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "X xs ys"   and "\<not> lnull xs" shows "\<not> lnull ys \<and> (\<not> lnull ys \<longrightarrow> r (lhd xs) (lhd ys) \<or> lhd xs = lhd ys \<and> (X (ltl xs) (ltl ys) \<or> llexord r (ltl xs) (ltl ys)))"
vertical_composite_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" shows "vertical_composite A B F G H \<sigma> \<tau> \<equiv> (category A \<and> category B \<and> functor A B F) \<and> (functor A B G \<and> functor A B H) \<and> natural_transformation A B F G \<sigma> \<and> natural_transformation A B G H \<tau>"
negligibleD_real: fixes f :: "nat \<Rightarrow> real"   and c :: "real" assumes "negligible f" shows "f \<in> o(\<lambda>x. inverse (real x powr c))"
AE_lborel_singleton: fixes c :: "'a" shows "AE x in lborel. x \<noteq> c"
const_OclOr: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and X' :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "const X"   and "const X'" shows "const (X or X')"
sfirst_0: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "P \<omega>" shows "sfirst P \<omega> = 0"
pred_sexpI1: fixes M :: "('c, unit) Old_Datatype.node set"   and N :: "('c, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "N \<in> sexp" shows "(M, Scons M N) \<in> pred_sexp"
simps_28: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Set_global i) ts = (if i < length (global \<C>) \<and> is_mut (global \<C> ! i) then type_update ts [TSome (tg_t (global \<C> ! i))] (Type []) else Bot)"
add_minus_cancel: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus a (plus (uminus a) b) = b"
\<gamma>_def: shows "\<gamma> = Sup (d ` sets M)"
has_bochner_integral_simple_bochner_integrable: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "Bochner_Integration.simple_bochner_integrable M f" shows "has_bochner_integral M f (Bochner_Integration.simple_bochner_integral M f)"
pos_zcount_in_zmset: fixes M :: "'b zmultiset"   and x :: "'b" assumes "0 < zcount M x" shows "x \<in>#\<^sub>z M"
FiniteLinorderEnvironment_axioms: shows "FiniteLinorderEnvironment jkbp agents"
embed_le_1: fixes G :: "'b \<Rightarrow> bool"   and s :: "'b" shows "\<guillemotleft> G \<guillemotright> s \<le> 1"
valid_not_I: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<upsilon> x" shows "\<tau> \<Turnstile> \<upsilon> not x"
delete_node_valid: fixes g :: "('c, 'd) graph"   and v :: "'c" assumes "valid_graph g" shows "valid_graph (delete_node v g)"
take_cols_mat_mul: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and inds :: "nat list" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * take_cols B inds = take_cols (A * B) inds"
m_translate_eq: fixes c :: "'a"   and a :: "'a" assumes "c \<in> carrier R"   and "a \<in> carrier R" shows "m_translate c a = c \<otimes> a"
compose_const_2: fixes a :: "'d"   and g :: "'c \<Rightarrow> 'e" shows "(\<lambda>x. a) \<circ> g = (\<lambda>x. a)"
is_ground_atm_mset_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AA :: "'b multiset" shows "substitution_ops.is_ground_atm_mset subst_atm AA = (\<forall>A. A \<in># AA \<longrightarrow> substitution_ops.is_ground_atm subst_atm A)"
leq_refl: fixes x :: "'a set" shows "x \<subseteq> x"
wtE_def: fixes \<xi> :: "'d \<Rightarrow> univ" shows "U.wtE \<xi> \<equiv> \<forall>x. intT any (\<xi> x)"
