hcju/isabelleps huggingface dataset:
simps_9: fixes f :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "map_option f (Some x2) = Some (f x2)"
freshen_def: fixes n :: "nat" shows "propositional.freshen n = id"
spr_repRels_def: fixes envObs :: "'e \<Rightarrow> ('e, 'f, 'g) BEState \<Rightarrow> 'h \<times> 'g option" shows "spr_repRels envObs \<equiv> \<lambda>a. rel_ext (\<lambda>((u, v), u', v'). envObs a u = envObs a u' \<and> envObs a v = envObs a v')"
lookup_pm_of_poly_single: fixes x :: "'b"   and p :: "'a poly"   and d :: "nat" shows "lookup (pm_of_poly x p) (monomial d x) = poly.coeff p d"
causal_program_history: fixes sb\<^sub>1 :: "'a memref list"   and sb\<^sub>2 :: "'a memref list" assumes "sb = sb\<^sub>1 @ sb\<^sub>2" shows "\<exists>is. instrs sb\<^sub>2 @ is\<^sub>s\<^sub>b = is @ prog_instrs sb\<^sub>2"
assms: shows "P \<longmapsto> \<tau> \<prec> P'"
length_pos_if_in_set: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> list.set xs" shows "0 < length xs"
sup_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(f \<squnion> g) x = f x \<squnion> g x"
conjugate_scalar_prod: fixes v :: "'a vec"   and w :: "'a vec" assumes "dim_vec v = dim_vec w" shows "conjugate (v \<bullet> w) = inner_prod w (conjugate v)"
intros_1: fixes \<eta> :: "nat" shows "S \<eta> (return_spmf (None, (), Void)) (return_spmf (None, Void))"
induct: fixes x1 :: "'a OCL_Types.type"   and x2 :: "'a OCL_Types.type"   and P :: "'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "element_type x1 x2"   and "\<And>\<tau>. P (Collection \<tau>) \<tau>"   and "\<And>\<tau>. P (Set \<tau>) \<tau>"   and "\<And>\<tau>. P (OrderedSet \<tau>) \<tau>"   and "\<And>\<tau>. P (Bag \<tau>) \<tau>"   and "\<And>\<tau>. P (Sequence \<tau>) \<tau>" shows "P x1 x2"
map_delete: fixes m :: "'q"   and a :: "'v" assumes "Q_invar m" shows "Q_lookup (Q_delete a m) = (Q_lookup m)(a := None)"
step_step': fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and S :: "'a st option"   and C :: "'a st option acom" assumes "Abs_Int num' plus' \<gamma>" shows "step (\<gamma>_option (\<gamma>_st \<gamma>) S) (ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) C) \<le> ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) (Abs_Int.step' num' plus' S C)"
real_root_le_0_iff: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "(root n x \<le> 0) = (x \<le> 0)"
\<phi>_def: fixes y :: "'d"   and f :: "'c" shows "\<phi> y f = (THE g. \<guillemotleft>g : y \<rightarrow>\<^sub>D G (C.cod f)\<guillemotright> \<and> \<psi> (C.cod f) g = f)"
assoc'_in_hom'_2: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<guillemotleft>\<a>\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] : local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<tau> \<Rightarrow> (cod \<mu> \<star> cod \<nu>) \<star> cod \<tau>\<guillemotright>"
intros_1: fixes t :: "('a, 'b, 'c, 'd) Tree"   and tset :: "('a, 'b, 'c, 'd) Tree set['a]" assumes "t \<in> set_bset tset" shows "(t, tConj tset) \<in> Tree_wf"
IH: fixes y :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "length y < length T"   and "(\<Gamma>, e, S) \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>y\<^esub> (\<Delta>, z, S)"   and "isVal z" shows "\<Gamma> : e \<Down>\<^bsub>upds_list S\<^esub> \<Delta> : z"
circlepath_add1: fixes z :: "complex"   and r :: "real"   and x :: "real" shows "circlepath z r (x + 1) = circlepath z r x"
