hcju/isabelleps huggingface dataset:
step_def: fixes g :: "'g"   and m :: "'node"   and s :: "'var \<Rightarrow> 'node option"   and v :: "'var" shows "step g m s v \<equiv> if v \<in> oldDefs g m then Some m else s v"
crel_vs_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and v :: "'d"   and s :: "('mem, 'e) state" shows "crel_vs R v s \<equiv> \<forall>M. cmem M \<and> P M \<longrightarrow> (case run_state s M of (v', M') \<Rightarrow> R v v' \<and> cmem M' \<and> P M')"
PROJ_DOM_IDEMPOT: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" shows "prob_proj PROB (prob_dom PROB) = PROB"
Hoare_dgr1_def: fixes P :: "'b \<Rightarrow> 'e"   and D :: "'b \<times> 'b \<Rightarrow> 'e \<Rightarrow> 'e"   and Q :: "'b \<Rightarrow> 'e" shows "\<turnstile>1 P{| D |}Q  \<equiv> \<exists>X. (\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (pair u i) j) \<and> P \<sqsubseteq> \<Squnion>range X \<and> \<Squnion>range X \<sqinter> - grd (step D) \<sqsubseteq> Q"
that: fixes M :: "V" assumes "vreal_ub A M"   and "\<And>T. vreal_ub A T \<Longrightarrow> M \<le>\<^sub>\<real> T" shows "thesis"
poly: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "poly (B $$ (i, j)) k = A $$ (i, j)"
fold_mset_empty: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'d" shows "fold_mset f s {#} = s"
igSubstAllIPresIGWlsAll_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstAllIPresIGWlsAll MOD \<equiv> igSubstIPresIGWls MOD \<and> igSubstAbsIPresIGWlsAbs MOD"
elims: fixes x :: "Monitor.args"   and xa :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and y :: "'a option list set" assumes "result_mmsaux x xa = y" obtains args :: "Monitor.args" and nt :: "nat" and gc :: "nat" and maskL :: "bool list" and maskR :: "bool list" and data_prev :: "(nat \<times> 'a option list set) queue" and data_in :: "(nat \<times> 'a option list set) queue" and tuple_in :: "('a option list, nat) mapping" and tuple_since :: "('a option list, nat) mapping" where "x = args" and "xa = (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)" and "y = Mapping.keys tuple_in"
j_smallest: fixes t :: "nat"   and i :: "nat" assumes "enat (t_C + t) < llength Sts"   and "(C, i) \<in># wP_of_wstate (lnth Sts (t_C + t))" shows "j \<le> i"
fn_norm_le_cong: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "Function_Norm.continuous E f norm"   and "linearform E f"   and "x \<in> E" shows "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
cases: fixes G :: "prog"   and a1 :: "state"   and a2 :: "state" assumes "G\<turnstile>a1 \<midarrow>sxalloc\<rightarrow> a2" obtains s :: "st" where "a1 = Norm s__" and "a2 = Norm s__"   | j :: "jump" and s :: "st" where "a1 = (Some (Jump j), sa__)" and "a2 = (Some (Jump j), sa__)"   | e :: "error" and s :: "st" where "a1 = (Some (Error e), sb__)" and "a2 = (Some (Error e), sb__)"   | a :: "loc" and s :: "st" where "a1 = (Some (Xcpt (Loc a__)), s)" and "a2 = (Some (Xcpt (Loc a__)), s)"   | s0 :: "st" and xn :: "xname" and a :: "loc" and x :: "abopt" and s1 :: "st" where "a1 = (Some (Xcpt (Std xn)), s0)" and "a2 = (Some (Xcpt (Loc a)), s1)" and "G\<turnstile>Norm s0 \<midarrow>halloc CInst (SXcpt xn)\<succ>a\<rightarrow> (x, s1)"
iso_char: fixes f :: "'a" shows "local.iso f = (C.iso f \<and> arr f \<and> arr (C.inv f))"
INTERP_def: fixes N :: "'b literal multiset set" shows "INTERP N = \<Union> (production N ` N)"
fresh_1: fixes a :: "name" shows "a \<sharp> TBase = True"
ir_valid_decomp: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_valid P c c' Q = (ir_valid P c SKIP (decomp P c c' Q) \<and> ir_valid (decomp P c c' Q) SKIP c' Q)"
vdw_def: fixes k :: "nat"   and l :: "nat"   and n :: "nat" shows "vdw k l n = (\<forall>a b col. a + int n \<le> b + 1 \<and> col \<in> {a..b} \<rightarrow> {..<k} \<longrightarrow> (\<exists>j start step. j < k \<and> 0 < step \<and> is_arith_prog_on l start step a b \<and> arith_prog start step ` {..<l} \<subseteq> col -` {j} \<inter> {a..b}))"
