hcju/isabelleps huggingface dataset:
InvariantsNoDecisionsWhenConflictNorUnitAfterApplyBackjump_2: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantUniqC (getC state)"   and "getC state \<noteq> [opposite (getCl state)]"   and "InvariantNoDecisionsWhenConflict (butlast (getF state)) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenUnit (butlast (getF state)) (getM state) (currentLevel (getM state))"   and "getF state \<noteq> []"   and "last (getF state) = getC state"   and "InvariantNoDecisionsWhenConflict [getC state] (getM state) (getBackjumpLevel state)"   and "InvariantNoDecisionsWhenUnit [getC state] (getM state) (getBackjumpLevel state)"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)" shows "let state' = applyBackjump state in InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state')) \<and> InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state'))"
soundI_memv': fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and R :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "\<And>I \<nu>. \<lbrakk>is_interp I; \<And>\<phi> \<nu>. List.member SG \<phi> \<Longrightarrow> \<nu> \<in> seq_sem I \<phi>\<rbrakk> \<Longrightarrow> \<nu> \<in> seq_sem I C"   and "R = (SG, C)" shows "sound R"
strict_partial_order_def: fixes r :: "('b \<times> 'b) set" shows "strict_partial_order r \<equiv> trans r \<and> irrefl r"
normal_form_zero_iff: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "F \<subseteq> P[X]" shows "(normal_form F p = 0) = (p \<in> ideal F)"
linear: fixes x :: "'a"   and y :: "'a" assumes "x \<in> U"   and "y \<in> U" shows "x \<le>\<^sub>o\<^sub>w y \<or> y \<le>\<^sub>o\<^sub>w x"
acquired_mono_in: fixes x :: "nat"   and pending_write :: "bool"   and xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "x \<in> acquired pending_write xs A"   and "A \<subseteq> B" shows "x \<in> acquired pending_write xs B"
weak_preservation_of_barbs: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_preserves_barbs Rel CWB" shows "weak_barbed_preservation_cond Rel CWB"
0: shows "\<forall>a\<in>A. a \<in> local.underS b"
induct: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "listsp A x"   and "P []"   and "\<And>a l. \<lbrakk>A a; listsp A l; P l\<rbrakk> \<Longrightarrow> P (a # l)" shows "P x"
insert_4: fixes W :: "'a set" assumes "W \<in> I"   and "\<forall>a\<in>insert U F. W \<subseteq> a" shows "thesis"
leadsto_classical: fixes F :: "'b \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> (Init F \<and> \<box>\<not> G \<leadsto> G) \<longrightarrow> (F \<leadsto> G)"
K_contractE: fixes r :: "comb"   and P :: "bool" assumes "K \<rightarrow>\<^sup>1 r" shows "P"
VectorSpaceHom_def: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" shows "VectorSpaceHom smult V smult' T \<equiv> ModuleHom smult V smult' T"
cardinal_cong: fixes x :: "V"   and y :: "V" assumes "x \<approx>\<^sub>\<circ> y" shows "vcard x = vcard y"
locally_sound_mode_use_def2: fixes lc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" shows "abs.locally_sound_mode_use lc \<equiv> \<forall>lc'\<in>abs.loc_reach lc. abs.respects_own_guarantees (fst lc')"
induct: fixes P :: "('b \<Rightarrow> 'c \<Rightarrow> assn) \<Rightarrow> 'b list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and a1 :: "'b list"   and a2 :: "'c list" assumes "\<And>Pa. P Pa [] []"   and "\<And>Pa a as c cs. P Pa as cs \<Longrightarrow> P Pa (a # as) (c # cs)"   and "\<And>uu_ v va. P uu_ (v # va) []"   and "\<And>uu_ v va. P uu_ [] (v # va)" shows "P a0 a1 a2"
normalize_1_iff: fixes a :: "'a" shows "(normalize a = (1::'a)) = is_unit a"