While_backwards_frontier_both_ir_valid': fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and k :: "nat"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and t :: "char list \<Rightarrow> int"   and t' :: "char list \<Rightarrow> int" assumes "\<And>n. \<forall>t t'. P (k + Suc n) t t' \<longrightarrow> (\<exists>s s'. P (k + n) s s' \<and> bval b s \<and> bval b' s' \<and> (c, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t')"   and "\<forall>t t'. Q t t' \<longrightarrow> (\<exists>s s'. (\<exists>n. P (k + n) s s') \<and> (WHILE b DO c, s) \<Rightarrow> t \<and> (WHILE b' DO c', s') \<Rightarrow> t')"   and "Q t t'" shows "\<exists>s s'. P k s s' \<and> (WHILE b DO c, s) \<Rightarrow> t \<and> (WHILE b' DO c', s') \<Rightarrow> t'"
vunion_in_VsetI: fixes a :: "V"   and i :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> Vset i"   and "b \<in>\<^sub>\<circ> Vset i" shows "a \<union>\<^sub>\<circ> b \<in>\<^sub>\<circ> Vset i"
nsderiv_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a" shows "NSDERIV f x :> D = (\<forall>h\<in>Infinitesimal - {0}. ((*f* f) (star_of x + h) - star_of (f x)) / h \<approx> star_of D)"
double_cadjoint: fixes U :: "'c \<Rightarrow> 'd" assumes "bounded_clinear U" shows "U\<^sup>\<dagger>\<^sup>\<dagger> = U"
empty_if_no_b_accessible: fixes TA :: "('b, 'a, 'c) tree_automaton_rec_scheme" shows "(ta_lang TA = {}) = (ta_initial TA \<inter> b_accessible (ta_rules TA) = {})"
simps_1: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "mult.Tensor f [] xs = []"
absolutely_integrable_spike_eq: fixes S :: "'c set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "(f absolutely_integrable_on T) = (g absolutely_integrable_on T)"
mono_POR_base_lf_rep: shows "mono POR_base_lf_rep"
le_less_linear: fixes x :: "'b word"   and y :: "'b word" shows "x \<le>s y \<or> y <s x"
zero_less_power2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (power.power one times a 2) = (a \<noteq> zero)"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end0 n (l, r) = (0 < n \<and> (l, r) = ([Bk], Oc \<up> n @ Bk # Oc \<up> n))"
max_x'_def: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "hlde_ops.max_x' a b y i = (if i < length a \<and> hlde_ops.Di' a b i y \<noteq> {} then Min (hlde_ops.Di' a b i y) else Max (set b))"
second_half_eq: shows "mod_arr = arr on {m..<r\<^sub>0}"
endofunctor_H\<^sub>L: fixes g :: "'a" assumes "ide g"   and "g \<star> g \<noteq> null" shows "endofunctor (local.Left g) (H\<^sub>L g)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs"   and "l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent"   and "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs" shows "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
that: fixes e :: "nat \<Rightarrow> real" assumes "y2 = aform_val e X"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "thesis"
add_invar_refineI: fixes f :: "'d \<Rightarrow> 'c nres"   and x :: "'d"   and R :: "('c \<times> 'e) set"   and f' :: "'f \<Rightarrow> 'e nres"   and x' :: "'f"   and I :: "'c \<Rightarrow> bool" assumes "f x \<le> \<Down> R (f' x')"   and "nofail (f x) \<Longrightarrow> f x \<le> SPEC I" shows "f x \<le> \<Down> {(c, a). (c, a) \<in> R \<and> I c} (f' x')"
that: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets M"   and "\<Union> (range A) = space M"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>"   and "disjoint_family A" shows "thesis"
IH: fixes \<rho> :: "complex mat" shows "denote_while_n M0 M1 DS n \<rho> \<in> carrier_mat d d"
norm_harm: fixes n :: "nat" shows "norm (harm n) = harm n"
iwlsFSbSwTR_qInit_respectsP_alphaGood: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSwTR MOD" shows "qInit MOD respectsP alphaGood"
char_poly_matrix_closed: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "char_poly_matrix A \<in> carrier_mat n n"