indexed_const_inj_on: shows "inj_on indexed_const (carrier R)"
expected_value_is_utility_function: fixes outcomes :: "'a set"   and x :: "'a pmf"   and y :: "'a pmf"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and u :: "'a \<Rightarrow> real" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "x \<in> \<P> outcomes"   and "y \<in> \<P> outcomes"   and "ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "(measure_pmf.expectation y u \<le> measure_pmf.expectation x u) = x \<succeq>[\<R>] y"
null_eqI: fixes n :: "'b" assumes "\<And>f. C' n f = n \<and> C' f n = n" shows "n = C'.null"
get_div: fixes p :: "nat"   and a :: "nat"   and m :: "nat" assumes "p < a" shows "m = (m * a + p) div a"
True: shows "\<exists>x\<le>n. 0 < f x"
max_input_Gt: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "GExp.max_input (Gt a1 a2) = max (AExp.max_input a1) (AExp.max_input a2)"
orderE: fixes b :: "'g multiset"   and a :: "'g multiset" assumes "b \<subseteq># a" obtains "a = a \<union># b"
LID\<^sub>GI: fixes s :: "'e \<Rightarrow> (('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr) option"   and r :: "'e"   and ls :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr"   and l :: "'d" assumes "s r = Some ls"   and "l \<in> LID\<^sub>L ls" shows "l \<in> LID\<^sub>G s"
UnI2: fixes c :: "'k"   and B :: "'k set"   and A :: "'k set" assumes "c \<in> B" shows "c \<in> A \<union> B"
le_supI2: fixes x :: "'a"   and b :: "'a"   and a :: "'a" assumes "x \<le> b" shows "x \<le> a \<sqinter> b"
secure2_def: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L" shows "secure2 p G c H = (\<forall>s t ss tt. s , c \<Down> t  \<longrightarrow> ss , c \<Down> tt  \<longrightarrow> EQ G p s ss \<longrightarrow> EQ H p t tt)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   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_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_new_character_data_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"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'b" shows "lookup (swap0 x xa xb) = (\<lambda>x. lookup xb (swap x xa x))"
mulex1_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "mulex1 P = (\<lambda>M N. (M, N) \<in> mult1 {(x, y). P x y})"
simps_9: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and v :: "'c match_expr"   and rs :: "'c rule list" shows "no_matching_Goto \<gamma> p (Rule v Empty # rs) = no_matching_Goto \<gamma> p rs"
ta_union_wrap_def: fixes TA :: "('a, 'c, 'd) tree_automaton_rec_scheme"   and TA' :: "('b, 'c, 'e) tree_automaton_rec_scheme" shows "ta_union_wrap TA TA' = ta_union (ta_remap USW1 TA) (ta_remap USW2 TA')"
simps_2: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and p :: "real mpoly" shows "substInfinitesimalLinear var b c (Less p) = liftmap (\<lambda>x A. fm.Atom (linear_substitution (var + x) (liftPoly 0 x b) (liftPoly 0 x c) A)) (convertDerivative var p) 0"
space: shows "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
ord0rel_def: shows "ord0rel = {(x, y). x \<le> y \<and> y \<le> x}"
transpose_carrier_mat: fixes A :: "'a mat"   and nc :: "nat"   and nr :: "nat" shows "(A\<^sup>T \<in> carrier_mat nc nr) = (A \<in> carrier_mat nr nc)"
subprob_space_kernel: fixes K :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and N :: "'b measure"   and a :: "'a" assumes "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "a \<in> space M" shows "subprob_space (K a)"
splus_pp_term: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and rep_nat_term :: "'b \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.nat_pp_term plus zero rep_nat_term splus" shows "splus = plus"