wf_strand_first_Send_var_split: fixes S :: "('b, 'a) strand_step list"   and t :: "('b, 'a) Term.term"   and \<I> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t {} S"   and "\<exists>v\<in>wfrestrictedvars\<^sub>s\<^sub>t S. t \<cdot> \<I> \<sqsubseteq> \<I> v" shows "\<exists>S\<^sub>p\<^sub>r\<^sub>e S\<^sub>s\<^sub>u\<^sub>f. \<not> (\<exists>w\<in>wfrestrictedvars\<^sub>s\<^sub>t S\<^sub>p\<^sub>r\<^sub>e. t \<cdot> \<I> \<sqsubseteq> \<I> w) \<and> ((\<exists>t'. S = S\<^sub>p\<^sub>r\<^sub>e @ send\<langle>t'\<rangle>\<^sub>s\<^sub>t # S\<^sub>s\<^sub>u\<^sub>f \<and> t \<cdot> \<I> \<sqsubseteq> t' \<cdot> \<I>) \<or> (\<exists>t' t''. S = S\<^sub>p\<^sub>r\<^sub>e @ \<langle>assign: t' \<doteq> t''\<rangle>\<^sub>s\<^sub>t # S\<^sub>s\<^sub>u\<^sub>f \<and> t \<cdot> \<I> \<sqsubseteq> t' \<cdot> \<I>))"
1: shows "\<forall>x. x \<in> X \<longrightarrow> (\<forall>xa. xa \<in> D x \<and> (\<forall>t. t \<in> D x \<longrightarrow> \<not> t < xa) \<longrightarrow> xa \<in> T u)"
filtermap_ln_at_right: shows "filtermap ln (at_right 0) = at_bot"
case_prodI2: fixes p :: "'b \<times> 'c"   and c :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
tailrec_body_def: fixes a :: "'e \<Rightarrow> 'c nres"   and b :: "'e \<Rightarrow> 'd nres" shows "tailrec_body a b \<equiv> \<lambda>D s. sup (a s \<bind> D) (b s)"
least_eq2: shows "(LEAST n. A' $ i $ n \<noteq> (0::'b)) = (LEAST n. A $ i $ n \<noteq> (0::'b))"
del_block_def: fixes b :: "('a \<times> 'b) set" shows "bkgdd.del_block b \<equiv> remove1_mset b (f bl)"
that_2: fixes b :: "bool" assumes "x = B\<cdot>(Discr b)"   and "y = CB\<cdot>(Discr b)" shows "thesis"
observable_transitionI: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and Q :: "'a"   and \<alpha> :: "'c"   and Q' :: "'a"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>"   and "weak_nominal_ts.tau_transition transition \<tau> P Q"   and "transition Q \<langle>\<alpha>,Q'\<rangle>"   and "weak_nominal_ts.tau_transition transition \<tau> Q' P'" shows "weak_nominal_ts.observable_transition transition \<tau> P \<alpha> P'"
simps_5: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int" shows "aval (Div a\<^sub>1 a\<^sub>2) s = aval a\<^sub>1 s div aval a\<^sub>2 s"
fix_clock_alt_def: fixes x1ba :: "'c state"   and xa :: "'a state \<times> 'b" shows "fix_clock x1ba xa = (case xa of (x3a, x2ba) \<Rightarrow> (update_clock (\<lambda>_. if clock x3a \<le> clock x1ba then clock x3a else clock x1ba) x3a, x2ba))"
weakPsiCongScopeExtSym: fixes x :: "name"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>"   and "x \<sharp> Q" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<doteq> \<lparr>\<nu>x\<rparr>P \<parallel> Q"
globs_set_locals: fixes l :: "lname \<Rightarrow> val option"   and s :: "st" shows "globs (set_locals l s) = globs s"
bound: fixes \<eta> :: "nat" shows "interaction_any_bounded_converter (sim_outer \<eta>) (bound_sim \<eta>)"
SD_preorderD_2: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b pmf"   and p :: "'b pmf" assumes "Order_Predicates.preorder_on carrier le"   and "SD le q p" shows "q \<in> lotteries_on carrier"
keys_monom_multE: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and v :: "'c"   and c :: "'f"   and t :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "term_powerprod pair_of_term term_of_pair"   and "v \<in> keys (term_powerprod.monom_mult pair_of_term term_of_pair c t p)" obtains u :: "'c" where "u \<in> keys p" and "v = term_powerprod.splus pair_of_term term_of_pair t u"
streams_sets: fixes X :: "'a set"   and M :: "'a measure" assumes "X \<in> sets M" shows "streams X \<in> sets (stream_space M)"
length_rows: fixes A :: "'c mat" shows "length (rows A) = dim_row A"