ex_complex_to_one: fixes z :: "complex" assumes "z \<noteq> 0" shows "\<exists>k. k \<noteq> 0 \<and> k * z = 1"
alpha_preserves_qGood1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "X #= X'"   and "qGood X" shows "qGood X'"
intros_2: fixes t' :: "real"   and t :: "real"   and S :: "'a set"   and s' :: "'a"   and \<omega> :: "(real \<times> 'a) stream"   and s :: "'a" assumes "t' \<le> t"   and "trace_in S t s' \<omega>" shows "trace_in S t s ((t', s') ## \<omega>)"
acc_iff: fixes x :: "'s option"   and y :: "'s option" shows "((x, y) \<in> K2.acc) = (\<exists>n. 0 < K2.p x y n)"
Conseq: fixes P' :: "state \<Rightarrow> bool"   and P :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool"   and Q' :: "state \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile>\<^sub>t {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile>\<^sub>t {P'} c {Q'}"
rOpen: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P (M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P') A\<^sub>P \<Psi>\<^sub>P"   and "x \<in> supp N"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> A\<^sub>P"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* yvec"   and "xvec \<sharp>* yvec"   and "distinct xvec"   and "distinct yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "yvec \<sharp>* \<Psi>\<^sub>P"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M"   and "A\<^sub>P \<sharp>* C"   and "x \<sharp> C"   and "xvec \<sharp>* C"   and "yvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) (M\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>\<langle>N\<rangle> \<prec> P') (x # A\<^sub>P) \<Psi>\<^sub>P"
wadjust_loop_erase_nonempty_5: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and b :: "cell list" assumes "wadjust_on_left_moving m rs (c, b)" shows "c \<noteq> []"
Not_Since: fixes \<phi> :: "formula"   and \<psi> :: "formula"   and I :: "\<I>" assumes "fv (Neg \<phi>) \<subseteq> fv \<psi>"   and "safe_formula \<phi>"   and "\<not> safe_formula (Neg \<phi>)"   and "safe_formula \<psi>"   and "P \<phi>"   and "P \<psi>" shows "P (Since (Neg \<phi>) I \<psi>)"
comp_mkArr: fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't"   and C :: "'t set"   and G :: "'t \<Rightarrow> 't" assumes "S'.arr (S'.mkArr A B F)"   and "S'.arr (S'.mkArr B C G)" shows "S'.mkArr B C G \<cdot>\<acute> S'.mkArr A B F = S'.mkArr A C (G \<circ> F)"
invol_dual_var: fixes x :: "'c \<times> 'd" shows "dual_prod (dual_prod x) = x"
subst_clause_fset_of_list: fixes \<Phi> :: "'b ltln list"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_clause (fset_of_list \<Phi>) m = min_set (list_dnf_to_dnf (subst_clause' \<Phi> m))"
simps_2: fixes p :: "fm"   and q :: "fm"   and k :: "int" shows "d_\<beta> (Or p q) k = (d_\<beta> p k \<and> d_\<beta> q k)"
distinct_69: fixes x3 :: "val"   and x111 :: "'a exp"   and x112 :: "'a exp" shows "Val x3 \<noteq> x111;; x112"
funct_def: fixes g :: "('b \<times> real) set" shows "funct g = (\<lambda>x. SOME y. (x, y) \<in> g)"
orthogonal_comp_self: fixes U :: "'a set" assumes "subspace U" shows "U\<^sup>\<bottom>\<^sup>\<bottom> = U"
fls_shift_of_nat_nth: fixes k :: "int"   and a :: "nat"   and n :: "int" shows "fls_nth (fls_shift k (of_nat a)) n = (if n = - k then of_nat a else (0::'a))"
le_pp_pp: fixes x :: "('a, 'b) pp"   and y :: "('a, 'b) pp" assumes "rep_nat_pp x \<le> rep_nat_pp y" shows "x \<le> y"
card_Pow_filter_one: fixes A :: "'b set" assumes "finite A" shows "card {x \<in> Pow A. card x = 1} = card A"
list_eq_iff: fixes xs :: "'d list"   and ys :: "'d list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
fun_join_def: shows "local.fun_join = (\<Lambda> f g. (\<lambda>x. f x \<squnion> g x))"