abs_summable_on_altdef: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "(f abs_summable_on A) = set_integrable (count_space UNIV) A f"
empty_correct: fixes q :: "('c, 'd) SkewBinomialTree list" shows "(q = []) = (queue_to_multiset q = {#})"
arr_char: fixes f :: "'a" shows "V.arr f = (f \<in> Collect arr)"
rword_is_strict_rewrite_ord: shows "is_strict_rewrite_ord rword_strict"
trace_zero: fixes n :: "nat" shows "trace (0\<^sub>m n n) = (0::'a)"
subset_UnderS_AboveS: fixes B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "B \<subseteq> Field r" shows "B \<subseteq> UnderS r (AboveS r B)"
invariantE: fixes P :: "'c \<Rightarrow> bool"   and succs :: "'c \<Rightarrow> 'c list"   and g' :: "'c"   and g :: "'c" assumes "invariant P succs"   and "g' \<in> set (succs g)"   and "P g" shows "P g'"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "subseq b a"   and "subseq c b" shows "subseq c a"
step: fixes p :: "('c, 'a, ('c, 'a, 'b) rpv) generat spmf" assumes "lossless_spmf p"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> lossless_gpv \<I> (c input)"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> P (c input)" shows "P (GPV p)"
var: fixes t :: "'trm" assumes "set (x # map snd txs) \<subseteq> var"   and "set (Var y # t # map fst txs) \<subseteq> trm"   and "set [] \<subseteq> fmla"   and "t \<in> set (Var y # t # map fst txs)" shows "set (getFrN (x # map snd txs) (Var y # t # map fst txs) [] (Suc (length txs))) \<inter> set (x # map snd txs) = {}"
subspB: shows "{x. a \<bullet> x = 0} \<subseteq> span B"
confs_conv_map: fixes P :: "'a prog"   and h :: "'heap"   and vs :: "'addr val list"   and Ts' :: "ty list" shows "(P,h \<turnstile> vs [:\<le>] Ts') = (\<exists>Ts. map typeof\<^bsub>h\<^esub> vs = map Some Ts \<and> P \<turnstile> Ts [\<le>] Ts')"
well_foundedI_pf: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>X. \<lbrakk>X \<subseteq> A; X \<subseteq> (less ``` X) \<inter> A\<rbrakk> \<Longrightarrow> X = {}" shows "well_founded A less"
derived_set_subset: fixes S :: "'b set"   and X :: "'b topology" assumes "S \<subseteq> topspace X" shows "(X derived_set_of S \<subseteq> S) = closedin X S"
S_satisfies_eqvt: fixes P :: "'state"   and \<phi> :: "unit"   and p :: "perm" assumes "P \<turnstile>\<^sub>S \<phi>" shows "p \<bullet> P \<turnstile>\<^sub>S p \<bullet> \<phi>"
components_are_iso: fixes a :: "'c" assumes "Cop.ide a" shows "S.iso (\<Phi>\<Psi>'.map a)"
partitioning_equi: fixes ts :: "'b set list"   and ss :: "'b set list" assumes "{} \<notin> set ts"   and "disjoint_list_rec ts"   and "\<Union> (set ss) \<subseteq> \<Union> (set ts)" shows "set (partitioning1 ss ts) = partitioning_nontail ss (set ts) - {{}}"
w: shows "(WHILE b DO c, ps') \<Rightarrow>\<^sub>A t' \<Down> ps''"
set_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "funs_term (Fun x21 x22) = insert x21 (\<Union> (funs_term ` set x22))"
lrev_work3_def: shows "lrev_work3 \<equiv> fix\<cdot>lrev_body3"
open_INT: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'b set" assumes "class.topological_space open"   and "finite A"   and "\<forall>x\<in>A. open (B x)" shows "open (\<Inter> (B ` A))"
bdd_above_bot: fixes A :: "'a list set" shows "prefix_order.bdd_below A"
admissible_fun: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "'a set \<Rightarrow> 'a"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "partial_function_definitions le lub"   and "\<And>x. ccpo.admissible lub le (Q x)" shows "ccpo.admissible (fun_lub lub) (fun_ord le) (\<lambda>f. \<forall>x. Q x (f x))"
cod_comp: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "V.seq g f" shows "V.cod (g \<bullet> f) = V.cod g"
arrI: fixes f :: "'a2"   and a :: "'a2"   and b :: "'a2" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>2 b\<guillemotright>" shows "A2.arr f"