h3: shows "eval (list_conj (map fm.Atom L @ F)) (take var xs @ x # \<Gamma>)"
Q_edges_def: shows "Q_edges \<equiv> \<Union> (H.edges_of_walk ` Q) \<inter> B"
star_subset_lists: fixes A :: "'b list set"   and S :: "'b set" assumes "A \<subseteq> lists S" shows "star A \<subseteq> lists S"
is_regular_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.is_regular R addition multiplication zero unit s U \<equiv> \<forall>\<pp>. \<pp> \<in> U \<longrightarrow> (\<exists>V. comm_ring.is_zariski_open R addition multiplication zero unit V \<and> V \<subseteq> U \<and> \<pp> \<in> V \<and> comm_ring.is_locally_frac R addition multiplication zero s V)"
Syntax_with_Numerals_and_Connectives_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" shows "Syntax_with_Numerals_and_Connectives var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi \<equiv> Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num \<and> Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"
sq_mtx_norm_times: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" shows "\<parallel>A * B\<parallel> \<le> (\<parallel>A\<parallel>) * (\<parallel>B\<parallel>)"
image_uminus_atMost: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.atMost less_eq x = ord.atLeast less_eq (uminus x)"
set_cases_2: fixes e :: "'g"   and a :: "('h, 'g, 'i) generat" assumes "e \<in> generat_outs a" obtains z2 :: "'i" where "a = IO e z2"
A: shows "\<forall>i. open (A i) \<and> z $ i \<in> A i"
f1: shows "memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s1) s1 \<noteq> None \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 \<noteq> None"
abs_eq: fixes xa :: "'c"   and x :: "'c" assumes "eq_onp in_p_image xa xa"   and "eq_onp in_p_image x x" shows "Abs_regular xa \<squnion> Abs_regular x = Abs_regular (xa \<squnion> x)"
r_both_3: fixes i :: "nat"   and f :: "recf"   and j :: "nat"   and g :: "recf"   and t :: "nat"   and x :: "nat"   and v :: "nat" assumes "\<And>x. eval r_phi [i, x] = eval f [x]"   and "\<And>x. eval r_phi [j, x] = eval g [x]"   and "eval r_result1 [t, i, x] \<down>= Suc v" shows "eval r_both [t, i, j, x] \<down>= prod_encode (0, the (eval f [x]))"
col_permutation_4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet B A C"
not_arg_cong_Inr: fixes x :: "'d"   and y :: "'d" assumes "x \<noteq> y" shows "Inr x \<noteq> Inr y"
that: fixes c :: "'b" assumes "homotopic_with_canon (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)" shows "thesis"
cos_monotone_minus_pi_0': fixes y :: "real"   and x :: "real" assumes "- pi \<le> y"   and "y \<le> x"   and "x \<le> 0" shows "cos y \<le> cos x"
atLeastatMost_psubset_iff: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset"   and d :: "'b multiset" shows "(subset_mset.atLeastAtMost a b \<subset> subset_mset.atLeastAtMost c d) = ((\<not> a \<subseteq># b \<or> c \<subseteq># a \<and> b \<subseteq># d \<and> (c \<subset># a \<or> b \<subset># d)) \<and> c \<subseteq># d)"
reindex_cong: fixes l :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'f"   and h :: "'c \<Rightarrow> 'f" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "sum g A = sum h B"
join_def: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and r :: "('b \<times> color) tree" shows "Set2_Join_RBT.join l x r = (if bheight r < bheight l then paint Black (joinR l x r) else if bheight l < bheight r then paint Black (joinL l x r) else B l x r)"
f: shows "f \<in> borel_measurable (lebesgue_on (g ` S'))"