continuous_on_interior: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "continuous_on s f"   and "x \<in> interior s" shows "isCont f x"
Collect_split: fixes P :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> bool" shows "{x. P (\<lambda>(xa, y). x xa y)} = curry ` {x. P x}"
comp_galcon_left_unit: fixes F :: "('a, 'b, unit, unit) galcon" shows "I\<^sub>g \<Y>\<^bsub>F\<^esub> \<circ>\<^sub>g F = F"
bag_cards_finite: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag" shows "finite {card (bag t) |t. t \<in> V\<^bsub>T\<^esub>}"
col_not_eq2: shows "column i (Gram_Schmidt_matrix A) \<noteq> column ia (Gram_Schmidt_matrix A)"
ascii_of_char_of_idem: fixes c :: "nat" assumes "c < 128" shows "String.ascii_of (char_of c) = char_of c"
reach4: fixes s :: "Store"   and l :: "Location"   and k :: "Location" assumes "\<not> s\<turnstile> l reachable_from ref k" shows "k \<noteq> l \<or> ref k = nullV"
LstSeqP_Mem_lemma: fixes r :: "tm"   and k :: "tm"   and y :: "tm"   and k' :: "tm"   and z :: "tm" shows "{LstSeqP r k y, HPair k' z IN r, k' IN k} \<turnstile> LstSeqP r k' z"
closed_translation_subtract: fixes S :: "'b set"   and a :: "'b" assumes "closed S" shows "closed ((\<lambda>x. x - a) ` S)"
mult_eq: fixes x :: "nat"   and y :: "nat" shows "x \<otimes>\<^bsub>R\<^esub> y = x * y mod p"
invertibleMtx2: shows "invertible_mat (M_mat_R signs2 subsets2)"
lex_ext_unbounded_iff: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list"   and ys :: "'a list" shows "lex_ext_unbounded f xs ys = ((\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys < length xs, (\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys \<le> length xs)"
thet_WhileFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com" assumes "\<not> bval b s" shows "\<down>\<^sub>t (WHILE b DO c, s) = 1"
point_rep_non_existance: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and x :: "'b" assumes "incidence_system point_set block_collection"   and "x \<notin> point_set" shows "block_collection rep x = 0"
set_intros_2: fixes y :: "'d"   and x31 :: "'d regex"   and x32 :: "'d regex" assumes "y \<in> atms x31" shows "y \<in> atms (Plus x31 x32)"
length_tl: fixes xs :: "'b list" shows "\<^bold>|tl xs\<^bold>| = \<^bold>|xs\<^bold>| - 1"
InvariantConflictClauseCharacterizationAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "\<not> getConflictFlag state" shows "let state' = applyUnitPropagate state in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
maj_fold: fixes f :: "'b \<Rightarrow> nat"   and bound :: "nat"   and w :: "'b list" assumes "\<And>l. l \<in> L \<Longrightarrow> f l \<le> bound"   and "set w \<subseteq> L" shows "foldr (\<lambda>x. (+) (f x)) w 0 \<le> length w * bound"
covifi: shows "ov\<inverse> O f\<inverse> \<subseteq> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse>"
homeomorphism_of_subsets: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and S' :: "'b set"   and T'' :: "'c set"   and T' :: "'c set" assumes "homeomorphism S T f g"   and "S' \<subseteq> S"   and "T'' \<subseteq> T"   and "f ` S' = T'" shows "homeomorphism S' T' f g"
drinks2_vend_invalid: fixes r :: "nat \<Rightarrow>f value option" assumes "\<nexists>n. r $ 2 = Some (Num n)" shows "possible_steps drinks2 2 r STR ''vend'' [] = {||}"
sym_Int: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "sym r"   and "sym s" shows "sym (r \<inter> s)"
is_at_def: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and v :: "nat"   and v' :: "nat" shows "is_at p i branch v v' \<equiv> \<exists>ps. branch !. v = Some (ps, i) \<and> ps !. v' = Some p"