of_int_1_le_iff: 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 z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq one (ring_1.of_int one plus zero minus z) = (1 \<le> z)"
return_node_determines_call_node: fixes n :: "'node" assumes "return_node n" shows "\<exists>!n'. \<exists>a a'. valid_edge a \<and> n' = sourcenode a \<and> valid_edge a' \<and> a' \<in> get_return_edges a \<and> n = targetnode a'"
PiE_E: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and x :: "'c" assumes "f \<in> Pi\<^sub>E A B" obtains "x \<in> A" and "f x \<in> B x"   | "x \<notin> A" and "f x = undefined"
encode_problem_parallel_complete_iv: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "is_valid_problem_strips \<Pi>"   and "is_parallel_solution_for_problem \<Pi> \<pi>"   and "\<forall>k op. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<longrightarrow> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"   and "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)"   and "\<forall>v l. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<le> l \<and> l < length \<pi> + 1 \<longrightarrow> \<A> (State l (index (\<Pi>\<^sub>\<V>) v)) = \<A> (State (length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1) (index (\<Pi>\<^sub>\<V>) v))" shows "\<A> \<Turnstile> encode_all_frame_axioms \<Pi> (length \<pi>)"
ActualActualI: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<A>\<phi> in v]" shows "[\<^bold>\<A>\<^bold>\<A>\<phi> in v]"
eq2: shows "adjoint (exexH_k k) * adjoint (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) = adjoint (exexH_k (Suc k))"
definition_of_range: fixes V1 :: "'b set"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" shows "endpoint_inV V1 ` matching_i 1 V E M L = {v. \<exists>e\<in>matching_i 1 V E M L. endpoint_inV V1 e = v}"
swap_general: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a \<noteq> b"   and "c \<noteq> d" shows "Fun.swap a b id \<circ> Fun.swap c d id = id \<or> (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id)"
h1: shows "\<forall>j\<le>D + z. str (t + j) = x"
butlast_take: fixes n :: "nat"   and xs :: "'g list" assumes "n \<le> length xs" shows "butlast (take n xs) = take (n - 1) xs"
Until: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula"   and \<psi> :: "'a mformula"   and \<psi>' :: "'a MFOTL.formula"   and pos :: "bool"   and \<phi>'' :: "'a MFOTL.formula"   and buf :: "'a option list set list \<times> 'a option list set list"   and nts :: "nat list"   and I :: "\<I>"   and aux :: "(nat \<times> 'a option list set \<times> 'a option list set) list" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j n R \<psi> \<psi>'"   and "if pos then \<phi>'' = \<phi>' else \<phi>'' = formula.Neg \<phi>'"   and "safe_formula \<phi>'' = pos"   and "fv \<phi>' \<subseteq> fv \<psi>'"   and "wf_mbuf2' \<sigma> j n R \<phi>' \<psi>' buf"   and "wf_ts \<sigma> j \<phi>' \<psi>' nts"   and "wf_until_aux \<sigma> n R pos \<phi>' I \<psi>' aux (Monitor.progress \<sigma> (formula.Until \<phi>'' I \<psi>') j)"   and "Monitor.progress \<sigma> (formula.Until \<phi>'' I \<psi>') j + length aux = min (Monitor.progress \<sigma> \<phi>' j) (Monitor.progress \<sigma> \<psi>' j)" shows "wf_mformula \<sigma> j n R (MUntil pos \<phi> I \<psi> buf nts aux) (formula.Until \<phi>'' I \<psi>')"
concI: fixes u :: "'c list"   and A :: "'c list set"   and v :: "'c list"   and B :: "'c list set" assumes "u \<in> A"   and "v \<in> B" shows "u @ v \<in> A @@ B"
liftE_handleE': fixes a :: "'c \<Rightarrow> ('e \<times> 'c) set \<times> bool"   and b :: "'f \<Rightarrow> 'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool" shows "(liftE a <handle2> b) = liftE a"
card_Suc_eq: fixes A :: "'b set"   and k :: "nat" shows "(card A = Suc k) = (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