store_typing_imp_mem_agree_None: fixes s :: "s"   and \<S> :: "s_context"   and i :: "nat" assumes "store_typing s \<S>"   and "i < length (s_inst \<S>)"   and "smem_ind s i = None" shows "memory (s_inst \<S> ! i) = None"
assms_2: shows "\<not> A \<longrightarrow> B"
LT12: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (F \<leadsto> H) \<longrightarrow> F \<leadsto> G \<or> H"
cases: fixes a :: "hf" assumes "a \<in> Induction_ax" obtains v :: "hf" and x :: "hf" and x0 :: "hf" and w :: "hf" and xw :: "hf" and xevw :: "hf" and allw :: "hf" and allvw :: "hf" and ax :: "hf" where "a = q_Imp x0 (q_Imp (q_All allvw) (q_All ax))" and "SubstForm v 0 x x0" and "SubstForm v w x xw" and "SubstForm v (q_Eats v w) x xevw" and "AbstForm w 0 (q_Imp x (q_Imp xw xevw)) allw" and "AbstForm v 0 (q_All allw) allvw" and "AbstForm v 0 x ax" and "v \<noteq> w" and "VarNonOccForm w x"
RG_aezdeltafun_closed: fixes g :: "'g"   and r :: "'a" assumes "g \<in> G" shows "r \<delta>\<delta> g \<in> group_ring"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_set_child_nodes_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
not_CONS_self: fixes N :: "('b, unit) Old_Datatype.node set"   and A :: "('b, unit) Old_Datatype.node set set" assumes "N \<in> list A" shows "\<forall>M. N \<noteq> CONS M N"
surjective: fixes r :: "('c, 'd, 'e) pre_digraph_scheme" shows "r = \<lparr>verts = verts r, arcs = arcs r, tail = tail r, head = head r, \<dots> = more r\<rparr>"
to_e_list_2: fixes a :: "b_e"   and b :: "b_e" shows "[$a, $b] = $* [a, b]"
subtermeq_imp_funs_term_subset: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "s \<sqsubseteq> t" shows "funs_term s \<subseteq> funs_term t"
tendsto_ereal: fixes f :: "'c \<Rightarrow> real"   and x :: "real"   and F :: "'c filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. ereal (f x)) \<longlongrightarrow> ereal x) F"
ide_cod: fixes f :: "'a" assumes "Right.arr f" shows "Right.ide (Right.cod f)"
rel_converter_mono: fixes A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and C' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "A' \<le> A"   and "B \<le> B'"   and "C \<le> C'"   and "R' \<le> R" shows "rel_converter A B C R \<le> rel_converter A' B' C' R'"
wcc_equivalence: fixes x :: "'a" shows "equivalence (wcc x)"
Image_Collect_case_prod: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set" shows "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
extremum: fixes a :: "'a upred" shows "a \<sqsubseteq> false"
P_def: shows "P = (\<lambda>I. distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. (T ^^ nat (i - Min I)) x))"
mertens_third_theorem_asymp_equiv': shows "(\<lambda>n. \<Prod>k<n. 1 - 1 / real (nth_prime k)) \<sim>[sequentially] (\<lambda>x. third_mertens_const / ln (real x))"
tensors_ht_l0': fixes j :: "nat"   and r0 :: "nat"   and M :: "nat" assumes "j < r0" shows "tensors_from_net (witness' r0 [M]) $ j = (if j < M then unit_vec M j \<otimes> unit_vec M j else tensor0 [M, M])"
path_connectedin_absolute: fixes X :: "'a topology"   and S :: "'a set" shows "path_connectedin (subtopology X S) S = path_connectedin X S"
maximal_literal_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and L :: "'b literal"   and C :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.maximal_literal trm_ord L C = (\<forall>x. x \<in> C \<longrightarrow> (L, x) \<notin> basic_superposition.lit_ord trm_ord)"
circline_set_def: fixes H :: "circline" shows "circline_set H = {z. on_circline H z}"
Nil: fixes x :: "real \<times> real" shows "ccw'.sortedP x []"