Conj\<^sub>\<alpha>_def': fixes tset\<^sub>\<alpha> :: "('b, 'c, 'd) Tree\<^sub>\<alpha> set['b]" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = abs_Tree\<^sub>\<alpha> (tConj (map_bset rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>))"
IH: fixes v :: "Vertex"   and w :: "Vertex" shows "((\<forall>v. s v \<noteq> v) \<longrightarrow> s v = w \<longrightarrow> s w = v) \<and> (s v = v \<longrightarrow> s w = w \<longrightarrow> v \<noteq> w \<longrightarrow> s = id)"
backward_terminating_iff2_aux: fixes x :: "'a" assumes "path x" shows "x ; 1 \<cdot> 1 ; x\<^sup>T \<cdot> - (1 ; x) \<le> x\<^sup>T\<^sup>\<star>"
stirling_integral_conv_stirling_integral_Suc: fixes s :: "complex"   and n :: "nat" assumes "s \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "0 < n" shows "stirling_integral n s = of_nat n / of_nat (Suc n) * stirling_integral (Suc n) s - complex_of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n)"
intersection_of_inc: fixes P :: "'b set set \<Rightarrow> bool"   and S :: "'b set"   and Q :: "'b set \<Rightarrow> bool" assumes "P {S}"   and "Q S" shows "(P intersection_of Q) S"
HStartBallot_valueChosen2: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HStartBallot s s' q"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
set_grouped_out_arcs: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" assumes "pair_wf_digraph (list_digraph G_list)" shows "sset (grouped_out_arcs G_list) = {out_arcs (with_proj (list_digraph G_list)) v |v. v \<in> pverts (list_digraph G_list) \<and> out_arcs (with_proj (list_digraph G_list)) v \<noteq> {}}"
claim2: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list"   and \<upsilon>' :: "'a list" assumes "trans r"   and "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<upsilon>'| -s r \<down>l (\<sigma> @ \<tau>') -s r \<down>l \<tau>, r|\<upsilon>'| -s r \<down>l \<sigma>' -s r \<down>l \<tau>) \<in> mul_eq r"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "isomorphic a a'" obtains f :: "'a" where "\<guillemotleft>f : a \<rightarrow> a'\<guillemotright> \<and> local.iso f"
restrict_PiE: fixes f :: "'f \<Rightarrow> 'g"   and I :: "'f set"   and S :: "'f \<Rightarrow> 'g set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
simps_2: fixes s :: "state"   and x :: "val"   and n :: "nat" shows "(s:x \<ge> Suc n) = (\<exists>a. x = Addr a \<and> heap s a \<noteq> None \<and> s:get_field s a pred \<ge> n)"
xt1_3: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a = b"   and "c \<le> b" shows "c \<le> a"
a_large: shows "\<forall>\<^sub>F n in sequentially. 1 < a n"
rel_fun_eq_onI: fixes A :: "'a set"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> R (f x) (g x)" shows "rel_fun (eq_on A) R f g"
canon_ang_uminus: fixes \<alpha> :: "real" assumes "\<downharpoonright>\<alpha>\<downharpoonleft> \<noteq> pi" shows "\<downharpoonright>- \<alpha>\<downharpoonleft> = - \<downharpoonright>\<alpha>\<downharpoonleft>"
isDERIV_Power_iff: fixes j :: "nat"   and fa :: "floatarith"   and n :: "nat"   and xs :: "real list" shows "isDERIV j (fa ^\<^sub>e n) xs = (if n = 0 then True else isDERIV j fa xs)"
distinct_3: fixes x3 :: "'a" shows "No_match \<noteq> Match x3"
k: shows "k = (GREATEST k. k \<le> n \<and> \<pi> k = y)"
preserves_cod: fixes f :: "'a" assumes "B.arr f" shows "B.cod (trg\<^sub>B f) = trg\<^sub>B (B.cod f)"
prod_filter_INF: fixes I :: "'b set"   and J :: "'c set"   and A :: "'b \<Rightarrow> 'd filter"   and B :: "'c \<Rightarrow> 'e filter" assumes "I \<noteq> {}"   and "J \<noteq> {}" shows "Inf (A ` I) \<times>\<^sub>F Inf (B ` J) = (INF i\<in>I. INF j\<in>J. A i \<times>\<^sub>F B j)"
Suc_2: shows "Suc k < min (nrows A) (ncols A)"