get_ancestors_also_parent: fixes h :: "(_) heap"   and some_ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and child :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_ancestors some_ptr \<rightarrow>\<^sub>r ancestors"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set ancestors"   and "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some parent"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h" shows "parent \<in> set ancestors"
assms_2: shows "list_all2 P ts\<^sub>1 ts\<^sub>2"
run_state_get: fixes s :: "'g" shows "run_state State_Monad.get s = (s, s)"
m_assoc: fixes x :: "int vec"   and y :: "int vec"   and z :: "int vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "x + y + z = x + (y + z)"
Factor_Node_2: fixes u :: "int poly factor_tree"   and f :: "int poly" assumes "u \<in> sub_trees (product_factor_tree p r)"   and "factor_node_info u = f"   and "\<And>x. x \<in># factors_of_factor_tree r \<Longrightarrow> Mp x = x" shows "Mp f = f \<and> f = Mp (\<Prod>\<^sub># (factors_of_factor_tree u)) \<and> factors_of_factor_tree (product_factor_tree p r) = factors_of_factor_tree r"
take_Suc: fixes xs :: "'d list"   and n :: "nat" assumes "xs \<noteq> []" shows "xs \<down> Suc n = hd xs # tl xs \<down> n"
shift_bounds_cl_Suc_ivl: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..Suc n} = (\<Sum>i = m..n. g (Suc i))"
er_def: fixes x :: "'a" shows "er x = x * x / n - (1::'a)"
simps: fixes list_rel :: "('b \<times> 'b option) set"   and oid :: "'b"   and ref :: "'b" shows "insert_alt list_rel (oid, ref) = (if \<exists>n. (ref, n) \<in> list_rel then {(p, n). (p, n) \<in> list_rel \<and> p \<noteq> ref} \<union> {(ref, Some oid)} \<union> {(i, n). i = oid \<and> (ref, n) \<in> list_rel} else list_rel)"
mat_inverse_def: fixes A :: "'c mat" shows "mat_inverse A = (if dim_row A = dim_col A then let one = 1\<^sub>m (dim_row A) in case gauss_jordan A one of (B, C) \<Rightarrow> if B = one then Some C else None else None)"
cases: fixes a :: "nat set"   and k :: "nat" assumes "a \<in> i_set_mult k" obtains n :: "nat" where "a = [n__\<dots>]"   | n :: "nat" where "a = [\<dots>na__]"   | n :: "nat" and d :: "nat" where "a = [n\<dots>,d]"   | r :: "nat" and m :: "nat" where "a = [ r__, mod m__ * k ]"   | r :: "nat" and m :: "nat" and c :: "nat" where "a = [ r, mod m * k, c ]"
mult_mat_vec_hom: fixes A :: "int mat"   and nr :: "nat"   and n :: "nat"   and v :: "int vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "of_int_hom.vec_hom (A *\<^sub>v v) = of_int_hom.mat_hom A *\<^sub>v of_int_hom.vec_hom v"
le: fixes y :: "'s" shows "measure_pmf.expectation N (\<lambda>x. pmf (K x) y) \<le> pmf N y"
diff_single_eq_union_zmset: fixes M :: "'b zmultiset"   and x :: "'b"   and N :: "'b zmultiset" shows "(M - {#x#}\<^sub>z = N) = (M = add_zmset x N)"
path_sourcenode: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" shows "hd (CFG.sourcenodes sourcenode as) = n"
extendEquivalentFormulaWithEntailedClause: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list"   and clause :: "Literal list" assumes "equivalentFormulae formula1 formula2"   and "formulaEntailsClause formula2 clause" shows "equivalentFormulae formula1 (formula2 @ [clause])"
dom_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "local.dom f = (if arr f then if f = One then One else Zero else null)"
pred_entailment_refl: fixes P :: "'BExp set" shows "P \<turnstile> P"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "L'.C.in_hom f a b"   and "L'.C.in_hom g b c" shows "L'.C.seq g f"
fst_special_simp: fixes a :: "'b"   and x :: "'a \<times> 'c" shows "fst (case x of (s, m) \<Rightarrow> (s, a, m)) = fst x"
read_only_unowned: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'sb"   and \<D> :: "'dirty"   and \<O> :: "nat set"   and \<R> :: "'rels" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<O> \<inter> read_only \<S> = {}"