CSC_Resolution_pre: fixes \<Gamma> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow>\<^sub>n"   and "\<forall>\<gamma>\<in>#\<Gamma>. is_cnf \<gamma>" shows "\<Union> (cnf ` set_mset \<Gamma>) \<turnstile> \<box>"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'b list"   and xs\<^sub>2 :: "'b list"   and ys\<^sub>1 :: "'b list"   and ys\<^sub>2 :: "'b list" shows "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if length xs\<^sub>1 \<le> length ys\<^sub>1 then xs\<^sub>1 = take (length xs\<^sub>1) ys\<^sub>1 \<and> xs\<^sub>2 = drop (length xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2 else take (length ys\<^sub>1) xs\<^sub>1 = ys\<^sub>1 \<and> drop (length ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
HStartBallot_HInv5_q1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "p \<noteq> q"   and "maxBalInp s (bal (dblock s q)) (inp (dblock s q))" shows "maxBalInp s' (bal (dblock s' q)) (inp (dblock s' q))"
ide_prod: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "ide (local.prod a b)"
arc_to_block_is_inj_out_arcs: fixes v :: "'a" shows "inj_on arc_to_block (out_arcs G v)"
u_def: shows "u = (\<lambda>n. SOME z. z \<in> A n \<and> x < z)"
neg: shows "negligible {x \<in> g ` S. x \<notin> g ` {x \<in> S. invertible (matrix (g' x))} \<and> f x \<noteq> 0}"
last_prog_hd_prog_append: fixes p\<^sub>s\<^sub>b :: "'b"   and sb :: "'b memref list"   and sb' :: "'b memref list" shows "last_prog (hd_prog p\<^sub>s\<^sub>b (sb @ sb')) sb = last_prog (hd_prog p\<^sub>s\<^sub>b sb') sb"
assms_3: shows "(p ^^ i) a = (p ^^ j) a"
rel_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp R" shows "transp (rel_option R)"
intros: fixes q :: "'d"   and l :: "'e"   and qs :: "'d list"   and \<delta> :: "('d, 'e) ta_rule set" assumes "q \<rightarrow> l qs \<in> \<delta>"   and "\<And>x. x \<in> set qs \<Longrightarrow> x \<in> b_accessible \<delta>" shows "q \<in> b_accessible \<delta>"
output_bind_resumption: fixes x :: "('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" assumes "\<not> Resumption.resumption.is_Done (x \<bind> f)" shows "Resumption.resumption.output (x \<bind> f) = (if Resumption.resumption.is_Done x then Resumption.resumption.output (f (the (Resumption.resumption.result x))) else Resumption.resumption.output x)"
intro: fixes set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')" shows "l_set_mode_get_shadow_root_axioms set_mode_locs get_shadow_root_locs"
less_eq_list_def: shows "(\<le>) = lexordp_eq"
ddvd_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ddvd y"   and "y ddvd z" shows "x ddvd z"
map_simp_ide: 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"   and a :: "'d" assumes "vertical_composite A B F G H \<sigma> \<tau>"   and "partial_magma.ide A a" shows "vertical_composite.map A B \<sigma> \<tau> a = B (\<tau> a) (\<sigma> a)"
path_in_V: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list" assumes "Digraph G"   and "Digraph.walk G xs \<and> distinct xs" shows "set xs \<subseteq> V\<^bsub>G\<^esub>"
select_convs_2: fixes ok :: "bool"   and wait :: "bool"   and tr :: "'c list"   and ref :: "'c set"   and more :: "'d" shows "tr \<lparr>ok = ok, wait = wait, tr = tr, ref = ref, \<dots> = more\<rparr> = tr"
pid_gstep_succI: fixes pid :: "nat"   and gc :: "('c, 'ls, 'gs) pid_global_config"   and a :: "'a"   and c' :: "'c"   and ls' :: "'ls"   and gs' :: "'gs" assumes "pid_valid gc pid"   and "cstep (cmd_of_pid gc pid) a c'"   and "en (fs_of_pid gc pid) a"   and "ex (fs_of_pid gc pid) a = (ls', gs')" shows "\<lparr>pid_global_config.processes = (pid_global_config.processes gc)[pid := \<lparr>command = c', state = ls'\<rparr>], state = gs'\<rparr> \<in> pid_gstep_succ gc"
member_remove: fixes x :: "'c"   and y :: "'c"   and A :: "'c set" shows "(x \<in> Set.remove y A) = (x \<in> A \<and> x \<noteq> y)"