set: fixes x :: "'b list" shows "set_array (Array x) = set x"
fun_eq_on_im: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "fun_eq_on f g A"   and "order.greater_eq A B" shows "f ` B = g ` B"
valid_lineI: fixes l :: "color list" assumes "\<And>i. i < length l \<Longrightarrow> valid_point l i" shows "valid_line l"
subset_succs_def: fixes A :: "bool list bdd list \<times> bool list"   and qs :: "bool list" shows "subset_succs A qs = add_leaves (subsetbdd (fst A) qs (nfa_emptybdd (length qs))) []"
indep_vars_iff_distr_eq_PiM: fixes I :: "'b set"   and X :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'd"   and M' :: "'b \<Rightarrow> 'd measure" assumes "I \<noteq> {}"   and "\<And>i. P.P.random_variable (M' i) (X i)" shows "P.P.indep_vars M' X I = (distr (Pi\<^sub>M UNIV (\<lambda>_. M)) (Pi\<^sub>M I M') (\<lambda>x. \<lambda>i\<in>I. X i x) = Pi\<^sub>M I (\<lambda>i. distr (Pi\<^sub>M UNIV (\<lambda>_. M)) (M' i) (X i)))"
le_inf_word: fixes w\<^sub>2 :: "'action stream"   and w\<^sub>1 :: "'action stream"   and p :: "'state" assumes "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1"   and "run w\<^sub>1 p" shows "run w\<^sub>2 p"
bij_swap_comp: fixes p :: "'a \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "bij p" shows "a \<rightleftharpoons>\<^sub>F b \<circ> p = Fun.swap (inv p a) (inv p b) p"
Inter_iff: fixes A :: "'c"   and C :: "'c set set" shows "s2p (\<Inter> C) A = (\<forall>X\<in>C. s2p X A)"
assms_2: shows "\<forall>n. \<phi> i (f \<triangleright> n) \<down>"
below_lubI: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "x \<sqsubseteq> lub S"
well_def_step: fixes subsets1 :: "nat list list"   and qs1 :: "real poly list"   and subsets2 :: "nat list list"   and qs2 :: "real poly list" assumes "all_list_constr subsets1 (length qs1)"   and "all_list_constr subsets2 (length qs2)" shows "all_list_constr (subsets_smash (length qs1) subsets1 subsets2) (length (qs1 @ qs2))"
linear_poly_sum: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "p \<lbrace> v \<rbrace> = (\<Sum>x\<in>vars p. Abstract_Linear_Poly.coeff p x *R v x)"
merge_length: fixes a :: "'a option list"   and n :: "nat"   and b :: "'a option list" assumes "length a = n"   and "length b = n" shows "length (merge a b) = n"
filter_inf: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and G :: "'b set" assumes "class.directed_semilattice_inf inf less_eq less"   and "ord.filter less_eq F"   and "ord.filter less_eq G" shows "ord.filter less_eq (F \<inter> G)"
Hyp_in_outPortsRule: fixes h :: "'form"   and c :: "('form, 'var) antecedent"   and r :: "'rule" shows "(Hyp h c |\<in>| outPortsRule r) = (c |\<in>| f_antecedent r \<and> h |\<in>| a_hyps c)"
enum_inj: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s"   and "i \<le> n"   and "j \<le> n" shows "(kuhn_simplex.enum base upd i = kuhn_simplex.enum base upd j) = (i = j)"
restrict_eq_linorder: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D' :: "'a set" assumes "eq_linorder_on D cmp"   and "D' \<subseteq> D" shows "eq_linorder_on D' cmp"
ncols_eq_card_columns: fixes A :: "(('a, 'b) vec, 'c) vec" shows "ncols_iarray (matrix_to_iarray A) = CARD('b)"
quasi_inverse_simps_2: fixes f :: "'b" assumes "D.equivalence_map f" shows "D.ide f\<^sup>~\<^sup>D"
dim_span: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set" assumes "vector_space scale" shows "vector_space.dim scale (module.span scale S) = vector_space.dim scale S"
subst_atm_mset_lists_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AAs :: "'b multiset list"   and \<sigma>s :: "'c list" shows "substitution_ops.subst_atm_mset_lists subst_atm AAs \<sigma>s = Map2.map2 (substitution_ops.subst_atm_mset subst_atm) AAs \<sigma>s"
simps_2: fixes t :: "('c, 'd) BinomialTree"   and v :: "('c, 'd) BinomialTree"   and va :: "('c, 'd) BinomialTree list" shows "getMinTree (t # v # va) = (if prio t \<le> prio (getMinTree (v # va)) then t else getMinTree (v # va))"
monoid_homomorphism_axioms_def: fixes \<eta> :: "'d \<Rightarrow> 'e"   and M :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d"   and composition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and unit' :: "'e" shows "monoid_homomorphism_axioms \<eta> M composition unit composition' unit' \<equiv> (\<forall>x y. x \<in> M \<longrightarrow> y \<in> M \<longrightarrow> \<eta> (composition x y) = composition' (\<eta> x) (\<eta> y)) \<and> \<eta> unit = unit'"
simps_1: fixes n :: "nat"   and m :: "nat"   and x :: "int \<times> int" shows "swap n m (C x) = C x"
add_nonneg_eq_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero x"   and "less_eq zero y" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
n_def: shows "n \<equiv> min (order x q) (order x p)"
ide_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.ide (trg\<^sub>B \<mu>)"
ub: shows "(\<forall>z. x < z \<and> z < ub \<longrightarrow> 0 < poly p z) \<or> (\<forall>z. x < z \<and> z < ub \<longrightarrow> poly p z < 0)"
distinct_17: fixes x2 :: "nat"   and x5 :: "msg" shows "Number x2 \<noteq> Hash x5"
atLeastAtMost_iff: fixes i :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and l :: "'d"   and u :: "'d" shows "(i \<in> ord.atLeastAtMost less_eq l u) = (less_eq l i \<and> less_eq i u)"
assms_4: shows "s1' = s1(V := \<lbrakk>e\<rbrakk>s1)"
ConjE: fixes A :: "triple set"   and C :: "triple set"   and c :: "triple" assumes "A |\<turnstile> C"   and "c \<in> C" shows "A |\<turnstile> {c}"
scaleC_add_left: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "complex"   and b :: "complex"   and x :: "'d" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC (a + b) x = plus (scaleC a x) (scaleC b x)"
subset_fst_snd: fixes A :: "('c \<times> 'd) set" shows "A \<subseteq> fst ` A \<times> snd ` A"
distinct_1: fixes x1 :: "'a tree23"   and x21 :: "'a tree23"   and x22 :: "'a"   and x23 :: "'a tree23s" shows "T x1 \<noteq> TTs x21 x22 x23"
insert_4: shows "dom F3 = dom (F(x \<mapsto> y))"
Tstruct_I_intT: shows "CM.Tstruct I_intT"
cons1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and top0 :: "top0"   and s2 :: "'b state"   and new_env :: "v sem_env"   and tops :: "top0 list"   and s3 :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_top ck env s1 top0 (s2, Rval new_env) \<and> evaluate_prog ck (extend_dec_env new_env env) s2 tops (s3, r)" shows "evaluate_prog ck env s1 (top0 # tops) (s3, combine_dec_result new_env r)"
hyperplane_face_of_halfspace_le: fixes a :: "'b"   and b :: "real" shows "{x. a \<bullet> x = b} face_of {x. a \<bullet> x \<le> b}"
FNCALL_Cons: fixes X :: "freeExp"   and X' :: "freeExp"   and Xs :: "freeExp list"   and Xs' :: "freeExp list"   and F :: "nat" assumes "X \<sim> X'"   and "(Xs, Xs') \<in> listrel exprel" shows "FNCALL F (X # Xs) \<sim> FNCALL F (X' # Xs')"
pos_pos_linear: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less zero a"   and "less zero b" shows "less a b \<or> a = b \<or> less b a"
unit_simps_2: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.trg (Span.unit a) = a"
sublist_append_front: fixes l1 :: "'b list"   and l2 :: "'b list" shows "subseq l1 (l1 @ l2)"
subset: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "max (Max B) (Max A) = Max A"
mat_of_rows_row: fixes i :: "nat"   and vs :: "'b vec list"   and n :: "nat" assumes "i < length vs"   and "vs ! i \<in> carrier_vec n" shows "row (mat_of_rows n vs) i = vs ! i"
dynProcModifyReturnSameFaults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
simps: fixes a1 :: "'b"   and a2 :: "'b \<Rightarrow> real"   and a3 :: "intv" shows "intv_elem a1 a2 a3 = ((\<exists>u x d. a1 = x \<and> a2 = u \<and> a3 = Const d \<and> u x = real d) \<or> (\<exists>d u x. a1 = x \<and> a2 = u \<and> a3 = Intv d \<and> real d < u x \<and> u x < real (d + 1)) \<or> (\<exists>c u x. a1 = x \<and> a2 = u \<and> a3 = Greater c \<and> real c < u x))"
nn_integral_fst: fixes f :: "'b \<times> 'a \<Rightarrow> ennreal"   and M1 :: "'b measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M count_space UNIV)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>count_space UNIV \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M count_space UNIV) f"
wb_lens_axioms_def: fixes x :: "'c \<Longrightarrow> 'd" shows "wb_lens_axioms x \<equiv> \<forall>\<sigma>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
fct_gen_subalgebra_sets: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c" shows "sets (fct_gen_subalgebra M N X) = sigma_sets (space M) {X -` B \<inter> space M |B. B \<in> sets N}"
ordIso_imp_ordLeq: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "ordIso2 r r'" shows "ordLeq3 r r'"
emeasure_eq_measure: fixes x :: "real \<times> 'a"   and A :: "(real \<times> 'a) stream set" shows "emeasure (K.lim_stream x) A = ennreal (L.prob x A)"
pos_add_strict: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.strict_ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less b c" shows "less b (plus a c)"
pequiv_pr: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<simeq> b" shows "a \<sqsubseteq> b"
orient_lit_inst_def: fixes L :: "'a literal"   and u :: "'a trm"   and v :: "'a trm"   and s :: "sign"   and \<sigma> :: "('a \<times> 'a trm) list" shows "orient_lit_inst L u v s \<sigma> = ((L = Pos (Eq u v) \<or> L = Pos (Eq v u)) \<and> (u \<lhd> \<sigma>, v \<lhd> \<sigma>) \<notin> trm_ord \<and> s = pos \<or> (L = Neg (Eq u v) \<or> L = Neg (Eq v u)) \<and> (u \<lhd> \<sigma>, v \<lhd> \<sigma>) \<notin> trm_ord \<and> s = neg)"
finite: fixes A :: "'a set" assumes "A \<in> set (weak_ranking le)" shows "finite A"
paste_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "weak_composition V H" shows "H \<nu> \<mu> = V (H \<nu> (partial_magma.cod V \<mu>)) (H (partial_magma.dom V \<nu>) \<mu>)"
insertion_polyadd: fixes e :: "nat \<Rightarrow> 'b"   and a :: "'b poly"   and b :: "'b poly" shows "insertion e (a +\<^sub>p b) = insertion e a + insertion e b"
partitioning1_subset: fixes a :: "'b set"   and ts :: "'b set list"   and ss :: "'b set list" assumes "a \<subseteq> \<Union> (set ts)" shows "a \<subseteq> \<Union> (set (partitioning1 ss ts))"
outputRemoveTau: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "\<tau>.P \<Longrightarrow>\<^sub>l a[b] \<prec> P'" shows "P \<Longrightarrow>\<^sub>l a[b] \<prec> P'"
porder_onE: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "porder_on A r" obtains "refl_onP A r" and "antisymp r" and "transp r"
natpermute_split: fixes h :: "nat"   and k :: "nat"   and n :: "nat" assumes "h \<le> k" shows "natpermute n k = (\<Union>m\<in>{0..n}. {l1 @ l2 |l1 l2. l1 \<in> natpermute m h \<and> l2 \<in> natpermute (n - m) (k - h)})"
Sup_le_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and b :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
map_cong: fixes x :: "'g +\<^sub>\<bottom> 'h"   and ya :: "'g +\<^sub>\<bottom> 'h"   and f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i" assumes "x = ya"   and "\<And>z. z \<in> set_sum_bot ya \<Longrightarrow> f z = g z" shows "sum_bot_map f x = sum_bot_map g ya"